ASTRONOMY, from αϋτηο or aerooy, a star, and vομος, a law, is the science which treats of the laws observed by the stars in their motions. By an extension of signification, it embraces every thing that is known relating to the nature and constitution, as well as to the motions, of the celestial bodies.

The present treatise is divided into Four Parts. In the First, which contains the History of Astronomy, the progressive advancement of the science from the times of the Chaldeans and Egyptians to the present day is briefly sketched, and the labours of those illustrious individuals commemorated, who have either theoretically or practically contributed most to its progress. The Second Part, which we have denominated Theoretical Astronomy, is devoted to a general view of the science, to the explanation of the different theories and methods by which the motions of the celestial bodies are represented, and their places computed; and the description of such facts as observation has made known respecting their nature and constitution. Part Third treats of Physical Astronomy; and Part Fourth of Practical Astronomy.

PART I.

HISTORY of astronomy.

Astronomy, if we dignify by that name the first rude attempts that were made to discover the order and connection of the celestial motions, may probably be regarded as the most ancient of all the sciences. In fact, a certain degree of attention to the heavenly bodies is forced even on the savage who inhabits the forest, and derives his subsistence from the spontaneous productions of the earth. The regular vicissitude of day and night inevitably compels him to observe the diurnal course of the sun; and he cannot fail soon to perceive, that the variety and succession of the seasons is equally dependent on the oblique annual course of the same great luminary. The moon, too, in the absence of the sun, is an object so conspicuous, so consoling, and so useful, that her motions must at all times have been watched with attention and interest; while her various phases, her alternate waxings and wanings, her regular disappearance and return after equal intervals of time, would be contemplated with admiration and delight. Nor are the wonders of the starry firmament less calculated to strike even the most heedless observer of the heavens. The magnificent spectacle of the sky bespangled with brilliant points, and revolving in obedience to eternal and unalterable laws, affords a constant succession of new objects of sublime and exalted contemplation. The occasional recurrence, also, of eclipses and other unusual phenomena, which seem to interrupt the general order and uniformity of the celestial motions, would stimulate to attentive observation; for the vanity of man has in all ages rendered him eager to connect his own destiny with the heavens, while his timidity has prompted him to regard every apparent deviation from the ordinary course of events as an emblem of the wrath, and a precursor of the vengeance, of superior beings.

But though mankind were probably first impelled by motives of mere curiosity to observe the courses of the stars, no great length of time could elapse ere they perceived that the regular and uniform revolutions of the heavens might be rendered subservient to their own wants and conveniencies. By the help of the stars the shepherd, during the night, could count the hours, the traveller track his course through the uniform wastes of the desert, and the mariner guide his bark over the ocean: the husbandman, also, learned to regulate his labours by the appearance of certain constellations, which gave him warning of the approaching seasons. The indications derived from the simple observation of such phenomena were doubtless extremely vague; but as civilisation advanced, the necessity of determining accurately the length of the solar year and of the lunar month, in order to regulate the calendar and the religious festivals, led to the accumulation and comparison of different observations, whereby errors were gradually diminished, and the foundations laid of a more perfect acquaintance with the heavenly motions.

Astronomy, presenting so many objects of interesting curiosity, and having so many practical uses, could not fail to be one of the sciences first cultivated by mankind. Its origin is consequently hid amidst the obscurity and traditions of the remotest ages, and is in fact coeval with the origin of society, and the earliest development of the human intellect. The records or traditions of almost every ancient nation furnish some traces of attention to the state of the heavens, and of some rude attempts· to discover the laws, the order, and the period of the most remarkable phenomena,—such as eclipses of the sun and moon, the motions of the planets, and the heliacal risings of the principal stars and constellations. The Chaldeans and Egyptians, Chinese and Indians, Gauls and Peruvians, equally regard themselves as the inventors of astronomy; an honour, however, of which Josephus deprives them all, in order to ascribe it to the antediluvian patriarchs. The fables relating to the two columns of brick and marble which these sages are said to have erected, and on which they engraved the elements of their astronomy, to preserve them from the universal destruction by fire and water to which, they are said to have learned from Adam, the earth was doomed, are not worth the trouble of repetition; nor is there any better próof than the assertion of that credulous historian, of their acquaintance with the annus magnus, or, as is most probably supposed, the astronomical cycle of 600 years, which brings back the sun and moon to the same points of the heavens so nearly, that its discovery implies a pretty correct knowledge of the solar and lunar motions. Passing over, therefore, those periods that present us only with a scanty detail of traditional observations or unimportant facts, we will proceed to give a brief account of the state of astronomy among some early nations who have undoubtedly contributed to the improvement of the science, or who, at least, have transmitted to future ages some monuments of their [7:3:722]astronomical labours, and of their attention to celestial observations.

Astronomy of the Chaldeans, Egyptians, Phoenicians, Chinese, and Indians.

According to the unanimous testimony of the Greek historians, the earliest traces of astronomical science are to be met with among the Chaldeans and Egyptians. The spacious level and unclouded horizon of Chaldea afforded the utmost facilities for observing the celestial phenomena; and its inhabitants, enjoying the leisure afforded by a pastoral life, and stimulated by the vain desire of obtaining a knowledge of the future from the aspects of the stars, assiduously cultivated astronomy and astrology. By a long series of observations of eclipses, extending, according to the testimony of some authors, over nineteen centuries, or even a longer period, they had discovered the cycle of 223 “lunations, or eighteen solar years, which, by bringing back the moon to nearly the same position with respect to her nodes, her perigee, and the sun, brings back the eclipses in the same order. This is supposed to be the period which they distinguished by the name of Saros. They had others, to which they gave the names of Sossos and Neros ; but nothing positive is known with regard to their nature or extent. One thing only is certain, which is, that these Chaldaic periods, whatever they were, were founded on no theoretical knowledge of the celestial motions. They were purely empirical, detected by the comparison of recorded observations, and suppose neither theory nor science, unless, indeed, a simple arithmetical operation is to be considered as such; nor is there any reason to suppose that the Chaldeans employed any process of computation whatever in their predictions of eclipses. Having once established their cycle, they were in possession of a simple means of predicting all those which occurred in the course of it, with as great a degree of accuracy as they considered requisite.

The knowledge of these lunisolar periods among the Chaldeans is doubtless of great antiquity. Simplicius, the commentator of Aristotle, asserts that Callisthenes transmitted to Aristotle from Babylon a collection of observations of all the eclipses which had happened during the nineteen centuries that preceded the conquest of Alexander. This relation, however, is at variance-with the accounts given by other historians. Epigenes, cited by Seneca and Pliny, who is supposed to have lived shortly before the time of Alexander, mentions observations of 730 years that had been found preserved on columns of brick. Ptolemy also makes mention of certain observations of eclipses that had been brought from Babylon, several of which he had calculated and verified; but the earliest of these ascends only to the year 720 before our era, or to the 26th of Nabonassar; and if either Hipparchus or himself had been acquainted with others of a more ancient date, they would doubtless have employed them in the determination of the mean motion of the moon. From this circumstance it appears probable that the Chaldeans had no observation sufficiently exact to be of any use to astronomy prior to the time of Nabonassar.

According to Apollonius of Myndus, the Chaldeans supposed the comets to be substances of the same nature as the plarets; that they are visible only during a portion of their revolutions, and that they re-appear after certain intervals. But this statement, which argues some just notions respecting the celestial bodies, is contradicted by Epigenes, who himself studied among the Chaldeans, and who affirms, that instead of regarding the comets as subjected like the planets to the operation of eternal laws, they attributed their formation to vortices of inflamed matter, supposing them to have only a temporary existence, and to move in random directions through the fields of space. Diodorus says, that they regarded the world as eternal and imperishable, and held that its order is due, not to chance, but to a divine providence; that the planets, which have peculiar motions, announce future events by their various aspects and colours; that they portend rains, tempests, excessive heats; sometimes also the appearance of comets, eclipses, earthquakes, and, in short, whatever has a beneficial or hurtful influence on the fortunes of nations or individuals.

From the few facts that can be gleaned from the vague accounts given by ancient authors of the astronomy of the ancient Chaldeans, it may be inferred that their boasted science was confined to observations of the simplest and rudest kind, neither guided by theory, nor assisted by instruments; for, notwithstanding the assertion of Herodotus, it is doubtful if they were acquainted even with the gnomon, the simplest of all instruments for determining the obliquity of the ecliptic, the altitude of the pole, and the length of the tropical year. If to the knowledge of their lunisolar periods, the result of ages of observation, we add the notion of a spherical revolution about an inclined axis, and an idea of the principal circles of the sphere and the position of the poles, the sum will comprehend all that constituted the science of a people regarded by antiquity as having made the greatest progress in the science of the stars. Astronomy, however, owes some obligations to their humble labours. The observations which they recorded served to correct the theories that were afterwards imagined by the more brilliant genius of the Greeks, and thereby furnished some materials for the edifice of the world.

The Egyptians were in ancient times the rivals of the Chaldeans in the cultivation of astronomy; and although they have left behind them still fewer monuments of their labours, they have obtained, through the exaggerated statements of the Greeks, even a greater reputation. The Greeks acknowledge themselves indebted to the Egyptians for their science and civilisation; but regarding themselves likewise as descendants of that ancient people, they indulged their vain-glory in magnifying the accounts of the antiquity and knowledge of their supposed ancestors. It is not improbable that some traditional observations of the heavens, along with some arts indispensable to society even in its earliest stages, were carried into Europe by tribes migrating from the banks of the Nile; and it is certain that the early philosophers of Greece travelled into Egypt for the purpose of acquiring a more perfect knowledge of astronomy than could be obtained in their own country. But the facts from which it can be inferred that the Egyptians had much to communicate, are few and ill attested. They are also blended with so much absurdity and fable, that no accurate notions can be formed, from the accounts that have been transmitted to us, of the real advances which that people had made in astronomical science. The priests were the depositaries of the national knowledge; and they carefully concealed it from the vulgar by shrouding it in allegories, traces of which, it has been remarked, may be detected in the institutions even of the present day.

According to Diogenes Laertius, the Egyptians reckoned 48,853 years from Vulcan to Alexander, during which they had observed 373 eclipses of the sun, and 832 of the moon. These numbers in fact nearly express the relative proportion of the eclipses of the two luminaries; but the enormous length of the period altogether exceeds [7:3:723]the bounds of credibility; and it has been remarked that 'the same number of eclipses might have been observed within the more probable period of twelve or thirteen centuries. Supposing the numbers to be accurately stated, it will follow that, as the observations terminated with the conquest of Alexander, the Egyptians must have been in the habit of observing eclipses at least 1600 years before the commencement of our era. By attentively observing the heliacal rising of the star Sirius, to which they gave the name of Thaat, or Thoth (the Watch-Dog), because its appearance shortly preceded the overflow of the waters of the Nile, the Egyptians had discovered that the year consists of 365 4 - days. This was their religious or sacred year. Their civil year consisted of only 365 days; consequently the sacrifices and feasts, which were regulated by it, successively corresponded to the different seasons. Instead of attempting to obviate this inconvenience by intercalation, they imposed an oath on their kings to maintain the use of the civil year, superstitiously imagining that each of the seasons would be blessed and rendered prosperous by enjoying in its turn the celebration of the feast of Isis. The difference between the lengths of the sacred and civil year suggested to them their famous sothic or canicular period of 1460 solar years, corresponding to 1461 civil years of 365 days, and which consequently brings back the months and festivals to the same seasons. Dion Cassius ascribes the week to the Egyptians, and says that they first dedicated a day to each of the planets; but it is sufficiently proved that this short cycle was in use among the Chinese and Indians from the remotest times, and was even known to the Druids of Gaul and Britain. It was more probably suggested to different nations by the phases of the moon. The Egyptians had likewise been attentive to the courses of the planets. Diodorus Siculus affirms that they could explain the phenomena of the stations and retrogradations; and Macrobius ascribes to them the knowledge of the real motions of Mercury and Venus, and says that they regarded these planets as satellites of the sun. This notion would do credit to their philosophy; but it is unfortunately not mentioned by any other author, and for this reason the testimony of Macrobius is suspected. The state of their practical astronomy may in some measure be inferred from the means they employed to determine the magnitude of the sun’s apparent diameter. By comparing the time, observed by means of a clepsydra, which the sun takes to mount above the horizon at the equinox, with that in which he makes a complete revolution of the sky, they estimated his diameter at 28' 48". An observation of this kind is liable to great uncertainty; and as there is no evidence that the Egyptians possessed the slightest knowledge of spherical trigonometry, they would probably make no allowance for the obliquity of the equator to their horizon; and if this correction was left out of the calculation, as it probably was, their diameter, already too small, ought to have been still farther reduced, and to have amounted only to 24'42". It has been conjectured by Goguet, that the obelisks of Egypt were intended to serve the purpose of gnomons; and this conjecture acquires some probability from their needle-shaped form, and the narrowness of their bases relatively to their heights. It has however been proved by MM. Jollois and Devilliers, in their description of Thebes, that the obelisks were connected with the walls of temples and palaces; a disposition which rendered them entirely unfit for the purposes of astronomical observation. Their summits were also of so unfavourable a form, that the Romans were obliged to surmount them with a ball in order to obtain a distinct and well-defined shadow. The pyramids have also been adduced as evidences of the early progress of astronomy among the Egyptians; for the faces of these stupendous masses are turned directly towards the four cardinal points, from which it is evident that the people by whom they were constructed were at least acquainted with the method of tracing a meridional line.

From this brief account it appears, that the only circumstances with which we are acquainted that imply the knowledge of astronomical methods among the Egyptians, are the length of the year, the doubtful discovery of the true motions of Mercury and Venus, and the position of the pyramids. The Chaldean observations were of use to Hipparchus and Ptolemy in the determination of some important elements; but those of the Egyptians exercised no influence whatever on the future progress of the science.

The Phoenicians are also generally enumerated among the nations who cultivated astronomy at a very early period, though it does not appear, from any facts mentioned by ancient authors, that they addicted themselves to the observation of the heavens, or made any discoveries relative to the motions of the planets. That they excelled in the art of navigation is certain, from the commercial intercourse which they carried on with many places on the coasts of Africa and Spain, and in the principal islands of the Mediterranean; and it may readily be allowed that in their long voyages they would direct their course during the night by the circumpolar stars. If they had any speculative notions of astronomy, these were probably derived from the Chaldeans or Egyptians.

In China, astronomy has been cultivated from the remotest ages, and always been considered as a science indispensably necessary to the civil government of the state. The Chinese boast of a series of eclipses, recorded in the annals of the nation, extending over a period of 3858 years, all of which, they pretend, were not only carefully observed, but calculated and figured previous to their occurrence. The same motives which led the Chaldeans and Egyptians to attend to the celestial phenomena, namely, the regulation and division of time, had equal influence among the Chinese; and we accordingly find the care of the calendar occupying the attention of their earliest princes. The emperor Fou-Hi, whose reign commenced about 2857 years before our era, is said to have assiduously studied the motions of the celestial bodies, and laboured to instruct his ignorant subjects in the mysteries of astronomy. But as they were yet in too rude a condition to be able to comprehend his theories, he was obliged to content himself with giving them a rule for the computation of time by means of the numbers 10 and 12, the combination of which produces the cycle of 60 years, which is the standard or unit from which they deduce their hours, days, and months. Tradition is silent with respect to the sources from which Fou-Hi derived his own knowledge. The Chinese attribute to him also the invention of arithmetic and music. In the year 2608 b. c., Hoang-Ti caused an observatory to be built, for the purpose of correcting the calendar, which had already fallen into great confusion, and appointed one set of astronomers to observe the course of the sun, another that of the moon, and a third that of the stars. It was then discovered that the twelve lunar months do not exactly correspond with a solar year; and that, in order to restore the coincidence, it was necessary to intercalate seven lunations in the space of nineteen years. If this fact rested on undoubted evidence, it would follow that the Chinese had anticipated the Greeks by 2000 years in the discovery of the Metonic cycle. The reign of Hoang-Ti is also rendered memorable by the institution of the Mathematical Tribunal, for promoting the science of [7:3:724]astronomy, and regularly predicting eclipses, to which 'an extraordinary importance has always been attached in China. The members of this celebrated tribunal were made responsible with their lives for the accuracy of their predictions, by a law of the empire, which ordained that, “whether the instant of the occurrence of •any celestial phenomenon was erroneously assigned, or the phenomenon itself not foreseen and predicted, either negligence should be punished with death.” In the reign of Tchong-Kang, the two mathematicians of the empire, Ho and Hi, were the victims of this absurd and sanguinary law; an eclipse having taken place which their skill had not enabled them to foresee.

The emperor Yao, who mounted the throne, according to the Chinese annals, about the year 2317 b. c., gave a new impulse to the study of astronomy, which had begun already to decline. He ordered his astronomers to observe with the utmost care the motions of the sun and moon, of the planets and the stars, and to determine the exact length of each of the four seasons. He sent Hi-Tchong to the east to observe the star situated at the point of the vernal equinox, Hi-Tchou to the south to examine that at the summer solstice, Ho-Tchong to the west, and Ho-Tchou to the north, to observe those situated respectively at the autumnal equinox and winter solstice. These docile observers found stars in the positions assigned by the emperor; but the extraordinary resemblance of their names imparts a fabulous air to the whole relation, and excites a very excusable incredulity even with regard to those statements which involve no improbability. To this emperor are attributed the Chinese division of the zodiac into 28 constellations, called the houses of the moon, and the severe laws already noticed in regard to the erroneous prediction of the celestial phenomena.

From the time of Yao the Chinese year consisted of 365^ days. They also divided the circle into 365£ degrees, so that the sun daily described in his orbit an arc of one Chinese degree. Their common lunar year consisted of 364¾-^- days; and by combining this number with 365į, they formed the period of 4617 years, after which the sun and moon again occupy the same relative positions.

The earliest Chinese observations we are acquainted with, sufficiently precise to afford any result useful to astronomy, were made by Tcheou-Kong, whose reign commenced about the year 1100 before our era. Two of these observations are meridional altitudes of the sun, observed with great care at the village of Loyang, at the time of the summer and winter solstices. The obliquity of the ecliptic thus determined at that remote epoch is 23° 54/ 3T5"; a result which perfectly agrees with the theory of universal gravitation. Another observation, made about the same time, relates to the position of the winter solstice in the heavens; and it also corresponds to within a minute of a degree with the calculations of Laplace. Laplace considers this extraordinary conformity as an indubitable proof of the authenticity of those ancient observations.

The golden age of Chinese astronomy extended from the reign of Fou-Hi to the year 480 b. c. ; that is, over a space of 2500 years. It is only, however, towards the latter part of this long period that the history of China becomes in any degree authentic; and the true date which must be assigned for the commencement of observations on which any reliance can be placed, is the year 722 B. c.; that is, 25 years posterior to the era of Nabonassar. From that period to the year 400 b. c. Confucius reckons a series of 36 eclipses, and of these 31 have been verified by modern astronomers, After this the science fell into great neglect, notwithstanding the inveterate tenacity with which the Chinese in general adhere to their ancient customs. The decline of their astronomy is ascribed, whether justly or not, to the barbarous policy of the emperor Tsin-Chi-Hong-Ti, who, in the year 221 B. c., ordered all the books to be destroyed, excepting those only which related to agriculture, medicine, and astrology, the only sciences which he considered as bein°∙ of any use to mankind. His fury, it is true, was princio pally directed against those of Confucius, the stern morality of which he felt to be a censure on his own profligacy; but those of science and astronomy were included in the general destruction. In this manner, it is said, the precious mass of astronomical observations and precepts which had been accumulating forages was irretrievably lost.

Lieou-Pang, the successor of Tsin-Chi-Hong, endeavoured to repair the disaster, by re-establishing the tribunal of the mathematics, and ordering a new series of observations to be undertaken. About the year 104 b. c. the astronomer Sse-Ma-Tsien gave some precepts for the calculation of eclipses, the motions of the planets, and the syzygies. He employed instruments of copper, the nature and construction of which are however not very well understood, for measuring the extent of the 28 zodiacal constellations; and he observed the meridional altitudes of the sun, by means of a gnomon 8 feet high. The differences of right ascensions, and the intervals between the risings, settings, and culminations of the stars, were measured by clepsydrae. It would appear that after this period astronomical observations continued for some time to be made in China with considerable regularity. In the 164th year of our era the astronomer Tchang-Heng constructed armillary spheres and a celestial globe. He also formed a catalogue of stars, which is said to have contained 2500, but without either latitudes or longitudes; a circumstance which gives us a very unfavourable idea of the state of practical astronomy at that time. About the eighth century of our era, all knowledge of the science seems to have been again lost. The predictions were erroneous; and the Chinese witnessed, with superstitious terror, eclipses of which their astronomers had given them no intimation. This induced the emperor Hieng-Tsong to call to his court the astronomer Y-Hang, by whose indefatigable activity a reform was speedily effected. With a view to determine the situations of the principal places of the empire, this astronomer constructed gnomons, spheres, astrolabes, quadrants, and other instruments; and sent one company of mathematicians to the south, and another to the north, with directions to observe daily the altitudes of the sun and the polar star. The latitudes of the cities were determined by observing the shadow of the gnomon, and the longitudes by eclipses of the moon. Y-Hang had the mortification of announcing two eclipses which did not take place. On these occasions he alleged the usual excuse, namely, that his calculus was not in error, but that the celestial bodies had deviated from their ordinary courses out of respect to the virtues of the emperor. The fate of Ho and Hi had probably suggested to the Chinese astronomers this ingenious mode of disarming the emperor’s resentment by flattering his vanity.

On considering attentively the accounts which have been given of the Chinese astronomy', we find that it consisted only in the practice of observations which led to nothing more than the knowledge of a few isolated facts. The missionaries who were sent out by the Jesuits about the end of the seventeenth century, to whom we are indebted for what is known of the early history [7:3:725]of China, either seduced by some appearances of truth, or thinking it prudent to conciliate the people whom they were attempting to convert, adopted their marvellous relations regarding the antiquity of their science, and spread them over Europe. As the history of the nation begins to become more authentic, their astronomy shrinks into its real but insignificant dimensions. Superstitiously attached to their ancient usages, and blindly adopting the habits of their ancestors, the Chinese continued to observe the heavens from century to century without making the slightest advances in theoretical knowledge. In later times they have adopted many improvements, for which they are entirely indebted to foreigners. During the time of the caliphs many Mahometans passed into China, carrying with them the astronomical methods and knowledge of the Arabians. The missionaries introduced the science of Europe; and the most that can be said in praise of the Chinese is, that their government sometimes relaxed so far its spirit of jealousy and exclusion, as to afford protection to these strangers, adopt their arts, and place them at the head of the mathematical tribunal.

The astronomy of the Indians forms one of the most curious problems which the history of science presents to the consideration of the learned, and one which, notwithstanding the numerous dissertations to which it has given rise, still continues involved in great uncertainty. Of the science of the ancient nations, of which we have already spoken, the accounts which have come down to our times are founded on conjecture and tradition; for few monuments remain to confirm or confute the glowing descriptions which authors have given of its high antiquity and great perfection. But the claims of the Indians rest on more solid foundations. We are in possession of the tables from which they compute the eclipses and places of the planets, and of the methods by which they effect the computation: we have, in short, an Indian astronomy committed to writing, which represents the celestial phenomena with considerable exactness, and which, therefore, could only be produced by a people far advanced in science. But the difficulty of the problem consists in determining the sources from which this science originated, and the epoch of its existence; whether it was created by the people who now blindly follow its preeepts without understanding its principles, or was communicated to them by another race of a bolder and more original genius, through channels with which we are unacquainted. Some authors regard India as the cradle of all the sciences, particularly of astronomy, which they suppose to have been cultivated there from the remotest ages; others date the origin of the Indian astronomy from the period when Pythagoras travelled into that country, and carried thither the arts and sciences of the Greeks; a third opinion is, that astronomy was conveyed to India by the Arabians in the ninth century of our era, and that the Brahmins are only entitled to the humble merit of adapting the rules and practices of that people to their own peculiar methods of calculation. We shall endeavour to describe very briefly the existing monuments of the Indian astronomy, which furnish the only data from which a rational conjecture can be formed relative to its antiquity and precision.

We possess four different sets of tables of Indian astronomy. The first which were known in Europe were brought from Siam by La Loubère, who had resided in that country as ambassador from Louis XIV. They were communicated by him to the celebrated Cassini, who, notwithstanding the difficulties arising from the complicated and useless operations which they directed, succeeded in detecting the principles on which they were constructed, and in explaining their use and signification. The date of these tables corresponds to the 21st of March in the year 638 of our era. They suppose two species of years, the solar tropical year, which they make to consist of 365 days 5 hours 50 min. and 4 sec., and the solar anomalistic year, that is, the period in which the sun returns to its apogee, which they estimate at 365 days 6 hours 12 min. 36 sec. This determination of the length of the solar year is too great only by 1 min. 15 sec. By means of the same tables the longitudes of the sun and moon are determined with considerable accuracy. They contain a correction for the sun’s mean place, which corresponds to the equation of the centre. At 90° from the apogee, where the inequality of the sun’s motion is greatest, they estimate the requisite correction at 2° 12', which is about 16’ too great. This determination deserves to be particularly remarked, because, on account of a secular inequality of the eccentricity of the sun’s orbit, there was once a time when the greatest value of the equation of the centre was nearly 2° 12'; and this fact is adduced as a proof of the remote antiquity of the observations from which the tables in question have been constructed. These tables suppose the apogee to retain always the same position relatively to the fixed stars; in reality it advances or gains on the stars about 10" annually; but the supposition is still much nearer the truth than in the system of Ptolemy, where the apogee is supposed to be absolutely at rest with regard to the plane of the sun’s orbit, and consequently to fall back among the stars by the whole quantity of the precession of the equinoxes, or about 50" annually. With regard to the motions of the moon, they are deduced from a period of 19 years, in whi ι ch are comprehended nearly 235 lunations; so that the cycle of Meton appears to have been known in Siam as well as in China. The moon’s apogee is supposed to have been in the beginning of the movable zodiac 621 days after the epoch of the 21st of March 638, and to make an entire revolution in the heavens in the space of 3232 days. The first of these suppositions agrees with Mayer’s tables to within a degree, and the second differs from them only by 11 hours 14 min. 31 sec. They contain only one correction for the two principal inequalities of the moon’s motion, the equation of the centre and the evection.

A second set of Indian tables was sent from Chrisnabouram, a town in the Carnatic, by Father Du Champ, to De Lisle, about the year 1750. They are fifteen in number. They give the mean motions of the sun, moon, and planets; equations of the centre for the. sun and moon; and two corrections for each of the planets, one of which corresponds to the apparent, the other to the real inequality. The epoch of these tables is not so ancient as that of the former. It corresponds to the 10th of March, at sunrise, in the year 1491 of our era, when the sun and moon were in conjunction.

A third set of astronomical tables was sent from India by Father Patonillet, and received by De Lisle about the same time with those of Chrisnabouram. These have not the name of any particular place affixed to them; but being calculated for the latitude of 16° 16', Bailly thinks it probable that they came from Narsapour. Their epoch is midnight, between the 17th and 18th of March 1569.

The fourth and last set of Indian tables which we possess have been published in the Memoirs of the Academy of Sciences. They were communicated by a learned Brahmin of Tirvalore, a small town on the Coromandel coast, to the French astronomer Legentil, who had gone to India to observe the transit of Venus in 1769. The tables [7:3:726]of Tirvalore, though somewhat different in form, present many points of resemblance with those formerly known in Europe. They suppose the same length of the year, the same inequalities of the sun and moon, and they are adapted nearly to the same meridian. But while they correspond with the other tables in these elements, they differ from them greatly in the antiquity of their epoch, which goes back to the famous era of the Calyougham, that is, the beginning of the year 3102 before Christ.

Now, the only question to be determined with regard to the antiquity of the Indian astronomy is, whether this epoch is real or fictitious; that is, whether the state of the heavens at the commencement of the Calyougham, as assumed in these tables, was actually determined by observation, or computed backwards from observations of more modern date. The solution to this question can only be obtained from the internal evidence afforded by the tables themselves; by examining whether the elements and precepts which they furnish are of sufficient accuracy to enable the places of the sun, moon, and planets to be calculated through a period of 44 centuries, without involving errors which the refined accuracy of the modern tables furnishes the means of detecting. A comparison of the Indian with the modern tables has been made at great length by Bailly, who imagines that he finds ample evidence of the reality of the era in question, and of the existence of an astronomy prior to that period, hardly yielding in accuracy to that which modern science has built on the theory of universal gravitation. The theory of Bailly has been adopted, and put forth with additional clearness and evidence, by the late Professor Playfair. One of the principal arguments which these illustrious authors bring forward in support of it is founded on the longitudes of the sun and moon. The mean place of the moon at the commencement of the Calyougham, that is, at midnight, between the 17th and 18th of February 3102 B. c., is stated by the Indian tables to be 306°. Her mean place, computed from Mayer’s tables, without taking into account the acceleration, with which the Indians in the 15th century were of course unacquainted, is 300° 51' 16". Hence there would be a discrepancy of 5° 8' 44". But, according to the theory and last tables of Laplace, the moon, in virtue of the acceleration of her mean motion, has passed over an arc of very nearly 6° more than she would have done had her mean motion continued uniform from the period of the Calyougham to the date of Mayer’s tables. This added to 300° 51' 16" gives 306° 51' 16" for the mean longitude of the moon at the epoch of the Calyougham, differing from the Indian determination by only 51' 16". Now, it is argued that this is a degree of accuracy which could have been reached only by actual observation, especially since, if the tables had been computed backwards, the error arising from the acceleration alone would have amounted to more than 5°. Bailly computes the place of the moon at the same epoch, from all the tables, Greek and Arabian, to which the Indians can be supposed to have had access, and the discrepancies are so great as to render his conclusion almost inevitable, that the Indian tables could not possibly have been drawn from such sources. The tables of Ptolemy make the moon’s longitude at that time 11° 52' 7" greater than the Indian tables; and those of Ulugh-Beigh, constructed at Samarcand in 1437, give a difference of 6° also in excess.

Similar results are obtained from the consideration of other elements. According to the tables of Tirvalore, the tropical year consists of 365 days 5 hours 50 min. 35. sec. Lacaille makes it 365 days 5 hours 48 min. 49 sec. The difference is 1 min. 46 sec. Now the tropical year, being affected by the precession of the equinoxes, is subject to a secular inequality, which, according tc√ the theory of Lagrange, renders it actually shorter by 40∙5 sec. at the present time than it was at the commencement of the Calyougham. The error of the Indian tables is thus reduced to 1 min. 5∙5 sec. In like manner, the obliquity of the ecliptic, which has been gradually diminishing during a great number of centuries, is supposed in the Indian tables to be greater than it is now found to be by observation. The Brahmins estimate it at 24°. The formula of Lagrange makes the variation, in 4800 years, amount to 22' 32". This therefore must be added to its obliquity in 1700, that is, to 23° 28' 41", in order to have the true obliquity at the commencement of the Calyougham. The sum is 23° 51' 13", and falls short of the Indian determination by 8' 47". We shall mention only another element, the equation of the centre of the sun. Bailly calculates that, according to the theory of Lagrange, the equation of the sun’s centre, at the epoch of the tables, was 2° 6' 28". The Indians make it 2° 10 ’ 32". The difference is only about 4', and incomparably less than could have resulted from calculation by any methods which we can suppose the Indians to have possessed.

These arguments, it must be admitted, are exceedingly specious, but they are not by any means convincing. Even with the best modern tables we could not, as Bailly himself acknowledges, answer for the accuracy of the places of the sun and moon computed for so remote an epoch. The corrections for the secular inequalities amount in that long period to considerable quantities; and these corrections are deduced by theory from elements with respect to which there exists great uncertainty. And if we cannot be sure of the true places by computing backwards from our own tables, with what degree of confidence can we pronounce upon the accuracy of the places assigned in the tables of the Indians? It may be said that comparisons of this kind can never be supposed to give results perfectly alike. Granted: but if the discrepancies are such that the lapse of a thousand years more or less is required to establish a rigorous conformity, what becomes of the famous epoch of the Calyougham? Some of the elements of the Indian tables could not have the values assigned to them but at a long period before that epoch. In order to find their equation of the sun’s centre, for example, it is necessary, according to the results of modern theory, to go back to 6000 years before our era. The argument, therefore, proves too much, and is consequently inconclusive. The different sets of tables of which we have spoken are closely allied with each other, and the most probable supposition is, that they are all derived from those of Chrisnabouram, of which the epoch is 1491. At that era the Indians were acquainted with the instruments, the geometry, and the researches of the Arabians and Greeks. Through this channel the tables seem to have come into their possession. The Brahmins adapted them to their own particular methods of computation, and threw back their epoch to the period when, according to these tables, all the planets were in conjunction with the sun. Every circumstance connected with the science of the Indians conspires to give us the humblest ideas of its value. Their methods of computation are encumbered with the unnecessary multiplications and divisions of enormous numbers, endless additions, subtractions, and reductions, for the purpose of obtaining numbers which could be put into technical verses, and even adapted to songs; so that the astronomer might be enabled to effect hi∙s calculations from memory alone, without its being necessary to have [7:3:727] recourse to tables or books. But simple and rude as these methods are, if they were really invented by the Brahmins, the science of that people must have greatly retrograded; for at present they merely follow a blind routine, utterly ignorant of theory, or the principles on which their processes are founded. Their astronomy, whether of ancient or recent origin, has produced no effect whatever on that of Europe; it has no filiation or connection with the science of the present day, and therefore has no other claim on our attention than such as may result from motives of mere curiosity.^[1. For an account of the Indian astronomy, see Bailly, Astronomie Indienne; also a Memoir by Professor Playfair, in the Edinburgh Transactions, vol ii., or in the 3d volume of his Works; and the Papers of Jones, Bently, and Davis, in the Calcutta Memoirs, the theory of Bailly is most satisfactorily refuted by Delambre. See his Histoire de l’ Astronomie Ancienne, tom. ι. ] Astronomy of the Greeks.

The origin of astronomy in Greece, as in other early nations, ascends beyond the period of authentic history, and is concealed amidst the fables and traditions of the remotest times. During the darkness of the heroic ages some gleams of an acquaintance with the motions of the stars occasionally burst forth; and some traces appear of astronomical observations, probably derived from Egypt, the country which also furnished Greece with its gods and its arts. The Greeks seem to have divided the heavens into constellations about 13 or 14∣ centuries before the Christian era; for the sphere of Eudoxus, which is probably one of the fruits of the famous voyage of the Argonauts, must be referred to that period. Their early attention to the appearances of the heavens is sufficiently attested by their mythological fables, the greater part of which are only allegories of the celestial motions, and of the operations of nature. The lively fancy and brilliant imagination of this ingenious people strewed flowers in the most rugged paths, and spread agreeable images over the driest and most uninviting subjects: hence the sky was quickly covered with legends of the loves and exploits of gods and heroes. It would be foreign to our present purpose to enter into an enumeration of these fables, or attempt to trace their connection with the first dawnings of astronomy: we shall content ourselves with barely alluding to Uranus, to Atlas and his son Hesperus, who gave his name to the planet Venus; also to his daughters the Atlantides, from whom the Pleiades received their appellation; to Endymion, who, on the summit of Mount Latmos, held nocturnal converse with the chaste Diana; to Hercules; and Chiron the centaur, who taught men the use of the constellations; Musaeus, who imagined the figures of men and animals which cover the celestial sphere; Orpheus and Linus, who explained the théogonies; Atreus, from whose banquet the sun fled back with horror; and Tiresias, who was struck blind for having witnessed some secret of the gods.

The true foundations of Grecian science were laid by Thales, who was born at Miletus 640 years before our era. He was descended from an illustrious family, which had formerly reigned in Phoenicia, and inherited an ample fortune, which he expended in collecting the expiring embers of oriental science. Instigated by the love of knowledge, he travelled first into Crete, and afterwards into Egypt, where he was initiated into the mysteries of the priests, to whom, in return, he is said to have taught the method of measuring the height of the pyramids by comparing their shadows with those of known objects. Returned to his own country, he publicly taught the truths he had collected during his travels, and formed a sect which has been distinguished by the title of the Ionian School. His doctrines regarding astronomy contain a few truths which do honour to his sagacity and observation, though they are mixed with much error and absurdity. He taught that the stars are formed of fire; that the moon receives her light from the sun, and is invisible at her conjunctions, because she is hid in the sun’s rays. He also taught the sphericity of the earth, which he placed at the centre of the world. He divided the sphere into five zones, by the arctic and antarctic circles, and the two tropics; and held that the equator is cut obliquely by the ecliptic, and perpendicularly by the meridian. He is also said to have observed eclipses; and Herodotus relates that he predicted the famous one which put a stop to the war between the Medes and the Lydians. It does not appear, however, that he ventured to assign either the day or the month of the eclipse, so that his prediction must have been confined to the year. According to Callimachus, he determined the positions of the stars which form the Lesser Bear, by which the Phoenicians guided themselves in their voyages. It is difficult, however, to conceive how Thales, unacquainted with instruments, could determine the positions of stars with so much accuracy as to render any essential assistance to the navigator. It is probable that he only pointed out the configuration, and some of the more brilliant stars of that constellation, among which he might remark that which is nearest the pole of the world.

Thales was succeeded by Anaximander, to whom is also attributed the invention of the sphere, and the knowledge of the zodiac. According to Diogenes Laertius, he supposed, like his master Thales, the earth to be spherical, and placed at the centre of the universe; but Plutarch ascribes to him the less philosophical opinion of its resemblance to a column. He supposed the sun to be of equal magnitude with the earth. He invented the gnomon, and placed one at Lacedaemon to observe the solstices and equinoxes. But the circumstance which does most honour to Anaximander, and which entitles him to the gratitude of posterity, is the invention of geographical charts. He is said also to have believed in the plurality of worlds,—a sublime idea, which was adopted by almost every succeeding philosopher of Greece.

Anaximenes succeeded Anaximander in the Ionian school, and maintained nearly the same doctrines. Pliny says he was the first who taught the art of constructing dials,—an invention which, as we have just seen, has also been ascribed to Anaximander. These two philosophers probably revived the knowledge of an instrument the use of which had been forgotten amidst the general rudeness and ignorance of their countrymen. Before their time the Greeks only marked the divisions of the day by the different lengths of the sun’s shadow.

Anaxagoras was the disciple and successor of Anaximenes. If this philosopher really entertained the ridiculous opinions ascribed to him by Plutarch, the Ionian school must rather have retrograded than advanced in sound philosophy from the time of Thales. He is said to have believed that the sun is a mass of red-hot iron, or of heated stone, somewhat bigger than the Peloponnesus,— that the heaven is a vault of stones, which is prevented from tumbling only by the rapidity of its circular motion,— and that the sun is prevented from advancing beyond the tropics by a thick and dense atmosphere, which forces [7:3:728]him to retrace his course. These absurd notions are pro- ' bably greatly exaggerated; but it does not appear that Anaxagoras contributed much to extend the knowledge of the heavens. A melancholy interest is, however, excited in his behalf, on account of the persecution which he suffered in consequence of his liberal opinions and his disregard for the superstitious notions of his age. Having shown the reason of the eclipses of the moon, he was accused of ascribing to natural causes the attributes and power of the gods. Having taught the existence of only one God, he was accused of impiety and treason towards his country. Sentence of death was pronounced on the philosopher and all his family; and it required the powerful interest of his friend and disciple Pericles to obtain a commutation of this iniquitous sentence into one of perpetual banishment.

While the Ionian sect was so successfully employed in cultivating and propagating a knowledge of nature in Greece, another, still more celebrated, was founded in Italy by Pythagoras. This renowned philosopher was in early youth a disciple of Thales. In quest of knowledge, which in those days could only be obtained by visiting the sages of foreign lands, he travelled into Egypt, Phoenicia, Chaldea, and India, where his memory is said still to subsist. Through the favour of Amadis, king of Egypt, to whom he was recommended by Polycrates, the tyrant of Samos, he was admitted into the sacred college at Memphis, though with great reluctance on the part of the priests. The severe ordeal through which these charlatans compelled him to pass, before they would consent to initiate him into their mysteries, was sufficient to have deterred the most courageous votary of knowledge; and Pythagoras was probably the only stranger who ever succeeded in fully exploring their secrets. After an absence of thirty years he returned to Greece, and began to give instructions in his native island of Samos. Soon after, he passed over to the Grecian colony established at Tarentum in Italy, and settled at Crotona, where he speedily acquired a splendid reputation. He was the first who assumed the modest title of philosopher, or lover of wisdom: formerly those who devoted themselves to the acquisition of learning were called sophists or sages.

Pythagoras is said to have acquired in Egypt the knowledge of the obliquity of the ecliptic, and that of the identity of the morning and evening stars. What he chiefly deserves to be commemorated for in the history of astronomy, is his philosophical doctrine regarding the motion of the earth. He taught publicly that the earth is placed at the centre of the universe; but among his chosen disciples he propagated the doctrine that the sun occupies the centre of the planetary world, and that the earth is a planet circulating about the sun. This system, which still retains his name, being called the old or Pythagorean system of the universe, is that which was revived by Copernicus. It is, however, only just to the memory of this last mentioned great man to observe, that there is a vast difference between the bare statement of the possibility of a fact, and the demonstration of its existence by irrefragable arguments. Pythagoras having remarked the relation which subsists between the tone of a musical chord and the rapidity of its vibration, was led by analogy to extend the same relation to the planets, and to suppose that they emit sounds proportional to their respective distances, and form a celestial concert too melodious to affect the gross organs of mankind. Another fancy into which he was led by his passion for analogies was the application of the five geometrical solids to the elements of the world. The cube symbolically represented the earth; the pyramid, fire; the octaedron, air; the icosaedron, or twenty-sided figure, water; and the do-decaedron, or figure with twelve faces, the exterior sphere of the universe. Pythagoras left no writings; and it is doubtful whether he really entertained many of the opinions and reveries which have usually been ascribed to him.

Philolaus of Crotona, a disciple of Pythagoras, embraced the doctrine of his master with regard to the revolution of the earth about the sun. He supposed the sun to be a disk of glass which reflects the light of the world. He made the lunar month consist of 29∣ days, the lunar year of 354 days, and the solar year of 365∣ days.

Nicetas of Syracuse seems to have been the first who openly taught the Pythagorean system of the universe. Cicero, on the authority of Theophrastus, the ancient historian of astronomy, gives hirtι the credit of maintaining that the apparent motion of the stars arises from the diurnal motion of the earth about its axis;^[2. “Nicetas Syracusius, uţ ait Theophrastus, coelum, solem, lunam, stellas, supera denique omnia, stare censet; neque, praeter terram, rem ullam in mundo moveri; quae cum circum axem se summa celeritate convertat et torqueat, eadem effici omnia, quasi, stante terra, coelum moveretur.” (Cicero, Acad. Qiioest. : Opera, tom. iv. p. 39, edit. Bipont.) Copernicus himself could not have stated the doctrine with greater precision. ] but this’rational doctrine seems to have been first broached by Heraclides of Pontus, and Ecphantus, a disciple of Pythagoras.

The introduction of the Metonic cycle forms an era in the history of the early astronomy of Greece. The Chaldeans, as we have already stated, established several lunisolar periods; and the difficulty of reconciling the motions of the sun and moon, or of assigning a period at the end of which these two luminaries again occupy the same positions relatively to the stars, had long embarrassed those who had the care of regulating the festivals. Meton and Euctemon had the honour of first obviating this difficulty, at least for a time; for the motions of the sun and moon being incommensurable, no period can be assigned which will bring them back to precisely the same situations. These two astronomers formed a cycle of nineteen lunar years, twelve of which contained each 12 lunations, and the seven others each 13, which they intercalated among the former. It had long been known that the synodic month consisted of 29∣ days nearly; and in order to avoid the fraction, it had been usual to make the twelve synodic months, which compose the solar year, to consist of 29 and 30 days alternately; the former being called deficient and the latter full months. Meton made his period to consist of 125 full and 110 deficient months, which gives 6940 days for the 235 lunations, and is nearly equal to 19 solar years. This cycle commenced on the 16th of July in the year 433 b. c. It was received with acclamation by the people assembled at the Olympic games, and adopted in all the cities and colonies of Greece. It was also engraved in golden letters on tables of brass, whence it received the appellation of the golden number, and has been the basis of the calendars of all the nations of modern Europe. It is still in ecclesiastical use, with such modifications as time has rendered necessary.

Eudoxus of Cnidus, about the year 370 b. c. , obtained; great reputation as an astronomer. According to Pliny,' he introduced the year of 365į-days into Greece. Archimedes says that he supposed the diameter of the sun to be nine times greater than that of the moon, which shows that he had in some degree overcome the illusions [7:3:729]of sense. The titles of three of his works have been preserved—the Period or Circumference of the Earth, the Phenomena, and the Mirror. His observatory was still standing at Cnidus in the time of Strabo. His memory deserves to be honoured for the contempt which he evinced for the Chaldean predictions, and for having contributed to separate true astronomy from the reveries of judicial astrology. Eudoxus seems to have been the first who attempted to give a mechanical explanation of the apparent motions of the planets. He supposed each planet to occupy a particular part of the heavens, and that the path which it describes is determined by the combined motion of several spheres performed in different directions. The sun and moon had each three spheres; one revolving round an axis which passes through the poles of the world, and which occasions the diurnal motion; a second revolving round the poles of the ecliptic, in a contrary direction, and causing the annual and monthly revolutions; the third revolving in a direction perpendicular to the first, and causing the changes of declination. Each of the planets had a fourth sphere to explain the stations and retrogradations. As new inequalities and motions were discovered, new spheres were added, till the machinery became so complicated as to be altogether unintelligible.

Although Plato can hardly be cited as an astronomer, yet the progress of the science was accelerated by means of the lights struck out by his sublime and penetrating genius. He seems to have had just notions of the causes of eclipses; and he imagined that the celestial bodies originally moved in straight lines, but that gravity altered their directions, and compelled them to move in'curves. He proposed to astronomers the problem of representing the courses of the stars and planets by circular and regular motions. Geometry was assiduously cultivated in the school of Plato; and on this account he claims a distinguished place among the promoters of true astronomy.

Astronomy is also under some obligations to Aristotle. In a treatise which he composed on this science, he recorded a number of observations which he had made; and, among others, mentions an eclipse of Mars by the moon, and the occultation of a star in the constellation Gemini by the planet Jupiter. As such phenomena are of rare occurrence, their observation proves that he had paid considerable attention to the planetary motions.

A great number of astronomers about this time appear on the stage, whose labours and observations prepared the way for the reformation of the science which was shortly after effected by Hipparchus. Helicon of Cizi-cene is renowned for the prediction of an eclipse, which took place, as Plutarch affirms, at the time announced. History records the names of only three individuals in ancient Greece who predicted eclipses, Thales, Helicon, and Eudcmus. Eudcmus composed a history of astronomy, a fragment of which, consisting of only a few lines, is preserved by Fabricius in the Bibliotheca Groeca. In this it is mentioned that the axes of the ecliptic and equator arc separated from each other by the side of a pente-decagon, which is equivalent to saying that they contain an angle of 24°. This is the first value which we find assigned by the Greeks to the obliquity of the ecliptic. It is given in round numbers, and may easily be supposed to contain an error of a quarter of a degree.

Calippus is celebrated for the period which he formed of four Metonic cycles. Having observed, by means of an eclipse of the moon which took place about six years before the death of Alexander, that the Metonic cycle contained an error of a fourth of a day, he introduced the period of 940 lunations, containing four Metonic cycles, diminished by one day. He likewise formed a collection of observations on the heliacal risings of the planets. Theophrastus wrote a history of astronomy, and supposed the milky way to be produced by the imperfect junction of the two hemispheres, which allowed the light to penetrate from the firmament beyond. Autolycus of Pitaneas wrote two books, one on the movable sphere, the other on the risings and settings of the stars. These are the most ancient of the astronomical works of the Greeks which have come down to our times.

Pytheas of Marseilles, about the time of Alexander the Great, determined the length of the solstitial shadows in various countries by means of the gnomon. He found the shadows equal at Marseilles and Byzantium—a circumstance which does not give a favourable idea of the accuracy of his observations, inasmuch as the difference of the latitudes of the two places amounts to 2∣ degrees. The observation is, however, interesting, as it is the most ancient of the kind which has been preserved after that of Tcheou-Kong, and as it confirms the successive diminution of the obliquity of the ecliptic. Pytheas undertook several voyages for the purpose of obtaining geographical and astronomical information, and advanced northwards as far as Iceland. His relations have been treated as fabulous by Strabo and Polybius, but the accuracy of the greater number of them has been confirmed by modern observation and experience. He was the first who distinguished the climates by the different lengths of the days and nights.

Astronomy in the School of Alexandria.

In the history of the various sects which have hitherto come under our review, we meet only with some useful remarks, with numerous hypotheses and conjectures, but with scarcely any appearance of regular and connected science. Up to this date the astronomical knowledge of the Greeks was confined to a few facts, the discovery of which implies no theory, and scarcely the aid even of the simplest instruments. The order and arrangement of the planets, the causes of eclipses, the identity of the morning and evening stars, the approximate length of the year, that of the lunar month, the obliquity of the ecliptic, and the cycles of Meton and Calippus, were almost the sole results of their astronomical speculations. In the Alexandrian school we meet for the first time with regular and systematic observations. We there find angular distances measured with appropriate instruments, and calculations made according to the rules of trigonometry.

After the premature death of Alexander, his principal generals shared among themselves his magnificent conquests, and Egypt fell to the lot of Ptolemy Soter. This prince was distinguished by an ardent love of science, and a desire to promote every species of liberal knowledge. He accordingly invited to his court, which he had established at Alexandria, the most eminent philosophers of Greece, and fixed them there by his liberality and munificent protection. His son, Ptolemy Philadelphus, who inherited his throne, also inherited his genius and love of science and learning. A superb edifice, styled the Museum, was assigned to the use of the men of science whom he had attracted to his capital, to which he also added an observatory, and the famous library, which had been collected with great care and at a vast expense by Demetrius Phalerius. The prince took great delight in the Museum; he visited it frequently, entered into familiar conversation with its inmates on the subject of their various pursuits, and by his own example stimulated their [7:3:730] zeal and encouraged their inquiries. This noble institution, which survived all the vicissitudes of nine centuries, was the means of conferring incalculable benefits on the human race; and the name of its founder, Ptolemy Philadelphus, will be gratefully remembered while science and learning occupy a place in the estimation of mankind.

The first astronomers of the Alexandrian school were Aristillus and Timocharis, who flourished under the first Ptolemy, about 300 years before Christ. The chief object of their labours was the determination of the relative positions of the principal stars of the zodiac, instead of merely announcing their risings and settings, as had been the practice of the orientals and the ancient Greeks. The observations of these two astronomers conducted Hipparchus to the important discovery of the precession of the equinoxes, and served as the basis of the theory which Ptolemy, some centuries afterwards, gave of that phenomenon.

Aristarchus of Samos, the next in order of the Alexandrian astronomers, composed a treatise on the Magnitudes and Distances of the sun and moon, which has been preserved to our times. In this treatise he describes an ingenious method which he employed to obtain the relative distances of the two luminaries. At the instant when the moon is dichotomized, that is, when the exact half of her disk appears to a spectator on the earth to be illuminated by the sun’s light, the visual ray passing from the centre of the moon to the eye of the observer is perpendicular to the line which joins the centre of the moon and sun. At that instant, therefore, he measured the angular distance of the two bodies, and finding it to be 87 degrees, he concluded, by the resolution of a right-angled triangle, that the distance of the sun is between eighteen and nineteen times greater than that of the moon. This method is perfectly correct in theory, but it is difficult to be assured of the exact instant of the moon’s dichotomy, and in an angle of such magnitude a very small error greatly affects the result. The error of Aristarchus is very considerable, the true angle being about 87° 50'. The estimated distance of the sun is by consequence far too small; yet the determination, faulty as it was, contributed to expand greatly the existing notions relative to the boundaries of the universe, for the Pythagoreans had taught that the sun is only three, or at most three and a half times more distant than the moon. Another delicate observation made by Aristarchus was that of the magnitude of the sun’s diameter, which, as we learn from Archimedes, he determined to be the 720th part of the circumference of the circle which the sun describes in his diurnal revolution. This estimate is not very far from the truth, and the observation is by no means an easy one. He embraced the doctrine of Pythagoras respecting the earth’s motion, and appears to have entertained juster notions than any of the astronomers who preceded him, on the magnitude and extent of the universe. The treatise on the Magnitudes and Distances is published in the third volume of the works of Dr Wallis, with a Latin translation by Commandine, and Some notes.

Eratosthenes, the successor of Aristarchus, was a native of Cyrene, and invited to Alexandria by Ptolemy Evergetes, who appointed him keeper of the royal library. He is supposed to have been the inventor of armillary spheres, a species of instrument extensively used by the ancient astronomers. Byjneans of an instrument of this kind he observed the distance between the tropics to be to the whole circumference of a great circle as 11 to 83; a ratio equivalent to 47° 42' 39", half of which gives 23° 51' 19∙5" for the obliquity of the ecliptic. This is a very important observation, and confirms the gradual diminution of the obliquity as indicated by theory. Eratosthenes is Celebrated for being the first who attempted, on correct principles, to determine the magnitude of the earth. Having remarked, by some means with which we are unacquainted, that Syene, the most southern of the cities of ancient Egypt, is situated nearly on the same meridian with Alexandria, he conceived the idea of determining the amplitude of the celestial arch intercepted between the zeniths of the two places, and of measuring at the same time their distance on the ground; operations which would afford data for the determination of the whole length of the terrestrial meridian. Syene was known to be situated exactly under the tropic; for at the summer solstice the gnomon had no shadow, and the sun’s rays illumined the bottom of a deep well in that city. On the day of the solstice he found the meridional distance of the sun from the zenith of Alexandria to be 7° 12', or a fiftieth part of the circumference. It had also been ascertained by the bematists or surveyors of Alexander and the Ptolemies, that the itinerary distance between Alexandria and Syene was 5000 stadia; therefore 5000 × 50 = 250,000 stadia form the circumference of a great circle of the earth, or the length of the terrestrial meridian. Unfortunately, on account of the uncertainty respecting the length of the stadium here employed, we possess no means of estimating the degree of approximation afforded by this rude though ingenious attempt; but the idea does immortal honour to Eratosthenes, and the moderns have added nothing to his method: their better success is owing solely to the progress of the arts and the perfection of astronomical instruments.

About this time the science of astronomy was enriched by the discoveries of some of the distinguished geometricians whose labours have so greatly extended the glory of the Alexandrian school. Euclid, the celebrated author of the Elements, lived in the reign of the first Ptolemy. He composed a book on the sphere, which probably served as a model for future works of the same kind, and was the first who treated in a geometrical manner of the phenomena of the different inclinations of the sphere. Conon of Samos, the friend of Archimedes, collected the records of eclipses, which had been observed by the ancient Egyptians; and Callimachus ascribes to him the constellation of Berenice’s hair. Archimedes, whose profound genius and deep knowledge of geometry and mechanics entitle him to the appellation of the Newton of the ancients, also claims a high rank among the cultivators of astronomy. His celebrated planetarium, which represented the motions of the sun, moon, planets, and starry sphere, has been a frequent theme of the admiration and praises of the poets:

Jura poli, rerumque fidem, legesque deorum, Ecce Syracusius transtulit arte senex.

Apollonius of Perga solved the important problem of the stations and retrogradations of the planets by means of epicyles and deferents; and he is entitled to the glory of having formed the alliance between the two sciences of geometry and astronomy, which has been productive of the greatest advantages to both.

Astronomy, which had as yet only consisted of a knowledge of isolated facts, acquired a systematic form, and almost a new existence, from the genius and assiduity of Hipparchus, one of the most astonishing men of antiquity, and perhaps the greatest of all in the sciences which are not purely speculative. This illustrious founder of astronomical science was born at Nice in Bythinia, and observed[7:3:731] at Rhodes. Flamsteed and Cassini, probably mis-Aed by some ambiguous expressions of Ptolemy, have related that his observations were made at Alexandria; and this opinion seems generally to have been adopted by historians. The question has been examined carefully, and at considerable length, by Delambre (Astronomie Ancienne), who comes to the conclusion that there is no reason whatever to infer that Hipparchus ever saw Alexandria. Ptolemy, in reporting the observations of Hipparchus, supposed Rhodes and Alexandria to be situated on the same meridian, and consequently does not find it necessary to mention the place at which the observations were made.

Hipparchus commenced his brilliant career by verifying the determination of the obliquity of the ecliptic made by Eratosthenes. He next directed his attention to the length of the tropical year. By comparing an observation of his own, of the summer solstice, with a similar one made by Aristarchus 140 years before, he found that the anciently received value of 365į days was too great by seven minutes. This leaves the tropical year a value still too great; but it is probable that the error arose from the inaccuracy of the observation of Aristarchus: for the observations of Hipparchus, compared with those of the moderns, make the length of the tropical year amount to 365 days, 5 hours, and 49 minutes, which is only 12 seconds greater than the truth. By a careful observation of the solstices and equinoxes, he discovered that the year is not divided by these points into four equal parts, the sun occupying 94⅜ days in passing from the vernal equinox to the summer solstice, and only 92į from the same solstice to the equinox of autumn. The sun, consequently, remained 187 days in that part of the ecliptic which lies between the equator and the north pole, and therefore only about 178 in the other part. This observation led Hipparchus to the great discovery of the eccentricity of the solar orbit. He accounted for the apparent inequality of the sun’s motion, by supposing that the earth is not placed exactly at the centre of the circular orbit of the sun, and that consequently his distance from the earth is subject to variation. When the sun is at his greatest distance, he appears to move more slowly; and when he approaches nearer, his motion becomes more rapid. The distance of the earth from the centre of the orbit is called the eccentricity : it produces an equation between the real and apparent motions, which is called the equation of the centre. He determined the magnitude of this equation in terms of the radius of the ecliptic, and fixed the position of the line of the apsides, or that which joins the two opposite points of the orbit which are at the greatest and least distance from the earth. With these data he formed the first tables of the sun which are mentioned in the history of astronomy. The discovery of the eccentricity also led Hipparchus to that of the inequality of the lengths of the solar days at different seasons of the year. In the interval which elapses between the sun’s passage over the meridian and his return to it the following day, the sun advances by his own proper motion towards the east nearly a degree. But the rate of this motion is unequal, varying between 57 and 61 minutes of a degree; and the accumulation of · the inequalities forms what iā called the equation of time, that is, the difference between the true time, as shown by the sun, and the mean time, shown by a well-regulated clock, the motions of which are equal and uniform.

The attention of Hipparchus was next directed to the motions of the moon; and on this subject his researches were attended with equal success. From the comparison of a great number of the most circumstantial and accurate-observations of eclipses recorded by the Chaldeans, he was enabled to determine the period of the moon’s revolution relatively to the stars, to the sun, to her nodes, and to her apogee. These determinations are among the most precious relics of ancient astronomy, inasmuch as they corroborate the results of theory in one of its finest deductions—the acceleration of the mean lunar motion—and thus furnish one of the most delicate tests of the truth of Newton’s law of gravitation. It was, indeed, by a comparison of the observations of Hipparchus with those of the Arabian and modern astronomers, that Dr Halley was led to the discovery of that curious and important phenomenon. Hipparchus also determined the eccentricity of the lunar orbit, and its inclination to the plane of the ecliptic; and the values which he assigned to these elements, making allowance for the evection and the inequalities of the moon’s motion in latitude, are to a few minutes the same as those which are now observed. He had also an idea of the second inequality of the moon’s motion, namely, the eveetion, and made all the necessary preparations for a discovery which was reserved for Ptolemy. He likewise approximated to the parallax of the moon, which he attempted to deduce from that of the sun, by determining the length of the frustum cut off from the cone of the terrestrial shadow by the moon when she traverses it in her eclipses. From the parallax he concluded that the greatest and least distances of the moon are respectively equal to 78 and 67 semi-diameters of the earth, and that the distance of the sun is equal to 1300 of the same semi-diameters. The first of these determinations exceeds the truth; the second falls greatly short of it, the distance of the sun being nearly equal to 24,000 terrestrial semi-diameters. It may, however, be remarked that Ptolemy, who undertook to correct Hipparchus with regard to the parallax, deviated still farther from the truth.

The apparition of a new star in the time of Hipparchus induced him to undertake the formation of a catalogue of all the stars visible above his horizon, to fix their relative positions, and mark their configurations, in order that posterity might have the means of observing any changes which might in future take place in the state of the heavens. This arduous undertaking was rewarded by the important discovery of the precession of the equinoxes, one of the fundamental elements of astronomy. By comparing his own observations with those of Aristillus and Timocharis, he found that the first point of Aries, which, in the time of these astronomers, or 150 years before, corresponded with the vernal equinox, had advanced two degrees, according to the order of the signs, or at the rate of 48 seconds a year. This determination is not very far from the truth; for, according to modern observations, the rate of the precession is about 50T seconds annually. His catalogue contained 1080 stars: it is generally, but erroneously, stated to have contained only 1022, after that of Ptolemy, in which the nebulous and some obscure stars are omitted. He also commenced a series of observations to furnish his successors with the means of forming a theory of the planets. Hipparchus likewise invented the planisphere, or method of representing the starry firmament on a plane surface, which afforded the means of solving the problems of spherical trigonometry in a manner often more exact and more commodious than the globe itself. He was the first who demonstrated the methods of calculating triangles, whether rectilineal or spherical; and he constructed a table of chords, from which he drew nearly the same advantages as we derive at present from the tables of sines. Geography is also indebted to him for the happy idea of fixing the position of places on [7:3:732] the earth by means of their latitudes and longitudes; 'and he was the first who determined the longitude by the eclipses of the moon.

These various labours and brilliant discoveries give a high idea of the industry and genius of Hipparchus. His writings have unfortunately all perished, excepting a commentary on the poem of Aratus; but the principal elements of his theories, together with a few observations, have been preserved in the Almagest of Ptolemy.

After the death of Hipparchus, nearly three centuries elapsed before any successor arose worthy of the name. During this long period astronomy gained no essential advancement. Some rude observations, scarcely superior to those of the Chaldeans, and a few meagre treatises, are the only monuments which exist to testify that science had not fallen into utter oblivion in an age so fertile of poets and orators. Geminus and Cleomedes wrote treatises, which have been preserved to our times; Agrippa and Menelaus are said to have observed; the Roman calendar was reformed by Julius Caesar and the Egyptian astronomer Sosigenes; and Posidonius measured a degree, and remarked that the laws of the tides depend on the motions of the sun and moon.

Ptolemy was born at Ptolemais in Egypt, and flourished at Alexandria about the 130th year of our era, under the reigns of Adrian and Antoninus. This illustrious ornament of the Alexandrian school is entitled by his own discoveries to the high rank among astronomers which has universally been assigned to him; but the most signal service which he conferred on science was the collection and arrangement of the ancient observations. Out of these materials he formed the Miyaλt; '∑υwaξις, or Great Composition, a collection which exhibits a complete view of the state of astronomy in the time of Ptolemy, and which contains the germ of most of the methods in use at the present day.

The hypothesis which Ptolemy adopted for the purpose of explaining the apparent motions, was that which had been followed by Hipparchus. We have already seen that the genius of Pythagoras, soaring above the illusions of sense, had conceived the sun to be situated at the centre of the universe, and the earth to circulate, like the other planets, about the sun; and that the same opinion was entertained and supported by Aristarchus and a few other astronomers. It would seem, however, that this philosophical idea never gained much ground in antiquity, even among the learned. The vulgar prejudice respecting the immobility of the earth continued to prevail; and it had become an inveterate axiom, that all the celestial motions must be circular and uniform. Ptolemy himself, who felt in its full force the difficulty of reconciling the appearances with the notion of a uniform circular motion, adopted the common opinion without scruple as a primordial law of the universe; for, says he, this perfection belongs to the essence of celestial things, which neither admit of disorder nor irregularity. To save this chimera—· the uniform circular motion—Apollonius imagined the ingenious apparatus of epicycles and deferents; and Hipparchus advanced a step farther, by placing the centre of the sun’s circle at a small distance from the earth. Ptolemy adopted both hypotheses, and supposed the planet to describe an epicycle by a uniform revolution in a circle, the centre of which was carried forward uniformly in an eccentric round the earth. By means of these suppositions, and by assigning proper relations between the radii of the epicycle and deferent circle, and also between the velocity of the planet and the centre of its epicycle, he was enabled to represent with tolerable accuracy the apparent motions of the planets, and particularly the phenomena of the stations and retrogradations, which formed the principal object of the researches of the ancient as- 1 tronomers. The notions of Apollonius and Hipparchus were thus reduced to a systematic form, and the proportions of the eccentrics and epicycles of all the planets assigned, by Ptolemy; on which account the system has been generally ascribed to him, and obtained the name of the Ptolemaic System of the universe. As a first attempt to bring the celestial motions within the grasp of geometry, it does infinite honour to the genius of its inventors. It is, however, totally irreconcilable with the precision of modern observations; for it is impossible to represent on this hypothesis the variations of the distances of the planets at the same time with their apparent motions. But this difficulty could scarcely be felt by Ptolemy, inasmuch as it was impossible, before the invention of the telescope and micrometer, to form any accurate estimate of the variations of the apparent diameter of a planet, and consequently of its distance. It must be admitted, however, that the Ptolemaic hypothesis might be sufficient for the wants of practical astronomy, that is, for calculating the places of the planets and forming tables of their motions, were it not for its extreme complication. The discovery of every new irregularity in the planetary motions exacts the addition of a new epicycle; and such was the confusion resulting from this circumstance, that Alphonso X., despairing of being able to comprehend the complicated machinery, was tempted to exclaim, that if the Deity had called him to his counsels at the creation of the world, he could have given good advice. Yet, notwithstanding all its defects, the system of Ptolemy gained a complete ascendency over the minds of mankind, and, so difficult is it to leave the beaten path, continued to be implicitly followed by every astronomer during fourteen centuries, having been only finally exploded by Kepler’s discovery of the elliptic orbit of Mars.

The most important discovery which astronomy owes to Ptolemy is that of the Evection of tfie moon. Hipparchus had discovered the first lunar inequality, or the equation of the centre, which serves to correct the mean motion at the syzygies, and had also remarked the necessity of another correction for the quadratures. He even undertook a set of observations, with a view to ascertain its amount and its law; but death put a stop to his labours before he had brought them to a successful issue. Ptolemy completed the investigation, and discovered that the eccentricity of the lunar orbit is itself subject to an annual variation, depending on the motion of the line of the apsides. The variation of the position of the apsides produces an inequality of the moon’s motion in her quarters, which has been technically denominated the evection. The equation given by Ptolemy, though of course empirical, is remarkably exact.

Ptolemy employed a very simple process for determining the moon’s parallax, which was probably suggested to him by the situation of Alexandria, where he observed. He determined the latitude of a place a little to the south of that city, over the zenith of which the moon was observed to pass when her northern declination was the greatest possible. But when the moon is in the zenith, or in the same’straight line with the observer and the centre of the earth, she has no parallax; consequently the obliquity of the ecliptic and the latitude of the station being known, the moon’s greatest northern latitude was also determined. The next step was to observe the moons meridian altitude fifteen days after the first observation, when her southern latitude was necessarily the greatest possible. This observation gave the apparent altitude of the moon, but her greatest northern and southern decli [7:3:733]nations being supposed equal, her true altitude, as seen from the centre of the earth, was easily computed from the previous observation, and the difference between the true and apparent altitudes gave the amount of the parallax.

The observations of Hipparchus relative to the motion of the stars in longitude, or the regression of the equinoctial points, were confirmed by Ptolemy, although he mistook its amount, and diminished a quantity which Hipparchus had already estimated too low. According to Hipparchus, the regression is at the rate of two degrees in 150 years. Ptolemy reduced it to one degree in 90 years. This disagreement would seem to indicate an error of more than a degree in the observations, which can with difficulty be admitted, considering the accordance which subsists among the different observations cited by Ptolemy in support of his own determination. For this and some other reasons Ptolemy has been accused of altering the observations of Hipparchus, and accommodating them to his own theory; but there does not appear to be any just ground for the imputation. The error with regard to the regression probably arose from the circumstance, that Hipparchus had assigned too great a value to the length of the year, whence the motion of the sun with regard to the equinoxes would be made too slow, and the longitudes employed by Ptolemy consequently diminished.

Ptolemy has been called the Prince of astronomers,—a title which may perhaps be justified by the universal and long-continued prevalence of his system, but to which he has no claim from the number or value of his own observations. After a laborious and minute examination of the Almagest, Delambre doubts whether any thing, saving the author’s declarations, is contained in that great work, from which it can be decisively inferred that Ptolemy ever observed at all. He indeed frequently makes mention of observations made by himself; but his solar tables, rate of the precession, eclipses, determination of the moon’s motion and parallax, and, above all, his catalogue of stars, render it impossible to doubt that the greater part of the results which he has given as observations are merely computed from the tables of Hipparchus. It is therefore difficult to allow to Ptolemy that good faith and “astronomical probity which forms one of the most indispensable qualities of an observer.” He never in any instance cites a single observation more than is just necessary for the object he has immediately in view, and consequently, by precluding all comparison of one observation with another, has deprived us of the means even of guessing at the probable amount of the errors of his solar, lunar, and planetary tables. If an astronomer, as Delambre justly remarks, were to adopt the same course at the present day, he would be certain of forfeiting all claim to confidence; but Ptolemy stood alone; he had neither judges nor rivals; he claimed admiration, and received it; and now no one condescends to calculate the few observations he has left us. (Delambre, Astronomie Ancienne, tom. i. Discours Préliminaire.) His catalogue contains only 1029 stars, and is therefore less extensive than that of Hipparchus, but it is exceedingly valuable on account of its details.

The name Almagest (Meγiorn, with the Arabic prefix) was bestowed on the Syntax by the Arabians, into whose language it was translated in the ninth century. The first Latin translation was from the Arabic, and published at Venice in 1515. It abounds in Arabic words and idioms, and is very inaccurate and barbarous. The second Latin translation was made from the original Greek by George of Trebizond, and is greatly superior to the first. It was published at Basle in 1541, and in 1551. The Greek text was published at the same time in 1538. Ptolemy was the author of numerous other works connected with astronomy, of which his Geography, ⅛ e ight books, is the best known. It contains a list of all the places of which the latitudes and longitudes had at that time been determined. His treatise on Optics was supposed to be lost, till an imperfect Latin translation, from an Arabic version, was lately discovered in the king’s library at Paris. The last book of this work contains a theory of astronomical refraction, more complete than any which existed before that of Cassini. It would seem that Ptolemy had not discovered the refraction at the time he composed the Almagest, no mention being made of the subject in that work. The explanation which he gives of the phenomenon is natural and satisfactory, indeed entirely conformable with that which is now universally adopted. The idea and explanation remained buried in the Optics till reproduced by Alhazen; but neither Ptolemy nor Alhazen attempted to estimate the amount of the refraction. His Planisphere and Analemma, in which he treats of the stereographic and orthographic projections of the sphere, show a perfect acquaintance with spherical trigonometry. In the last-mentioned work he makes use of the sines, and his constructions comprehend three of the four general theorems in modern use. Divers treatises also on music, dialling, chronology, and mechanics, attest the universality of Ptolemy’s genius, and his unremitting application to the pursuits of science. Like Archimedes, he had a desire to transmit to posterity the history of his labours by a public monument. In the temple of Serapis, at Canopus, he is said to have consecrated a marble pillar, with an inscription containing the principal elements of his astronomy, such as the length of the year, the eccentricity of the solar and lunar orbits, the dimensions and forms of the epicycles of the planets, &c.

On the death of Ptolemy astronomy ceased to be cultivated among the Greeks. The Alexandrian school subsisted indeed for some centuries after; but genuine science had fled, and its place been usurped by the vain wranglings of theologians and grammarians. During the long period of six or seven centuries, the labours of those who assumed the name of astronomers were confined to needless or trifling commentaries on the works of Hipparchus and Ptolemy, and were productive of no observations, or even remarks, having a tendency to enlarge the boundaries of the science. The genius of the Roman dominion was unfavourable to the development or exercise of the higher faculties of the human intellect; and the natural sciences, with the liberal arts, faded away under the withering influence of military despotism.

From the brief account which has now been given, it will be easily inferred that the Greeks cultivated astronomy rather as a speculative than a practical science. None of their numerous sects ever evinced any taste for observation or experiment; and hence, while geometry made great and rapid advances in their hands, physics and experimental philosophy were entirely neglected. The prevailing passion for speculation pervaded even their astronomy. They explained the doctrine of the sphere, and the apparent motions of the planets; and framed ingenious theories to account for such phenomena as came immediately under the cognizance of their senses; but if we except the observations of Hipparchus and Ptolemy, and perhaps two solstitial distances of the sun from the zenith, observed by Eratosthenes, we remark among them no observations made with instruments capable of measuring angular distances. Before Hipparchus, no mention is made of the astrolabe; and the recorded determinations do not [7:3:734] give us a very favourable idea of the accuracy of that instrument. On casting our eyes over the catalogue of Ptolemy, we scarcely ever meet with a fraction of a degree smaller than one-twelfth, that is to say, less than five minutes; whence we may infer that the astrolabe only measured twelfth parts of a degree. Occasionally, indeed, the fractions one-fourth and three-fourths occur; but these were most probably inserted by estimation. The Greeks of Alexandria committed an error of no less than 15' with regard to the altitude of the pole, one of the most essential elements to an observer; and it does not appear that they were ever able to determine the time to within a quarter of an hour. Yet notwithstanding these circumstances, which indicate that the art of observation was still in its infancy, the science of astronomy is vastly indebted to the labours and speculations of the Greeks. The complicated but ingenious hypotheses of Ptolemy prepared the way for the elliptic orbits and laws of Kepler, which, in their turn, conducted Newton to the great discovery of the law of gravitation.

Astronomy of the Arabians.

While the nations of western Europe were involved in the thickest shades of ignorance and barbarism, the torch of science was rekindled, and blazed forth with extraordinary splendour, among the Saracens. The burst of fanaticism which enabled the followers of Mahomet to carry their religion and their arms over the fairest portion of the ancient world subsided, in a great measure, after a century and a half of uninterrupted conquest, and was succeeded by a period of repose, during which they cultivated the arts of peace and civilisation with the same ardour which had characterized their achievements in arms. Under the enlightened and munificent protection of the caliphs, Bagdat became what Alexandria had been under the Ptolemies, the centre of politeness and knowledge.

The accounts which we possess of the Saracen literature are imperfect and scanty; but the first of the caliphs who appears to have encouraged the study of astronomy was Abougiafar, surnamed Almansor, or the Victorious, who reigned in the eighth century. His grandson Almarnon, the seventh of the Abassides, and second son of the famous Haroun Al Raschid, who reigned at Bagdat from 813 to 833, is celebrated for the protection which he gave to learning, and the zeal with which he laboured to propagate the sciences of the Greeks among his subjects. In granting peace to the emperor Michael HL, he stipulated for liberty to collect in Greece all the writings of the philosophers. These he transported into his own country, and caused to be translated into Arabic. Finding it mentioned in the geography of Ptolemy that a degree of the earth was equivalent to 500 stadia, he resolved to have this fact verified by a new measurement; and in obedience to the commands of the caliph, a company of mathematicians assembled in the spacious plain of Sinaar, where, having observed the altitude of the pole, they separated themselves into two parties, and proceeded in opposite directions along the meridian, measuring the distance they passed over till the altitude of the pole varied one degree. Being unacquainted with the nature of the instruments made use of in these geodetic operations, we cannot estimate the probable accuracy of the result; but as it agreed perfectly with the statement of Ptolemy, we have a right to infer that the measurement was executed in a very inadequate manner, and that the mathematicians of the caliph adopted the ancient determination from want of confidence in their own.

The Syntax of Ptolemy was translated into Arabic under the reign of Almamon, by Isaac Ben Honain. The translation was afterwards revised by Thabet or Thebith Ben Korah, and it was about this time that it received the appellation of Almagest. Astronomical observations, which, as we have had occasion to remark, had been greatly neglected by the successors of Hipparchus, formed a principal object of the attention of the Arabians. By the orders of Almamon, the obliquity of the ecliptic was observed, and found to be 23° 33'. According to the modern tables, the obliquity at that time was 23° 36' 34", so that the error was less than that of Hipparchus and Ptolemy, in their determination of the same element. This observation supposes instruments of some accuracy. Among the astronomers whom Almamon drew to his court, we find the names of Habash of Bagdat, who composed three books of astronomical tables; Ahmed, or Mohammed Ben Cothair, better known by' the name of Alfragan, or Alfranius, who, from his great expertness in computing, was styled the calculator. He composed an elementary treatise on astronomy, which was only an abridged extract of the works of Ptolemy; and likewise wrote on sun-dials, and gave a description of the astrolabe. The Jew Meshala, whose treatise on the elements was published at Nuremberg in 1549, also lived in the time of Almansor or Almamon.

The most celebrated of the Arabian astronomers was Albategnius, or Muhammed Ben Geber Albatani, so called from Batan, a city of Mesopotamia, where he was born. He was a prince of Syria, and resided at Aracte or Racha, in Mesopotamia; but many of his observations were made at Antioch. Having studied the Syntax of Ptolemy, and made himself acquainted with the methods practised by the Greek astronomers, he began to observe, and soon found that the places assigned to many of the stars in Ptolemy’s tables were considerably different from their actual situations, in consequence of the error which that great astronomer had committed with regard to the precession of the equinoxes. Albategnius measured the rate of the precession with greater accuracy than had been done by Ptolemy; and he had still better success in his attempt to determine the eccentricity of the solar orbit, his value of which differs extremely little from that which results from modern observations. In assigning the length of the year, however, he fell into an error of more than two minutes; but this proceeded, as has been shown by Dr Halley, from too great confidence in the observations of Ptolemy. Albategnius also remarked that the place of the sun’s apogee is not immovable, as former astronomers had supposed, but that it advances at a slow rate, according to the order of the signs,—a discovery which has been confirmed by the theory of gravitation. A new set of astronomical tables, more accurate than those of Ptolemy, likewise resulted from the indefatigable labours of Albategnius; and his observations, important in themselves, are doubly interesting on account that they form a link of connection between those of the astronomers of Alexandria and of modern Europe. The works of Albategnius were published in 1537, under the title of De Scientia Stellarum.

Thebith Ben Korah, another Arabian, acquired celebrity by proposing an explanation of the motions of the stars, which, under the name of the “System of Trepidation, was eagerly received by the astronomers of the middle ages, and disfigured the tables of Alphonse, and even those of Copernicus. He ascribed to the eighth sphere, or that of the fixed stars, two motions; one the diurnal motion, the other that of trepidation, performed in small circles round the first points of Aries and Libra, and. of [7:3:735]which the radii were 4° 1 S' 33". He therefore supposed two ecliptics, one fixed in the ninth sphere, the other movable in the eighth. According to this construction, the motion of the stars is sometimes direct and sometimes retrograde.

The Arabians have been said to be not only the cultivators but the apostles of the sciences, on account of the activity with which they propagated them among all the nations subjected to their dominion. The Fatimite caliphs, who reigned in Egypt during two centuries, rivalled their predecessors the Ptolemies in the encouragement which they gave to astronomy. Under the caliph Hakem, who reigned from 996 to 1021 of our era, Ebn Jounis acquired a splendid reputation. He constructed a set of tables, and composed a sort of celestial history, in which he has recorded numerous observations of his own and of other astronomers belonging to the same country. This work, imperfectly known through some extracts, long excited the curiosity of astronomers, as it was supposed to contain observations tending to establish the acceleration of the mean motion of the moon. A manuscript copy of it, belonging to the university of Leyden, was, in 1804, transmitted to the French Institute, and translated by Professor Caussin. It contains 28 observations of eclipses from the year 829 to 1004; seven observations of the equinoxes; one of the summer solstice; one of the obliquity of the ecliptic made at Damascus, by which the value of that element is found to be 23° 35'; and likewise a portion of tables of the sun and moon, with some other matter illustrative of the state of astronomy among the Arabians. The observations which regard the acceleration of the mean lunar motion are two eclipses of the sun and one of the moon, observed by Ebn Jounis, near Cairo, in the years 977, 978, and 979, and they agree with theory in confirmiug the existence of that phenomenon.

The Saracen conquests in Spain were attended with the same happy results as in Egypt, and science flourished in that country while the rest of Europe was involved in the darkest shades of ignorance. Arzachel is supposed to be the author of the Toledo Tables, constructed about the year 1180, but which, on account of the established reputation of those of Albategnius, were never in great estimation. He made some changes in the dimensions which had been assigned by Hipparchus and Ptolemy to the solar orbit, and deserves the praise of having been an exact and attentive observer. Alhazen, who flourished in the same country about the same period, contributed to the progress of astronomy by a treatise on Optics, in which he clearly indicated the necessity of making an allowance for the celestial refraction in astronomical observations. His treatise contained a theory of reflection and refraction, an explanation of the cause of the twilight, and of the magnitude of the horizontal moon. Averroes, a physician of Cordova, made an abridgement of the Almagest in the twelfth century, and Almansor found the obliquity of the ecliptic to be 23° 33' 30", which proves that practical astronomy had now attained to a tolerable degree of exactness.

If we inquire what effect the labours of the Arabians and their disciples had on the progress of astronomy, we shall find that their services were confined entirely to the practical part In point of theory they did absolutely nothing. They admitted all the hypotheses of Ptolemy without the slightest alteration, even with timid and superstitious respect, and did not advance a single step towards the discovery of the solar system. But with regard to instruments and methods of calculation, their improvements were numerous and important. They constructed instruments on a larger scale, and divided them with greater care; and, even from the time of Almamon, we remark among them new and more exact determinations of the obliquity of the ecliptic, of the positions of some stars, of the precession, of the length of the year, and of the eccentricity of the sun’s orbit. To these fundamental points they added numerous observations of eclipses and conjunctions; they industriously sought out and corrected the errors of Ptolemy’s tables; they perceived the necessity of marking the instant of each phenomenon with greater care; and their determinations of the commencement and end of eclipses are in general accompanied with the altitude of a star, which afforded them the means of calculating the hour, angle, and the true time. In cases where less precision was wanted, they made use of their clepsydrae and solar dials, to the construction of which they paid particular attention. Trigonometry derived signal advantages from their constant care to facilitate the calculations of spherical astronomy. Albategnius substituted the sines for the chords,—a most important improvement, the idea of which was probably suggested to him by the Analemma of Ptolemy. By this happy substitution the solution of all rectangular spherical triangles was reduced to four general formulae, of which the Greeks had the equivalent in a much less commodious form. The same astronomer also appears to have invented a very remarkable rule for the oblique-angled triangles, perfectly identical with one of the four general formulae now in use. Ebn Jounis, and his contemporary Aboul Wefa, were acquainted with the tangents and secants, and employed them very dexterously in reducing complicated binomial expressions to a single and simpler term. They also employed subsidiary arcs and other artifices in the calculus of the sines, in order to facilitate the labour of computation. These substitutions are now common; but they remained long unknown in Europe; and 700 years after they were employed by the Arabians, we first meet with some examples of their use in the writings of Thomas Simpson.

The zeal of the Arabians for astronomical observations was communicated by them to the Persians and Tartars. About the year 1072, Omar Cheyam determined the length of the tropical year, and introduced the calendar which has ever since been used in Persia. Holegu-Ilecou-Khan, who conquered that country about the year 1264, caused an observatory to be built at Maragha, near Tauris, where he assembled the most celebrated astronomers who could be found within his dominions, and employed them in forming new astronomical tables. This work was directed by the famous Nassireddin, and brought to a conclusion in the year 1269. With the exception of some trifling corrections of the mean motions, the whole of these tables are copied from Ptolemy.

Ulugh Beigh, a Tartar prince, and grandson of the great Tamerlane, not only encouraged the study of astronomy, but was himself a diligent and successful observer. At Samarcand, the capital of his dominions, he established an academy of astronomers, and caused the most magnificent instruments to be constructed for their use. By means of a gnomon 180 feet in height, he determined the obliquity of the ecliptic to be 23° 30' 20", the precession of the equinoxes at 1° in 70 years, and obtained elements for the construction of tables which have been found to be scarcely inferior in accuracy to those of Tycho Brahĕ. The ancient astronomy had produced only one catalogue of the fixed stars, that of Hipparchus. Ulugh Beigh has the honour of having formed a second, after an interval of sixteen centuries. This learned and munificent prince, whose virtue and talents deserved the esteem of mankind, was assassinated by his own son in the 58th year of his age.
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Revival of Astronomy in Europe.

After the death of Ulugh Beigh, astronomy received no farther accessions in the east. But the seeds of knowledge had now begun to take root in a more propitious soil, and Europe, destined to carry the development of the human energies to its fullest extent, began to awake from the lethargy in which it had continued during so many ages. The first dawnings of returning day appeared in Spain. In spite of the horror inspired by the Moslem religion, the Christians began to perceive and acknowledge the superiority and utility of the science of the Moors; and the schools of Cordova became the resort of all those whom curiosity, or love of knowledge, induced to seek abroad for that information which could not be obtained in their own countries. Geber, afterwards Pope Silvester IL, acquired the knowledge of arithmetic from that source; and John of Halifax, better known by the name of Sacrobosco, after having studied some time in Spain, made an abridgement of the Almagest, which was long famous under the title of a Treatise of the Sphere.

The emperor Frederick II. is no less celebrated for his protection of the sciences, than for the continual struggles in which he was involved with the popes. He founded the university of Naples, and caused Latin translations to be made of the works of Aristotle and the Almagest of Ptolemy. About the same time astronomy was zealously encouraged and cultivated by Alphonso X., king of Castile. This monarch, whose liberal mind seems to have been far superior to the age in which he lived, formed a college or lyceum at Toledo, the capital of his dominions, whither he assembled the most eminent astronomers that could be found, whether Christians, Moors, or Jews, and engaged them in the task of correcting the errors of the ancient tables. From their united labours were produced the Alphonsine Tables, which obtained great celebrity, and were, in some respects, superior in accuracy to airy which had preceded them. They are supposed to have been chiefly the work of the Rabbi Isaac Aben Sid, surnamed Hazan, inspector of the synagogue of Toledo. They are said to have cost the king 40,000 ducats,—a sum certainly far exceeding their real value, which is confined to the correction of some epochs, and a more accurate determination of the sun’s motions and the length of the year. The same century gave birth to several other individuals distinguished by their attachment to the sciences. Campanus of Nivari translated Euclid, and left a treatise on the sphere. Vitello, a native of Poland, composed a treatise on optics, in ten books; and Albert, bishop of Ratisbon, whom his contemporaries styled the great, was the author of some works on arithmetic, geometry, astronomy, and mechanics. But the greatest luminary of that age was Roger Bacon, a Franciscan friar, whose numerous works contain many indications of a powerful and inventive genius. He made many important discoveries in optics; but his knowledge of natural philosophy and chemistry, uncommon in those days, had nearly proved fatal to him; for he was suspected of necromancy, and thrown into a dungeon, from which he did not escape till he had satisfied his superiors and the pope that he had never held unlawful intercourse with the devil. He composed a work on the utility of astrology, the places of the stars, and the aspects of the moon; and he had the merit of perceiving the necessity of reforming the calendar.

The fourteenth century produced no astronomer from whose labours the science gained any accessions. George Purbach, or Beurbach, so named from a small town in Austria, where he was born in 1423, obtained great celebrity as a professor. He studied at Vienna, and after giving proofs of distinguished talents, he travelled into Italy, where he was favourably received by the cardinal of Cusa, who himself cultivated astronomy. On his return to Vienna he undertook a translation of the Almagest; and although ignorant both of Greek and Arabic, his perfect acquaintance with the subject enabled him to correct many errors which had been introduced through the carelessness or ignorance of former translators. He published a table of sines for every ten minutes to a radius of 6,000,000 parts, which was afterwards extended by his scholar Regiomontanus to every minute of the quadrant. The most celebrated of his works is his Theoricce Nova Planetarum, which was published in 1460. He constructed á celestial globe, on which was represented the motion of the stars in longitude from the time of Ptolemy to the year 1450. He also measured the obliquity of the ecliptic, and is considered as the inventor of decimal arithmetic. He died in 1461, having only reached his 38th year, with the reputation of being, at that time, the first astronomer in Europe.

Purbach had the good fortune to form a disciple who' executed many of the plans which had been interrupted 1 by his premature death. This was the celebrated John' Muller of Königsberg, better known by the name of Regiomontanus. Attracted in his youth to Vienna by the great reputation of Purbach, he continued to study there during ten years, and on the death of his master repaired to Rome for the purpose of acquiring tire Greek language, and of making himself, through it, acquainted with the Almagest. At Rome he continued his observations, and translated into Latin the works of Ptolcmy, the Conics of Apollonius, and some other treatises of ancient science. In 1471 he retired to Nuremberg, where, with the aid of Bernard Walther, a wealthy burgess, he founded an observatory, and furnished it with excellent instruments, principally of his own invention, by means of which he was enabled to detect many errors in the ancient tables. On the invitation of Pope Sixtus IV., who wished to reform the calendar, he again repaired to Rome; but after a few months’ abode there he died suddenly, according to some accounts, of the plague, according to others, through the effects of poison administered to him by the sons of George of Trebizond, who adopted this execrable method of revenging the exposure which he had made of their father’s errors in the translation of the Almagest. Regiomontanus was a learned and skilful man, but the great expectations which his early labours gave of future services to astronomy were disappointed by his Untimely death. He paid great attention to trigonometrical calculation; and, although he did not reach the point which had been attained by the Arabians, he had the merit of introducing some useful theorems which till then were entirely unknown in Europe. His genius, however, did not enable him to rise above the prejudices of his age, for he was an astrologer as well as an astronomer, and is said to have most lamented the errors of the Alphonsine Tables on account of the uncertainty which they occasioned in the calculation of genitures or horoscopes.

After the death of Regiomontanus, Walther continued to observe at Nuremberg during thirty years. His observations were collected by order of the senate of Nuremberg, and published by Schoner in 1544, a second time by Snellius, and, lastly, along with those of Tycho Brahé. In 1484 Walther began to make use of clocks, then a recent invention, to measure time in celestial observations. He was also the first who employed the planet Venus as a term of comparison for determining the longitudes of the stars.

Nuremberg had the honour of producing another astro [7:3:737]nomer of some celebrity. John Werner was the first who 'explained the method which was afterwards brought into general use by Maskelyne, of finding the longitude at sea, by observing the distance between a fixed star and the moon. He published some mathematical and geographical treatises, and made a number of observations to determine the obliquity of the ecliptic and the precession of the equinoxes.

• We are now come down to a period in the history of astro- , nomy when the science was destined to undergo a totalrenovation, and the system which had been so laboriously established by Ptolemy, and blindly followed as an article of religious belief by Arabs,’Persians, Tartars, and Europeans, during so many centuries, was about to be exploded for ever. In proportion as the observations became more numerous and accurate, the difficulty of representing them by the Ptolemaic system became greater; and astronomers were obliged to have recourse to the most violent and improbable suppositions in order to explain the phenomena, and, in the language of Ptolemy, save the appearances. We have mentioned that Pythagoras and his disciples entertained an idea very different from that which commonly prevailed, and supposed the sun to be the immovable centre of the celestial motions. It appears, however, from Aristotle, that this opinion was not founded on any analysis of the phenomena, but on certain metaphysical notions respecting the comparative dignity of the several elements. For example, fire, being a nobler substance than earth, ought to occupy the centre or place of honour. But such arguments could have little weight except in the schools, and accordingly were rejected by Ptolemy as too absurd to require a serious refutation. In order to give any probability to the ∙ Pythagorean doctrine, it was necessary to explain the succession of the seasons and the precession of the equinoxes on the hypothesis of the annual revolution of the earth about the sun; to show how the unequal motions of the planets in concentric orbits would give rise to the phenomena of the stations and retrogradations; to account, in short, for all the appearances, and point out their coherence and mutual connection. All this was effected by Copernicus, who had thereby the glory of first making known the true system of the universe, and of leading the way in that career of astronomical discovery in which the genius of the human race has gained its noblest trophies.

Nicholas Copernicus was born at Thorn, a city of Prussia, on the confines of Poland, according to Junctinus on the 19th of January 1472, and according to others on the 19th February 1473. From his earliest years he displayed a great fondness and aptitude for mathematical studies, and pursued them with corresponding success at the university of Cracow. Stimulated by the desire of acquiring a reputation equal to that of Regiomontanus, he set out for Italy at the age of 23 years, in order to study astronomy at Bologna under the celebrated Dominic Maria. He afterwards removed to Rome, where he employed himself in studying and teaching the mathematics, and where he made several astronomical observations about the year 1500. On his return to his native country he was made a canon of Ermeland by his uncle the bishop of Worms, and took up his residence at Frauenberg, a small Prussian town near the mouth of the Vistula, where he passed 36 years of his life in observing the heavens and meditating on the system of the world. In this retirement he composed his famous work entitled Astronomia Restaurata, sive de Revolutionibus Orbium Cceles- tium, in which he explained the celestial motions in a manner as simple and connected as the system of Ptolemy was complicated and incoherent. The system which Copernicus adopted in this work is now so familiar to every one, that it is almost unnecessary to describe it. The heaven, composed of stars perfectly at rest, occupies the remotest bounds of space, then the orbit of Saturn, next Jupiter, Mars, the Earth accompanied by its moon, Venus, Mercury, and, lastly, the Sun immovable at the centre. By this arrangement the stations and retrogradations of the planets became simple mathematical corollaries, following from the differences of the radii of their orbits and their unequal motions. The diurnal rotation of the earth explained more simply and rationally the apparent daily revolution of the heavens; and the precession of the equinoxes was referred to a small variation in the inclination of the earth’s axis to the plane of the ecliptic. But the simplicity of the system, and its consequent probability, were the only arguments which Copernicus was able to bring forward in proof of its reality. The motion of the earth can indeed never be made an object of sense; but after Richer’s discovery of the diminution of gravity towards the equator, it was impossible to doubt longer of the existence of its rotatory motion; and when Roemer had measured the velocity of light, and Bradley observed the phenomena of the aberration, the evidences of its annual revolution were rendered equally convincing. Great, however, as were the merits of Copernicus, it must be acknowledged that he left his system in a very imperfect state. After the example of the ancients, he assumed as an axiom the uniform circular motion of the planets; and as the only motions which are observed are in a state of incessant variation, he was obliged, in order to save the inequalities, to suppose a different centre to each of his orbits. The sun was placed within the orbit of each of the planets, but not in the centre of any of them, consequently he had no other office to perform than to distribute light and heat; and, excluded from any influence on the system, he became as it were a stranger to all the motions. Yet notwithstanding these and other imperfections, the establishment of the doctrine of the earth’s motion, with an evidence which dissipated the illusions of sense, was a great step towards the true knowledge of the planetary system; and when we consider the ignorance and prejudices of the age, and that Copernicus was moreover a priest, we cannot hesitate to admit his claim to a high rank among philosophers. But whether the actual services which he rendered to astronomy are commensurate with the great fame he has obtained, may admit of doubt. He revived an ancient opinion opposed to the prejudices and religious dogmas of his times, and fortified it with new and strong, though not absolutely convincing, proofs. It seldom happens, however, with regard to those sciences which ultimately appeal to experience, that general reasoning, even of the soundest kind, tends much to their real advancement; and there is little reason for thinking that astronomy would have been less perfect, or that any discoveries since made in it would have been retarded a single day, even if Copernicus had never lived. His great merit, like that of Lord Bacon, consists in the sound views which he took of nature, and in advancing so far before the general attainments of his age.

Fearing the opposition which was likely to arise from religious bigotry to opinions so much at-variance with vulgar prejudice, Copernicus long delayed the publication of his great work; and it was only at the urgent request of his friends that he at last allowed it to be printed. He is said to have received the last sheet of it only on the day of his death. He was buried in the cathedral of Frauenberg, and his only epitaph consisted of some spheres cut out in relief on his tombstone.
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The ideas of Copernicus soon spread over Germany, 'where astronomy was at that time diligently cultivated; but they do not seem to have met with general favour before the commencement of the seventeenth century. The art of observing was, however, gradually receiving improvement; instruments were constructed on better principles, and more accurately divided; and the methods of computation were rendered much less laborious. Nonius, or Nunez, a Portuguese, invented the ingenious method of subdividing the small divisions of instruments which still retains his name; Reinhold extended the table of tangents of Regiomontanus to every minute of the quadrant, reformed the tables of Copernicus, and composed many works of practical utility; but of the immediate successors of Copernicus, no one deserves to be more honourably mentioned than William IV., landgrave of Hesse. This prince built a magnificent observatory on the top of his palace at Cassel, which he furnished with excellent instruments of copper, and is said to have calculated himself the positions of no less than 400 stars. He was aided in the labours of the observatory by some astronomers of great merit whom his liberality drew to his court; among others, by Rothman, and Justus Byrgius, a distinguished artist, to whom Kepler has ascribed some idea of the logarithms.

Tycho Brahe, one of the best and most indefatigable observers of whom practical astronomy can boast, was born at Knudstorp, in Scania, the 13th of December 1546. His family was one of the most ancient and noble in Denmark; and his father, probably thinking that illustrious birth superseded the necessity of education, refused his consent to have his son instructed in Latin. Through the kindness of a maternal uncle, however, he was rescued from the state of barbarous ignorance to which he had been doomed by his parent; and, after having received the requisite preliminary instruction, he was sent to the university of Leipsic to study jurisprudence and scholastic philosophy. The tastes of the future astronomer were first excited by an eclipse of the sun which happened in 1560. Struck with astonishment at the accuracy of the prediction, he conceived a vehement desire to become acquainted with the principles of so certain a science, and exerted his utmost ingenuity to elude the obstacles which were interposed by his parents and governor to prevent him from acquiring the elementary notions of mathematics and astronomy. By placing the hinge of a common compass near his eye, he contrived to guess at the distances of the planets from the stars, and by this means, according to his own account, detected several errors in the Ephemerides of Stadius. By his persevering efforts he at last obtained the consent of his family to study according to his own inclinations; and from that moment he divided his time between the observation of the heavens, and chemical experiments. He visited the different cities of Germany where he hoped to meet with astronomers and skilful mechanicians, and was received with flattering attention by the landgrave of Hesse-Cassel, with whom he contracted an intimate friendship. On his return to Denmark he obtained from Frederick II. a grant of the small island of Huen, in the strait of Sunda, together with a pension and some presents, by means of which, and an expenditure of 100,000 crowns of his own patrimony, he was enabled to build the castle of Uraniburg, and procure a magnificent collection of the largest and most accurate instruments which could then be constructed. In this celebrated retreat he passed twenty-five years, actively employed in making observations, and attracting by his discoveries the attention of the learned throughout Europe. On the death of his protector Frederick he fell under the ' displeasure of the government, and a storm of persecution was raised against him, from causes which he has not explained, by a minister named Walckendorp, who, on this account, has been devoted by Lalande to the infamy and execration of all ages. He was deprived of his pension, compelled to leave the castle of Uraniburg, and to banish himself from Denmark. He retired first to Wandesburg, near Hamburg; he afterwards sought an asylum in Bohemia, and ultimately settled at Prague under the protection of the emperor Rudolph II. Here he resumed his observations, assisted by the illustrious Kepler and Longomontanus. The causes of his exile are involved in mystery; but to whatever it was owing, it turned out fortunate for the progress of astronomy. Had he remained in his island, his observations would not probably have fallen into the hands of Kepler, and the discovery of the laws of the planetary motions might have been deferred to another age. He died at Prague on the 24th of October 1601.

As an indefatigable and skilful observer, Tycho is justly considered as far superior to any astronomer who had preceded him since the revival of the science in Europe. His ample fortune gave him the means of procuring the best instruments which the age could produce; and by his ingenuity and persevering application, he was admirably qualified to employ them to the best advantage. He computed the first table of refractions, and if it extended only to 45°, the reason was, that the effects of refraction, at a higher altitude, were altogether insensible to his instruments. His solar tables were brought to so great a degree of exactness, that he affirms he could never detect an error in them exceeding a quarter of a minute; but there is reason to suspect some exaggeration in this statement, particularly as Cassini, a century after, with much better means, could scarcely answer for errors of a whole minute. He contributed greatly to the improvement of the lunar tables, and detected a considerable inequality in the moon’s motion in longitude, to which he gave the name of the Variation, by which it has ever since been distinguished. He also discovered an equation in latitude similar to the evection which had been observed by Hipparchus, and fixed its amount with great accuracy. He remarked the fourth inequality of the moon in longitude, although he failed in his attempt to ascertain its amount, or assign its law. He represented the inequalities of the motions of the nodes, and in the inclination of the lunar orbit, by the motion of the pole of that orbit in a small circle round the pole of the ecliptic. He demonstrated that the region of the comets is far beyond the orbit of the moon, and determined the relative and absolute positions of 777 fixed stars with a scrupulous attention, which gave his catalogue an immense superiority over those of Hipparchus and Ulugh Beigh; and he left to his successors a regular series of observations on the planets, amassed for the purpose of establishing the truth of his own system, but of which Kepler made a better use by employing them to establish the system of Copernicus.

These are some of the important benefits which resulted to astronomy from the labours of Tycho. As a philosopher he ranks low. Alchemy and judicial astrology, in the reveries of which he was a firm believer, engrossed as much of his attention as astronomy. Yet his errors, or rather weaknesses, ought to be viewed with indulgence. He was seated, to use the simile of Bailly, on the confines of two ages; partaking of the darkness which preceded, and the light which came after him. He rejected the simple system of Copernicus, and, whether from participating in the religious scruples of his age, or from the [7:3:739]ambition of appearing as the author of a new system, h¢ ’ restored to the earth its immobility, and placed it in the centre of the motions of the sun and moon. The Tycho-nic system was an unsuccessful attempt to reconcile the incongruous hypotheses of Ptolemy and Copernicus. It never enjoyed any real estimation; and its followers were only found among those who dreaded the anathemas of the church, or who, belonging to some university or religious corporation, were deprived of the liberty of expressing their real opinions.

Tycho was assisted in his observations at Huen, during eight years, by Longomontanus, who afterwards be- ’ came professor of the higher mathematics at Copenhagen. This astronomer composed a large work, entitled Astronomia Danica, in which he deduced the elements of the different theories from the observations made at Uranibourg, and gave formulae for computing the planetary motions, according to the three systems of Ptolemy, Copernicus, and Tycho.

The great mass of accurate observations accumulated by Tycho furnished the materials out of which his disciple Kepler may be said to have constructed the edifice of the world. This great man, the true founder of modem astronomy, was born at Wiel, in the kingdom of Wurtemberg, on the 27th of December 1571. He studied philosophy at Tubingen, and was instructed in mathematics and astronomy by Moestlin, whose name deserves a place in the history of science, on account of his having been one of the first who had the courage to adopt and to teach the system of Copernicus. The philosophical mind of Kepler, disgusted with the improbabilities and absurdities of the ancient system, received with transport the novel doctrines explained by Moestlin. The appointment of mathematician to the emperor, which he procured on the death of Stadius, confirmed him in the resolution which he had taken to devote himself to astronomical pursuits; and the energy of his character enabled him in a very short time to make himself thoroughly master of the different hypotheses and principal discoveries which had been made prior to his time. In the year 1596 he published a Prodromus of Dissertations on the properties and causes of the celestial orbits, which procured him the friendship of Tycho, and an invitation to take part in the observations and researches of that great astronomer at Prague. On the death of Tycho, which happened soon after, Kepler obtained possession of his invaluable collection of observations, and was charged with the task of completing and publishing the Rudolphine Tables.

During his short residence with Tycho, Kepler learned to check the fanciful suggestions of his brilliant imagination, and to draw his conclusions from observations alone, by rigorous and patient induction. The observations of the Danish astronomer had furnished him with the means of establishing with certainty the truth or inaccuracy of the various hypotheses which he successively imagined; and the diligence with which he laboured in comparing and calculating these observations during 20 years, was finally rewarded by some of the most important discoveries which had yet been made in astronomy. Deceived by an opinion which had been adopted by Copernicus, and had never been called in question by the ancients, that all the celestial motions are performed in circles, he long fruitlessly endeavoured to represent, by that hypothesis, the irregular motions of Mars; and after having computed with incredible labour the observations of seven oppositions of that planet, he at length succeeded in breaking down the barrier which had so long obstructed the progress of knowledge, and found that the motions could only be accurately represented by supposing the planet to move in an ellipse, having the sun in one of its foci. Having arrived at this important result, he next proceeded to consider, the angular motion of the planet, and finding that it was not uniform in respect of any point situated within the orbit, he concluded that the uniform motion, till then universally received as an axiom, was a vain chimera, which had no existence in nature. He perceived, however, that the areas described by the radius vector of the planet, at its greatest and least distances, were equal in equal times; and subsequent observations enabled him to demonstrate that this equality extended to every point of the orbit. It was therefore discovered that Mars moves in an elliptic orbit, of which the sun occupies a focus, and in such a manner, that the area described by a line drawn from the centre of the planet to that of the sun is always proportional to the time of description. The same conclusions he found to be true in respect of the orbit of the earth; and therefore he could no longer hesitate to extend them by analogy to the other planets. These are two of the three general principles which are known by the name of the Laws of Kepler.

It was some years later before Kepler arrived at the knowledge of the analogy which subsists between the distances of the several planets from the sun, and the periods in which they complete their revolutions. To the discovery of this analogy he attached the greatest importance, and regarded all his other labours as incomplete without it. After having imagined numberless hypotheses, it at last occurred to him to compare the different powers of the numbers which express the distances and times of revolutions; and he found, to his infinite satisfaction, that the squares of the periodic times of the planets are always in the same proportion as the cubes of their mean distances from the sun. This is the third law of Kepler. He demonstrated it to be true of all the planets then known. It has been found to be equally true in regard to those which have been since discovered, and likewise to prevail in the systems of the satellites of Jupiter and Saturn. It is indeed, as can be shown mathematically, a necessary consequence of the law of gravitation, directly as the masses, and inversely as the squares of the distances.

By these brilliant discoveries, the solar system was reduced to that degree of beautiful simplicity which had been conceived by Copernicus, but from which that great astronomer had found himself constrained to depart. The sun could not occupy the common centre of the circular orbits, but his place is in the common focus of the elliptic orbits of all the planets; and it is to this focus that every motion is to be referred, and from which every distance is to be measured. The discovery of the elliptic motion, of the proportionality of the areas to the times, and the method of dividing an ellipse, by straight lines drawn from the focus to the periphery, into segments having a given ratio, formed the solution of a problem which had been the constant object of the labours of all astronomers from Ptolemy to Tycho, namely, to assign the place of a planet at any instant of time whatever. The tables which he computed for the elliptic motions form the model of those in present use. Some additions have been made in consequence of the perturbations, which the geometry of Kepler was inadequate to estimate, and which were only partially detected by the genius of Newton. It has been considered matter of surprise that Kepler did not think of extending the laws of the elliptic motion to the comets. Prepossessed with the idea that they never return after their passage to the sun, he imagined that it would only be a waste of time to attempt the calculation of the orbits of bodies which had so transitory an existence. He supposed [7:3:740]the tail to be produced by the action of the solar rays, which, in traversing the body of the comet, continually carry off the most subtile particles, so that the whole mass must be ultimately annihilated by the successive detachment of the particles. He therefore neglected to study their motions, and left to others a share of the glory resulting from the discovery of the true paths of the celestial bodies. .

The observations of eclipses had formed the principal object of the earliest astronomers, but it was Kepler who first showed the practical advantages which may be derived from them, by giving an example of the method of calculating a difference of meridians from an eclipse of the sun. The method extends to occultations of the stars, and is deservedly considered as the best we possess for determining geographical longitudes and correcting the tables. He composed a work on optics, replete with new and interesting views, and gave the first idea of the telescope with two convex glasses, which has since been advantageously substituted for that of Galileo. Prompt to seize every happy idea of his contemporaries, he perceived with delight the advantages which practical astronomy would derive from the new invention of the logarithms, and he immediately constructed a table, from which the logarithms of the natural numbers, sines, and tangents could be taken at once.

Kepler was not merely an observer and calculator; he inquired with great diligence into the physical causes of every phenomenon, and made a near approach to the discovery of that great principle which maintains and regulates the planetary motions. He possessed some very sound and accurate notions of the nature of gravity, but unfortunately conceived it to diminish simply in proportion to the distance, although he had demonstrated that the intensity of light is reciprocally proportional to the surface over which it is spread, or inversely as the square of the distance from the luminous body. In his famous work De Stella Martis, which contains the discovery of the laws of the planetary motions, he distinctly states that gravity is a corporeal affection, reciprocal between two bodies of the same kind, which tends, like the action of the magnet, to bring them together, so that when the earth attracts a stone, the stone at the same time attracts the earth, but by a force feebler in proportion as it contains a smaller quantity of matter. Further, if the moon and the earth were not retained in their respective orbits by an animal or other equipollent force, the earth would mount towards the moon one fifty-fourth part of the interval which separates them, and the moon would descend the fifty-three remaining parts, supposing each to have the same density. He likewise very clearly explains the cause of the tides in the following passage. “If the earth ceased to attract its waters, the whole sea would mount up and unite itself with the moon. The sphere of the attracting force of the moon extends even to the earth, and draws the waters towards the torrid zone, so that they rise to the point which has the moon in the zenith.” It is not difficult to imagine how much these views must have contributed to the immortal discovery of Newton.

It is afflicting to consider how frequently the just rewards of true merit are usurped by charlatanism and pretension. While the fire-eaters and astrologers of Rudolph were basking in the sunshine of imperial favour, Kepler, from whose labours the sciences derived the most signal benefits, passed his life in extreme indigence. Born without fortune, the only revenue he possessed, and out of which he had to maintain a numerous family, arose from the precarious produce of his writings, and his pension of mathematician to the emperor,—a pension which, owing to the calamities of the times, was seldom duly paid.

On this account he was obliged to prefer frequent solicitations, and undertake long journeys, whereby he lost his time, always precious to genius, and exhausted his mind in anxiety. He died on the 15th of November 1630, at Ratisbon, whither he had gone to solicit the arrears that were due to him. In the present century a marble monument has been erected to his memory by an enlightened prince, Charles of Alberg. It contains his bust and the ellipse of Mars; a monument more glorious and more imperishable than brass or marble.

Contemporary with Kepler was the illustrious Galileo, whose discoveries, being of a more popular nature, and far more striking and intelligible to the generality of mankind, had a much greater immediate effect on the opinions of the age, and in hastening the revolution which was soon about to change the whole face of physics and astronomy. Galileo-Galilei, a Florentine patrician, was born at Florence in the year 1564. He passed his youth at Venice, where he continued till he was appointed to a professor’s chair at Padua. After a residence of eighteen years in that city, he was induced to remove to Pisa by Cosmo II, who assigned him a pension, and conferred on him the tide of his first mathematician. While residing at Venice, he heard it reported that Metius, a Dutch optician, had discovered a certain combination of lenses, by means of which distant objects were approximated to the sight. This vague and scanty intelligence sufficed to excite the curiosity of Galileo, who immediately set about inquiring into the means whereby such an effect could be produced. His researches were attended with prompt success, and on the following day he had a telescope which magnified about three times. It was formed by the combination of two lenses, a plano-convex and plano-concave, fitted in a leaden tube. In a second trial he obtained one which magnified seven or eight times; and subsequent essays enabled him to increase the magnifying power to 32 times. On directing his telescope to the moon, he perceived numerous inequalities on her surface, the diversified appearances of which led him to conclude almost with certainty that the moon is an opaque body similar to the earth, and reflecting the light of the sun unequally, in consequence of her superficial asperities. The planet Venus exhibited phases perfectly similar to those of the moon. These phases had been formerly announced by Copernicus as a necessary consequence of his system; and the actual discovery of their existence made it impossible to doubt of the revolution of Venus round the sun. He detected the four satellites or moons of Jupiter, and, in honour of his patron, gave them the name of the Medicean Stars. The discovery of these little bodies circulating round the huge orb of Jupiter afforded him a strong analogical proof of the annual revolution of the earth, accompanied by its moon. He perceived spots on the disk of the sun, from the motions of which he concluded the rotation of that body about its axis in the space of 27 days. The singular appearances of Saturn were beheld by him with no less pleasure than astonishment. His telescope was not sufficiently powerful to separate the ring from the body of the planet; and to explain the appearances he supposed Saturn to be composed of three stars almost in contact with one another. These discoveries proved that the substances of the celestial bodies are similar to that of the earth, and demolished the Aristotelian doctrine of their divine essence and incorruptible nature. They enlarged the ideas of mankind respecting the planetary system, and furnished the most convincing arguments in favour of the doctrines of Copernicus.

The discoveries of Galileo excited the envy of his [7:3:741] contemporaries, and stirred up against him a persecution which embittered the last days of his life. The motion of the earth, which he had proved so triumphantly, was considered contrary to many express declarations of Scripture; it was also considered as a heresy in the schools, where the doctrines of Aristotle were followed with implicit submission. In defending the Copernican system, Galileo had incurred the bitterest resentment, both of theologians and peripatetics. His great reputation, his title of professor and first mathematician, gave them reason to dread that the new doctrines, recommended by such an advocate, would spread too rapidly, and finally overthrow the altars of Aristotle. They therefore combined to check their progress, and to procure the ruin of Galileo by injurious representatives to the Grand Duke and the court of Rome. Soon after his first telescopic discoveries he was cited to appear before the Inquisition, and a promise extorted from him that he would never, either by word or writing, support the opinion of the motion of the earth. But in a great mind the love of truth is the most imperious of all passions, and cannot be restrained by the frowns of despotism, or the persecution of a fanatical tribunal. The evidences of the motion of the earth burst forth from every point of the heavens; and Galileo, in his celebrated Dialogues on the system of the world, exposed them in so clear and forcible a manner, that, although he appeared to put them forth for the purpose of refuting them, it was not difficult to perceive that he regarded them as complete and indubitable. For this relapse into heresy he was again brought before the tribunal of the Inquisition, and, in the seventieth year of his age, condemned formally to retract and abjure the doctrine of the earth’s motion, and to be imprisoned during the pleasure of the Inquisition. This scandalous proceeding has called forth the indignant reprobation of every lover of truth and freedom. “What a spectacle,” exclaims Bailly, “an old man, whose hairs were blanched with study, watchings, and benefits to mankind, on his knees before the sacred Scriptures, abjuring the truth in the eyes of Italy, which he had enlightened, in opposition to the testimony of his conscience, and in spite of the manifestations given by nature through all her works.”

The sentence passed on Galileo was not inflicted with great severity. At the end of a year the Grand Duke had the influence to procure his release from prison; but he was prohibited from returning to Florence, and obliged to confine himself to the Tuscan territory. He retired to the village of Arectri, where he resumed his observations, and shortly afterwards discovered the libration of the moon. The satellites of Jupiter continued also to engage his attention, and he commenced a table of their motions, and pointed out the method of determining the longitude by means of their eclipses. The states of Holland, aware of the great benefit of his researches to commerce and navigation, sent two astronomers, Hortensius and Blaeu, to present him with a gold chain, and encourage him to persevere in his useful labours. A short time after receiving this honourable mark of esteem from a foreign country, Galileo suddenly became blind; and the task of forming the tables of the satellites was reserved for Cassini. He survived this misfortune only a few years, and expired in 1642, in the seventy-eighth year of his age..

Science is indebted to Galileo for two other discoveries of a different kind, less brilliant perhaps, but of far greater importance than those which we have yet enumerated. These are the isochronism of the vibrations of the pendulum, and the law of the acceleration of falling bodies. His telescopic discoveries could not have remained long unknown; in fact, with the exception of those of the phases of Venus, and of the triple form of Saturn, they were alI fiercely disputed, even during his own lifetime. It is now' universally admitted that he was the first who discovered the satellites of Jupiter, and the spots of the sun; but the very circumstance of other claimants to these discoveries having arisen, proves that they were within the reach of ordinary attention. No one ever thought of disputing with Kepler the discovery of the laws of the planetary motions. Those of Galileo required only eyes, and may be regarded as following of course from the discovery of the telescope; but his persecution, his condemnation, and his being compelled to retract and abjure a doctrine of which he had given the physical proof, inspire a hallowed interest in his history, and contributed much to the great reputation which he acquired throughout Europe.

While astronomy was making these rapid advances in the hands of Kepler and Galileo, an event occurred in Scotland which contributed, though less directly, no less powerfully, to the acceleration of its progress. This was the invention of the logarithms by Lord Napier, baron of Merchiston; “an admirable artifice,” says Laplace, “which, by reducing to a few days the labour of many months, doubles the life of the astronomer, and spares him the errors and disgust inseparable from long calculations; an invention of which the human mind has the more reason to be proud, inasmuch as it was derived exclusively from its own resources.” It may be added, that without this, or some equivalent artifice, the computations rendered necessary by more correct observations would far exceed the limits of human patience or industry, and astronomy could never have acquired that precision and accuracy by which it is now distinguished above all the other branches of human knowledge.

The same epoch presents to us a great number of excellent observers, who, although they did not produce any revolution in the state of astronomy, still rendered it useful service. Scheiner is celebrated for his observations on the solar spots, and his disputes with Galileo. John Bayer of Augsburg published a description of the constellations, accompanied by maps, in which the stars are marked by a Greek letter; a simple idea, which has been universally adopted. Lansberg, a Flemish mathematician, published in 1632 a set of astronomical tables, which, though filled with inaccuracies, rendered good service to science by apprizing Horrox of the transit of Venus over the sun’s disk, which that young astronomer and his friend Crabtree had the satisfaction of observing on the 24th of November 1639. They were the first who ever witnessed that interesting but rare phenomenon. Snellius is celebrated for his measure of the earth. Gassendi, who had the merit, along with Descartes, of hastening the downfall of the Aristotelian philosophy in France, made some useful observations, particularly one of a transit of Mercury in 1631. His works, which fill six folio volumes, abound with curious and useful researches. Riccioli, a Jesuit, born at Ferrara in 1598, contributed to the progress of astronomy, not so much by his own discoveries, as by collecting and rendering an account of those of others. He rejected the system of Copernicus, and was more zealous in maintaining the doctrines of the. church than in investigating nature; but his works form a· vast repertory of useful information. His Novum Almagestum is a collection of the observations, opinions, and. physical explanations of the phenomena, together with all the methods of computation then known. He was assisted in his labours by Grimaldi, who discovered the inflection of light, and gave the names to the principal spots of the moon which are now used by astronomers.
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The most accurate observations that were ever made prior to the adaptation of the telescope to astronomical instruments were those of Hevelius, a rich citizen of Dantzic, who devoted his life and a large fortune to the service of astronomy. Having fitted up an observatory, and furnished it with the best instruments which could be procured, he commenced a course of observations, which he followed assiduously upwards of forty years. In his Selenographia he has given an accurate description of the face and spots of the moon, accompanied with excellent delineations of her appearance in her different phases and librations. The idea of making drawings of the different phases of the moon had previously occurred to Gassendi and Peiresc, but they had not been able to execute the project; indeed the difficulty attending it was such, that it occupied Hevelius, who was an excellent draughtsman, as well as observer, during a great number of years. Hevelius made an immense number of researches on comets; and finding that the observations could not be represented by rectilinear or circular orbits, he supposed them to move in parabolas. During a temporary absence from Dantzic, this indefatigable astronomer had the misfortune to lose, in a great fire which occurred in the city, his observatory, instruments, manuscripts, and almost the entire edition of the second volume of his Machina Caelestis, which contained the results of his long labours and numerous observations. He was now in his old age, but his zeal did not give way under the terrible calamity. He patiently recommenced all his calculations, reconstructed tables of the sun, and prepared for publication his Firmamentum Sobiescianum, or celestial chart, which did not appear till after his death. Towards the latter part of his life, the use of telescopic sights began to be generally adopted. Hevelius, however, resisted the innovation, and continued to employ plain sights. This preference given to the ancient method by so skilful an observer induced Dr Halley to visit him at Dantzic, for the purpose of ascertaining, by a comparison of observations made at the same time and plaĉe, which of the two methods gave the most correct results. Dr Halley observed with the telescope, and Hevelius with his own instruments; but such was the dexterity he had acquired through long practice, that the difference of their observations seldom amounted to more than a few seconds, and in no case to so much as a minute. Notwithstanding this agreement, it is to be regretted that Hevelius did not adopt the new method; for, on account of the greater precision given to instruments by the use of the telescope, his observations, which were made without it, cannot now be admitted in the construction of tables, and consequently are for the most part useless to astronomy.

Few individuals have rendered more important services to science than Huygens. Born at the Hague in 1629, he studied geometry under Schooten, the commentator of Descartes, and gave early proofs of proficiency in that science by a treatise on the quadrature of the conic sections. Having passed into France, he studied law at the university of Angers; but his principal attention was directed to the physical sciences, particularly to optics. He employed himself in grinding and polishing lenses; and constructed a telescope of ten feet, with which he discovered one of the satellites of Saturn. His application of the pendulum to clocks deserves to be considered as one of the best gifts which genius has ever conferred on science. He seems to have conceived the idea of this application in 1656; and he presented the first description of the pendulum clock to the states of Holland in 1657. He endeavoured to make this invention subservient to the problem of the longitudes; and if his efforts were not attended with the desired success, it may be said, that without another invention, in which also he had a principal share,' that of the spiral spring, the object would never have been accomplished so nearly as it was a hundred years later. By means of his excellent telescopes he discovered that the extraordinary appearance exhibited by Saturn was occasioned by a ring surrounding the body of the planet, and inclined to the ecliptic in an angle which he estimated at 21°. He published his observations on this planet in a work entitled Systema Saturnium, which still shows some traces of that species of reasoning from final causes which so greatly disfigures the writings of Kepler. For example, on discovering the satellite, he conceived that as the number of satellites now equalled the number of planets, it was in vain to look for more, the equality being necessary to the harmony of the system. He lived, however, to witness the discovery of four more satellites belonging to the same planet. Huygens was invited to settle in France by Colbert, the patriotic minister of Louis XIV., who assigned him a pension and a seat in the academy of sciences. He continued in that country till the revocation of the edict of Nantz in 1681, when he resigned his pension and retired to Holland. After this he contributed several papers to the Philosophical Transactions, and in 1690 published a treatise on light and gravitation. Geometry, mechanics, and optics, are indebted to the genius of Huygens for many important discoveries. His theorems on central forces, his researches on the doctrine of probabilities and continued fractions, and his theory of involutes and evolutes, raise him to the highest rank among the mathematicians of his age. He died on the 8th of June 1695, at the age of 66 years.

The application of telescopes and micrometers to graduated instruments forms an important epoch in the history of astronomy. This happy improvement was first brought into use by Picard in 1667. Morin, indeed, had applied a telescope to the quadrant so early as 1634, and perceived the stars in full day in 1635. In 1669 Picard began to observe the stars on the meridian in the daytime, with a quadrant, to which, in concert with Azout, he had applied an astronomical telescope having cross wires in its focus. Huygens invented the plate micrometer in 1650; Malvasia that with the fixed wires in 1662; and Azout that with the movable wire in 1666. (SeeDe-lambre, Astronomie du Moyen Age, p. 618. Note by Bouvard.) It is principally to the happy discoveries and ingenious inventions just referred to, and the fine application of the pendulum to clocks by Huygens in 1656, that we must attribute the rapid progress since made in practical astronomy, and the extreme precision of modern observations. Picard was also the first who introduced the modem method of determining the right ascensions of the stars, by observing their meridional passages, and employed the pendulum for that purpose. He likewise introduced the method of corresponding altitudes, and is entitled to be regarded as the founder of modern astronomy in France.

Roemer, the friend and pupil of Picard, discovered the progressive motion of light in 1675, and measured its velocity by means of the eclipses of Jupiter’s satellites. He was the first who erected a transit instrument, which gave a new accuracy to observations of right ascension.

The Royal Observatory of Paris was completed in 1670, and its direction intrusted to Dominic Cassini. This ce-ĵ lebrated astronomer was born at Perinaldo, in the county of Nice, and educated in a college of the Jesuits at Genoa. He acquired an early passion for astronomical observations; and in 1644 was invited to Bologna by the marquis Malvasia, where, in 1650, he succeeded Cavalleri [7:3:743] as professor of astronomy. In this situation he continued 'till 1669, when he went to Paris on the invitation of Louis XIV. He was naturalized in France in 1673, and continued in the charge of the observatory till his death, which happened in 1712.

Cassini enriched astronomy with a great number of curious observations and discoveries. He determined the motions of Jupiter’s satellites from observations of their eclipses, and constructed tables of them, which were found to be remarkably exact. He observed that the ring of Satum is double, and discovered four of the satellites of that planet. He also determined the rotation of Jupiter and Mars, and made a number of observations on Venus with the same view. He observed the zodiacal light, and made a near approximation to the parallax of the sun. We also owe to him the first table of refractions, calculated on correct principles; and a complete theory of the libration of the moon. Galileo had only observed the libration in latitude; Hevelius explained the libration in longitude, by supposing that the moon always presents the same face to the centre of her orbit, of which the earth occupies a focus. Cassini made the important remark, that the axis of rotation of the moon is inclined to the ecliptic, and that its nodes coincide with those of the lunar orbit, so that the poles of the orbit, ecliptic, and equator of the moon, are on the same circle of latitude, the pole of the ecliptic being situated between the other two. The greater number of these discoveries are, however, only of secondary importance; and it must be confessed that Cassini took no part in the great and permanent improvements which astronomy received in that age. He has, nevertheless, obtained an extraordinary reputation. Lalande remarks, that in his hands astronomy underwent the most signal revolutions, and that his name is, in France, almost synonymous with that of creator of the science. Delambre has, however, taken a different and far more accurate view of the real services of Cassini. “The revolution in Astronomy,” this judicious critic observes, “was brought about by Copernicus, by the laws of Kepler, by the pendulum of Huygens, the micrometers of Azout and Picard, by the sectors and mural of Picard, and his method of corresponding altitudes, by the transit instruments of Roemer; and Cassini appears to us an entire stranger to all these innovations. He followed another route; he devoted a long life to painful observations, which at last deprived him of sight. Let us not refuse him the praise which he has so well merited, but let us reserve a place in our esteem for labours less brilliant perhaps, but of greater and more permanent utility, and which evince at least equal talent and sagacity.”

Cassini was assisted in his observations by his nephew, James Philip Maraldi, who determined the regression of the nodes, and the progressive motion of the apsides of the orbit of Jupiter. This astronomer also corrected the theory of Mars, and observed the sun’s parallax. He rejected the hypothesis of the progressive motion of light, as being insufficient to explain the inequalities of Jupiter’s satellites; and he conceived the design of forming a new catalogue of the stars, which, however, was never executed. He died in 1729.

There is no period in the history of mankind so distinguished by great and important discoveries, or so remarkable for the rapid development of the human intellect, as the seventeenth century. We have already noticed the invention of the pendulum, and its application to regulate the motion of time-keepers; of the telescope, and some of the phenomena of the new worlds it has exposed to view; of the logarithms, by which computations are so much abridged; and of the mechanical contrivances for measuring minute angles in the heavens. The same century witnessed the application of algebra to geometry, the discovery of the laws of the planetary motions, of the infinitesimal calculus, the acceleration of falling bodies, the sublime theory of central forces, and the great principle of gravitation which connects the celestial orbs, and regulates the motions which it had been the business of the astronomer to observe since the earliest ages of the world. The different steps which conducted to this important discovery, and the immediate consequences deduced from it by its immortal author, are so fully developed in the admirable Dissertation on the Progress of the Physical Sciences, in this Encyclopaedia, that it is unnecessary to enter into any detail respecting them here; we may only remark, that if observation has furnished the data for the discovery of the mechanical principle and primordial laws of the universe, the knowledge of these laws has been, in turn, of the most essential service to observation, by guiding and directing it to its most important objects. Many of the inequalities of the planetary motions, in consequence of their minuteness and the slowness with which they vary, could not have been detected by observation; others might perhaps have been perceived, but we should still have been ignorant whether their constant accumulation might not ultimately change the state of the system, and, by destroying all confidence in the tables, demolish the fabric which had been reared at such a vast expense of time and labour. But when these inequalities are detected by theory, and separated from the mean motions with which they were blended, it becomes an object of the highest interest to confirm their existence by the most delicate and accurate observations. Hence, a more refined practice has constantly followed every theoretical discovery. Besides, it is the perfection of theory, and not the mere knowledge of isolated facts, which gives astronomy its greatest value in the eyes of the philosopher. Numerous and important as its applications are, they have but a subordinate interest, in comparison of the knowledge of those general laws to which every particle of matter in the universe is subject, and by the discovery of which man has penetrated so deeply into the mysteries of nature.

By the discovery of the law of gravity Newton laid the foundations of physical astronomy; and by the consequences which he deduced from that law, proceeded far in the erection of the superstructure. He showed that the motions of all the bodies of the planetary system are regulated by its influence; he determined the figure of the earth on the supposition of its homogeneity; he gave a theory of the tides, discovered the cause of the precession of the equinoxes, and determined some of the principal lunar inequalities and planetary perturbations. Many of his theories were left in an imperfect state; for it is not in matters of science that it is given to the same individual to invent and bring to perfection: their complete development required that several subsidiary sciences should be farther advanced; but it has been the triumph of his system, that every subsequent discovery has only tended to strengthen and confirm it. This bright ornament of the human genius was born on the 25th of December 1642, the day of the death of Galileo, and died on the 20th of March 1727, in the 85th year of his age.

While physical astronomy was undergoing a complete revolution in the hands of Newton, the practical part was receiving great improvement from Flamsteed, the first astronomer royal, who conducted the Greenwich Observatory. This celebrated institution, from which so many important discoveries have emanated, was erected under the reign of Charles II. in 1675. Flamsteed was appoint; [7:3:744]ed to it in 1676, and continued with indefatigable zeal to discharge the duties of the office during the long period of 33 years. In the course of this time he made an immense number of excellent observations, the results of which are given in the Historia Caelestis, the first edition of which was published in 1712, at the expense of Prince George of Denmark, the husband of Queen Anne. The second appeared in 1723, some time after the death of the author, in three volumes folio. The first volume contains the observations which he made, first at Derby, and afterwards at Greenwich, on the fixed stars, planets, comets, spots of the sun, and Jupiter’s satellites. The second volume contains the transits of the planets and stars over the meridian, and the places of the planets deduced from these observations. The third contains an historical notice, in which he gives a description of the instruments used by Tycho and himself; catalogues of fixed stars by Ptolemy, Ulugh Beigh, Tycho, the landgrave of Hesse, and Hevelius; together with the British Catalogue, containing the places of 2884 Stars. The labours of Flamsteed were, however, confined entirely to the practical part of astronomy. He made no improvements in theory; but he is entitled to the merit of having been the first who brought into common use the method of simultaneously observing the right ascension of the sun and a star, a method by means of which the determination of the positions of the stars is reduced to the observation of meridional transits and altitudes. He was likewise the first who explained the true principles of the equation of time; and he improved the lunar tables by introducing into them the annual equation which had been suggested by Horrox. The Atlas Caelestis, another posthumous work of Flamsteed, was published in 1753.

Flamsteed was succeeded in the observatory by Dr Halley, a philosopher whose inventive genius and indefatigable activity rendered him one of the brightest ornaments of his country. Halley was the son of a wealthy citizen of London, where he was born on the 8th of November 1656. From his earliest years he applied himself with ardour to the study of mathematics and astronomy; and having procured a few instruments, he began to make observations, by which he was led to remark the inaccuracy of the tables of Jupiter and Saturn. In his 19th year he published a direct and geometrical method of finding the eccentricities and aphelia of the orbits of the planets; and in the year following he undertook a voyage to St Helena, with a view to form a catalogue of the stars in the southern hemisphere. The station was unfortunately chosen, for, owing to the incessant rains and foggy atmosphere of that island, he was able to determine the places of only 360 stars in the course of a whole year. He had, however, the satisfaction of observing a transit of Mercury over the sun’s disk, a phenomenon which suggested to him the important remark, that the transits of the inferior planets might be advantageously employed in determining one of the most essential elements of the planetary system, viz. the parallax of the sun, and consequently the diameters of the orbits. The method has since been successfully employed in the case of Venus: the transits of Mercury, though much more frequent, are not so well adapted to the purpose. On his return from St Helena he was commissioned by the Royal Society to visit Hevelius at Dantzic, and determine, by a direct comparison of observations, the dispute which had arisen between that astronomer and Dr Hooke respecting the relative advantages of plain and telescopic sights. After this Halley for some time travelled on the Continent, and on his return to England devoted himself entirely to scientific pursuits. The fruits of his leisure soon began to appear in the multitude of treatises which he published from time to time in every department of the mathematical and physical sciences. The’ most important of those connected with astronomy was his Synopsis Astronomies Cometicae, — a work abounding in profound and original views, and which, in respect of theory, formed perhaps the most remarkable accession to the science that had been made since the time of Kepler. In this work he revived an ancient opinion, that the comets belong to the solar system, and move in very eccentric orbits round the sun, returning after stated but long intervals. He also ventured to predict that the comet of 1681 would again return to its perihelion in 1759,—the first prediction of the kind that was verified. In 1720 Dr Halley was appointed to succeed Flamsteed in the Royal Observatory; and though now in the 64th year of his age, he undertook, with a view to improve the lunar theory, to observe the moon through a whole revolution of her nodes, erroneously supposing, that after such a revolution the errors of the tables would again appear in the same order. He was the first who, by a comparison of ancient and modern observations, remarked the acceleration of the mean motion of the moon, and thus called the attention of mathematicians to an important and curious phenomenon, the physical cause of which was at length detected by the powerful analysis of Laplace. He was also the first who pointed out the secular inequalities of Jupiter and Saturn, occasioned by their mutual perturbations,—a theory that formed the subject of several profound memoirs of Euler and Lagrange, and for the complete development of which astronomy is likewise indebted to Laplace. Besides these important discoveries in astronomy, the labours of Dr Halley also greatly contributed to the promotion of geometry and navigation. He undertook two long voyages, and traversed the Pacific Ocean to observe the deviations of the magnetic needle; and posterity will gratefully recollect that it v¢as through his pressing solicitations that Newton consented to the publication of the Principia. In 1703 he succeeded Dr Wallis as professor of geometry at Oxford. He became secretary of the Royal Society in 1713, and died at Greenwich in 1742.

The discoveries of Bradley, who succeeded Halleyas] astronomer royal, form a memorable epoch in the history t of the science. This great astronomer was born at Sher- “borne in Gloucestershire in 1692, and acquired from his uncle, Mr Pound, an early taste for astronomical observation. He was destined for the church, and for some time held a curacy, which he resigned in 1721, on being appointed to succeed Keill as Savilian professor of astronomy at Oxford. His first essays indicated no extraordinary talent; but an opportunity having presented itself of engaging in more important researches, he embraced it with ardour; and by his sagacity and perseverance was conducted to two discoveries which have entirely changed the face of astronomy.

A singular motion of the polar star had been observed] by Picard, of which, however, that astronomer could nei- c ther assign the law nor give any satisfactory explanation. 1 He only remarked that the inequality was annual, and amounted to about 40 seconds. Hooke, in 1674, a few years after the observations of Picard, imagined that he had discovered a parallax in some of the stars; and Flamsteed, following the ideas of Hooke, explained, by means of parallax, the minute changes of position which he had observed in Polaris and some circumpolar stars. Manfredi and Cassini demonstrated the error of Flamsteed, but were not more successful in their attempts to explain the motion in question. Samuel Molyneux conceived the idea of verifying all that had been said respecting the [7:3:745]supposed parallaxes, and for this purpose commenced a 'series of observations at Kew, with an excellent .24 feet sector constructed by Graham. Bradley, who happened at that time to reside at Kew, took a part in these observations, the result of which was, that the remarks of Picard were confirmed beyond the possibility of doubt. It was, however, abundantly evident that the apparent motions alluded to were not connected in any manner with parallax; it therefore became an object of the greatest interest to determine their physical cause, and assign their law and period. The first idea that occurred was to inquire whether they arose from a change of position in the earth’s axis; but this supposition was found to be inadequate to the explanation of the phenomena. Molyneux having been in the mean time appointed a lord of the admiralty, the observations were discontinued at Kew; they were, however, shortly after resumed by Bradley at Wanstead, with a smaller but more convenient instrument; and after they had been continued several years, it was found that the star (γ draconis) on which they were principally made, appeared to describe annually a small ellipse, the transverse axis of which amounted to 40". This was an important determination; for the ellipse afforded the means of computing at all times the aberration of any star whatever, whether in longitude, latitude, declination, or right ascension. Bradley also pointed out the physical cause of the aberration, and demonstrated that it resulted from the combination of the motion of light with the annual motion of the earth. This capital discovery was made in 1728.

Bradley, anxious to verify his ingenious theory, conti- nued his observations, and soon felt the difficulty that had so much embarrassed Picard. The places of the stars, 'calculated according to his formula for the aberration, could not be reconciled with the observations. The errors continued to augment during nine years, after which they went on diminishing during the nine years following. This inequality, of which the period, like that of the nodes of the moon, was 18 years, was readily explained by supposing a slight oscillation of the earth’s axis, occasioned by the action of the moon on the protuberant parts surrounding the equator of the terrestrial spheroid. After assiduously observing its effects during twenty years, Bradley found that the phenomena could be accurately represented by giving the pole of the equator a retrograde motion about its mean place in an ellipse whose axes are 18" and 16", and completing its revolution in the period of 18 years. This result was communicated to the Royal Society in 1748. To these two grand discoveries of Bradley, the aberration and nutation, modern astronomy is wholly indebted for all its accuracy and precision; and, as Delambre remarks, they assure to their author a distinguished place, after Hipparchus and Kepler, above the astronomers of all ages and all countries.

Bradley was appointed astronomer royal in 1741, and from this time to the period of his death made an immense number of observations, which were presented by his heirs to the university of Oxford, on condition that they should forthwith be published. The first volume appeared only in 1798, edited by Dr Hornsby. The rest were committed to the care of the late Dr Abraham Robertson, and appeared in 1805. Dr Bradley was chosen a corresponding member of the Academy of Sciences in 1748. In 1752, twenty-four years after his great discovery of the aberration of light, he was admitted into the Royal Society. He died on the 13th of July 1762, in the 70th year of his age.

While England was deriving so much glory from the brilliant discoveries of Bradley, .France produced a multitude of excellent astronomers, by whose successful labours every department of the science was signally promoted. Among these Lacaille is pre-eminently distinguished, both by the variety and importance of his observations, and the indefatigable zeal with which, during twenty-two years, he prosecuted the most laborious researches. He commenced his astronomical career in 1739, by assisting Cassini de Thury (grandson of the first Cassini) in the verification of the measurement of the meridian through France. In 1751 he undertook a voyage to the Cape of Good Hope, the primary objects of which were to determine the sun’s parallax, by means of observations on the parallaxes of Mars and Venus, while similar observations were made in Europe; and to form a catalogue of the southern circumpolar stars. No undertaking for the benefit of science was ever more successfully executed. In the course of a single year, Lacaille, without assistance, observed upwards of ten thousand stars, situated between the tropic of Capricon and the pole, and computed the places of 1942 of them; a labour which would scarcely be credited, if the details of his observations had not been published in the Coelum Australe Stelliferum, a work which was given to the world in 1763. Our admiration of the rapid execution of this vast undertaking will be increased, when we consider, that during the same time he measured a degree of the meridian, and made numerous observations on the moon simultaneous with those of Lalande, who observed at Berlin, in order to determine the moon’s parallax, by means of direct observations made at the extremities of a meridional arc of upwards of 85 degrees. Before his return to Europe, he visited the isles of France and Bourbon, where he measured the length of the pendulum, and made numerous remarks on the natural and civil history of those countries. Astronomy is likewise indebted to Lacaille for a table of refractions which he computed from a comparison of above 300 observations made at the Cape and at Paris. In 1757 he published his Astronomies Fundamenta, in which he gave rules and tables for computing the apparent motions of the stars, which continued to be employed till Lambert supplied the corrections depending on the nutation, and Delambre those depending on the aberration. To defray the expense attending the publication of this important work, which he distributed in presents to the different observatories and astronomers in Europe, he submitted to the drudgery of calculating ephemerides during ten years. Lacaille composed several elementary works for the use of the students in the college of Mazarin, where he occupied the chair of astronomy, and inserted a great number of memoirs in the volumes of the Academy of Sciences. This great astronomer, distinguished as much by the excellence of his moral qualities as his profound knowledge and indefatigable zeal, died suddenly in 1762.

The Royal Observatory of Paris continued under the direction of the family of Dominie Cassini during 120 years. James Cassini, the second of that name, is principally known by a work on the magnitude and figure of the earth, and his Elements of Astronomy. He seems not to have duly appreciated the new discoveries which were daily making around him. His Elements, published in 1740, contains no mention of the aberration; and he adopted the opinion that the earth is elongated instead of being flattened at the poles. His son, Cassini de Thury, was chiefly occupied with the meridian, and the geometrical survey of France. The last astronomer of the family, the Count de Cassini, was obliged to resign the observatory at the revolution.

The question of the figure of the earth furnished ample materials for the practical as well as the speculative as [7:3:746]tronomer during the last century. The results of the measurement of the meridian by Cassini were at variance with the theories of Newton and Huygens; and the Academy of Sciences resolved on making a decisive experiment by the actual measurement of the lengths of two degrees, one at the equator, and another in as high a latitude as could be reached. In the year 1735, three astronomers, Godin, Bouguer, and La Condamine, were commissioned by the French government to accomplish the first of these objects in Peru; and the year following, Maupertuis, Clairaut, Camus, and Lemonnier, went to Lapland to execute the second under the polar circle. Notwithstanding the greater difficulties they had to contend with, the first party were the most successful; but the result of both operations established the compression of the earth at the poles. Bouguer published the details of the Peruvian measurement in an admirable work on the Figure of the Earth, in which he has also inserted an account of a great number of experiments made by him in the same country to determine the length of the seconds’ pendulum, and the effects of the attraction of mountains on the plumb-line. Bouguer is likewise the author of an excellent treatise on light. This accomplished mathematician and experimenter did not adopt the Newtonian theory of gravitation, but he was the last apostle of the Cartesian philosophy in the Academy of Sciences.

It would greatly exceed the limits of this article to give even an abridged account of tfie numerous observers who, about this period, contributed to the improvement of every department of practical astronomy. We must therefore content ourselves with merely noticing the names of some of the most distinguished, leaving the details of their labours to be given in the biographical articles interspersed throughout the work.

Delisle formed a school of astronomy in Russia, and has left a method of computing the heliocentric places of the sun’s spots, and of Mercury and Venus in their transits over the sun’s disk, and likewise of determining, by means of the stereographic projection, the directions of their path when they enter and leave the disk. Lemonnier introduced the discoveries of Bradley into his Astronomical Institutions, and was the instructor of Lacaille and Lalande. He published a Histoire Celeste, containing a collection of observations from 1666 to 1685, and a number of other works and memoirs connected with astronomy. He made an immense number of observations, but their accuracy is far inferior to those of Bradley. Wargentin, secretary of the Academy of Sciences of Stockholm, devoted the whole of his life to the correction of the tables of the satellites of Jupiter. The theory of the satellites was not then far advanced; but when theory failed him, he profited by the remarks of others and by his own reflections, and endeavoured by repeated trials to find empirical equations capable of conciliating the tables with the best observations. By confining himself almost exclusively to this subject, he acquired a high reputation, and was ranked among the first astronomers of an epoch which abounds in great names. His tables of the satellites have, on account of their superior accuracy, been employed in determining the masses, and other elements, which serve as the basis of the analytical theories.

One of the most celebrated astronomers of the last century was Lalande. He commenced his early career by a set of lunar observations at Berlin, undertaken simultaneously with those of Lacaille at the Cape, for the purpose of determining the moon’s horizontal parallax. On his return to Paris he was received, though only twenty-one years of age, into the Academy of Sciences; and on the resignation of Delisle was appointed professor of astronomy m the college of France. By his extraordinary zeal, indefatigable activity, and the care which he took to have his name constantly before the public, Lalande soon became one of the most distinguished men of his day; but his reputation, acquired in a great measure from attention to subjects which had only an ephemeral interest, and not through any permanent or fundamental additions to science, has already begun to wane; and his works, many of which are of considerable utility, seem to have fallen into unmerited neglect. His character as an astronomer is fairly and impartially summed up by Delambre in the following terms :—“If Lalande did not renew the foundations of astronomical science, like Copernicus and Kepler,— if he did not, like Bradley, immortalize himself by two brilliant discoveries,—if he was not so learned and accurate a theorist as Mayer,—if he was not to the same decree as Lacaille an exact, expert, scrupulous, industrious, and indefatigable observer and calculator,—if he had not the constancy to attach himself, like Wargentin, to a single object, in order to become the first in a particular department,—and if, in all these respects, he was only an astronomer of the second rank,—he was the first of all as a professor. No one ever did more to propagate the knowledge and love of astronomy. It was his object to be useful and celebrated; and he succeeded, through his labours, his activity, his credit, and his solicitations: by means of a most extensive correspondence he incessantly laboured for the benefit of science: he even endeavoured to serve it after his death, by founding a medal, which the Academy of Sciences adjudges annually to the astronomer who has made the most interesting observation, or written the most useful memoir.” (Delambre, Astronomie du Moyen Age.)

Bradley was succeeded in the Greenwich Observatory by Dr Bliss, who died in the course of a few years after his appointment. Dr Bliss was in turn succeeded by Dr Maskelyne, the late astronomer royal,· under whose care the observatory maintained the high character which it acquired from the immortal labours of his illustrious predecessors. Dr Maskelyne began his astronomical career in 1761, when he was appointed to observe the transit of Venus at the island of St Helena, and endeavour to verify the existence of a small parallax of the star Sirius, which seemed to be indicated by the observations of Lacaille at the Cape. Unfortunately the state of the weather prevented him from observing the transit; and his observations on Sirius were abandoned in consequence of the discovery of a defect in the zenith sector which he had carried out with him for the purpose of making the observations in question. The main objects of his voyage were thus frustrated; but some indirect advantages resulted from it, which compensated in some measure for the disappointment. The attention of Ramsden was called to the sector, and a better method of constructing these instruments was devised. At St Helena he made several interesting observations on the tides, the variation of the compass, the moon’s horary parallaxes, &c. In going out and returning home, he paid particular attention to the different methods of finding the longitude at sea, and practised that which depends on observations of the lunar distances from known stars, taken with a Hadley’s sextant, or other reflecting instrument. He also composed a set of rules for the use of the seamen, which he published on his return, first in the Philosophical Transactions, and afterwards in the British Mariners Guide. In the year 1765 he was appointed astronomer royal, and soon after recommended to the Board of Longitude the general adoption in the navy of the lunar method of finding the [7:3:747]longitude, and proposed that tables for facilitating that method should be calculated and published in the Nautical Almanac. This recommendation was adopted, and the Nautical Almanac continued to be published under his superintendence during forty-eight successive years. He also published a set of tables requisite to be used with that long-celebrated ephemeris; and by these means contributed to bring nautical astronomy to its present state of precision. Dr Maskelyne was most assiduous in the discharge of his important duties as astronomer royal. On one occasion only did he consent to leave the observatory, viz. when at the request of the Royal Society he proceeded to Scotland for the purpose of determining, by experiments on Schehallien, the deflection of the plumb-line occasioned by the attraction of the mountain. The result of the observations made on this occasion furnished data for the determination of the mean density of the earth, which Dr Hutton, by a laborious process of calculation, found to be about five times greater than that of water. Dr Maskelyne had the credit to procure, at the expense of the Royal Society, the regular publication of his observations, by which he rendered a great service to all the astronomers of Europe. The lunar method of determining the longitude, the Nautical Almanac, the catalogue of 36 stars, and some improvements in the construction of instruments, are the principal benefits for which astronomy is indebted to Dr Maskelyne.

Sir William Herschel, born in Hanover in 1738, has rendered his name immortal by the discovery of a new planet without the orbit of Saturn, and thereby doubling the ancient boundaries of the solar system. Having settled in England, at Bath, he began to devote his leisure to the construction of telescopes and the polishing of reflecting mirrors. Endowed with equal skill and patience, he soon obtained instruments superior to any that had been known before, by means of which he was led to the most brilliant discoveries that have been made in the heavens since the time of Galileo. Being employed in making a review of the sky with a powerful telescope, he perceived, on the 13th of March 1781, near the feet of Gemini, a star of the fifth magnitude, having a disk perfectly well defined, and differing in appearance from the other stars which afforded the same quantity of light. On observing it with a telescope whose magnifying power was 932, he perceived its diameter was enlarged, while that of the stars underwent no change. These circumstances were sufficient to draw his attention to the star, and nothing more was requisite to enable him speedily to discover that it had a slow motion. He at first supposed it was a comet, and acquainted Dr Maskelyne with the discovery. The circumstance was soon made known at Paris; and it was gradually perceived, that as the distance of the star did not sensibly vary, it was necessary to regard it as a seventh planet. Herschel, in honour of his royal patron, gave it the name of the Georgium Sidus ; but the mythological appellation of Uranus has prevailed. On the 11th of January 1787 he discovered two satellites revolving round the new planet, and subsequently found that it was accompanied by four others. It was soon noticed that Uranus had been observed by Flamsteed, Mayer, and Lemonnier, who had each supposed it to be one of the fixed stars. Their observations enabled Delambre to correct the elements of the orbit, and calculate tables of its motion. By means of his powerful telescopes Herschel determined the figure and rotation of Saturn, discovered the parallel belts on his surface, and perceived that the ring was double. In 1789 he discovered two new satellites belonging to this planet, and revolving within the ring. From some appearances indicated by the fixed stars,

Herschel was led to conclude that the whole solar system is in motion about some distant centre, and that its direction is at present towards the constellation Hercules, a conclusion, however, which succeeding observations have not verified. His observations on nebulae and double stars have opened up a new field of research, boundless in extent, and interesting by reason of the variety of the objects it presents to the attention of the observer. The extraordinary activity with which he pursued his favourite occupations is attested by 67 memoirs communicated from time to time to the Royal Society. A great part of these, however, is filled with speculations of no value to astronomy; and his taste was rather to observe astronomical phenomena than to engage in computations, or the more arduous and essential, though less fascinating, labours through which the science can be really benefited. In the course of the following chapters we shall have frequent occasion to allude to his discoveries, and his ingenious speculations concerning the constitution of the sun and the sidereal heavens.

Few individuals have contributed so eminently to the perfection of modern astronomy as Delambre, the late perpetual secretary of the Academy of Sciences. The scientific life of this illustrious astronomer did not commence till he had attained his 40th year; but from that time till his death it was occupied by a series of unremitting labours to enlarge the boundaries, and ameliorate the practice and theory of the science. Associated with Mechain, he was employed during the troubles of the revolution in measuring the meridian from Dunkirk to Barcelona,—a labour which was prosecuted with admirable zeal in the face of innumerable difficulties, and even dangers of the most formidable kind. By an immense number of excellent observations he determined the constants which enter into the formulae deduced from theory by the profound researches of Lagrange and Laplace, and also formed a set of tables much more exact than any that had appeared before them. His Astronomic Théorique et Pratique, in three quarto volumes, contains the best rules and methods which have yet been devised for the guidance of the practical astronomer; and his Histoire, in six large volumes 4to, gives an account of every successive improvement which has been made in the science, and a full abstract of every work of celebrity which has been written respecting it, since the first rude observations of the Greeks to the end of the last century. It is invaluable to the historian, and will ever remain a proud monument of the profound learning and laborious research of its author.

The observatory which was established at Palermo about the year 1790, under the active superintendence of Piazzi, holds a distinguished rank among the similar institutions of Europe. Piazzi, born in 1746, took the habit of the religious order of the Theatins at Milan, and finished his noviciate in the convent of St Anthony. Among his preceptors he had the advantage of counting Tiraboschi, Beccaria, Le Soeur, and Jacquier; and from these illustrious masters he speedily acquired a taste for mathematics and astronomy. After filling several professors’ chairs in the colleges of the Jesuits at Rome and Ravenna, he was appointed in 1780 professor of the higher mathematics in the academy of Palermo. Here his first care was to reform the general system of education; and, by the alterations which he introduced, he contributed powerfully to dissipate the shades of ignorance, which, under the double influence of the Jesuits and the Inquisition, still lowered over the soil of Sicily. After having rendered this service to literature, he obtained from the prince of Caramanico viceroy of the island, permission to found an observatory, and undertook a voyage to France and [7:3:748]England, in order to provide the instruments necessary for the new establishment. Having procured a vertical circle, a transit, and some other instruments from Ramsden, he returned to Palermo and commenced his observations. His first care was to prepare a new catalogue of stars, the exact positions of which he justly considered as the basis of all true astronomy. In prosecuting this object he did not content himself with a single observation, but before he fixed the position of any star, observed it several times successively; and, by this laborious but accurate method, he constructed his first great catalogue of 6748 stars, which was crowned by the Academy of Sciences of France, and received with admiration by the astronomers of all countries. His constant practice of repeating his observations led to another brilliant result, the discovery of an eighth planet. On the 1st of January 1801, Piazzi, searching for the star 87 of the catalogue of Mayer, cursorily observed a small star of the eighth magnitude, between Aries and Taurus. On the following day he remarked that the star had changed its position, and accordingly supposed it to be a comet. He communicated his observations to Oriani, who, seeing that this luminous point had no nebulosity like the comets, and that it had been stationary and retrograded within comparatively small limits like the planets, computed its elements on the hypothesis of a circular orbit. He found that this hypothesis agreed with the observations, and other astronomers soon confirmed its accuracy. He gave the planet the name of Ceres Eerdinandea, in honour of Ferdinand, king of Naples, in whose dominions it was discovered, and who proposed to consecrate the event by a gold medal, struck with the effigy of the astronomer; but Piazzi, nobly preferring the interests of science to vain honours which could add nothing to his glory, requested that the money destined for this purpose should be employed in the purchase of an equatorial, which was still wanting to his observatory. In 1814 he published a new catalogue, extended to 7646 stars,—a splendid monument of indefatigable zeal and activity. He made an uninterrupted series of solstitial observations from 1791 to 1816, for the purpose of determining the obliquity of the ecliptic, which, compared with those of Bradley, Mayer, and Lacaille, in 1750, give a diminution of 44" in a year. Besides these labours, sufficient to occupy a life of ordinary industry, Piazzi composed a number of memoirs for the different societies of which he was a member, and was intrusted by the Neapolitan government with several important commissions respecting the public instruction and the regulation of weights and measures in Sicily. He died in 1826, after having bequeathed his library and all his instruments to the observatory at Palermo, and assigned a liberal annuity to be devoted to the instruction of young men who evince a decided taste for astronomy.

The discovery of Ceres led to that of three other little planets, circulating at nearly the same distance from the sun,—a circumstance unique in the construction of the solar system. On account of the smallness of the new planet, and the nebulosity by which it is surrounded, the difficulty of finding it after it had for some time ceased to be observed, was so considerable, that Dr Olbers of Bremen was induced to examine, with particular care, the configurations of all the small stars situated in the vicinity of its path, in order to have the means of detecting it at any time with greater facility. While engaged in making observations for this purpose, he discovered, on the 28th of March 1802, and nearly in the same place where he had before observed Ceres, another planet similar in size and appearance, to which he gave the name of Pallas. The extraordinary circumstance of the discovery of two planets having nearly the same mean distance, and performing their revolutions nearly in the same time, led’ Dr Olbers to imagine that Ceres and Pallas were fragments of a larger planet which had revolved in the same place, and been shattered by some external force or internal convulsion. The immediate consequence of this hypothesis was the probability of the existence of other fragments of the original planet, hitherto undiscovered; and that if such fragments existed, the planes of their orbits would pass through the points in which the orbits of Ceres and Pallas intersect each other. This bold idea acquired some probability from the subsequent discovery of two other planets in the very quarter of the heavens to which Dr Olbers had directed the attention of astronomers. Juno was discovered by Μ. Harding of Lilienthal, on the 2d of September 1804; and Vesta, by Dr Olbers himself, on the 29th of March 1807. Diligent observation has not since added to their number.

Astronomy is a science which borrows the aid of several other sciences, with which its advancement is simultaneous. Instruments capable of appreciating or measuring' the exceedingly minute quantities about which the observer now concerns himself, could only be produced in a very advanced state of mechanics. In the accurate graduation of large instruments, our English artists have long maintained an unrivalled superiority. Graham, a celebrated watch-maker, united to great dexterity in the mechanical arts a decided taste for observation; and the extraordinary improvements which he effected in the art of dividing form an era. in the history of practical astronomy. The sector used by Bradley, in the observations which led to the discovery of the nutation, was of his construction, and he was also the inventor of the first compensation pendulum. Sisson, who succeeded him, maintained in this department the honour and pre-eminence of ∙ England. Bird was originally a weaver in Durham, and first gave proof of his mechanical genius by dividing dials for the watch-makers in a manner far superior to what had been commonly practised. He was employed by Sisson in the division of mathematical instruments, and recommended by that artist to Graham, who instructed him in his methods. He constructed the 8 feet quadrants employed in the Greenwich and Paris observatories, and another of 6 feet for Mayer at the observatory of Göttingen. Ramsden invented a machine for the more accurate division of instruments, on account of which he received a premium from the Board of Longitude. The mural quadrants of this artist were held in high estimation, and the transit instrument, sextant, and refracting micrometer, in passing through his hands, received considerable ameliorations. His astronomical instruments in general were considered the best that could be procured in Europe. Mr Troughton, who at present stands the foremost of our' British opticians, has more than rivalled his predecessors. In the hands of this distinguished artist and astronomer, the division of instruments has been carried to a degree of precision and accuracy which probably will never be surpassed; while the great improvements he has introduced into the methods of constructing and mounting large instruments, and his ingenious inventions to elude natural obstacles, and guard against the accidental derangements from which it is impossible altogether to protect them, entitle him to the high place which he holds by universal consent among the most eminent philosophers and original mechanicians of the age. The numerous circles and other instruments of his manufacture, whether in the hands of private individuals or deposited in the different public observatories, are considered as the finest specimens of [7:3:749] what art has yet accomplished for the advancement of 'astronomical science.

Progress of Physical Astronomy.

Having now endeavoured to give an account of the labours of those astronomers who have principally contributed to make us acquainted with the state of the heavens, and the order and succession of the various phenomena they exhibit, we will conclude this part of the article by briefly adverting to the profound researches of some illustrious mathematicians who have developed the theory of Newton, and raised the fabric of physical astronomy to its present proud elevation.

: Although the law of gravitation, as proposed by Newton, had from the first been admitted by all the most eminent astronomers of Britain, it was for a long time either opposed or neglected on the Continent. In fact, great improvements were required both in analysis and mechanics before it admitted of other applications than had been made by its great author, or could be regarded as any thing more than a plausible hypothesis. Newton demonstrated, that if two bodies only were projected in space, mutually attracting each other with forces proportional directly to their masses, and inversely to the squares of their distance, they would each accurately describe an ellipse round the common centre of gravity; and the spaces described by the straight line joining that centre and the moving body, would be proportional to the time of description, according to the second law of Kepler. But when it is attempted to apply Newton’s law to the case of the solar system, great difficulties immediately present themselves. Any one planet in the system is not only attracted by the sun, but also, though in a greatly smaller degree, by all the other planets, in consequence of which it is compelled to deviate from the elliptic path which it would pursue in virtue of the sun’s attraction alone. Now, the calculation of the effects of this disturbing force was the problem which geometers had to resolve. In its most general form it greatly transcends the power of analysis; but there are particular cases of it, and those too the cases presented by nature, in which,by reason of certain limitations in the conditions, it is possible to obtain an approximate solution to any required degree of exactness. For example, the Sun, Moon, and Earth form in a manner a system by themselves, which is very slightly affected by the aggregate attractions of the other planets. In the same way the Sun, Jupiter, and Saturn form another system, in which the motions are very little influenced by the action of any other body. In these two cases, then, the number of bodies to be taken into consideration is only three; and in this restricted form, the problem, celebrated in the history of analysis under the denomination of the Problem of the Three Bodies, is susceptible of being treated mathematically. With the hope of ameliorating the lunar tables, and of completing the investigations which Newton had commenced in the Principia, three distinguished geometers, Clairaut, D’Alembert, and Euler, about the middle of the last century, undertook, simultaneously, and without the knowledge of each other, the investigation of the problem of the three bodies, and commenced that series of brilliant discoveries which it is the glory of our own times to have seen completed.

Clairaut’s solution of the problem of the three bodies was presented to the Academy of Sciences in 1747, and was applied to the case of the moon. From this solution he deduced with great facility, not only the inequality of the variation, which Newton had obtained by a more complicated, though at the same time a very ingenious method, but also the evection, the annual equation, and many other inequalities which Newton had not succeeded m connecting with his theory. It happened, however, curiously enough, that in the calculation of one effect of the disturbing force, namely, the progression of the moon s apogee, Clairaut was led into an error which produced a result that threatened to overturn the system of gravitation. The error consisted in the omission of some of the terms of the series expressing the quantity in question, which he wrongly supposed to have only an insensible value; and by reason of this omission, his first approximation gave only half of the observed progressive motion of the apogee. As this result was confirmed by D’Alembert and Euler, who had both fallen into the same error, it seemed to follow, as a necessary consequence, either that the phenomenon depended on some other cause than the disturbing force of the sun, or that the law of gravity was not exactly proportional to the inverse square of the distance. The triumph which this result gave to the Cartesians was not of long duration. Clairaut soon perceived the cause of his error; and by repeating the process, and carrying the approximations farther, he found the computed to agree exactly with the observed progression,—a result which had the effect of dissipating for ever all doubt respecting the law of gravity. The researches of Clairaut were followed by a set of lunar tables, much more correct than any which had been previously computed.

The return of the comet of 1682, which Halley had predicted for the end of 1758, or beginning of 1759, afforded an excellent opportunity for putting to the test both the theory of gravity and the power of the new calculus. Clairaut applied his solution of the problem of the three bodies to the perturbations which this comet sustained from Jupiter and Saturn, and, after calculations of enormous labour, announced to the Academy of Sciences, in November 1758, that the comet would return in the beginning of the following year, and pass through its perihelion about the 15th of April. It returned according to the prediction, but passed its perihelion on the 13th of March. The correction of an error of computation reduced the difference to nineteen days; and if Clairaut had been aware of the existence of the planet Uranus, he might have come still nearer the truth.

Besides these important researches on the system of the world, Clairaut composed an admirable little treatise on the figure of the earth, in which he gave the differential equations, till then unknown, of the equilibrium of fluids, whether homogeneous or heterogeneous, supposing an attractive force, following any law whatever, to exist among the molecules. He applied these equations to the earth; demonstrated that the elliptic figure satisfies the conditions of equilibrium; assigned the ellipticity of the different strata of which the earth may be supposed to be formed, together with the law of gravitation at the exterior surface. He likewise discovered the important theorem which establishes a relation between the oblateness of the terrestrial spheroid and the increase of gravitation towards the poles, on every supposition which can be imagined relative to the interior construction of the earth. By means of this theorem the ellipticity of the spheroid is deduced from observations of the lengths of the seconds’ pendulum at different points of the earth’s surface.

D’Alembert, as has already been mentioned, presented a solution of the problem of the three bodies to the Academy of Sciences at the same time with Clairaut. In the year 1749 he published his treatise on the precession of the equinoxes,—a work remarkable in the history of analysis and mechanics. By means of his newly invented [7:3:750]calculus of Partial Differences, and the discovery of a ʹ fertile principle in dynamics, he determined from theory the rate of the precession, which is nearly 50" in a year. He also determined the nutation of the earth’s axis, which had been discovered by Bradley, and assigned the ratio of the axes of the small ellipse which the true pole of the earth describes around its mean place, in the same time in which the nodes of the lunar orbit complete a revolution. The solution of .this problem led to the determination of the ratio of the attractive forces of the sun and moon, which D’Alembert found to be that of seven to three very nearly; whence he inferred that the mass of the earth is 70 times greater than that of the moon. He proved likewise that the precession and nutation are the same in every hypothesis concerning the interior constitution of the earth. In 1754 he published the first two volumes of his Researches on the System of the World. In this work he applied the formulae by which he had calculated the motions of the moon to the motions of the planets disturbed by their mutual attraction, and pointed out the simplest method of determining the perturbations of the motions, of a planet occasioned by the action of its own satellites. D’Alembert also treated the subject of the figure of the earth in a much more general manner than had been done by Clairaut, who had. confined his investigations to the case of a spheroid of revolution. He determined the attraction of a spheroid of small eccentricity, whose surface can be represented by an algebraic equation of any order whatever, and even supposing the spheroid to be composed of strata of different densities.

The works of D’Alembert, which are extremely numerous, abound generally in profound and original views, and contributed greatly to the advancement of the physical and mathematical sciences; but it is to be regretted that they are very frequently deficient in that perspicuity and order so necessary in abstruse speculations, and that the course of his reasoning can be followed with difficulty when he descends into the detail of analytical operations. He had a horror of calculation, and delighted in general considerations and speculations which frequently turned on matters of pure curiosity.

The first memoir of Euler on the planetary perturba- 1 tions was transmitted to the secretary of the Academy of Sciences in July 1747, some months before Clairaut and D’Alembert had communicated their solutions of the problem of the three bodies, and it carried off the prize which the academy had proposed for the analytical theory of the motions of Jupiter and Saturn. In this memoir Euler gave the differential equations of the elements of the disturbed planet, but suppressed the analysis by which he had been conducted to them. This analysis, however, he subsequently expanded in two memoirs, the first of which appeared in the Berlin Memoirs in 1749, and the second in those of Petersburg in 1750. Of these supplementary memoirs the first is remarkable on several accounts. It contains the first example of a method which has been fruitful of important consequences—namely, that of the variation of the arbitrary constants in differential equations, and the development of the radical quantity which expresses the distance between two planets in a series of angles, multiples of the elongations. The expressions which he gave for the several terms of this series were simple and elegant; and he demonstrated a curious relation subsisting among any three consecutive terms, by means of which all the terms of the series may be calculated from the first two. He was thus enabled to develop the perturbing forces in terms of the sines and cosines of angles increasing with the time, and thereby to surmount a very great analytical difficulty. Notwithstanding, however, the great merit of Euler’s memoir several of the formulae expressing the secular and periodic' inequalities were found to be inaccurate; and in order to procure a correction of these errors, and give greater perfection to so important a theory, the academy again proposed the same subject for the prize of 1752. This prize was also carried off by Euler. In the memoir which he presented on this occasion, he considered simultaneously the motions of Jupiter and Saturn, and determined, in the first instance, the amount of their various inequalities, independently of the consideration of the eccentricities of their orbits. Pushing the approximations farther, and having regard to the inequalities depending on the eccentricities, he arrived at a most important result relative to the periodic nature of the inequalities occasioned by the mutual perturbations of the planets; which laid the foundation of the subsequent discovery by Lagrange and Laplace of the permanent stability of the planetary system. He demonstrated that the eccentricities and places of the aphelia of Jupiter and Saturn are subject to constant variation, confined, however, within certain fixed limits, which it never exceeds; and he computed that the elements of the orbits of the two planets recover their original values after a lapse of about 30,000 years. In the year 1756 the Academy of Sciences crowned a third memoir of Euler on the same subject as the two former, namely, the inequalities of the motions of the planets produced by their reciprocal attractions. This memoir, analytically considered, is also of great value. The method which he followed and illustrated has since been generally adopted in researches of the same nature, and consists in regarding as variable, in consequence of the disturbing forces, the six elements of the elliptic motion, viz. Isí, the greater axis of the orbit, which, by the law of Kepler* gives the ratio of the differential of the mean longitude to the element of the time; 2cζ the epoch of this longitude; 3cζ the eccentricity of the orbit; 4iΛ, the motion of the aphelion; ĩdh, the inclination of the orbit to a given fixed plane; and, Qth, the longitude of the node. By considering separately the variations introduced into each of these elements by the disturbing forces, Euler obtained some important results; but even in this memoir his theory was not rendered complete. He did not consider the variation of the epoch; and the expression which he gave for the motion of the aphelion did not include that part of it which depends on the ratio of the eccentricities of the orbits of the disturbed and disturbing planet. Besides, the present memoir, like the two former, contained several errors of computation, which, by leading to results known to be wrong, probably prevented the author himself from being aware of the full value of the ingenious methods of procedure which he had exposed. Euler concluded this important memoir by making an extended application of his formulae to the orbit of the earth as disturbed by the action of the planets. From some probable suppositions, first employed by Newton, relative to the ratios of the masses of the planets to that of the sun, he determined the variation of the obliquity of the ecliptic at 48" in a century,—a result which agrees well with observation. By this determination the secular variation of the obliquity of the ecliptic, which had been regarded by Lahire, Lemonnier, D’Alembert, and other eminent astronomers, as uncertain, was placed beyond doubt.. The three memoirs which we have mentioned contain the principal part of Euler’s labours on the perturbations; but physical astronomy is indebted to him for many other researches. He gave a solution of the problem of the precession of the equinoxes, and made several important steps in the lunar theory, with which he seems to have [7:3:751]occupied himself before he undertook the investigation of the planetary perturbations. In the year 1772, when entirely blind, he directed his son, Albert Euler, and two illustrious pupils, Krafft and Lexell, in the composition of a work of enormous labour on the same subject, which was undertaken with a view to discover the cause of the moon’s acceleration. This work was concluded with a set of lunar tables deduced entirely from theory; but they were found to be far inferior to those of Mayer, and in some respects hardly equal to those of Clairaut.

The labours of Euler form an epoch in the history of the mathematical sciences. From the arithmetic of sines, and the simplest formulae of trigonometry up to the variation of the arbitrary constants of differential equations, there is hardly an analytical theory which has not received extension or improvement from the creative powers of his great mind. The transactions of the Berlin and Petersburg Academies are filled with his inestimable productions; and if fecundity were not itself one of the attributes of genius, the number of the profound and laborious researches in which he was engaged would appear altogether incredible.

The first theory of Euler formed the basis of the excellent lunar tables which were calculated by Tobias Mayer, and first published in the Memoirs of the Academy of Göttingen in 1753. Mayer was a skilful astronomer, and determined the co-efficients of the arguments of the different lunar inequalities from his own observations. He continued to correct and improve his tables till the time of his death, which happened in 1762, when a copy of them, containing his last corrections, was presented by his widow to the Board of Longitude in London. Bradley ascertained their accuracy by comparing them with a great number of his own observations, and made so favourable a report concerning them, that the Board of Longitude presented the widow of Mayer with a gratuity of L.30.00. They were printed along with the author’s lunar theory in 1765. Subsequently, the Board of Longitude directed Mason, who had been assistant to Bradley, to revise them, under the superintendence of Dr Maskelyne. Mason compared them with about 1200 of Bradley’s observations; he corrected the co-efficients of Mayer, and introduced some new equations which had been indicated by that astronomer, but which he had considered as too uncertain, or of too small a value, to render it necessary to load his tables with them. Mayer’s tables, thus corrected, were published in 1784, and for a long time continued to be the most accurate that had appeared.

The solution of the problem of the three bodies by Clairaut, D’Alembert, and Euler, gave rise to many other important works relative to the theory of the moon, into the merits of which, however, our limits will not permit us to enter. Thomas Simpson, Walmesley, Frisi, Lambert, Schulze, and Matthew Stewart, treated the subject with more or less success; but the complete explication of the theory of the lunar and planetary perturbations was reserved for two mathematicians, whose discoveries perfected the theory of gravitation, and explained the last inequalities which remained to be accounted for in the celestial motions,—we mean Lagrange and Laplace.

In the year 1764, the Academy of Sciences of Paris, which had so successfully promoted the great efforts that had already been made to perfect the theory of attraction, proposed for the subject of a prize the theory of the libration of the moon. This was considered as an appeal to the genius of Lagrange, whose splendid talents had suddenly shone forth in two profound papers on the theory of sound, inserted among the Memoirs of the Turin Academy, and in which the calculus of variations was first explained and made use of. Lagrange had the honour of carrying off the prize; but although he treated the subject in a manner altogether new, and with extraordinary analytical skill, he did not on this occasion arrive at a complete solution of the problem. In 1766 he obtained another prize for a theory of Jupiter’s satellites. In the admirable memoir which Lagrange presented to the academy on this subject, he included in the differential equations of the disturbed motion of a satellite, the attracting force of the sun, as well as of all the other satellites, and thus, in fact, had to consider a problem of six bodies. His analysis of this problem is remarkable, inasmuch as it contained the first general method which was given for determining the variations which the mutual attractions of the satellites produce in the forms and positions of their orbits, and pointed out the route which has since been so successfully followed in the treatment of similar questions.

Of all the grand discoveries by which the name of La-1 grange has been immortalized, the most remarkable is i that of the invariability of the mean distances of the pla- 1 nets from the sun. We have already mentioned that* Euler had perceived that the inequalities of Jupiter and Saturn, in consequence of their mutual actions, arc ultimately compensated, though after a very long period. In prosecuting this subject, which Euler had left imperfect, Laplace had discovered that, on neglecting the fourth powers in the expressions of the eccentricities and inclinations of the orbits, and the squares of the disturbing masses, the mean motions of the planets, and their mean distances from the sun, are invariable. In a short memoir of 14 pages, which appeared among those of the Berlin Academy for 1776, Lagrange demonstrated generally, and by a very simple and luminous analysis, that whatever powers of the eccentricities and inclinations are included in the calculation of the perturbations, no secular inequality, or term proportional to the time, can possibly enter into the expression of the greater axis of the orbit, or, consequently, into the mean motion connected with it by the third law of Kepler. From this conclusion, which is a necessary consequence of the peculiar conditions of the planetary system, it results that all the changes to which the orbits of the planets are subject in consequence of their reciprocal gravitation, are periodic, and that the system contains within itself no principle of destruction, but is calculated to endure for ever.

In 1780 Lagrange undertook a second time the subject of the moon’s libration; and it is to the memoir which he now presented to the Berlin Academy that we must look for the complete and rigorous solution of this difficult problem, which had not been resolved before in a satisfactory manner, either on the footing of analysis or observation. In the same year he obtained the prize of the Academy of Sciences on the subject of the perturbations of comets. In 1781 he published, in the Berlin Memoirs, the first of a series of five papers on the secular and periodic inequalities of the planets, which together formed by far the most important work that had yet appeared on Physical Astronomy since the publication of the Principia. This series did not, properly speaking, contain any new discovery; but it embodied and brought into one view all the results and peculiar analytical methods which had appeared in his former memoirs, and contained the germs of all the happy ideas which he afterwards developed in the Mécanique Analytique.

On account of the brilliant discoveries and important labours which we have thus briefly noticed, Lagrange must be considered as one of the most successful of those illustrious individuals who have undertaken to perfect the theory of Newton, and pursue the principle of gravitation[7:3:752] to its remotest consequences. But the value of his services to science are not limited to his discoveries in physical astronomy, great and numerous as they were. After Euler, he has contributed more than any other individual to increase the power and extend the applications of the calculus, and thereby to arm future inquirers with an instrument of greater power, by means of which they may push their conquests into new and unexplored fields of discovery.

With the name of Lagrange is associated that of Laplace, whose rival labours divided the admiration of the scientific world during half a century. Like Newton and Lagrange, Laplace raised himself at an early age to the very highest rank in science. Before completing his 24th year, he had signalized himself by the capital discovery of the invariability of the mean distances of the planets from the sun, on an hypothesis restricted, indeed, but which, as we have already mentioned, was afterwards generalized by Lagrange. About the same time he was admitted into the Academy of Sciences, and thenceforward devoted himself to the development of the laws which regulate the system of the world, and to the composition of a series of memoirs on the most important subjects connected with astronomy and analysis. His researches embraced the whole theory of gravitation; and he had the high honour of perfecting what had been left incomplete by his predecessors.

Among the numerous inequalities which affect the motion of the moon, one still remained which no philosopher as yet had been able to explain. This was the acceleration of the mean lunar motion, which had been first suspected by Dr Halley, from a comparison of the ancient Babylonian observations, recorded by Hipparchus, with those of Albategnius and the moderns. The existence of the acceleration had been confirmed by Dunthorne and Mayer, and its quantity assigned at 10" in a century, but the cause of it remained doubtful. Lagrange demonstrated that it could not be occasioned by any peculiarity in the form of the earth; Bossut ascribed it to the resistance of the medium in which he supposed the moon to move; and Laplace himself at first explained it on the supposition that gravity is not transmitted from one body to another instantaneously, but successively in the manner of sound or light. Having afterwards remarked, however, in the course of his researches on Jupiter’s satellites, that the secular variation of the eccentricity of the orbit of Jupiter occasions a secular variation of the mean motions of the satellites, he hastened to transfer this result to the moon, and had the satisfaction to find that the acceleration observed by astronomers is occasioned by the secular variation of the eccentricity of the terrestrial orbit. This was the last celestial phenomenon which remained to be accounted for on the principle of gravitation.

Another discovery relative to the constitution of the planetary system, which does infinite honour to the sagacity of Laplace, is the cause of the secular inequalities indicated by ancient and modern observations in the mean motions of Jupiter and Saturn. On examining the differential equations of the motions of these planets, Laplace remarked, that as their mean motions are nearly commensurable (five times the mean motion of Saturn being nearly equal to twice that of Jupiter), those terms of which the arguments are five times the mean longitude of Saturn, minus twice that of Jupiter, may become very sensible by integration, although multiplied by the cubes and products of three dimensions of the eccentricities and inclinations of the orbits. The result of a laborious calculation confirmed his conjecture, and showed him that in the mean motion of Saturn there existed a great inequality, amounting at its maximum to 48' 2"∙3, and of which the period is 929 years; and that in the case of Jupiter there exists a corresponding inequality of nearly the same period, of which the maximum value is 19' 46", but which is affected by a contrary sign, that is to say, diminishes while the first increases, and vice versa. He also perceived that the magnitude of the co-efficients of these inequalities, and the duration of their periods, are not always the same, but participate in the secular variations of the elements of the orbits.

The theory of the figures of the planets, scarcely less interesting than that of their motions, was also greatly advanced by the researches of Laplace. He confirmed the results of Clairaut, Maclaurin, and D’Alembert, relative to the figure of the earth, and treated the question in a much more general way than had been done by those three great mathematicians. From two lunar inequalities depending on the non-sphericity of the earth, he determined the ellipticity of the meridian to be - 3 ⅛; very nearly.

Newton, in the Principia, explained the cause of the phenomena of the tides, and laid the foundations of a theory which was prosecuted and extended by Daniel Bernoulli, Maclaurin, Euler, and D’Alembert; but as no one of these geometers had taken into account the effects of the rotatory motion of the earth, the subject was in a great measure new when it was taken up by Laplace in 1774. Aided by D’Alembert’s recent discovery of the calculus of Partial Differences, and by an improved theory of hydrodynamics, he succeeded in obtaining the differential equations of the motion of the fluids which surround the earth, having regard to all the forces by which these motions are produced or modified, and published them in the Memoirs of the’Academy in 1775. By a careful examination of these equations, he was led to the curious remark, that the differences between the heights of two consecutive tides about the time of the solstices, as indicated by Newton’s theory, are not owing, as Newton and his successors had supposed, to the inertia of the waters of the ocean, but depend on a totally different cause, namely, the law of the depth of the sea, and that it would disappear entirely if the sea were of a uniform and constant depth. He also arrived at the important conclusion, that the fluidity of the sea has no influence on the motions of the terrestrial axis, which are exactly the same as they would be if the sea formed a solid mass with the earth. The same analysis conducted him to the knowledge of the conditions necessary to insure the permanent equilibrium of the waters of the ocean. He found that if the mean density of the earth exceeds that of the sea, the fluid, deranged by any causes whatever from its state of equilibrium, will never depart from that state but by very small quantities. It follows from this, that, since the mean density of the earth is known to be about five times greater than that of the sea, the great changes which have taken place in the relative situation of the waters and dry land must be referred to other causes than the instability of the equilibrium of the ocean.

Closely connected with the problem of the tides is that of the precession of the equinoxes, which also received similar improvements in passing through the hands of Laplace. He demonstrated, as has just been mentioned, that the fluidity of the sea has no influence on the phenomena of precession and nutation. He considered some of the effects of the oblate figure of the earth which had not been attended to by D’Alembert, and showed that the annual variation of the precession causes a corresponding variation in the length of the tropical year, which at present is about 9 or 10 seconds shorter than it was in the time of Hipparchus. He proved that the secular inequa [7:3:753]lities of the motions of the earth and moon have no sensible effect in displacing the axis of the earth’s rotation; and he determined the nutation of the lunar orbit corresponding to the nutation of the terrestrial equator.

Physical astronomy is also indebted to Laplace for a complete theory of the system of Jupiter’s satellites, from which Delambre constructed a set of tables which represent the motions of these bodies with all desirable accuracy. And when to these numerous and most important researches we add the mathematical theories of molecular attraction, and the propagation of sound, together with many great improvements in analysis,—and reflect,besides, that he is the author of the Mécanique Céleste, the Système du Monde, and the Théorie des Probabilités, —∙we shall not hesitate to rank him next to Newton among the greatest benefactors of the mathematical and physical sciences.

By the brilliant discoveries of Laplace, the analytical solution of the great problem of physical astronomy was completed. The principle of gravity, which had been discovered by Newton to confine the moon and the planets to their respective orbits, was shown to occasion every apparent irregularity, however minute, in the motions of the planets and satellites; and those very irregularities which were at first brought forward as objections to the hypothesis have been ultimately found to afford the most triumphant proofs of its accuracy, and have placed the truth of the Newtonian law beyond the reach of all future cavil. Such is the state to which analysis has now attained, that the geometer embraces in his formulae every circumstance which affects the motions or positions of the different bodies of the planetary system; and the conditions of that system being made known to him at any given instant of time, he can determine its conditions at any other instant in the past or future duration of the world. He ascends to remote ages to compare the results of his theories with the most ancient observations; he passes on to ages yet to come, and predicts changes which the lapse of centuries will hardly be sufficient to render sensible to the observer. But notwithstanding the proud elevation to which the theory of astronomy has been raised, it is still far from having reached the limit beyond which further refinement becomes superfluous. The masses of the plañets, and some other elements, remain to be determined with still greater precision, by a diligent comparison of the analytical formulae with good observations; and the labours of the geometer may still be beneficially employed in giving greater simplicity to the calculus, or in extending its power over subjects which have hitherto eluded its grasp. The recent discovery of two periodic comets completing their revolutions in comparatively short intervals of time, opens up an interesting field for speculation and research, and will doubtless be the means of throwing light over some curious, and as yet very obscure, points, respecting the appearances, motions, and physical constitution of those strange bodies.

In the other departments of the science, also, numerous questions still remain to be discussed, the solution of which will occupy and reward the future labours of the astronomer. The curious phenomena of double and multiple stars, some of which appear to form connected systems of bodies revolving about one another, or a common centre of motion,—the variable stars,—the proper motions of the stars,—the translation of the solar system in space,—the progressive condensation of nebulae,—are subjects still in a great measure new; for it is only of late years that observers have begun to direct the requisite attention towards them, or indeed have been in possession of instruments of sufficient power and delicacy to observe and measure the minute changes which take place beyond the boundaries of our own system. The observations of some living astronomers, particularly of Mr Herschel and Mr South, have made known some important facts regarding the nebulae and double-stars, and laid the foundations of interesting discoveries to be realized by their successors in future ages of the world.

Among the events auspicious to the future progress of astronomy in our own country, we cannot forbear to regard the establishment of the Astronomical Society of London, an association which, though it dates only from 1820, includes in its list of Members the names of the greater part of the most distinguished cultivators of the science in Europe. The society has already given the best proofs of its energy and usefulness, by the publication of a catalogue of the reduced places of 2881 of the principal stars visible above our horizon, and four splendid volumes of Memoirs, replete with information and remarks of the most useful kind to the observer. The number of public observatories recently established, or at present in a state of progress, in various parts of the country and its colonies, also holds out a cheering prospect to the friends of astronomical discovery. Nor if we turn our eyes abroad shall we perceive less reason for congratulation. The same zeal for extension and refinement is manifested in every country of Europe; and while some of the states, among which Prussia and Denmark deserve to be honourably mentioned, are gaining themselves glory by the encouragement they hold out to the cultivation of this department of science, the exertions of individual observers appear to increase in proportion as the field is narrowed by their success. Armed with instruments to which former times had nothing to compare, the astronomer of the present day not only anticipates a more perfect knowledge than he yet possesses of the nature and number of the bodies belonging to the solar system, but aspires to set at rest the disputed question of parallax, to determine the proper motions of the stars, and trace the effects of gravity in the remotest regions of the universe.

Such are the present prospects of astronomy. The mind delights to contemplate unlimited advancement in a walk in which its efforts have been so signally triumphant; yet in consequence of what has already been done, the future progress of the science must necessarily be slow; and it must be acknowledged that greater precision than has already been attained is unnecessary to navigation or geography, or any practical application of which astronomy admits. The future results of observation may add to the stock of speculative knowledge, but can only be remotely useful to mankind.

The following works may be consulted on the history of 1 astronomy :—Ricci oli, Almagestum Novum, Bononiae, 1653, t 2 vols. folio; Sherburn’s Translation of the Astronomiam of c Manilius, London, 1675; Weidler, Programma de Veteris et r Novae Astronomiae Discrimine, Wittembergae, 1720; Souciet, Observations Mathématiques, Astronomiques, &c. 1729, 2 vols. 4to. The second volume of this work, by Gaubil, contains a history of the Chinese Astronomy, with Dissertations. Weidìer, Historia Astronomiae, Wittemb. 1741; Idem, Commentatio de Mechanica Astronomica Medii Aevi, 1742; Long’s Astronomy, London, 1742; Costard’s Letter to Martin Folkes concerning the Rise and Progress of Astronomy among the Ancients, London, 1746; Foulques’ History of Astronomy, London, 1746; Heathcote, Historia Astronomiae, Cantab. 1747, 8vo; Esteve, Histoire Générale et Particulière de ïAstronomic, Paris, 1755, 3 vols. 12mo; Goguet, Origine des Lois, des Arts, et des Sciences, various editions; Jablonow, De Astronomiae Ortu ac Progressu, &c. Romae, 1763, 4to; Bailly, His- [7:3:754]foire de l’Astronomie Ancienne et Moderne, 1775, 1779, • 1782, 4 vols. 4to. This work has been published without the notes and calculations, in 2 vols. 8vo, Paris, 1805. Gentil, Sur l’Astronomie des Indiens, Hist. Acad., Paris, 1772; Gaubil, Histoire de l’ A str. Clιτnoise, in the Lettres Edifiantes et Curieuses, tom. xxvi.; Bailly, Traite de l’Astr. Indienne, Paris, 1787, 4to; Playfair’s Remarks on the Astronomy of the Brahmins, in the Edinburgh Transactions, republished in the 3d volume of his Works ; Asiatic Researches, vols. ĩi. vi. vin.; Adam Smith’s Fragment on the History of Astronomy ; Lalande, Astronomie, Paris, 1792, 3 vols. 4to; Idem, Bibliographie Astronomique, avec l’ Histoire de l’ Astronomie depuis 1781 jusques à 1802; Vince’s Astronomy, vol. ii. Cambridge, 1799; Kaestner, Geschichte der Mathematik, Göttingen, 1796; Schaubach, Geschichte dei' Griechischen Astronomie bis auf Era tosthenes, Göttingen, 1802, 8vo; Montucla, Histoire des'. Mathématiques, Paris, an 7, an 10 (1802), 2d edit. 4 vols. 4to; Laplace, Exposition du Système du Mondes and Pond’s Translation of the same work; Small’s History of the Discoveries of Kepler, 1803; Bossut, Histoire Generale des Mathématiques, Paris, 1810, 2 vols. 8vo;

Voiron, Histoire de l’ Astronomie depuis 1781 jusques a 1811, Paris, 1811, 8vo, a continuation of Bailly; Delambre, Histoire de l’Astronomie Ancienne, Paris, 1817, 2 vols. 4to; Idem, Histoire de l’Astronomie du Moyen Age, 1819, 4to; Idem, Histoire de l’Astronomie Moderne, 1821, 2 vols. 4to; Idem, Histoire de l’Astronomie du χυiii. Siècle, 1827, 4to. These six volumes of Delambre’s Histoire contain copious extracts of all the principal works which have been published on the subject of astronomy, interspersed with much enlightened criticism.

PART II.

THEORETICAL ASTRONOMY.

CHAP. I.

GENERAL PHENOMENA OF THE HEAVENS.

Sect. I.— Of the Celestial Sphere. When a spectator on a clear evening directs his attention to the sky, he perceives a concave hemisphere studded with innumerable brilliant points, all which appear to move in parallel directions, constantly preserving the same distances and relative positions. New groups of stars incessantly follow one another, rising in the east, mounting to a certain height, and then gradually sinking till they disappear in the west. On turning his face towards the north he observes some groups of stars which remain visible during the whole night; wheeling round a certain fixed point, and describing circles of greater or less magnitude, according as they are at a greater or less distance from that point. The same phenomena are observed every successive night; and as the stars present always the same configurations, and succeed each other in the same order, the mind is irresistibly led to the hypothesis of the diurnal revolution of the whole heavens about a fixed axis. This hypothesis implies that the stars move in parallel planes, and that their mutual distances remain invariable. In order, however, to be assured that such is the case, it is necessary to have recourse to more accurate observations, and to refer their successive positions to something that is fixed,—to certain points or planes which do not participate in the general motion.

What is most obviously adapted for this purpose is the Horizon, or the plane which appears extended all round us, and bounded by a circle of which the eye of the spectator occupies the centre. Let the circle SENW (fig. 1) represent the horizon, C being the centre or place of the observer. Suppose now that a star is observed to rise at a certain point of the horizon A, and, after tracing its circuit in the sky, to set at the point A'; suppose, also, that another star rises at B, and sets at B'; then, if the distances AB, A'B' are measured by means of some angular instrument, it will be found that AB is equal to A'B', and that this is the case with regard to any two stars whatever. Since therefore the arcs AB and A'B' are equal, it follows that the chords A A', B B’, joining their extremities, are parallel; whence we infer, that if two stars rise successively at the same point of the horizon A, they will also set at the same point A', a circumstance which is of itself strikingly indicative of the uniformity of the motion of the celestial bodies.

Through C let a perpendicular be drawn to the chord A A', meeting the boundary of the horizon in S and N. These two points are the South and North points of the horizon; and another straight line drawn through C, parallel to A A', will meet the horizon in E and W, the East and West points. The four together, namely, S, E, N, W, are called the four Cardinal points of the horizon. The horizontal circle is itself denominated the Azimuth Circle. Azimuth distances are measured from the North and South points, so that SA is the azimuth distance of a star’s rising at A, and SA' is that of its setting at A'. The complement of the azimuth, or its defect from a right angle, is called the Amplitude, which is consequently measured from the East and West points; thus the arc EA is the amplitude of a star’s rising at A, or its ortive amplitude, and WA' is that of its setting, or its οccasive amplitude.

Conceive a straight line to pass through C, perpendicular to the horizon. This line, which is called the Vertical, will meet the visible hemisphere in a point directly above the observer, which is denominated the Zenith; the point diametrically opposite, in which the prolongation of the vertical meets the invisible hemisphere, is the Nadir. The plane determined by the vertical and the straight line NS is called the Meridian, because when the sun reaches it, he is equally distant from the points at which he rises and sets, or, it is mid-day. The great circle formed by the intersection of the meridian with łhe celestial sphere is the Meridional circle; it divides the sphere into the eastern and western hemispheres. Any circle passing through the Zenith and Nadir is called a Vertical Circle ; that which intersects the meridional circle at right angles; or which passes through the points E and W, is called the Prime Vertical.

Let HPZH' (fig. ,2) represent a meridional circle passing through P, that point of the sphere which appears to remain immovable; and let Z be the zenith, and HH' the horizon. Join PC', and let, it meet the circle in P'. The points P and P' are denominated the Poles of the, world, and the straight line PP∖is the Axis about which the whole heavens appear to revolve. Let also, uv, Hy, kl, AEAE, ý'H', be projections of the paths of different stars [7:3:755]1 on the vertical plane PZH. One of the first circum- • stances which strikes the attention of the observer is the 'inequality of the times during which different stars continue visible above the horizon. A star which describes the circle AEAE continues visible during a shorter time than one which describes hl, and still shorter than one which moves in ^Αg. It will be remarked, however, that the interval between two successive risings of the same star is always the same, in whatever point of the horizon it makes its appearance. Those which describe AEAE are peculiar in this respect, that the interval between their rising and setting is exactly equal to that between their setting and rising; hence the great circle which they describe is called the Equator, and when the sun describes that circle, the days and nights are exactly of the same length. In general, the time during which a star continues visible is greater in proportion as its path is nearer to the visible pole. Those, however, which describe the circle Hg, or any other parallel, as uv, nearer to P, remain always above the horizon; hence such circles are called circles of Perpetual Apparition, and the stars describing them are called Circumpolar Stars.

There are some other technical terms, chiefly employed in the old treatises of astronomy, to denote particular circles of the sphere. A Secondary to any great circle, is a circle passing through the poles of its primary: thus the meridian is a secondary to the horizon HH', and to the equator AEAE'. The Colures are two secondaries to the equator, one of which passes through the equinoctial points, and is called the Equinoctial Colure ; the other intersects the equator at the distance of 90° from the former, and is denominated the Solstitial Colure. The small circles parallel to the equator are simply termed Parallels ; those which are parallel to the horizon are called Almicantars ; and the different meridians are likewise termed Hour Circles.

The hypothesis of a spherical revolution implies, as has already been mentioned, that the stars move in parallel planes, or describe parallel circles round the pole. This point, in reference to any particular star, must be determined by observing whether its angular distance from the pole remains invariable during its whole revolution. Let Z (fig. 3) be the zenith, P the pole, and S the place of a star. The arcs PZ, ZS, PS, which join these points on the sphere, will form a spherical triangle, of which the sides PZ and ZS, or the distances of the pole and the star from the zenith, are given by direct observation. The angle PZS is also given in the same manner, for it is equal to the star’s azimuth counted from the north point of the horizon. In the triangle ZPS we have therefore given the two sides ZP and ZS, together with the included angle PZS, from which data the polar distance PS is found by the following formula of spherical trigonometry:

cos. PS = sin. ZS sin. ZP cos. PSZ + cos. ZS cos. ZP. In order to adapt this formula to logarithmic computation, an auxiliary angle φ may be assumed such, that tan. φ = cos. PZS tan. ZP, by the substitution of which the formula becomes cos. ZP. cos. (PZS — <ρ) cos. PS =; cos. φ from which the polar distance PS of the star can be easily computed.

On performing the computation for any particular star, PS is always found to be of a constant magnitude, whatever may be the value of PZS, that is, at whatever part of its course the star is observed. Hence wė infer that the diurnal motion of the stars round the pole is exactly circular.

. After having recognised that the celestial bodies move in parallel planes, the next object of the astronomer is to determine whether they revolve with velocities uniform or variable. For this purpose it is necessary to compare, by means of a well-regulated clock, the angular distances between the meridian and the successive positions of a star, with the times elapsed since the star’s passage over the meridian. Let mno (fig. 4) be the parallel described by a star, and ssl two of its successive positions. The arcs ns, ns t , intercepted between the meridional plane and the planes passing, through the axis PP' and the points s, s', are measured by the arcs of the equator Ca, Cδ. Now, it is the result of constant experience, that the number of degrees in the arc Ca or Cb is to the number in the whole circumference as the time employed by the star in describing na or nb is to the time in which it completes a revolution. The time occupied by the star in completing its revolution is called a sidereal day; and the uniformity of the diurnal revolution enables us to calculate the angle CPs, which is measured by Ca, without having recourse to any other observation than that of the star’s meridional passage. Denoting a sidereal day by T, the interval of time which elapses while the star describes the arc ns by t, and the angle at the pole by P, we shall have the analogy T: t : : 360: P; therefore P= 370°; a formula which gives the angle at P, or its measure Co. The angles CPs, CPs', and sPs ’ are called Hour Angles, because the arcs which they intercept on the star’s parallel of declination correspond to the hours and fractions of an hour into which the sidereal day is divided. For example, if the hour angle sPs' contain 15°, the two planes passing through s and s' will intercept on mn, the diurnal circle of the star, an arc equal to 15°; consequently there will be 24 such arcs in the whole circumference, each of which will be described in the 24th part of a sidereal day, or in one hour of sidereal time. In consequence of this perfect uniformity of the diurnal motion, the arc which a star describes on its parallel is conveniently measured by the time of its description; and the sidereal day, or the interval between two successive returns of the same star to the plane of the meridian, offers the most perfect unit of time, inasmuch as it is exactly the same to all the inhabitants of the earth, and remains absolutely unalterable in all ages, being one of the very few invariable elements of the system of the world.

In fixing the position of a point on a plane surface, geometers usually refer it to two straight lines at right angles to each other, the positions of which are supposed to be known. The same practice is imitated by astronomers, who refer the position of any point on the surface of the sphere to two great circles, which are employed as a system of rectangular co-ordinates, and from which all distances are measured. The equator being a circle of the sphere not subject to any arbitrary condition, but determined by the very nature of the spherical revolution, is obviously well adapted for this purpose. It is therefore universally employed as a term of comparison in assigning the places of the celestial bodies. Suppose an hour circle passing through the star 5 to intersect the equator at a; the arc sa, which measures on the sphere the distance of s from the equator, is called the Declination of the star. All other stars situated on the same parallel mo have the same declination; and it is invariable for all places of the earth, because it is not affected by the diurnal motion which is performed in the parallel mo. The declination is northern or southern, according as the star is situated in the northern or southern hemisphere. It. is expressed in degrees from 0 to 90, proceed[7:3:756]ing from the equator to the pole. The polar distance ; Pä is the complement of the declination, because Ps + sa ·=. 90°. In order to distinguish s from all other stars which may be situated on the same parallel, we must also know the point a of the equator to which it corresponds; and for this purpose it is necessary to select a point w of the equator for the origin of the co-ordinates. This point ought to be independent of the diurnal motion, otherwise its position could only be determined for a single instant of time; and it must be such that it can be readily found, in whatever part of the earth the observer may be situated. Astronomers have agreed to employ for this purpose the point determined by the intersection of the equator with another very remarkable circle of the sphere, of which we shall speak hereafter, and to count the degrees on the equator, setting out from ɤ, from 0 to 360, in the direction opposite to that of the diurnal motion, or from west to east. The arc ɤ a, which measures the distance between ɤ and a, is called the Right Ascension of the star: it is also the right ascension of any other star situated in the hour circle PsP'. It is measured by the interval of time which elapses between the transit of the star s over the meridian, and that of the point ɤ, which has been chosen as the point of departure. It may be remarked that the point ɤ is also the origin of sidereal time; that is to say, the time is counted 0 hour 0 min. 0 sec. when that point passes the meridian.

From these definitions it is easy to conceive the general method of forming a catalogue of stars. A clock being regulated to sidereal time, arid marking 0 hour at the instant of the transit of the point ɤ (or any star chosen arbitrarily), the hours, minutes, and seconds, are noted at which the stars successively pass the axis of the transit instrument, together with their respective altitudes at the same instant. These observations give the right ascensions and declinations; for the time is easily converted into degrees, at the rate of 15° an hour, or by the formula given above for determining the hour angle. If the star s pass the meridian one hour after st, the arc a b, which measures the distance between their horary circles, PαP', PZ*P', is 1 hour or 15°; and the declination of a star s which moves in the parallel hl (fig. 2) is AEZ= H'Z— H'AE, that is, equal to the difference between its meridional altitude and the complement of the altitude of the pole H P. Modern celestial charts are constructed on the principle of assigning to each star the place indicated by the values of these two co-ordinates; and as observation proves that the mutual distances and relative positions of the stars scarcely undergo any sensible variation, a globe or chart once constructed will serve to represent the state of the heavens for at least a long series of ages.

Having now considered the general phenomena of the diurnal revolution, and explained the terms that are technically employed in assigning the positions of the celestial bodies, we may next proceed to inquire how these phenomena are to be accounted for,—whether the stars are really in motion, as they seem to be, or if the apparent motion is only an illusion occasioned by the revolution of our own earth. The perfect uniformity of the motions of the different stars renders it exceedingly improbable that they are disconnected; hence the simplest view of the phenomena is obtained by substituting for each star its projection on the celestial sphere, at an infinite distance, and supposing this sphere, with the projections of all the stars, to revolve in 24 hours from east to west round an immovable axis. But it is easy to see that the phenomena will be equally well explained by supposing that the starry firmament is absolutely at rest, and that the earth revolves in the same time, round the same axis, but in an opposite direction, from west to east. In both cases the stars remain immovable, the phenomena are exactly the same, and, relatively to Spherical Astronomy, in which we only concern ourselves with the apparent motions, it is absolutely indifferent which of the two hypotheses we adopt. They are both only modes of explaining certain appearances; and the one may be employed which renders the explanation most simple and perspicuous. The proofs of the earth’s motion will go on accumulating as we proceed. At present we need only remark, that as the organ of sight makes us acquainted with the existence of relative, and not of absolute motion, it is impossible to decide, merely from appearances, whether the motion we perceive is real or otherwise; for whatever motion one body may have in respect of another, it is always possible to explain the phenomena by supposing it to be perfectly at rest, and the other to move in an opposite direction. The conclusions which we draw from the optical effects of motion afford no mathematical certainty with regard to the cause of that motion. The hypothesis of the revolution of the sphere is attended with innumerable and insurmountable difficulties. The distance of the nearest fixed star from the earth is not less than 350,000 times the distance of the sun; a distance which light, prodigious as its velocity is, would not traverse in less than five years. This immense line, therefore, supposing the heavens to revolve round the earth, would form the radius of a circle, the circumference of which, six times larger, would be passed over by the star in the space of 24 hours; its velocity must therefore be 6 × 5 × 365= 10950 times greater than that of light. This velocity, which is nearly equivalent to 2100 millions of miles in a second of time, is so enormous that it baffles the utmost efforts of the imagination to form any conception of it; and the supposition of its existence will appear still more revolting when we reflect that the distance of the nearest star is probably many thousand times less than that of the Milky Way. If we ascribe the motion to the earth, its velocity, though it may still be supposed great, is moderate in comparison of many well-known phenomena of nature. A point on the equator will describe in 24 hours a circle of about 25,000 miles, or about 17 miles in a minute; a velocity somewhat exceeding that of sound, but 7400 millions of times less rapid than the preceding.

In all that has hitherto been said respecting the apparent motion of the starry sphere, we have only had regard to the innumerable multitude of stars which constantly maintain the same relative situations, and which are in consequence called A⅛ei∕ Stars. There are, however, several other bodies, some of them remarkable on account of their splendour, which, besides participating in the general motion, have peculiar motions of their own, and are incessantly shifting their positions among the fixed stars. An attentive observation of the state of the sky during a few successive evenings, will suffice to show that there are some stars which have in the mean time changed their places; and, on continuing to observe them, they will be found to separate themselves from particular constellations, and gradually but imperceptibly to approach others, till they at length appear, after unequal intervals of time, in an opposite quarter of the heavens. From this circumstance they were designated by the Greeks Planets, I that is, wandering stars, in contradistinction to those which obey only the law of the diurnal motion. Besides the sun and moon, there are five discernible by the naked eye, and which have consequently been known from the remotest ages. These are Mercury, Venus, Mars, Jupiter, and Saturn. Five others, namely, Uranus, Ceres, Pallas, Juno, [7:3:757]land Vesta, have been discovered by the aid of the tele- ’ ∙ scope. Astronomers, with a view to abbreviate their descriptions, have appropriated a certain symbol to each of these planets, as well as to the sun and moon: thus, in the order of distance from the sun, the names and characteristics are as follows :—

The Sun 0 Ceres p

Mercury £ Pallas £

Venus Ç Jupiter Iţ.

Mars $ Saturn T 2

Vesta g Uranus Ķ

Juno $

The Earth, which, as we shall afterwards see, is also a planet, and takes its place between Venus and Mars, has for its symbol θ, and the moon 5 . Venus, Jupiter, and sometimes Mars, are distinguished by their extraordinary brilliancy. Mercury, on account of his proximity to the sun, is rarely visible to the naked eye. Uranus, discovered by Sir W. Herschel in 1778, can with difficulty, by reason of his great distance, be perceived without a telescope. Ceres, Pallas, Juno, and Vesta, discovered since 1801, are extremely small in size, and can only be seen with the aid of the telescope.

f of all the celestial bodies, the most interesting to us are the sun and moon; and their peculiar motions have accordingly, in every age of astronomy, been studied with the greatest attention. The proper motion of the moon is particularly remarkable. In the course of a single night she separates herself very sensibly from the stars in her vicinity, moving over a space nearly equal to her own breadth in an hour, and completing a whole circuit in about 27 days. The sun moves with much less velocity, but his motion is still sufficiently apparent. If we take notice of the stars which immediately follow him when he sinks under the horizon, we shall find that in the course of a few nights they will be no longer visible. Others which, some time previously, did not set till long after him, have taken their places and now accompany the sun. In the morning similar appearances present themselves, but in a contrary order. The stars which appear in the eastern horizon at sun-rise, are, after a few days, considerably elevated above it at the same time. Thus the sun seems to fall daily behind the stars, by insensible degrees, till at last he appears in the east when they are about to set in the west. To account for these appearances, the ancients supposed the sun’s diurnal motion to be really slower than that of the stars; hence they supposed him to be attached to a different sphere. For like reasons they ascribed particular spheres to the moon and each of the planets; and as no trace of these imaginary spheres is perceptible in the heavens, they next supposed them to be crystalline and transparent. The appearances are explained equally well, and with infinitely greater simplicity, by ascribing to the sun a proper motion, in a direction opposite to that of the diurnal rotation of the sphere, in consequence of which he advances to meet the stars, instead of falling behind them.

The greater part of the observations of the early astronomers had for their object the determination of the positions of the stars relatively to the sun at his rising and setting, by which they fixed the seasons, and regulated the operations of agriculture. They distinguished all these phenomena by technical terms, which occur very frequently in the works of the ancient poets, particularly in Hesiod and in Ovid’s Fasti. A star which rises at the same time with the sun is effaced by his light, and is said to rise cosmically (orths cosmicus). Soon after, when the sun by his proper motion has advanced so far to the east that the star can be perceived on the eastern horizon in the morning twilight, it disengages itself from the sun’s rays, and is said to rise heliacally ( 0r tus heli α cu s Y At the end of six months, the sun being diametrically opposite to the same star, it sets as the sun rises, and in this case it is said to set cosmically (occasus cosmicus) : at nearly the same time it rises when the sun sets, and is said to rise acronically (ortus acronyctus). The sun afterwards begins to approach the star, till he advances so near that it is again about to be effaced by his light; it is now said to set heliacally, or just so long after the sun as to be visible when he has disappeared (occasus heliaaus). At the end of a year, the star again rises and sets at the same moment with the sun; it is now said to set acronically (occasus acronyctus). These distinctions, and the ancients had several others of the same kind, which are all defined by Ptolemy, are now scarcely ever mentioned. They have lost the whole of their interest since, in the progress of astronomy, more certain methods have been discovered of determining the commencement of the year and the seasons.

The proper motions of the planets are in general, like that of the sun, in a direction opposite to the diurnal motion, or from west to east, among the stars: but they do not preserve the same character of uniformity; the planet sometimes becomes stationary among the fixed stars, and even advances from east to west in the direction of the diurnal motion. In this case it is said to retrograde. This retrograde motion is, however, not of long continuance. After having been accelerated during a short time, it begins to relax. The planet again becomes stationary, and then resumes its direct motion from west to east. These phenomena were observed in the remotest antiquity, and their explanation formed the principal part of the rational astronomy of the Greeks and the Arabians.

Besides the planets, other bodies occasionally make their appearance in the heavens, which, by reason of the extraordinary phenomena they exhibit, have been frequently contemplated with terror and dismay, and regarded by superstitious ignorance as harbingers of calamity, and precursors of the divine vengeance. These bodies shoot down, from the remote regions of space, with inconceivable velocity, towards the sun. At their first appearance they are small; their light is feeble and dusky; and they are generally accompanied by a sort of nebulosity or luminous tail, from which they have derived their appellation of Cornets (coma, hair). As they approach the sun, their apparent magnitudes and brilliancy greatly increase, and the nebulosity sometimes occupies a large portion of the heavens, presenting a magnificent and astonishing spectacle. Having attained the point of their orbits nearest the sun, they again recede to enormous distances, and vanish by insensible degrees. They differ from the planets not only by the appearances they present, but also by the diversity of their motions; for, instead of being confined to a particular zone, and moving from west to east, they traverse the sky indifferently in all directions. They are visible only in a small part of their orbit, which, being near the sun, is passed over with prodigious rapidity. They seldom continue visible longer than six months. Their number is entirely unknown, but during the last two centuries upwards of 150 have been observed, and their orbits computed.

Thus we recognise three distinct classes of celestial bodies; the planets, the comets, and the fixed stars. It is the business of the astronomer to determine the positions of these bodies in the heavens; to observe their motions, and measure them with precision; to discover the laws by which their courses are regulated; and from these laws to assign the past or future state of the heavens [7:3:758] at any given instant of time. The fixed stars form infi-; nitely the most numerous class; but their motions being sensibly uniform, and their mutual distances invariable, the principal object with regard to them is to determine their places and relative positions. The comets in general return only after long periods; and observations are not yet sufficiently numerous to allow any certain deductions to be made with respect to their nature and constitution. The planets present a great variety of curious phenomena; they are always within our view; observations on them may be multiplied indefinitely; and for this reason, as well as on account of their proximity, and their forming a system of which the earth constitutes a part, they present by far the most interesting objects of astronomical study.

Sect. II.— Of the Globular Form of the Earth, Parallax, and Refraction.

Having now considered the general phenomena of the diurnal motion, we proceed next to inquire into the form of the earth, and our situation with respect to the centre round which the celestial sphere appears to revolve. The plane of the horizon seems to be stretched out indefinitely till it actually meets the sky; but this illusion is quickly dissipated by transferring ourselves from one place to another on the surface of the earth, and attending to the phenomena which such a change of place gives rise to. Let an observer, for example, set out from any given point in the northern hemisphere, and proceed directly south. In proportion as he advances, the stars in the southern region of the heavens will be elevated more above the horizon, and describe larger segments of their diurnal circles, while new ones come into view which were invisible at the station he left. On the other hand, the polar star, with those in its vicinity, will be depressed, and some stars which before continued above the horizon during the whole time of their revolution will now rise and set. The planes of the diurnal circles become also more perpendicular to the horizon, so that the aspect of the heavens is entirely changed. If, instead of advancing in the direction of the meridian, the spectator proceeded towards the east or west, he would in this case also remark that his horizon constantly shifted its position. A star will pass his meridian sooner as he advances eastward, or later as he travels westward; in short, by a change of place in any direction whatever, the perpendicular to the horizon, or the plumb-line, will correspond to a different point of the heavens. The plane of the horizon is therefore variable, and its variation by insensible degrees indicates with the greatest evidence the rotundity of the earth. Experience also shows that a spectator sees more of the terrestrial surface in proportion as he is elevated above it; and that on a mountain surrounded by the sea, or standing in the middle of a level plain, the horizon appears equally depressed all round, which is the distinctive feature of a spherical curvature. In fig. 5 let 0 be the place of a spectator on the summit of a mountain, the straight line H H will represent his horizon, and Oh, Ohl the directions of visual rays to the remotest visible points on the surface of the earth. The inclinations of these lines to the plane of the horizon, or the angles HOa, ΗΌΛ' are called the apparent depression ; and these being always observed to be equal to each other, it follows that the curve hAh' is a circle. On the Peak of Teneriffe, Humboldt observed the surface of the sea to be depressed on all sides in an angle of nearly 2°.. The sun arose to him 12 minutes sooner than to an inhabitant of the plain; and from the plain, the top of the mountain appeared enlightened 12 minutes before the rising or after the setting of the sun. The same phenomena, though on a smaller scale, are observed with regard to every mountain or elevation on the earth. Another familiar illustration of the globular figure of the earth is derived from the successive and gradual disappearance of a ship which leaves the shore and stands out to sea. The hulk disappears before the sails and the rigging, and the top of the mast is the last part that is visible. The ship thus appears gradually to sink under the horizon, exactly in the same manner as she must necessarily do on the supposition that the surface of the sea is spherical. The appearance of the moon at the time she is eclipsed is also demonstrative of the roundness of the earth. When the moon penetrates the shadow of the earth, the line which separates the illuminated from the eclipsed portion of the disk is circular; evidently proving the conical form of the shadow, and consequently the roundness of the body by which the shadow is projected. From all these considerations, it is inferred with the highest certainty that the earth with its waters forms a round mass, isolated in space.

The globular form of the earth is a property common to it with the other bodies which compose the planetary system. The sun and moon are evidently round bodies; and when the planets are examined through a telescope their disks appear sensibly circular; and as they are known to have a motion of rotation, in consequence of which they successively present different points of their surfaces to the earth, the uniform roundness of their disks may be taken as a conclusive proof of their sphericity. We shall by and by have occasion to remark other striking analogies between the planets and the earth.

Since the figure of the earth is spherical, the horizontal plane cannot coincide with any considerable portion of its surface. It may be defined to be the plane which touches the earth at any given point, and is perpendicular to the vertical line, or the direction of gravity at that point. The Sensible Horizon of any place is the plane which passes through the eye of the spectator, perpendicular to the plumb-line at that place. The Rational Horizon is a plane parallel to this, passing through the centre of the earth. The particular phenomena of different places depend on the position of their horizon with respect to the planes of the apparent diurnal motion of the sun and stars. The rational horizon of a place on the equator passes through the poles, and divides equally die equator and its parallels. Hence the days and nights arc always equal in such places, and each of the stars performs one half of its revolution above, and the other below its horizon. The circles of diurnal motion are all perpendicular to the horizon, and therefore the inhabitants are said to be under a Right Sphere. If a spectator could place himself directly under the pole, his horizon would coincide with the equator, and the whole of the northern celestial hemisphere would be within his view, while no part of the southern hemisphere would be visible to him, on account of its being always beneath the horizon. The circles of the diurnal motion being parallel to the equator, and consequently also to the horizon, the fixed stars would never either rise or set. A spectator thus situated is said to be under a Parallel Sphere· In Intermediate places the circles of the diurnal motion are oblique to the horizon, one pole being always elevated above it, and the other equally depressed below it. The stars whose distances from the elevated pole are not greater than the arc of its elevation above the horizon never set, while those within the same distance ol the depressed pole never rise. These phenomena belong to all places situated between the equator and the poles, and the inhabitants of such places are said to be under an Oblique Sphere.
[7:3:759]

The situation of a place on the surface of the earth is determined by two co-ordinate circles, in the same manner as that of a star on the celestial sphere. Let Z (fig. 2) be a place of which the position is required to be assigned with reference to circles of the terrestrial sphere, analogous to those which we have described as belonging to the sphere of the heavens. Since the axis of the celestial sphere passes through the centre of the earth, we may suppose the poles of the world, the equatorial circle, and the meridians, to be transferred to the earth’s surface, so that the same figure may represent either the celestial or terrestrial sphere. The intersection of the plane P Z AE with the surface of the earth is the terrestrial meridian of the place Z, and the straight lines C Z, C AE, in that plane, intercept an arc of the terrestrial meridian between Z and the equator, containing as many degrees as the arc of the celestial sphere which measures the declination of the zenith of the place Z. The angle AE C Z which is measured by the meridional arc AE Z is called the Geographical Latitude of Z. But in order to particularize the point at which the equator is intersected by the meridian, it is necessary to assume some point of the equator to which all the other points of that circle may be referred. The meridian passing through the assumed point is called the First Meridian ; and the angular distance between the planes of the first and any 3ther meridian measured by the equatorial arc intercepted by these planes on the equator, is called the Geο- ∣ rαρhical Longitude of the place through which the last neridian passes. The longitude is said to be east or west, ιccording as the degrees of the equator are counted from he first meridian towards the east or west. It is usual to eckon the degrees towards the east, all round the globe, or rom 0 to 360°. From these definitions it is evident that he geographical latitudes and longitudes are exactly ana-ogous to the declinations and right ascensions on the ce-estial sphere. The first point of the terrestrial equator ;annot be determined by any star or fixed point in the icavens, on account of the diurnal motion; it is therefore ecessary to fix its position by means of known places on he earth; and geographers are in the habit of assuming s the first meridian that which passes through the capi-ιl city or principal observatory of their country. The hoice is of no importance; for in geography, as in astro-omy, what is essential to be known is only the difference f longitudes or meridians, in order to reduce the situ-tion of a place, or observations which have been made at t, to any indicated meridian.

,1 The length of the terrestrial radius is a very im-ortant element in astronomy, inasmuch as it furnishes ıe observer with the only scale by which he can esti-ιate the distances of the sun, moon, and planets. On ıe supposition that the figure of the .earth is perfectly i Aerical, the general, principles on which its magnitude i ιay be determined are sufficiently obvious; but the acuate determination of its actual dimensions is attended ith great practical difficulties, which can only be overcome by the perfect instruments and refined science of modern times. Eratosthenes seems to have been the l∙'st who made use of astronomical methods to determine l’te circumference of the earth, or the length of the median. He remarked that, at Syene, in the Thebais, the fin on the meridian, at the time of the summer solstice, fas vertical; and that at Alexandria, at the same time, is zenith distance was 7° 12 ’. Now let S (fig. 6) be the, l ιn, vertical to Μ, and Z the zenith of Alexandria, C iiing the centre of .the earth, and 0 Alexandria. The i gle Z 0 S is the sun’s zenith distance, which was ob-rved to be 7° 12'. But Z 0 S = Z C S + 0 S C; whence, as the angle 0 S C, which is extremely small on account of the sun’s great distance, may be neglected in the calculation, the angle Z C S at the centre is also equal to 7° 12', and therefore the whole circumference eauals 360° *

rļo 12' × Μ 0. To obtain the length of the meridian, it is therefore only necessary to measure the arc Μ 0. Eratosthenes assumed the distance between Syene and Alexandria to be 5000 stadia; hence the circumference of

360°

the earth = l -L-^ × 5000 stadia = 250,000 stadia. The uncertainty which exists respecting the length of the Egyptian stade prevents us from deriving any precise information from this rude attempt to estimate the dimensions of the globe.

The method which has just been described takes for granted that the meridian is exactly circular,—an assumption which, even supposing the ’method perfect in all other respects, would lead to erroneous results, especially in an arc of so great a magnitude as seven degrees. In a small arc, of 1° for example, the error arising from the non-sphericity of the earth will be insensible; for it is certain, from the phenomena before explained, that the deviation from the spherical figure is not very considerable; and besides, whatever the nature of the meridional curve may be, it will sensibly coincide with its osculating circle, throughout the extent of an arc of 1°. It has been found by numerous and accurate experiments, that the lengths of arcs of 1° on the same meridian are longer in proportion as we advance nearer the pole. Hence, on account of the similarity of the isosceles triangles of which these arcs form the bases, their sides, or the terrestrial radii, must also be longer, and consequently the convexity of the earth is less towards the pole than at the equator. The surface of the earth is extremely irregular, even independently of the inequalities occasioned by mountains and cavities; yet it has been discovered that the meridional curves differ almost insensibly from ellipses; whence it is concluded that the figure of the earth is an ellipsoid of revolution about its shortest axis. In comparing the results of the various measurements which have been made with the formulae belonging to the dimensions of such a body, this conclusion has been fully verified; and the lengths of the arcs, the ellipticity, the distance of the pole from the equator, and, in short, all the elements of the spheroid, have been determined. The results of theory and observation give an ellipticity amounting very nearly to - 5 ∙⅛- c -; that is to say, the equatorial is to the polar diameter in the ratio of 306 to 305. The following may be regarded as a very near approximation to the dimensions of the earth in English miles :—
[table]

The figure of the earth is one of the most difficult and most important questions of astronomy; and as our limits will not permit us to treat it in this place with all the details which its importance renders necessary, we shall reserve, for a separate article, an account of the different geodetical operations which have been undertaken with a view to determine it, together with the development of the mathematical theory of its equilibrium. See Figure of the Earth.

In describing the phenomena of the diurnal revolution, we have supposed the eye of the spectator to be placed [7:3:760]in the centre of motion, on which supposition the appa- rent and true places of the celestial bodies will be the ’same. This, however, as is evident, can only be rigorously true in respect of one point of the earth’s surface, and is therefore most probably not true of any. Astronomers have on this account agreed to refer all the motions to the centre of the earth; that is, to consider the centre of the earth as the centre of the celestial sphere, and to regard as the true geocentric place of a star, that at which it would appear to a spectator placed at the centre. But the spectator is necessarily placed at the surface. In order, therefore, to render observations made at different stations on the earth’s surface comparable with one another, we must consider what changes will be produced in the apparent positions of the celestial bodies, in consequence of their being observed from a point not situated in the centre of motion.

Let O f and 0 (fig. 6) be two stations on the earth’s surface on the same meridian, from both of which a star S is observed at the same time. As an object is always seen in the direction of the visual ray, the spectator at 0 will refer the star S to a certain point I on the surface of the celestial sphere, while the spectator at O' will refer it to a different point I' in the direction O' I'; and the difference of the apparent places I and I' depends on the magnitude of the angle OSO', which is subtended by the chord of the arc OO'. If a third spectator could be placed at C, the centre of the earth, the same star would appear to him in the direction CS, and be referred to its true place on the sphere at Q. Now, the difference between the angles ZOS and ZCS, that is, the angle OSC, under which the radius of the earth is seen by an observer at S, is called the parallax of S. On account of the great distance of the fixed stars, this angle is altogether insensible with regard to them; it is, however, very sensible in the case of the moon, with regard to which it amounts to about 1°. The greatest parallax of the nearest planets does not exceed 30".

From the definition which has just been given, it appears that the parallax of a star is the angle comprised between two lines drawn from the star, the one to the centre of the earth, and the other to any point whatever on its surface. But it is evident, that while the distance of S from the earth remains constant, the angle OSC will vary with the angle ZOS, that is, with the distance of the star from the zenith, or its altitude above the horizon. Let OH (fig. 7) represent the horizon. It is easy to see that the angle OSC will have its greatest value when OS touches the surface of the earth, or coincides with the horizon OH, that is, when the angle ZOS = 90°. It is equally evident that the parallax OSC vanishes entirely when the line CS coincides with CZ, that is, when the star is in the zenith of the observer at 0; hence the parallax depends on the altitude of the star, according to a law which we now proceed to determine.

In the triangle OSC (fig. 7), we have sin. OSC: sin.

CO

SOZ;: CO: CS, whence sin. OSC = ⅛⅛- sin. SO Z. Let the angle SOZ, or the zenith distance of the star = Z, the semidiameter of the earth OC = α, the distance of the star CS = r, and the parallax OSC = p, we have then sin. p = f sin. Z. On account of the smallness of the angle p, which, as we have mentioned, in no case exceeds 1°, the arc may be substituted for the sine (for their difference even in an arc of 1° does not exceed 0"∙18): the above expression therefore becomes P — ~ sin ∙ Z; that is to say, the parallax of a star is proportional to the ' sine of its zenith distance. At the zenith, Z = 0, there-- fore sin. Z = 0, and the parallax vanishes: at the horizon, Z = 90°, and sin. Z = 1; the parallax, therefore, becomes equal to p In this case it is at its maximum, and is denominated the Horizontal Parallax; in all other cases it is called the Parallax of Altitude. Denoting the horizontal parallax by P, we have P = -, whence p = P sin. Z; that is to say, the parallax of altitude is equal to th e ho r i. zontal parallax, multiplied by the sine of the apparent zenith distance. It is evident that the apparent altitude SOH is always less than the true altitude SBH, by the whole amount of the parallax; the effect of the parallax is therefore to depress the object, or increase its zenith distance; hence, if θ be the apparent altitude of the star, and ff its true altitude, ff will be found from the equation il — Û +P sin. Z.

From the above results it is manifest, that if the horizontal parallax can be by any means determined, the parallax at any other altitude will be found at the same time. The determination of the horizontal parallax is, however, attended with considerable difficulty, and various methods have been proposed and practised to ascertain its exact amount, modified by particular circumstances in the cases of the different celestial bodies. The method, however, which serves as the basis of all the others is extremely simple, and exactly analogous to that by which the distance of a remote object is determined on the surface of the earth. Suppose two observers to be stationed at the points 0 and O' (fig. 6), of which the] latitudes are known, and which are both situated on the same meridian, and let them simultaneously observe the zenith distances of the star S; these observations will give the angles ZOS, Z'O'S, whence their supplements SOC and SO'C become known at the same time. A third angle of the quadrilateral figure SOCO', namely OCO,' is also known, being measured by the meridional arc OMO ’, the difference or sum of the latitudes of the two observers, according as they are on the same or opposite sides of the equator. The two sides also, CO and CO', being semidiameters of the earth, are supposed to be known; every part of the quadrilateral figure is therefore determined, and its diagonal CS may be calculated by the rules of plane trigonometry. But when CS is determined, the horizontal parallax is obtained immediately from the formula P = -, that is, P = On this principle the ho-

T L/ð

rizontal parallax of the moon was determined by Lacaille and Ļalande, the former observing at the Cape of Good Hope, and the latter at Berlin; and the parallax of Mars by Lacaille at the Cape, and Wargentin at Stockholm.

Let us suppose two lines to be drawn from the centre of a planet touching the surface of the earth in points diametrically opposite; the inclination of these two straight lines is the double of the horizontal parallax; but the same angle also measures the diameter of the earth as seen from the planet: hence the horizontal parallax of a planet is equal to the apparent semidiameter of the earth at the distance of the planet. From this consideration the true diameter, and consequently the volume, of a planet maybe found by measuring its apparent diameter, that is, the optical angle under which its diameter appears when seen from the earth. Let d be the a PP a ^ rent, and D the true diameter of a planet; the angle įd is comprised between two straight lines drawn from the [7:3:761]earth, the first of which is directed to the centre of the planet, and the last is a tangent to it: hence į D = r tan. ⅛d or r sin. ∙⅜√, according as τ is the line which touches the planet or that which is directed to its centre. Now, on account of the smallness of the angle l,d, the sine and tangent are sensibly equal to the arc; we may therefore suppose in all cases D = τ d. Combining this with the equation P = -, we deduce — = ~; which gives the ratio between the true diameters of the planet and the earth in terms of the apparent magnitude and horizontal parallax of the planet. It follows also from this last equation that -p-, or the ratio of the apparent diameter and parallax of a planet, is constant; the horizontal parallax may therefore be found at any time whatever by measuring the planet’s diameter.

From the equation P=~, we have ʹ-J√ that is, the distance of a planet from the earth is known in terms of its parallax and the earth’s semidiameter. The parallaxes, therefore, give the ratio of the distances of all the planets from the earth, and consequently of their distances from one another, and from the sun; hence the radius of the earth furnishes the scale by which the astronomer measures the dimensions of the whole solar system, and the magnitudes or volumes of all the bodies of which it is composed. On this account the accurate determination of the parallaxes of the celestial bodies is a problem of great importance in practical astronomy.

It is evident from the mere inspection of the figure, that the plane in which the straight lines C S and O S are situated is the vertical plane passing through S, consequently the whole effect of the parallax is to diminish the altitude of a planet in its vertical circle. When the observation, therefore, is made in the meridian, the effect of the parallax is to alter the declination, without producing any change whatever in the right ascension of the planet. Out of the meridian it is necessary to have regard to the azimuth or hour angle, as well as to the altitude, in calculating the correction due to the parallax. When we sup-Îiose the earth to be spherical, the formulae for the calcu-ation of the parallax are extremely simple, because the radius is constant, and the vertical line, or perpendicular to the surface of the earth, passes . through its centre. But in the case of the true or elliptical figure of the earth neither of these circumstances takes place. The radius in this case is variable, and must be determined by a particular process of computation for every point on the meridian; and the vertical line, with reference to which the latitude of the place and the altitudes are determined, does not pass through the centre of the earth, but makes different angles with the axis. These circumstances render the calculation of the parallax a matter of much greater complication and difficulty.

The term parallax in its general signification properly denotes change of place. There are consequently various kinds of parallaxes, such as parallax of right ascension, of declination, longitude, latitude, &c. In what has preceded, we have .supposed the earth to be at rest in the centre of the universe, and therefore have had regard solely to the variations produced in the apparent diurnal motions by the eccentric position of the observer on the surface of the earth, and which are comprehended generally under the denomination of the diurnal parallax.

The effects of the diurnal parallax are only sensible with regard to those bodies of which the distance from the earth is not so great as to be incomparable with the earth’s semidiameter. That the apparent place of an object must be changed in consequence of a change in the situation of the observer, is a simple geometrical truth which no experiment was required to discover. The first observers must accordingly have anticipated a parallax in all the celestial bodies; and it was doubtless only after considerable experience that they admitted the existence of any exception to the general law. There is, however, another cause of variation in the apparent positions of the celestial bodies also connected with the earth, the existence of which could not be known a priori, but must have been discovered by experience alone. We allude to the refraction of the rays of light in passing through the earth’s atmosphere.

The angular distance between two stars is found to undergo very sensible variations at different hours of the day. This phenomenon cannot be explained by any proper motion of the stars, because it evidently depends on their altitude above the horizon; and the differences are found to be the same daily at the same altitude. It is most striking when we compare a star which, without setting, passes the meridian twice a day, once near the zenith, and the second time near the horizon, with another star situated very near the pole, and of which the altitude is consequently nearly invariable. It will be found that at the time of the first transit the distance between the two stars is greater than at the second by nearly half a degree. It is evident, therefore, that the phenomenon consists in diminishing the distance between a star situated in the horizon and the visible pole, that is to say, in elevating the stars, whereas the effect of the parallax is to depress them. The refraction also takes place in an equal degree with regard to the fixed stars, and even the moon and planets, without being in any degree modified by the great differences in the distances of the celestial bodies, contrary in. this respect likewise to the parallax, which depends entirely on the distance. The reason of this will be evident from the consideration of the physical cause of the phenomenon.

According to the known principles of optics, a ray of light, in passing obliquely from one transparent medium into another of a different density, does not hold on .in its rectilinear course, but is refracted, or bent towards the denser medium. Now, the atmosphere which surrounds the earth may be regarded as composed of an infinity of concentric spherical strata, the densities of which are greater in proportion as they are nearer to the earth’s surface. When a ray of light, therefore, proceeding from a star enters the atmosphere, it is inflected towards the earth, or bent so as to form a smaller angle with a perpendicular to the surface of the earth; and this inflection will be increased by every successive stratum of the atmosphere through which the light passes. In fig. 8, let AA', BB', CC', represent the boundaries of the successive strata, which, for the sake of illustration, we here suppose to have a finite thickness. A ray of light proceeding from S comes in contact with the highest stratum of the atmosphere AA' at a. The molecular attraction of this atmospherical stratum, acting in the direction of a normal to A A' at a, causes the luminous ray to deviate from the direction S x, and assume another, a y, in which it would continue to move if the atmosphere were equally dense from A A' to the earth. But in the course of its progress the ray penetrates another denser stratum at b, and consequently suffers another inflection; so that instead of proceeding in the direction a y, it is bent into a new direction bz, more nearly perpendicular to the concentric strata. A similar effect is produced at c, so that the luminous ray, when it finally reaches the observer at 0, has as [7:3:762]sumed the direction c 0. In its progress from a to O, it has therefore successively moved in the direction of the sides of the polygon a, b, e, 0; and to the spectator at 0, the star from which it proceeded, instead of appearing in its true place at S, will appear to be at S', or in the last direction of the visual ray. Now, if A A' is the most elevated stratum of the atmosphere into which the ray enters in the direction S a, it is clear that the whole effect is produced by the atmospherical strata situated below A A', and that the length of S a is perfectly indifferent; hence the refraction is entirely independent of the distance of the stars, provided they are beyond the limits of the earth’s atmosphere.

The decrease of the density of the atmosphere, from the surface of the earth upwards, follows the law of continuity, or takes place by insensible degrees; so that the luminous ray, in traversing the atmosphere, enters at every instant into a denser medium, and is therefore continually brought nearer and nearer to the vertical direction. Hence the true path of the ray is curvilinear, and concave towards the earth, as represented in fig. 9. This is equivalent to the supposition that the thickness of the different concentric strata of uniform density is infinitely small, and that the light, as it successively penetrates each, deviates from its former path by an infinitely small angle, which may be considered as the differential of the refraction, the total amount of which will therefore be obtained by integration. The direction of the ray, when it reaches the eye of the observer, is the tangent to the last portion of its curvilinear path; and the apparent zenith distance of the star will be ZOS', while the real zenith distance is ZOS. The difference of these two angles, namely S'OS, is what is denominated the Astronomical Refraction. It is evident that the whole path of the ray is confined to the vertical plane, in which the star and the eye of the observer are situated; for the earth and its atmosphere being very nearly spherical, that plane will divide the strata symmetrically; the attracting forces will therefore be equal on each side of it, and consequently produce no effect in a lateral direction. When the observed star is due north or south, the vertical plane is the plane of the meridian; hence, in meridional observations, the whole of the refraction, like that of the parallax, takes place in declination, while the right ascension remains unaltered.

It is evident that the amount of the refraction is greater in proportion as the observed star is nearer to the horizon; for in this case the luminous rays strike the tangent planes of the atmospherical strata more obliquely, and have besides to traverse a greater extent of atmosphere before they arrive at the eye of the observer. On determining by experiment the refraction at every altitude from zero to 90°, tables of Refraction may be constructed, which will furnish the means of discovering the law of its diminution; but as such a process would be exceedingly tedious, and likewise subject to lead to erroneous results on account of the inevitable errors of observation, it is found more convenient to assume some hypothesis for a basis of calculation, and to verify the results which it leads to by comparing them with observation. In regard to media which may be said to be permanent, such, for instance, as water and glass, the determination of the refraction is not attended with great difficulty; but the circumstances are greatly altered when we come to make experiments on the atmosphere. In this case the difficulty arises from the incessant changes which the atmosphere is undergoing relatively to its refringent powers; changes which it is impossible for the observer fully to appreciate, inasmuch as he can only determine its physical state within a short distance of the earth, while that of the upper strata remains wholly unknown to him. The refringent power of the atmosphere is affected by its· density and temperature. The effects of the humidity are insensible; for the most accurate experiments seem to prove that the watery vapours diminish the density of the air in the same ratio as their refractive power is greater. It is therefore only necessary, even in delicate experiments, to have regard to the state of the barometer and thermometer at the time the observation is made. At a medium density, and at the temperature of melting ice, it was found by Biot and Arago, from a great number of exact experiments, that at any altitude between 10° and the zenith the refraction is very nearly represented by the formula r = 60"∙6 tan. (Z — 3 - 25 × r), in which τ is the refraction corresponding to a given zenith distance Z. With the exception of the numerical co-eñĩcients, this formula was first given by Bradley; but whether it was deduced from theory by that great astronomer, or was only empirical, is uncertain. Bradley’s formula was τ = 57" tan. (Z — 3 × r). When the direction of the luminous rays makes a smaller angle than 10° with the horizon, it becomes indispensable to take into account, in the calculation of the refraction, the law of the variation of the density of the atmosphere at different altitudes; a law which is subject to incessant variation, from the operation of winds, and other causes which agitate the atmosphere, as well as the decrease of temperature in the superior regions. For this reason all astronomical observations which have not refraction directly for their object, or which are by their nature independent of its influence, are made at an elevation exceeding 10°. For lower altitudes, it is to be feared that no theory will ever be found sufficiently exact to entitle the observations to much confidence.

The existence of the atmospherical refraction was not unknown to the ancient astronomers; but it is only in modem times that the subject has been studied with the requisite care to admit of its influence being calculated in the reduction of observations. Ptolemy does not allude to the subject in the Almagest, but he has given a sufficiently exact idea of it in his work on Optics. He mentions experiments made to ascertain how far a ray of light is bent from its rectilinear course in passing from air into glass and water; and observes that the astronomical refraction brings a star nearer to the zenith, and that it is feebler in proportion as the star is more elevated. He likewise indicates a method of measuring its effects, although he does not seem to have attempted to practise it. The same notions, but in a less precise form, were reproduced in the Optics of Alhazen. Walther began to estimate the effects of refraction near the horizon; and Tycho, a century after, found the means of measuring them with greater accuracy, and was thereby enabled to construct a table. Tycho estimated the horizontal refraction at 34 ’, and supposed it to vanish at the altitude of 45°; which proves that his ideas on the subject were less accurate than those of Ptolemy, who says expressly that it vanished only at the zenith. Dominic Cassini was the first who proposed an hypothesis for calculating the refraction at any altitude; and he computed a table, which was published in 1662. Since that time the subject of refraction has been investigated, theoretically and practically, with the most scrupulous and delicate attention, and tables constructed to exhibit its amount at every altitude, and at the different seasons of the year. (See Mécanique Celeste, tom. iv. p. 231; Mr Ivory’s Paper in the Phil. Trans, for 1823, p. 409; Delambre, Astronomie du xviii. Siècle, p. 774.)

The refringent power of the atmosphere gives rise to a number of curious phenomena. When the sun appears in [7:3:763]I the horizon, the rays of light which issue from the extremities of his vertical diameter are refracted unequally on ' account of the difference of altitude, so that the disk, at other times circular, then assumes an oval appearance; , and on measuring the horizontal and vertical diameters by means of the micrometer, the former is sometimes found to exceed the latter by four or five minutes of a degree. This effect is particularly remarkable when the sun is observed at his rising or setting from the top of a mountain, or an elevation near the sea-shore, in which cases the difference of the two diameters sometimes, in particular states of the atmosphere, amounts to upwards of 6', or a fifth of the whole apparent diameter of the sun. In the case of the moon this oval appearance is seldom so perceptible; for unless she happens to be exactly full, or in opposition, at the same time that she is in the horizon—and this can rarely be the case—the defect of the illumination of her disk reduces her figure to those rounder but somewhat irregular shapes which it often exhibits. The horizontal refraction being equal to the apparent diameter of the sun or moon, it happens also that either of these luminaries, and consequently any other of the celestial bodies, may be entirely visible above the plane of the horizon, when in fact it has set below it. A curious instance of this was observed at Paris in 1750. The moon rose eclipsed, while the sun, whose place at the time of a lunar eclipse is diametrically opposite to that of the moon, still appeared above the horizon. Another phenomenon occasioned by the refraction of the atmosphere is the appearance of the moon when she is eclipsed, or immersed in the earth’s shadow, and consequently receives no light directly from the sun. On these occasions her disk continues distinctly visible, and is of a dusky red colour, exhibiting the appearance of tarnished copper, or iron nearly red-hot. This phenomenon occasioned great perplexity to the ancient astronomers who were unacquainted with the refractive powers of the atmosphere; and some of them ascribed it to the native light of the moon, in virtue of which she continues visible when hid from the sun’s rays. Plutarch ingeniously supposed it to be occasioned by the light of the stars reflected from the dark surface of the moon; but the quantity of this light is greatly too small to produce the effect in question. It is now explained by the scattered beams of light bent into the earth’s shadow by refraction in their passage through the atmosphere.

The refraction of the rays of light in traversing the earth’s atmosphere is the cause of Twilight, which sensibly lengthens the duration of the day, and prevents a sudden transition from light to darknesS on the disappearance of the sun. When the sun is more than 33' below the horizon, the refraction is not powerful enough to bring his rays sufficiently near the earth to reach our eyes; they pass over our heads, and are irregularly reflected by the molecules of the atmosphere. By this means a portion of the celestial vault is enlightened, while the sun is invisible. This illumination of the upper regions is called the twilight. It commences as soon as objects can be distinguished before sun-rise, and terminates when they cease to be visible after the sun has set. The time, however, at which the twilight commences and terminates cannot be assigned with any degree of precision. It is generally supposed to be limited by the depression of the sun 18° below the horizon. Lacaille found the limit in the torrid zone to be between 16° and 17°. According to Lemonnier, it varies in France between 17° and 21° ∙ The duration of the twilight will evidently be longer or shorter according as the inclination of the plane of the sun’s orbit to the horizon is more or less oblique.

To assign the time at which it is a maximum or minimum, or of a given length at a given latitude, is a problem of pure geometry, which has been frequently solved since the time of the Bernoullis.

The scintillation or twinkling of the stars is another phenomenon produced by certain modifications of atmospherical refraction. The atmosphere is at all times more or less agitated; and on this account the clusters of molecules of which it is composed are constantlyundergoing momentary compressions and dilatations, which occasion minute differences of refraction, and consequent changes in the directions of the luminous rays. The minute and rapid variations thus occasioned in the apparent places of the stars produce the twinkling or tremulous appearances which, in certain states of the atmosphere, particularly on the approach of rain after a long drought, are very remarkable. The agitations of the atmosphere are manifest to the eye in the tremulous motion of the shadows cast from high towers, and in looking at objects through the smoke of a chimney, or over beds of hot sand. The scintillation of the planets is much less than that of the stars. This proceeds from the magnitude of their disks, which, although apparently very small, are still far greater than those of the fixed stars, and indeed so considerable, that the accidental variations in the directions of the rays of light are too minute to displace them entirely. The borders of their disks are affected only by a slight undulation; whereas the stars, which are merely brilliant points of insensible magnitude, undergo a total displacement.

The apparent enlargement of the sun and moon in the horizon is an optical illusion, connected in some measure with the atmosphere, of which various explanations have been given since the time of Ptolemy. According to the ordinary laws of vision, the celestial bodies, particularly the moon, which is nearest to the earth, ought to appear largest in the meridian, because their distance is then less than when they are near the horizon; and yet daily experience proves that the contrary takes place. To an observer placed at E (fig. 10), the visual angle subtended by the moon in the horizon at Μ is somewhat less than that under which she appears in the zenith at 0; and this fact, a consequence indeed of her circular motion, is proved by accurate measurements of her diameters in those circumstances by the micrometer. The mean apparent diameter of the moon, at her greatest height, is 31' in round numbers, but in the horizon she seems to the eye two or three times larger. The commonly received explanation of this phenomenon was first given by Descartes, and after him by Dr Wallis, James Gregory, Malebranche, Huygens, and others, and may be stated as follows. The opinion which we form of the magnitude of a distant body does not depend exclusively on the visual angle under which it appears, but also on its distance; and we judge of the distance by a comparison with other bodies. When the moon is near the zenith there is no interposing object with which we can compare her, the matter of the atmosphere being scarcely visible. Deceived by the absence of intermediate objects, we suppose her to be very near. On the other hand, we are used to observe a large extent of land lying between us and objects near the horizon, at the extremity of which the sky begins to appear; we therefore suppose the sky, with all the objects which are visible in it, to be at a great distance. The illusion is also greatly aided by the comparative feebleness of the light of the moon in the horizon, which renders us in a manner sensible of the interposition of the atmosphere. Hence the moon, though seen under nearly the same angle, alternately appears very large and very small. Desaguliers illustrated the doctrine of the horizontal moon on the [7:3:764] supposition of our imagining the visible heavens to be only a small portion of a spherical surface, as mnop (fig. 10), in which case the moon, at different altitudes, will appear to be at different distances, and therefore seem to vary in magnitude, as at m, n, o. It is evident, however, that this affords no explanation of the phenomenon; for why does the sky, it may be asked, appear to be a smaller segment than a hemisphere? In the solution of this question the whole difficulty is contained.

It may be remarked that these illusions disappear as soon as the intermediate objects are concealed from view. They may be destroyed by regarding the moon through a tube, which permits her disk alone to be seen, and in this manner she will appear no larger at the horizon than near the zenith. The same effect may be produced by viewing her through a smoked glass, because the dark tint permits only the luminous object to be seen, and conceals all the rest. The only precaution necessary to be observed in making this experiment, is to place the eye in such a position that the surrounding bodies may not be visible.

CHAP. II.

OF THE SUN.

Sect. I.— Of the Apparent Circular Motion of tire Sun in the Ecliptic, and Position of the Ecliptic in Space.

The consideration of the diurnal motion common to all the celestial bodies, and the succession of day and night resulting from it, forms the first object of astronomy. The second is to pass from the diurnal motion of the sphere to the proper motion of the sun, and from the vicissitudes of the day to the seasons of the year. The sun, as has already been remarked, is constantly shifting his place among the stars. If we observe the altitude of any star, or group of stars, above the eastern horizon at sun-set, we shall find, on making the same observation a few days afterwards, that its elevation is considerably increased, and that it has approached nearer to the meridian. At the end of three months it will appear at sun-set on the meridian, and from that time continue to advance nearer and nearer to the sun, till it is at last concealed by the splendour of his rays. After remaining for some time invisible, it will again make its appearance in the morning to the westward of the sun, and its distance from him will continue to jncrease daily, till, at the end of a year, it has made a complete circuit of the sky, and regained the position it occupied at the time of the first observation. The earliest observers explained this phenomenon by supposing the diurnal motion of the sun to be less rapid than that of the stars, in consequence of which he constantly falls behind them. This supposition would be admissible if the sun remained constantly in the plane of the equator; but as he deviates considerably from that plane, on both sides of it, it is infinitely more simple to ascribe to the sun a motion of his own, independent of the diurnal motion, and performed in a contrary direction, in virtue of which he traces an oblique route among the stars.

The oblique path of the sun may be determined by observing the positions of the stars near which he successively passes in the course of the year. The same object is however accomplished with much greater precision by the modern practice of observing his declinations and right ascensions, that is to say, his distances from the equator, and an arbitrary meridian. In this manner his place may be accurately assigned every day; and, by a repetition of similar observations, a series of points will be obtained on the surface of the celestial sphere, which mark out his annual course. The result of constant experience shows, that the declination reaches its maximum on the south side of the equator about the 22d of December, when it amounts to 23∙465 degrees. From this time it gradually diminishes till about the 21st of March, when the sun reaches the plane of the equator. At this time the days and nights are of equal length all over the earth, and the instant of time at which the sun’s centre is in the equatorial plane is called the instant of the equinox. The sun then appears on the opposite side of the equator, and his declination or meridional altitude continues to increase till about the 22d of June, when he becomes stationary, and then again shapes his course towards the equator. His maximum declination on the north side of the equator is exactly equal to that on the south, amounting to 23° ∙465. The sun now continues to approach the equator till about the 24th of September, when he again reaches that plane, and a second equinox succeeds. Continuing still to move in the same direction, he declines from the equator southward, till he reaches his former limit about the 22d of December, after which he resumes his former course.

The two small circles of the sphere, parallel to the equator, which pass through the two points where the declination is great, are called the Solstices, or the Tropics ; that on the northern hemisphere is called the Tropic of Cancer, and the other is called the Tropic of Capricon. These two parallels, which mark the extreme limits of the sun’s declination, are, as has just been stated, equally distant from the equator, with regard to which the variations of declination on either side are perfectly symmetrical and uniform.

If at the time of the vernal equinox we remark the stars which set in the true west while the sun is rising in the east, and which are then separated from him by a semi-circumference, it will be found that the difference of their right ascensions, and the right ascensions of those which have precisely a similar situation relatively to the sun at the time of the autumnal equinox, is exactly 180°. It follows, therefore, that the solar orbit intersects the equator in two points diametrically opposite; but we have seen that its northern and southern declinations are equal; hence the orbit projected on the sphere must be a great circle, provided it lies wholly in the same plane. Whether this is the case or not it will be easy to prove by means of a few observations, in the following manner.

Let AQ (fig. 11) be the equator, AE the orbit of the sun, PSM, PTN two circles of declination, drawn through any two points S and T of the orbit. If the sun’s path is confined to a plane, then AE must be a great circle, and 1 11 1 1 . . , τ tan. MS 1 we shall have the equation sin. AM= . ~, accord-

Ί tan. MAS

ing to the well-known properties of spherical triangles. Let the cotangent of the unknown but constant angle MAS = n, the declinations MS = D, NT = D', the right ascensions AM=AE, AN=AE'; then, according to the above formula, we must have sin. AE = n tan. D, and sin. AE' = η tan. D', at whatever points of the orbit S and T may be situated. From these two last equations there result also cos. AE = V1 — n 2 tan. 2 D, cos. JE' = √1—n 2 tan. 2 D', which, being substituted in the trigonometrical formula sin. (AE'—√E)=sin. AE'cos. AE—cos. AE' sin. AE, we shall have sin. (AE'—AE)=n tan. D' √1 —n 2 tan. 2 D — n tan. D √1 — n 2 tan. 2 D'. The observations of the meridional altitudes will give the declinations D and D'; and the difference of right ascensions, AE'—AE, will be found by comparing the time of the sun’s culmination, or transit over [7:3:765] the meridian, with that of a star. If, therefore, it is • found that, by assigning a certain constant value to n, this ' equation will satisfy all the observations, combined by pairs, of the sun’s right ascensions and declinations, it will follow that the plane determined by any two points in the sun’s course and the point in which it intersects the equator, has always the same inclination to the equator; in other words, all the planes so determined are identical.

Now the observations of the sun’s right ascensions and meridional altitudes, which have been made daily during so great a number of years, and under so many different meridians, are found to conform entirely with the preceding formulae: they therefore furnish so many proofs that the projection of the sun’s orbit is a great circle of the celestial sphere, and that the orbit itself is wholly confined to the same plane.

The great circle which the sun describes in virtue of his proper motion is called the Ecliptic. It has received this name from the circumstance that the moon, during eclipses, is either in the same plane or very near it. These phenomena can in fact only happen when the sun, earth, and moon are nearly in the same straight line, and, consequently, when the moon is in the same plane with the earth and the sun. The angle formed by the planes of the ecliptic and equator, and which is measured by the arc of a circle of declination intercepted between the equator and tropic, is called the Obliquity of the Ecliptic. The two points in which the equator and ecliptic intersect each other are called the Equinoctial Points : they are also denominated the Nodes of the Equator ; and the straight line conceived to join them is the Line of the Equinoxes, or the Line of the Nodes. The node through which the sun passes on coming from the south to the north of the equator is called the Ascending Node, and is usually distinguished by the character &; the opposite node is the Descending Node, and is marked by ö. A straight line passing through the centre of the earth, perpendicular to the plane of the ecliptic, is called the Axis, and the points in which its prolongation meets the sphere are called the Poles, of the Ecliptic ; these denominations being analogous to those of the axis and poles of the equator. The two small circles of the sphere which pass through the poles of the ecliptic, and are parallel to the equator, are called the Polar Circles.

The ecliptic has been divided by astronomers, from time . immemorial, into twelve equal parts, called Signs, each of which consequently contains 30 degrees. The names and symbols by which they are characterized are as follows :—
[table]

In each of these signs the ancients formed groups of stars, which they denominated asterisms, constellations, animals (ζωδ∕α), not confined to the ecliptic, but included within an imaginary belt, extending 8° on each side of it, to which they gave the name of Zodiac (ξωδ∕ακoς κύκλος, circle or zone of the animals'). The term sign is now employed only to denote an arc of 30°, and will probably soon be banished entirely from the astronomical tables. It is already confined to the tables of the planets. Thus, to denote that the longitude of a planet is 276° 12', it is usual to write 9 s 6° 12'. Formerly it was usual to employ the characteristic symbol, and to write >y 6° 12', meaning that the planet was 12' in the 6th degree of Capriconus, or the tenth sign. This inconvenient practice is now laid aside, and the signs, when they are employed, are simply - distinguished by the ordinal numbers.

As the greater part of the celestial phenomena connected with the planetary system take place either in the ecliptic or in planes not greatly inclined to it, it is found to be most convenient to refer the positions of the planets, and frequently those of the stars also, to that plane. The first point of Aries, which is the technical expression for the intersection of the ecliptic and equator, or the place of the sun at the vernal equinox, is assumed as the origin from which the degrees of the equator, as well as of the ecliptic, are counted from west to east, or in the direction of the sun’s annual motion. The angular distance of the sun from this point is called his Longitude; and the longitude of a star is the arc intercepted on the ecliptic between the same point and a great circle passing through the star perpendicular to the ecliptic. The arc of this circle intercepted between the star and the ecliptic, or, which is the same thing, the complement of the star’s distance from the pole of the ecliptic, is called the Latitude of the star; so that longitude and latitude are with regard to the ecliptic what right ascension and declination are with regard to the equator. Thus, let O<r Q (fig. 12) be the equator, IɤL the ecliptic, P and: E the north poles of these two circles respectively, and S the place of a star. Having drawn through the pole of the ecliptic and the star the great circle ESM perpendicular to the ecliptic in Μ; then ɤM is the longitude of S, counted on the ecliptic from the vernal point <p towards the east; and SM, the distance of the star from the ecliptic, is its latitude; and for this reason all great circles passing through the poles of the ecliptic are called Circles of Latitude. The place of a star is thus determined by its longitude cγ> Μ, and its latitude SM, as well as by its right ascension ɤ R, and its declination SR; and when the angle Lɤ Q is known, it is easy to pass from the one system of co-ordinates to the other by means of the formulae of spherical trigonometry. These formulae are of constant use, for it is the declinations and right ascensions only which are directly observed.

The plane of the sun’s orbit will be determined completely when its inclination to the equator and the position of the line of the nodes in space have been made known by observation. The declination of the sun twice a year, namely, at the summer and winter solstice, is equal to the obliquity of the ecliptic; whence, if the solstice happened exactly at mid-day, the obliquity would be given directly by an observation of his meridional altitude. This circumstance, however, can happen only for one terrestrial meridian; but as the declination of the sun when he approaches the tropics varies little from one day to another, his greatest observed declination will be a very near approximation to the obliquity, at whatever part of the earth the observation may have been made. It is easy, however, to correct the error which results from the observation being made on a meridian different from the solstitial colure. Taking an example from Woodhouse, let us suppose the sun’s declination to be observed on three successive days (the 20th, 21st, and 22d of June), and found to be on these days respectively, 23° 27' 37", 23° 27' 41", 23° 27' 20"; then it is obvious, that if the middle observation gave the greatest inclination exactly, the other two would differ from it equally, which they do not. The maximum declination is therefore a quantity somewhat different from 23° 27' 41"; and it is easy to conclude, from the inspection of the numbers, that it is nearer to 23° 27' 41" than to either of the other two. It is obvious, therefore, that that observation must have been made within 12 hours of the [7:3:766]time when the sun was exactly in the solstitial point. In order to form an exact notion of the amount of error which may possibly arise from this circumstance, let S (fig. 13) be the place of the sun at the time of the observation, and X the true but as yet unknown solstitial point, and S s, Xx meridional arcs intersecting the equator in 5 and x. The arc S X must be less than 30', for in 12 hours the variation of the sun’s longitude does not exceed an arc of that magnitude. Suppose it 30': then, by Napier’s rules, rad. × sin. Ss = sin. <y> × sin. S ɤ, and rad. × sin. X a?—sin. <γ, × sin. X<v ’; whence, on eliminating sin. ɤ, and observing that sin. X ɤ = sin. 90° = 1, we shall have sin. S ä sin. S 5 sm. X x = -. — 5 — = . sin.⅛ɤ cos. SX

By taking, according to the observation, S s = 23° 27'41", and S X = 30 , , we shall find from the logarithmic tables Ar = 23° 27' 44"∙5. It will be observed that 30 , is the maximum error in longitude; if instead of 30 ’ it had been supposed only 3', the corresponding error in declination would have amounted only to 0"∙035. In the example chosen, the error of longitude is about 20', whence the error of declination is 1"∙5 nearly, and consequently the resulting obliquity differs little from 23° 27' 42"∙5. This result is, however, to be understood of the apparent obliquity, which is subject to slight variations, depending on the longitudes of the moon’s nodes: the mean obliquity, deduced from the comparison of a great number of observations, both of the summer and winter solstice, may be regarded as amounting to 23° 27' 41" at the commencement of the year 1830. We shall see, when we come to speak of the Nutation, in what the difference between the apparent and mean obliquity consists.

When the mean value of the obliquity of the ecliptic, as determined by the delicate instruments of the present day, is compared with that given by ancient observations, it appears to have undergone a progressive diminution, and is always greater as the observation is more remote. The ancient observers were not, indeed, possessed of the means of determining an element of this sort with great precision; but as all the observations recorded in history agree in making the obliquity greater in former times than it is now, the probability is almost infinite that the angle formed by the planes of the equator and ecliptic has really diminished; for, had the differences of the values assigned to it arisen solely from errors of observation, they would have been in excess and defect indifferently, instead of being, as they are, uniformly in excess. The various observations and traditions by which the progressive diminution of the obliquity is confirmed have been collected by Bailly; in the following table we have inserted those which appear to be the best authenticated, and have added the results of some recent observations, which exhibit the present value of the obliquity, and the rate of its diminution.
[table]

Although the comparison of these observations with one another gives very discordant results relatively to the law according to which the obliquity varies, their totality places the fact of its progressive diminution beyond all manner of doubt. Lacaille, who was followed by the greater number of astronomers, estimated the diminution of 44" in a hundred years; a result to which he was led chiefly by a comparison of his own observations with those of Walther. Lalande, after comparing an immense number of modern observations with those of the 17th, 16th, and 15th centuries, and also with those of the Arabians and Chinese, found the secular diminution to be 50". Bessel, rejecting the ancient observations as too uncertain, and comparing those only which have been made since the time of Bradley, has fixed the secular diminution at 45"∙7, which differs inconsiderably from the determination of Lacaille. It cannot yet be determined by observation whether this diminution is uniform, or accelerated, or retarded; but so slow is the rate at which it proceeds, that it may, without any sensible error, be regarded as uniform for many centuries to come.

The gradual diminution of the obliquity of the ecliptic might lead us to suppose that a time will ultimately arrive when that plane will coincide with the equator, and the earth be deprived, in consequence, of the agreeable vicissitude of the seasons. But the theory of universal gravitation, which has revealed the cause of the diminution, has also shown that there are certain limits which the angle of the two planes can never exceed, and between which it must continue for ever to oscillate. Geometers have not yet ventured to assign the precise extent of these limits, but their existence is certain; and the planes of the ecliptic and equator, which have been approaching to each other during the last 2000 years, will, in the course of some thousands of years more, begin to recede.

In what has yet been said respecting the diminution of. the obliquity of the ecliptic, no fact has been mentioned ∣ from which it can be inferred, whether the phenomenon isι [7:3:767]occasioned by the displacement of the plane of the eclip- ∙tic or that of the equator. This question may also be de- ' cided by a comparison of modern with ancient observations; for it is evident, that if the inclination of these two planes becomes less, the stars which are situated between them, particularly those near the solstitial colure, will appear to approach to that plane which changes its position; so that if the ecliptic is displaced, the latitudes of those stars will be diminished, or their declinations if the displacement belongs to the equator. It was first observed by Tycho, and the observation has been confirmed by succeeding astronomers, that the latitudes of the southern stars situated near the solstitial colure, that is, of those stars whose longitudes are nearly 90°, have diminished upwards of 20' since the time of Hipparchus and Ptolemy, while the latitudes of the northern stars have undergone a corresponding augmentation. From this fact it is proved that the diminution of the obliquity is occasioned by the displacement of the ecliptic; and theory has shown that the cause of the displacement is the action of the planets, particularly of Jupiter and Venus, on the earth, by virtue of which the plane of the earth’s orbit is drawn nearer to the planes of the orbits of these two planets. This, however, though by far the most considerable, is not the sole cause of the phenomenon; for theory also shows that a slight motion of the plane of the equator is produced by the attraction of the sun and moon, but so very minute that its effects will only become appreciable after a long series of ages.

After determining the inclination of the plane of the ecliptic to that of the equator, the only element requisite to fix its position absolutely in space, is the situation of the straight line formed by its intersection with that plane, that is to say, the line of the nodes. The longitudes of the stars, as has already been mentioned, are counted on the ecliptic from the vernal equinox; and therefore, if the line of the equinoxes, which is the same as the line of the nodes, is invariable, the longitude of any star will always be the same, whatever interval of time may elapse between two observations of that longitude. But on comparing the actual state of the heavens with the observations recorded by ancient astronomers, it is perceived that the longitudes of all the stars are considerably increased; whence we must infer, either that the whole firmament has advanced in the order of the signs, or that the equinoctial points have gone backwards, or retrograded. The latter supposition is infinitely the more probable; for it is inconceivable that the innumerable multitude of stars should have a common motion relatively to points which depend solely on the motion of the earth. The phenomenon is therefore to be explained by attributing to the equinoctial points a retrograde motion from east to west, in consequence of which, the sun, whose motion is direct, arrives at them sooner than if they remained at rest; and therefore the equinoxes, spring, autumn, and the other seasons, happen before the sun has completed an entire circuit. On this account the motion has been denominated the Precession of the Equinoxes. As this motion is extremely slow, its exact amount can be discovered only by a comparison of observations separated from each other by long intervals of time; but the imperfection of instruments prior to the sixteenth century renders the ancient observations of little authority where quantities so minute are concerned, and therefore some discrepancies may be expected in the different determinations of the amount of the precession. The comparison of modern observations with those of Hipparchus gives its annual amount equal to 50∣ seconds, and with those of Ptolemy somewhat greater. The mean result of the observations of Tycho, compared with those of Lacaille, gives 50"∣. On com- ' paring modern observations with one another, we find- 50"∙06. Delambre, in his solar tables, supposes the annual precession to be equal to 50"∙1. According to this estimate the equinoctial points go backwards at the rate of one degree in 71∙6 years nearly, and therefore will make a complete revolution of the heavens in about 25,868, or nearly 26 thousand years.

The discovery of the precession of the equinoxes is generally attributed to Hipparchus, who, on comparing his own observations with those of Timocharis, more ancient by 160 years, perceived that in this· interval the longitudes of the stars had been augmented by about two degrees. It would seem, however, from many proofs collected by Bailly, that this motion, slow as it is, was known to all the ancient nations who cultivated astronomy, long before the time of Hipparchus. It is indeed easy to conceive, from the great attention which they gave to the heliacal risings of the remarkable stars, that they might observe a gradual change of the seasons at the occurrence of these phenomena, from which they would necessarily be led to conclude a variation of the star’s longitude. In consequence of this regression of the equinoctial points, the sun’s place among the zodiacal constellations at any given season of the year is now greatly different from what it was in remote ages. Some time prior to Hipparchus, the first points of Aries and Libra corresponded to the vernal and autumnal equinoxes; those of Cancer and Capricon to the summer and winter solstices: at present these constellations have receded 30 degrees from the same points of the ecliptic. The vernal equinox now happens in the constellation Pisces, the summer solstice in Gemini, the autumnal equinox in Virgo, and the winter solstice in Sagittarius. Astronomers, however, still count the signs from the vernal equinox, which, therefore, always corresponds to the first point of the Sign of Aries. On this account it is necessary to distinguish carefully between the Signs of the Zodiac, which are fixed with regard to the equinoxes, and the Constellations, which are movable with respect to those points.

The diminution of the obliquity of the ecliptic arises from the displacement of the ecliptic itself; the precession of the equinoxes is, on the contrary, occasioned by the ¢ continual displacement of the plane of the terrestrial equator. This displacement results from the combined action of the sun and moon (for the influence of the planets amounts only to a fraction of a second, and is consequently scarcely sensible,) on the mass of protuberant matter accumulated about the earth’s equator, or the matter which forms the excess of the terrestrial spheroid above its inscribed sphere. The attracting force of the sun and moon on this shell of matter may be resolved into two; one parallel to the plane of the equator, the other perpendicular to it. The tendency of this last force is to diminish the angle which the plane of the equator makes with that of the ecliptic; and if the earth had no motion of rotation, it would soon cause the two planes to coincide. In consequence, however, of the rotatory motion of the earth, the inclination of the two planes remains constant; but the effect produced by the action of the force in question is, that the plane of the equator is constantly shifting its place, in such a manner that the line of the equinoxes advances in the direction of the diurnal motion, or contrary to the order of the signs, its pole having a slow angular motion about the pole of the ecliptic, so slow indeed, that it requires nearly 26,000 years to complete its revolution.

If the sun and moon moved in the plane of the equator, there would evidently be no precession; and the effect of [7:3:768]their action in producing it varies with their distance from that plane. Twice a year, therefore, the effect of the sun in causing precession is nothing; and twice a year, namely at the solstices, it is a maximum: on no two successive days of the year is it exactly the same, and consequently the regression of the equinoctial points, which results from· the sun’s action, must be unequable. On this account the obliquity of the ecliptic is subject to a semi-annual variation; for the sun’s force, which tends to produce a change in the obliquity, is variable, while the diurnal motion of the earth, which prevents the change from taking place, is constant. Hence the plane of the equator is subject to an irregular motion, which is technically called the Solar Nutation. The existence of the solar nutation is, however, only a deduction from theory, for its amount is too small to be perceptible to observation; but a similar effect of the moon’s action is sufficiently appreciable, and was, in fact, discovered by Dr Bradley before theory had indicated its existence. Its period, however, is different, and depends on the time of the revolution of the moon’s nodes, which is performed in 18 years and about 7 months. During this time the intersection of the lunar orbit with the ecliptic has receded through a complete circumference; and the inequality of the moon’s action will consequently, in the same time, have passed through all its different degrees. Bradley observed that the declinations of the stars continued to augment during nine years, that they diminished during the nine years following, and that the greatest change of declination amounted to 18". He remarked further, that this motion was connected with an irregularity of the precession of the equinoxes, which followed exactly the same period; whence he concluded that the motion of the poles of the equator, occasioned by this vibration of its plane, was not confined to the solstitial colure. A series of observations on stars differently situated proved that all the phenomena could be explained on the hypothesis that the pole of the equator describes in 18 years a small circle of 18" diameter, contrary to the order of the signs; or that the axis of the world, following the circumference of this circle, describes the surface of a cone, the axis of which forms with its side an angle of 9". This apparent vibratory motion is significantly denominated the Nutation of the Earth’s Axis.

In consequence of the two motions which occasion the precession and the nutation, the true path of the pole of „the equator round that of the ecliptic is an epicycloidal curve, which will be understood by referring to fig. 14. Let E be the pole of the ecliptic, round which the pole of the equator P describes, in virtue of the precession, and in a direction contrary to the order of the sines, the circle PQR, of which the radius EP is equal to the obliquity of the ecliptic, or the mean distance of the two poles. While P, the mean place of the pole, moves in the circle PQR, with a velocity equal to the regression of the equinoctial points, or at the rate of 50"∙1 a year, the true pole p describes at the same time round P a small circle of 18" diameter in the same direction pqr. The true path of the pole is therefore along the circumference of a circle pqr, the centre of which retrogrades on the circumference of another circle, and consequently moves in an epicycloid abcdefg i the curve which results from the composition of the two motions. Suppose the mean pole at Q, the true pole at a, and αQ=9". In the course of nine years the mean pole will have retrograded from Q to R, making QR=9 X 50"∙1, while the true pole will have accomplished a semi-revolution in its circle. It will’therefore be at c (Re being=9"), and have described the epicycloidal arc abc, the greatest distance of which from the circle PQR is at b, and equal to 9". At the end of the following nine years the mean pole has retrograded from R to S, or 9×50"∙1, and the true pole has returned to the same point a of its epicycle; it will therefore be found at e Se being =9". In this interval it has necessarily been within the circle PQR, its greatest distance from which at d is 9", so that in 18 years it has traced the curve abede. But ac = QR + a Gt + Re = 9 × 50"∙1 + 18" = 7'49", andce=RS-Re—Se = 9 × 50"∙1 —18" = 7' 13". From a to c the motion of the true pole in the epicycle is in the same direction as that of the mean pole; from c to e it is in an opposite direction, but as it is always much slower, being only about 3" in a year, while that of the mean motion is 50"∙1, the true motion which results from the combination of both will always be in the direction ce, and at d its velocity will be equal to their difference. At b and d the difference between the latitudes of the true and mean pole is a maximum, while the difference of their longitudes is nothing; in other words, the correction of the obliquity is greatest when that of the precession vanishes: at a, c, and e the correction of the obliquity vanishes, and that of the precession is a maximum.

Dr Bradley remarked that the effects of the nutation would be represented still more accurately by supposing 1 the curve described by the pole of the equator about its* mean place to be an ellipse instead of a circle, the transverse and conjugate axes being 18" and 16" respectively. This is also confirmed by theory, from which Laplace calculated the semi-axes of the ellipse at 9"∙63 and 7"T7. The semi-transverse axis of the ellipse described by the pole in virtue of the sun’s action alone does not exceed half a second, and is consequently totally inappreciable. The sensible part of the nutation, therefore, follows exactly the period of the revolution of the nodes of the moon. From a series of 810 observations, embracing three complete revolutions of the moon’s node, the major and minor semiaxes of the ellipse have been found by Μ. Lindenau to be 8"∙977 and 6"∙682 respectively.

It is now easy to see the reason of the distinction drawn above between the mean and the true or apparent obliquity. The mean obliquity is represented by the radius EP of the deferent circle PQR, along the circumference of which the centre of the small circle or ellipse is carried, while the true obliquity is that quantity increased or diminished by the nutation. The calculation of the mean obliquity from the true is performed by the aid of the astronomical tables.

The progressive diminution of the mean obliquity and the nutation of the earth’s axis are inequalities distinguished from each other, not only by their being derived from different and distinct causes, but still more by the very great difference of time required for their full development. Almost every other element of the planetary system is affected in a similar manner by inequalities of two kinds, which are distinguished by the terms secular and periodic. The secular inequalities proceed with extreme slowness, and continue progressive in the same sense during many centuries; while the periodie are much more rapid in their march, and run through the whole period of. their changes in comparatively short intervals of time. The inequalities of the first kind are also periodic; but their periods are vastly longer, and may be reckoned by centuries instead of years; and from this circumstance they derive their name of secular inequalities. Sect. II.— Of the Orbit of the Sun.

Having now considered the situation of the plane in which the sun is observed to move relatively to the fixed stars, and also the small secular and periodic variations [7:3:769]to which it is subject, ŵe proceed next to inquire into the •nature of his orbit, or the curve which he describes on ’that plane. The means which we have of determining the solar orbit are the observed variations of the sun’s angular velocity, and of his distance from the earth; hence two kinds of observations are necessary,—the first, of his daily meridional altitudes, from which, as the inclination of the orbit is known, his angular velocity is easily found; and the second, of his apparent diameter, the variations of which follow the inverse ratio of his distance.

That the angular velocity of the sun’s motion in his orbit is not uniform, is obvious from the fact that he remains 7f days longer in the northern than in the southern signs, or, which is the same thing, that the interval between the vernal and autumnal equinoxes is 7^ days longer than the interval between the autumnal and vernal. It is proved by numerous observations that the sun moves with the greatest velocity when at a point situated near the winter solstice; while at the opposite point of the orbit, or near the summer solstice, his velocity is the least. At the first point the diurnal motion is 1° ∙01943, and at the second only 0° ∙95319. It is constantly varying between these two points; and the variation is observed to be nearly proportional to the sun’s angular distance from the point of his orbit where his velocity is a maximum or minimum. The mean velocity is 0° ∙98632, or nearly 59' 11", which is the rate of the sun’s daily motion about the beginning of April and October.

The point of the solar orbit, which is .the most remote from the earth, is called the apogee (a r zο γr i ς, away from the earth); that which is nearest is called the perigee (πs%ι γrrς, near the earth). The same points are also called respectively the superior and inferior Apsis of the orbit, and the straight line which joins them is called the Line of the Apsides.

The exact determination of the sun’s diameter is a problem which engaged the earliest astronomers, but of which, before the invention of the telescope and micrometer, it was impossible to obtain a solution sufficiently accurate to make known its variations. Archimedes, by an extremely ingenious though imperfect method, demonstrated that it must be included between the limits s ⅜ σ and of a right angle, that is, between 27' and 32' 55"∙7. The Egyptians, by observing the time which the Sun takes to rise above the horizon, found it to be between a 750th and a 700th part of a circumference, that is, between 28 ’ 48" and 30' 51"∙5. Aristarchus of Samos supposed it to be 30 ’. The precision of modern observations shows that the apparent diameter is greatest about the time of the winter solstice, and least about the summer solstice; but there is some discrepancy among the results of different astronomers with respect to its actual magnitude. According to Lalande, the greatest apparent diameter, about the end of December, is 32' 35"∙5; the least, about the end of June, 31' 30"∙5. According to the tables of Delambre, its greatest value is 32' 35"∙6, and its least 31'31"; the mean apparent diameter, or the diameter at the sun’s mean distance, is equal to 32' 2"∙9.

From these remarks it is obvious, that if the orbit of the sun be a circle, the earth is not situated in the centre of that circle, otherwise the distance of the sun from the earth would remain always the same, which is contrary to fact. It is possible, however, that the variation in his angular velocity may not be real, but only apparent. Thus, in fig. 15, let AMPN be the orbit of the sun, C the centre of that orbit, and E the position of the earth at some distance from the centre. It is obvious that P is the sun’s perigee, and A his apogee. Now, as the sun’s apparent orbit is a circle having the earth in its centre, it is evident that this orbit must be αMpN, and that the angular motion of the sun will be measured upon that circle. Suppose now that the sun in his apogee moves from A to A', it is obvious that his apparent or angular motion will be the segment a α' of the apparent orbit, considerably smaller than AA'; so that at the apogee the angular motion of the sun will be less than his real motion. Again, let the sun in his perigee move from P to P, describing a segment préciser ly equal to the segment A A'. This segment, as seen from the earth, will be referred to ppt, which in that case will be the sun’s angular motion, evidently considerably greater than his real motion.

Hence it is obvious, that even on the supposition that the sun moved equably in his orbit, his angular motion as seen from the earth would still vary, that is, would be smallest at the apogee and greatest at the perigee, and that the angular and real motion would only coincide in the points Μ and N, where the real and apparent orbits intersect each other. From the figure, it is obvious also that the angular velocity would increase gradually from the apogee to the perigee, and diminish gradually from the perigee to the apogee, which likewise corresponds with observation.

But if the variation in the angular motion of the sun were owing alone to the eccentric position of the earth within the solar orbit, it is easy to demonstrate that in that case the diminution of his angular velocity would follow the same ratio as the diminution of his diameter. The fact however is, that the angular velocity diminishes in a ratio twice as great as the diameter of the sun does. The variation of the angular velocity cannot then be owing to the eccentricity alone. Hence it follows that the variation of the motion of the sun is not merely apparent, but real, and that its velocity in its orbit actually diminishes as his distance from the earth increases. Twq causes, then, combine to produce the variation in the sun’s angular velocity; namely, 1. the increase and diminution of his distance from the earth, and, 2. the real increase and diminution of his velocity in proportion to this variation of distance. These two causes combine in such a manner that the daily angular motion of the sun diminishes as the square of his distance increases, so that the product of the angular velocity multiplied into the square of the distance is a constant quantity.

The observation, that the sun’s angular motion in his orbit is inversely proportional to the square of his distance from the earth, is due to Kepler. The discovery was made by a careful comparison of the sun’s diurnal motion with his apparent diameter, which is inversely proportional to his distance from the earth. Let ASB (fig. 16) be the sun’s orbit, E the earth, and S' the sun. Suppose a line ES joining the centres of the earth and sun to move round along with the sun. It is obvious that when S moves to S', ES, moving along with it, is now in the situation ES', having described the small sector SES'. In the same time that S performs one revolution in its orbit, the radius vector ES will describe the whole area ABS inclosed within the sun’s orbit. Let ŞŞ' be the sun’s angular motion during one day. It is evident that the small sector SES' is proportional to the square of ES, multiplied by SS'; for the radius vector is the sun’s distance from the earth, and SS' his angular motion. Hence this sector: is a constant quantity, whatever the angular motion of the! sun be; and the whole area SEA increases as the number ļ of days which the sun takes in moving from S to A., Hence results that remarkable law, first pointed out by Kepler, that the areas described by the radius vector are proportional to the times of descriptiori. Suppose the sun to describe SS' in one day, and SA in twenty days; then [7:3:770]the area SES' is to the area SEA as 1 to 20, or the area SEA is 20 times greater than the area SES'.

The knowledge of these facts enables us to draw upon paper, from day to day, lines proportional to the length of the radius vector of the solar orbit, and having the same relative position. If we join the extremity of these lines, by making a curve pass through them, we shall perceive that this curve is not exactly circular. Let E (fig. 17) represent the earth, and E a, E b, E c, E d, E e, &c. the position and length of the radius vector during every day of the year; if we join together the points a, b, c, cl, e, f g, It, i, k, l, m, n, o, by drawing the curve aeim through them, it is obvious that this curve is not a circle, but elongated towards a and i, the points which represent the sun’s greatest and least distance from the earth. The resemblance of this curve to the ellipse induced Kepler to compare them together; and he ascertained their identity, and thus proved that the orbit of the sun is an ellipse, having the earth in one of its foci.

Having arrived at the knowledge of the true nature of the sun’s orbit, it becomes necessary, in the next place, to determine its position on the plane of the ecliptic, that is, to assign the position of the transverse axis, or line of the apsides, with reference to some other straight line given by position on that plane. The line which it is most convenient to assume for this purpose is the line of the equinoxes; and the position of the apsides may be determined from observations of the time which the sun occupies in performing a semi-revolution, counting from different points of his orbit. Thus, if among the observations of the sun’s longitude made daily during the course of a year, we compare, two and two, all those which are diametrically opposite, or which differ by 180°, it will be found that the interval between them will be somewhat less than half a year, if the sun, during that interval, has passed through his perigee, or longer if he has passed through his apogee; and that the time between the two observations will differ less from half the time of a whole revolution, in proportion as the sun’s place, at the time of the two observations, was nearer to the apsides. If the difference of time between the two observations is exactly half a year, then the place of the apsides would be obtained at once, because the sun must have been in those points when the observations were made. The probability is, however, infinitely small, that this can ever happen exactly; but as the position of the apsides is known very nearly from the diurnal observations of the variation of the sun’s angular velocity, the necesssary corrections can easily be supplied by computation.

From the comparison of observations made in different ages, it appears that the position of the apsides is not fixed on the plane of the ecliptic, but that the greater axis of the solar ellipse revolves in the direction of the sun’s annual motion. The observations of Hipparchus, compared with those of the present times, show that the apsides have a direct motion at the rate of 65" in a year. According to the observations of Walther, the longitude of the apogee in 1496 was 93° 57' 57"; and according to Lacaille, the same element in 1750 was 98° 37' 28"; whence, dividing the difference by 254, the interval between the two epochs, the annual motion of the line of the apsides amounts to 66". Delambre found, by the comparison of a great number of modern observations, that the annual motion is 6l"∙95 a year. The theory of attraction, which, in respect of such slow and minute variations, must be considered as giving more accurate results than can be obtained directly by observation, gives 61"∙9 for the yearly progressive motion of the apsides.

In the above determination, the position of the ap sides is referred to the line of the equinoxes, and their motion compared with the sun’s tropical revolution. But if we wish to determine their motion with reference to the fixed stars, it is necessary to have regard to the retrograde movement of the equinoctial points; for it is obvious, that if the line of the equinoxes is not fixed, the displacement of the apsides with respect to the stars will be increased or diminished by the whole amount of its motion, according as the two motions are in the same or opposite directions. Now the motion of the line of the apsides is direct, and has just been stated to amount to 61"∙9 in a solar year; that of the equinoxes is, on the contrary, retrograde, and amounts to 50"∙1 in the same time. Hence the displacement of the apsides, with reference to a star or fixed point in the ecliptic, is 61"∙9 — 50"∙1 = 11"∙8 a year, in the direction of the sun’s annual motion. The time, therefore, in which the solar perigee completes a revolution in the κ κ ∙ 360°

heavens, or returns to the same star, is τ-—, equal near- 11"∙8

ly to 110,000 years.

Since the greater axis of the solar ellipse has a progressive motion on the plane of the ecliptic, it forms a variable angle with the line of the equinoxes, and at distant epochs will coincide with that line, or be perpendicular to it. The epochs at which these phenomena happen may be easily found by simple proportions, when the longitude of the perigee at any given time, and its annual motion, are known. According to the observations of Lacaille, already quoted, the longitude of the perigee in 1750 was 278° 37' 28"; but when the longitude of the perigee was 270°, the greater axis of the solar ellipse must have been perpendicular to the line of the equinoxes. The difference of these two longitudes is 8° 37' 28", and the number of years requisite to describe that arc, at the rate of 61"∙9 8° 37' 28"

annually,is—- q — = 500 years nearly; whence the

O JL V

major axis was perpendicular to the line of the equinoxes in the year 1250, when the perigee of the orbit coincided with the winter solstice.

Suppose it were required to assign the epoch at which the major axis coincided with the line of the equinoxes. At the occurrence of this phenomenon the longitude of the perigee was 180°, consequents from that time to 1750 the perigee had advanced 278° 37'28"-180° = 98° 37'28".

98° 37' 28"

Now —— = 5735; therefore 5735 is the number

61"∙9

of years intervening between the occurrence of the phenomenon and 1750. Hence about 4000 years before the commencement of our era, the transverse axis of the solar orbit coincided with the line of the equinoxes; and it is a singular coincidence that this epoch is considered by chronologists to be that of the beginning of the world, or, to speak more correctly, of the first traces of the existence of the human race; for numerous physical circumstances attest that the earth itself has existed during an infinitely longer period. The same phenomenon will again occur when the longitude of the perigee shall have reached 360°, or become zero; and by calculating as above, this will be found to take place in the year 6485 of our era. The solar perigee will then coincide with the vernal equinox; in the former case it coincided with that of autumn.

By reason of the progressive motion of the perigee, the i length of the seasons is continually varying. Let AP (figs.i 18, 19, 20) represent the line of the apsides, S and W the summer and winter solstices, V and O the vernal and autumnal equinoxes. When the greater axis was perpendicular to the line of the equinoxes, as happened in the year 1250 of our era, the perigee P (fig. 18) coincided [7:3:771]with W the winter solstice; and on that account the time between the autumnal equinox 0, and the winter solstice W, was equal to the time between W and V, or between the winter solstice and the vernal equinox. But in this position the equator, which is here represented by VO, divides the ellipse into two unequal portions, the smaller of which must be described in less time than the greater, because the times of description are proportional to the spaces passed over. The summer was therefore, at that time, longer than the winter, and both were divided into equal parts by the solstices. In any other position the seasons differ in length from each other. In the time of Hipparchus, the longitude of the perigee was less than in 1750 by 32° ∙292, or was 246° ∙330. The position of the ellipse with regard to the equinoxes was therefore such as is represented in fig. 19, the angle PEW being 270° — 246° ∙33 = 23° 40' 12". The interval between V and S was then 94⅜ days, and that between S and 0 only 92∣ days. The spring was therefore at that time longer than the summer, and the winter longer than the autumn. The position of the ellipse in 1800 is represented in fig. 20. The angle PEW was then 9° 29', and the following were nearly the lengths of the seasons :—
[table]

The spring is therefore at present shorter than the summer, and the autumn longer than the winter.

After describing the position of the transverse axis of the solar ellipse on the plane of the ecliptic, we come next to consider the species of that ellipse, or its eccentricity, on which depend the apparent inequalities of the sun’s angular motion. It is evident that this element will be made known if we can by any means determine the ratio of the two segments into which the major axis is divided by the focus. This ratio might seem to be obtainable without any difficulty by comparing the sun’s apparent diameters at the perigee and apogee, to which the distances are inversely proportional; but such observations, on account of the irradiation and other difficulties, are liable to considerable uncertainty: it is therefore necessary to have recourse to other methods of estimating the eccentricity, and the variations to which it is subject. These methods are derived from the fundamental law of the elliptic motion, namely, the proportionality of the areas described by the radius vector to the times of description, by which the inequalities of motion and the ellipticity pf the orbit are connected with each other; so that when the inequalities are made known by direct observation, the species of the ellipse can be computed from the principles of geometry. It will, however, be necessary to attend more closely than we have yet done to the phenomena resulting from the eccentric position of the earth in the sun’s orbit, and to explain some terms technically employed by astronomers to abbreviate and simplify their descriptions.

The sun’s motion may be regarded as composed of two distinct parts, namely, a circular uniform motion, which constitutes the principal part of it; and a correction depending on the deviation of the orbit from a circle, which modifies the first, and alternately accelerates and retards the mean angular velocity. To illustrate this, let E (fig. 21) be the earth, PSA the orbit of the sun, AP the line of the apsides, and «γ» λ the intersection of the plane of the orbit∙with that of the equator. While the true sun S moves round his elliptic orbit, describing areas proportional to the times, conceive a fictitious sun $ to move round the earth in the circumference of a circle of which the radius is equal to EP the sun’s distance at the perigee, and with a uniform motion such, that if the two suns set out at the same instant from P, they may also return to the same point together, after having completed each a revolution. At the perigee the radius vector of the real sun is a minimum; his velocity is consequently a maximum, and he therefore advances before the fictitious sun. But his motion relaxes in proportion as his distance from P becomes greater; and at a certain point S of the orbit it becomes equal to that*of the fictitious sun, which has then only reached the point s. After passing the point S, the angular distance SEs continues to diminish on account that the motion of the real sun continues to relax, and it vanishes altogether when he reaches his apogee at A, the fictitious sun arriving at a in the same straight line with A and E at the same instant of time. From the apogee to the perigee the phenomena are reversed. The motion of the real sun being then the slowest possible, he at first falls behind the fictitious sun; but his motion continuing to be gradually accelerated, he has again acquired the same velocity when he reaches S', after which his motion is more rapid than that of the fictitious sun, which he at last overtakes at P. If, therefore, we conceive a radius vector to be drawn from the earth to each of the two suns, those lines will form with each other a variable angle, having its maximum value when the velocities are equal, and vanishing at the perigee and apogee. This angle, namely, SEs, is called the Equation of the Centre, or Equation of the Orbit. It is the correction necessary to be made to the longitude of the sun deduced from his equable motion, in order to have his real longitude. Those quantities which form the difference between the true and mean results are,-in astronomy, denominated equations. The ancients employed the term Prosthaphaeresis in the same sense.

From the perigee to the apogee, the true longitude of the sun is found by adding the equation of the centre to the mean longitude, and in the other half of the orbit by taking the difference of the same quantities. The equation of the centre is a maximum when the sun is at the points S and S', or when the mean and true motions are equal; and if its value can be obtained from observation at those times, the eccentricity of the orbit may be deduced by means of the geometrical properties of the ellipse. Now the points S and S' may be found very nearly by observing when the diurnal velocity of the sun is equal to the mean motion, or to 0∙985647 parts of a degree. By these observations the angle SES', which is the difference of the real longitudes of the sun, or of his distances from tγ>, will be given. But $ E √ is also given; for it is the angle which would be described by the sun, in virtue of his mean motion, during the interval between the two observations. Hence the difference between sE√ and SES' is given, being equal to twice SEs the maximum equation of the centre, in consequence of the symmetrical position of the points S and S', and also of $ and √, in respect of the line of the apsides AP. The accuracy of a result obtained in this manner may appear questionable, on account of the impossibility of determining by observation the exact moment at which the sun’s true motion in longitude is equal to his mean motion; but as the real motion varies very little during a few days before and after that epoch, the equation of the centre will be scarcely affected by a slight error in the time. Besides, any error in the result is corrected by taking the mean of a great number of similar observations.
[7:3:772]

By a comparison of numerous observations of this kind, Delambre found that the greatest equation of the centre, in the year 1776, amounted to 1° ∙9254, or 1° 55' 31"∙4. In 1801 the same equation was 1° 55' 27"∙ 3. (See Baily’s Astronomical Tables, p. 66.)

Having obtained the value of the greatest equation of the centre, the eccentricity of the orbit may be computed from the first two terms of the following series, e = ⅜E — 24?3 E 3 — 21 θ 3 5 E 5 — &c.

in which e represents the eccentricity, and E the greatest equation of the centre. By reversing the series the following expression is found for the greatest equation in terms of the eccentricity, viz.

E = 2e +Ä e3 + ÄP + &c ·

For an investigation of the above formulae the reader may consult Biot, Astronomie Physique, tom. ii. p. 185.

It has been discovered by observation, that the equation of the centre of the sun’s orbit is subject to a nearly uniform secular diminution. This fact is confirmed by theory, from which also the secular diminution has been assigned. It amounts to 8° ∙0047 in a century. This phenomenon implies a corresponding diminution of the eccentricity of the solar orbit amounting to ∙000037495 in the same time, the semiaxis major being unit. This will be conceived more distinctly by converting the above fraction into some familiar expression of linear magnitude. Supposing the mean distance of the sun to be 96,000,000 of miles, the fraction ∙000037495 will represent nearly 3700 miles, so that the annual diminution of the eccentricity is nearly at the rate of 37 miles,—a line which is considerable on the surface of the earth, but which has scarcely an appreciable ratio to the immense distance of the sun. If, however, this diminution, small as it is, were to be continued indefinitely, the eccentricity would ultimately vanish, and the sun’s orbit would be changed into a circle; but the theory of universal gravitation proves that, like all the other variations of the elements of the solar system, the variation of the eccentricity is subject to periodic laws. After having continued during a certain time to diminish, the eccentricity will again begin to increase, and will successively pass through all its former values. Thus it will continue to oscillate within certain limits, of which the extent, though not precisely known, cannot be very great: and the solar orbit will eternally preserve its elliptic form, unless the application of some external force shall derange the system of the world, or modify the laws by which it is at present governed.

The true nature of the solar orbit being known, together with its situation on the plane of the ecliptic, and the amount of its eccentricity, as also the time of a revolution, the sun’s longitude may be determined at any assigned epoch. This determination is what is usually termed Keplers Problem, having been first proposed by that great astronomer on the hypothesis of an elliptic orbit. Its solution, which is not susceptible of being exhibited under a finite form, is derived from the principle of the equable description of areas. Let ASP (fig. 22) represent the semi-orbit of the sun, C the centre of the orbit, E the focus occupied by the earth, and let the time and motion be counted from the perigee P. On AP describe a semicircle, on the circumference of which let a point Μ be supposed to move uniformly, and to describe the semicircle PMA, in the same time in which the sun describes the semi-ellipse PSA. Now, suppose that at the end of a given time t, the sun has moved from P to S; then, denoting by T the time of a complete revolution, and putting π for the semi-circumference, the place of Μ, at the end of the same time t, will be given by the equation PM = ∙ψ 2 or. Through S let a perpendicular be drawn to AP, meeting the semicircle in D and AP in N, and join EM, ES, ED, CM, and CD. The different angles of this figure have received certain technical names, which it is necessary to explain. The angle PES, which is the true longitude of the sun, is called the True Anomaly· PEM is called the Mean Anomaly, and PCD is denominated the Eccentric Anomaly. This last is measured by the arc PMD, which is the perigean distance of a star describing a circle circumscribed about the solar ellipse, and having constantly the same absciss PN as the sun. The word anomaly originally signified inequality ; it now simply designates the angular distance of a planet from its perihelion, seen from the sun, or, as in the present case, the angular distance of the sun from his perigee.

According to the law of the equable description of areas, we have the sector PCM to PES as the surface of the semicircle to the surface of the semi-ellipse; that is, as the semi-transverse to the semi-conjugate axis, or as ND to NS, according to a well-known property of the ellipse. Hence PCM: PES:: PED: PES, and therefore PCM = PED = PCD — ECD. Now, if we express the arc PM by z, PD by x, the eccentricity EC by e, and make the radius equal to unity, we shall have the sector PCM = ±z, the sector PCD = ⅛x, and the triangle ECD = ļe. DN = ⅜e sin. x. Substituting, therefore, these expressions in the above equation, it will become, on multiplying both sides of it by 2,

Z = a? — e sin. X............... (1.) where z denotes the mean, and x the eccentric, anomaly.

If in this equation x and e were both known, it would be easy to deduce from it the mean anomaly z ; but the data are z and e, and the value of x can only be obtained in a series, the equation being a transcendental one. Its resolution gives the value of the eccentric in the mean terms of anomaly; and if another equation can be found in which the eccentric anomaly is expressed in terms of the true, it will only remain to put these two expressions equal to one another in order to have an equation between the mean and true anomaly.

For this purpose it will be necessary to equate two values of the radius vector ES, expressed in terms of the eccentric and true anomalies respectively. Putting ES ≡= r, we shall have, in the first place, r 2 = EN 2 + NS 2 ; but if the semiaxis minor is represented by b, the common equation of the ellipse gives

NS 2 = -į (ω 2 — CN 2 ) = 0 2 --Ş CN 2 ; and we have also

EN 2 = EC 2 + 2 EC . CN + CN 2 = e 2 + 2e . CN + CN 2 ; therefore, by addition,

NS 2 +EN 2 = 0 2 +e 2 +2e . CN+(1--^ŋCN 2 . ⅛2 e 2

Now, 5 2 +e 2 = α 2 , and 1 — = —; therefore

NS 2 +EN 2 = r 2 = α 2 +2e. CN+-J CN 2 ; consequently r = a+-CN. 1 a

But CN = a cos. X ; therefore ultimately r = aJ r e cos. X.................. (2.) [7:3:773]To obtain a second value of τ, let SF be drawn from S ’ to the other focus of the ellipse, and let SF = r ʹ. The rectangular triangles ESN, FSN give SN 2 = r 2 — EN 2 , and SN 2 = ∕ 2 — (2 e — EN) 2 ; therefore r 2 — EN 2 = f 2 — (2 e — EN) 2 , whence we derive r 2 — ∕ 2 = 4 e. EN — 4c 2 . But by the property of the ellipse r-j-r f = 2 α, from which it is easy to deduce r 2 — P 2 = 4« 2 — 4αP = 4ar — 4ß 2 ; and hence, by equating these two values of r 2 — r t2 , e. EN — e 2 = ar — cd. Now, if we denote the angle PES, which is the true anomaly, by v, then EN = r cos. v, and consequently by substitution, er cos. v — (?— ar — a 2 , whence r = ..................(3.)

a — e cos. v

If we now compare the two values of r given by the equations (2) and (3), we shall find a cos. v — e cos. x = , a — e cos. v and therefore, 1 — cos. χ = (α- ∖ ∙e) (1 — cos. v)

1 + cos. x = (α— e) (1 + cos. ν) ; whence, by dividing, 1 — cos, v a —e 1 — cos. a;

1 + cos. v - a +e ’ 1 + cos. x ' consequently, by the trigonometrical formulae, tan. ļv = 2 ∙ tan ∙ iae ('M

The equations (1) and (2) were first given by Kepler; the last is due to Lacaille. The problem of finding the sun’s place after any given time is solved analytically by (1) and (4).

To express υ the true anomaly, in a series involving the powers of x the mean anomaly, is a problem requiring the aid of the higher analysis, and which it is unnecessary to investigate here. The following are a few of the first terms of the series, in which nt is substituted for x, n being the mean motion in the unit of time, and t the time elapsed since the passage through the perihelion :—

v~ηt — ^2e — ⅛ e3 +c^> e5 y sin. wt so η 11 i 17 ,∖ . n

+ (√~≡ ’ * + 192 i J sm ∙ 2,ii -(⅛ e5 -S e, ) sin ∙ 3 " i + , &c.

The form of the sun’s orbit is discovered from the variations of his apparent diameter in passing from the perigee to the apogee; but in order to obtain a knowledge of its real dimensions, or of the mean distance of the sun from the earth, it is necessary to determine his parallax, that is to say, the angle subtended by the earth’s semidiameter as seen from the sun. This angle is too small to be measured directly in the manner which was pointed out when treating of parallax; it is most accurately deduced from observations of the time occupied by an inferior planet in passing over the sun’s disk, as will be explained afterwards. From such observations it has been found to be less than 9", an angle of which the sine is to radius in the ratio of 1 to 23,000 nearly; whence the distance of the sun is about 23,000 times the semidiameter of the earth. The following table exhibits more accurately the numerical results of calculation.
[table]

When the sun’s parallax is known, we are enabled not only to estimate his distance, but also to compare his diameter and magnitude with those of the earth. For example, the mean parallax of the sun is 8"∙56; his mean apparent semidiameter, seen from the earth, is 961"*4; the semidiameter of the sun and earth are therefore to each other in the ratio of 961∙4: 8∙56, or of 112 to unity nearly. Supposing the sun and earth to be both spherical bodies, which is a supposition sufficiently accurate for the present purpose, their volumes will be to one another as the cubes of their radii; hence the volume of the sun is to that of the earth as the cube of 112 to unity, or the sun’s volume is about 1,300,000 times greater than that of the earth.

Sect. III.— Of the Motion of Translation of die Earth, and the Aberration of Light.

Hitherto we have regarded the sun as being actually in motion round the earth in an elliptic orbit, of which the earth occupies one of the foci. This supposes the motion which we observe to be real. But we have already seen that all the phenomena of the diurnal revolution of the celestial bodies may be equally well explained by supposing these motions to be only optical illusions, of the same nature as those by which a person sailing rapidly along a river, or the sea-shore, is almost irresistibly led to ascribe the motion to the banks on the shore, even when fully aware that k the appearance is only occasioned by his own motion in a contrary direction. In the same manner all the phenomena of the annual motion are susceptible of an equally simple explanation, on the hypothesis that the earth revolves in an elliptic orbit, and that the sun is at rest in one of the foci of the ellipse. In fact, if there were no other bodies in the universe than the earth and the sun, it would be matter of absolute indifference which of the two hypotheses we should adopt in order to explain the appearances. Supposing, for example, the earth to revolve in a circular orbit (the eccentricity, on account of its smallness, being here neglected), and the sun to be placed at the centre; a spectator at E (fig. 23) sees the sun S in the heavens at the point A, and corresponding to the star a on the celestial sphere. If the earth now take the place of the sun, and the sun be placed on the curve at A, the spectator will still see the sun correspond to the star a. Let the earth be transferred from E to E ’; the spectator will then see the sun in A'. The sun therefore appears to him to have moved from A to A', which is exactly the same appearance as would be presented to a spectator at the centre by a real translation of S from A to A'; so that, in respect of the annual motion, the appearances, when referred to the celestial sphere, are precisely the same on either hypothesis.

If we now consider the phenomena with reference to the earth, we shall find that they may be all equally well explained on the hypothesis of the earth’s motion, and the immobility of the sun. The most remarkable phenomenon connected with the annual revolution is the variation of the seasons; and’in order to explain their cause, it is only necessary to suppose that the earth, in describing its oblique orbit, always preserves its axis parallel to the same straight line. Let A, B, C, D (fig. 24) repre[7:3:774]sent the earth in four different positions of its orbit, n s being its axis, and n and s its north and south poles respectively. While the earth goes round the sun in the order of the letters A, B, C, D, its axis ns preserves its obliquity, and always continues parallel to its first direction. At A the north pole inclines towards the sun, and brings all the northern places more into the light than at any other season of the year. But when the earth is at C, the opposite point of the orbit, the north pole declines from the sun, and a less portion of the northern hemisphere enjoys the blessings of his light and heat. At B and D the axis is perpendicular to the plane of the orbit, so that the poles are situated in the boundaries of the illuminated hemisphere, and the sun, being directly over the equator, makes the days and nights equal at all places. These phenomena are illustrated in fig. 25, which represents the situation of the north pole with regard to the limits of illumination, in eight different positions of the orbit. In this figure, AE is the terrestrial equator, T the tropic of Cancer, the dotted circle the parallel of London, U the arctic or north polar circle, and P the north pole, where all the meridians or hour-circles meet. The spectator is supposed to be placed at the pole of the ecliptic.

WTιen the earth is at the beginning of Libra, about the 20th of March, the sun, as seen from the earth, appears at the beginning of Aries in the opposite part of the heavens, the north pole is just coming into light, and the sun is vertical to the equator, which, with all its parallels, is divided into two equal parts by the circle which forms the boundary between the dark and illuminated hemispheres, and therefore the days and nights are equal all over the earth. As the earth moves in the ecliptic according to the order of the letters A, B, C, D, &c., the north pole P comes more and more into the light, and the days increase in length at all places north of the equator AE. When the earth comes to the position between B and C, or the beginning of Capricon, the sun, as seen from the earth, appears at the beginning of Cancer, about the 21st of June; and the north pole of the earth inclines towards the sun, so as to bring into light all the north frigid zone, and more of each of the northern parallels of latitude in proportion as they are farther from the equator. As the earth advances from Capricon towards Aries, and the sun appears to move from Cancer towards Libra, the north pole recedes from the light, which causes the days to decrease and the nights to increase in length, till the earth comes to the beginning of Aries, and then they are equal as before; the boundary of light and darkness cutting the equator and all its parallels equally. The north pole then goes into the dark, and does not emerge till the earth has completed a semi-revolution of its orbit, or from the 23d of September till the 20th of March. All these phenomena will be readily understood from the bare inspection of the figure; and it will be perceived that what has been said of the northern hemisphere is equally true of the southern in a contrary sense, that is, at opposite seasons of the year.

The only objection against the annual motion of the earth, which at first sight creates some difficulty, is the enormous distance which, on account of the want of annual parallax, that hypothesis makes it necessary to assign to the fixed stars. Abstracting from the precession, which in the region of the poles amounts only to about 20", the axis of the earth, in every part of its orbit, appears to be directed towards the same point of the starry sphere. It is certain, however, that the radius of the terrestrial orbit is upwards of 90 millions of miles; and therefore, when the earth is at opposite points of this immense circle, its pole ought to be directed to points in the heavens 18Q millions of miles distant from each other. Suppose the sun to be at rest in S (fig. 26), the centre of the orbit EαZ>, and the earth to be at the point E. Suppose also A to be the projection of S on the sphere of the fixed stars, and w the first point of the ecliptic. The longitude of A, as seen from E, is the angle AEɤ; but when seen from S, the longitude of the same point is ASɤ∙ This last is the true longitude of A; Aew is the apparent longitude, and their difference is the angle Sɤ E. Now, in the first half of the ecliptic, the apparent longitude will be less than the true, and in the second half greater; hence there ought to result an apparent annual inequality in the sun’s longitude. But as no such inequality is indicated by observation, it becomes necessary either to abandon the hypothesis of the earth’s motion, or to suppose the distance of the fixed stars to be so great, that, in comparison of it, the radius ES, which exceeds 90 millions of miles, is altogether insensible; and that the whole orbit of the earth, or the ellipse E«ô, is a mere point in comparison of a great circle of the starry sphere.

The most convincing proof of the earth’s motion is not to be found in any circumstance of which the senses can take immediate cognizance, but is afforded by the full development of the planetary system, and the mutual connection of all the truths of rational astronomy,—by the clearness, simplicity, and coherence which this hypothesis gives to the most complicated phenomena,—and can therefore only be fully appreciated after an attentive study of the whole series of facts which astronomy makes known. There exists, however, one direct proof of the earth’s annual motion, in a phenomenon discovered by the accurate observation and patient sagacity of Bradley, although it is one which, we are almost tempted to think, ought to have been perceived a priori, after Roemer’s discovery of the progressive motion of light. It is known by the name of the Aberration of Light, and is manifested in a small dif-∙ ference between the apparent and true places of a star,' occasioned by the motion of light combined with that of the earth in its orbit.

To illustrate this effect, conceive a body to move in the direction EE' (fig. 27), and another to impinge on it in the direction SE. To find the direction of the resulting motion, take EC and EA proportional to the two velocities respectively, and, having completed the parallelogram EABC, draw the diagonal EB. The combination of the two motions produces an impression on the eye exactly similar to that which would have been produced if the eye had remained at rest in the point E, and the molecule of light had come to it in the direction ES'; the star, therefore, whose real place is at S, will appear to the spectator at E to be situated at S'. The difference between its true and apparent place, that is, the angle SES', is the aberration, the magnitude of which is obtained from the known ratio of EA to Efe, or the velocity of light to that of the earth in its orbit. Now, we know from the phenomena of the eclipses of Jupiter’s satellites, that a ray of light describes a line equal to the mean radius of the ecliptic in 8 min. 13∙2 sec. or 493∙2 seconds of time. But the arc described by the earth in that time is found from the proportion

365 d ∙256: 493≡∙2:: 360°: x, whence x = 20''∙25. It is evident, therefore, that when the directions of the two motions are at right angles, the star S always appears in advance of its real place, in a direction parallel to that of the motion of the earth, by a quantity equal to 20"∙25. This quantity, 20"∙25, is called the Constant of Aberration ; but as it has been obtained on the assumption that the earth moves uniformly in a circular orbit, it is evidently not altogether exact, and recourse [7:3:775]lmust be had to observation to determine its precise •amount. Bradley supposed it to be 20"; Dr Brinkley, from the mean of 2633 comparisons of various stars, has deduced the value 20"∙37. From a series of 2000 observations of circumpolar stars, made for the purpose with the utmost care at the Greenwich Observatory by Mr Richardson, its value was found to be 20"∙302. On account of the number and accuracy of these observations, the constant of aberration may be regarded as one of the best determined elements of astronomy. (See Memoirs of the Astronomical Society, vol. iv.)

It is easy to see that, on the same principles, there ought also to be a diurnal aberration; but the diurnal rotation of the earth being sixty-five times less rapid than the orbital motion, its effect in producing aberration does not amount to more than 0 s ∙0208, when reduced to seconds of time. It is consequently altogether insensible in the observations.

The magnitude of the angle of deviation SES' depends on the relative directions of the earth and the visual ray, and may have any value from 0 to 20"∙3. Suppose, for example, we observe a star situated in the plane of the ecliptic. When the earth is at that point of its orbit, between the sun and the star, where the tangent to the orbit is perpendicular to the visual ray (which, on account that the star has no sensible parallax, always maintains a parallel direction), the apparent place of the star will be 20 ’ '∙3 to the eastward of its true place. When the earth is in the opposite point of its orbit the same star will appear to be 20"∙3 to the westward of its true place; so that it will appear to have an oscillatory motion on the ecliptic, the range of which is 4O"∙6, and the period exactly a year. Half-way between these two points the tangent of the orbit is parallel to the direction of the ray of light, and consequently there is no aberration. When the star is not situated in the ecliptic, it will suffer a displacement in latitude as well as in longitude. To render this still more sensible, let EEE (fig. 28) be the ecliptic, E the earth, and A the true place of a star situated at any altitude above the ecliptic. In the direction EA take Eα to represent the velocity of light, ab that of the earth, and in a parallel direction, that is, parallel to the tangent to the ecliptic at E; the line Eó will now be the apparent visual ray, and the star will seem to be situated at B. Suppose the earth to be placed at different points of its orbit; the lines Eα will be all parallel to each other on account of the infinite distance of the star A; the lines ab will vary little in magnitude, because they are very small in comparison of Ea, but their directions will undergo every possible change, being always parallel to the tangent at E. At the two points of the orbit where the tangent is parallel to E A, the two lines Eα and ab coincide, and consequently there is no aberration. Let us next suppose the star to be situated in the pole of the ecliptic. In this case the visual ray is constantly perpendicular to the direction of the earth’s motion, so that the star will always appear at a distance of 20"∙3 from its true place, or appear to describe a small circle about the pole of the ecliptic. In all other situations, out of the ecliptic, the star’s apparent path will be an ellipse, the major axis of which, parallel to the plane of the ecliptic, is always 4O"∙6, while the minor axis varies with the latitude.

• The cause of the aberration being known, we have two methods of measuring its quantity, or the extent of the apparent oscillations, viz. by calculating the angle of deviation SAS' (fig. 27), the tangent of which is to radius as AB to AE, that is, as 1 to 10313; or, by direct observation, when the star is in opposition to the sun. The two methods give exactly the same result, and the dimensions of the ellipse described by any star not situated in the ecliptic, in consequence of the aberration, are found to be the same when computed from theory or determined by direct observation. The motion of translation of the earth, therefore, receives a mathematical demonstration from this agreement; and the phenomenon of the aberration, otherwise unimportant on account of its minuteness, thus becomes one of the most interesting discoveries ever made in astronomy. The fact of the earth’s orbital motion is, however, rendered so probable by other phenomena, that it must have been universally admitted, although the direct proof had never been discovered.

Sect. IV.— Of the Measure of Time. Equation of Time.

The notion of time, or of succession, is generally said to be acquired from the aspect of natural phenomena; but from whatever source it may be derived, it is certain that time itself can only be measured by comparing it with something of which the senses can take cognizance. If we have a series of events, such as the oscillations of a pendulum, or the flux and reflux of the sea, uniformly succeeding each other, then time may be measured by the number of such events that have been observed. In like manner, if a body move uniformly in a certain direction, time may be measured by the spaces over which it successively passes. In strict language, motion and time, being heterogeneous quantities, cannot measure one another; but different times are compared with each other by means of the motions that have taken place in those times respectively; for the motions being supposed uniform, equal spaces are passed over in equal times, or, which is the same thing, the times are directly proportional to the spaces described. If, therefore, we assume for unit the time which is absolved while a certain uniform motion takes place, we may obtain, by a simple proportion, the corresponding time of any other similar motion. In this manner we find the ratio of one portion of time to another, although we can form no idea whatever of its absolute quantity.

In order that motion may be employed as a measure of time, it is indispensably requisite that it be perfectly uniform. The only motions having this property, with which we are acquainted, are those of the rotation of the celestial bodies about their own axes. The motion of the sun and planets in their orbits is irregular from various causes: even the successive returns of the fixed stars to the meridian are rendered unequal on account of the precession and nutation; but the diurnal rotation of the earth ap- peaiŝ to be. affected by no cause of irregularity whatever. Time, therefore, may be properly measured by this uniform motion; and the method of proceeding is as follows. The culmination of a star, or of a given point of the equator, marks the instant at which the day commences, and at any other instant its horary angle determines the portion of the day which has elapsed at the moment of the observation. In order, therefore, to determine the time of an observation, we must find the horary angle formed by the meridian of the observed star with that which passes through the given point where the motion is supposed to begin; in other words, it is necessary to determine the star’s right ascension. This, however, when attempted by direct measurement, is a determination attended with difficulty, and liable to considerable uncertainty. Happily the invention of the pendulum has rendered it unnecessary; and astronomers, instead of deducing the time from the right ascensions, determine, on the contrary, the right ascensions by means of the time indicated by the clock. The clock, of which the motion is supposed to be perfectly regular, is adjusted in such a [7:3:776]manner that its index describes 24 hours, while a point of the equator describes an arc of 360°, or makes a complete revolution. This is called a sidereal day. The sidereal hour is divided in the usual manner into minutes, seconds, &c.; whence the corresponding arcs are easily found at the rate of 15° to an hour. The point whose culmination marks the origin of time is arbitrary; but astronomers have agreed to choose for that purpose the equinoctial point of Aries, from which the right ascensions are reckoned, so that the hours of the clock and the degrees of the equator may commence at the same instant. This point, it is true, is not, and cannot be, permanently marked by any star; but the right ascension in time of any star whatever being the hour of its transit over the meridian, the star will be in the plane of the meridian at the instant denoted by its right ascension in time. On this account sidereal time expresses the actual right ascension of the zenith; or, as it is frequently termed, the right ascension of the mid-heaven.

We have already seen, that on account of the precession of the equinoxes, a star employs somewhat more time than the first point of Aries in returning to the meridian. It is therefore not without some violation of language that the interval between two successive transits of a given point of the equator over the meridian is called a sidereal day, which, in its strict acceptation, denotes the time which elapses between two successive transits of the same star; but in this case the difference is so small as to be totally inappreciable. The annual precession in longitude is = 50"∙1, and that in right ascension is nearly the same, excepting in regard to the circumpolar stars, which, therefore, are not employed in regulating the clocks. This arc of 50"∙l, converted into time, gives 3*3 seconds as the time in which a star will pass the meridian later than the equinoctial point, at the end of the year; and this small quantity being distributed over the whole year, is altogether insensible in short intervals of two or three days. The successive transits of a star, therefore, if we abstract from the nutation and aberration, will mark the sidereal day within the hundredth part of a second of time; and the sidereal year, though not immediately ascertainable by observation, becomes a quantity which may be easily computed. But the regularity of the motion of the stars is deranged by the effects of aberration and nutation; so that in order to measure time with the precision required by modern observers, it is necessary to be acquainted with the minute displacements of the stars. If they seem to return to the meridian after equal portions of absolute time, it is only because our organs are unable to distinguish the hundredths of a second.

The sidereal day is a measure of time which, on account of its uniformity and the facility of observing it, is excellently well adapted for astronomical purposes; but relatively to the ordinary wants of life it is not sufficiently marked;—the culmination of the stars is an event entirely unconnected with civil occupations, and which, for any given star, is even invisible during a great part of the year. The proper motion of the sun causes the sidereal day to commence sometimes by day and sometimes by night, so that great confusion and embarrassment would arise from regulating time and civil affairs by the motions of the stars. On this account the diurnal revolution of the sun has been universally adopted as the measure of time. This is called the civil day, and denotes the interval of time which elapses between two successive transits of the sun over the same hour-circle. Most nations have agreed in reckoning it from the inferior semicircle of the meridian, so that the civil day commences and terminates at midnight; but astronomers, in imitation of Hipparchus and Ptolemy, usually reckon the commencement of the day from the instant of the sun’s culmination, that is, from noon; and count through the 24 hours from one noon to the following. Thus 9 o’clock in the morning of February 14th is by astronomers called February the 13th at 21 hours. The day thus determined is called the astronomical or solar day ; and being regulated by the true motion of the sun, the time which is measured by it is called true or apparent time.

Astronomical or solar days are not equal. Two causes, in particular, conspire to produce their inequality, namely, the unequal velocity of the sun in his orbit, and the obliquity of the ecliptic. The effect of the first cause is sufficiently sensible. At the summer solstice, when the sun’s motion is slowest, the astronomical day approaches nearer the sidereal than at the winter solstice, when his motion is most rapid.

To conceive the effect of the second cause, it is necessary to have regard to the motion of the sun in reference to the equator. The sun describes every day a small arc of the ecliptic. Through the extremities of this arc suppose two meridians to pass; the arc of the equator, which they intercept, is the sun’s motion for that day referred to the equator, and the time which that arc takes to pass the meridian is equal to the excess of the astronomical day above the sidereal. But it is obvious that at the equinoxes the arc of the equator is smaller than the corresponding are of the ecliptic, in the proportion of the cosine of the obliquity of the ecliptic to radius: at the solstices, on the contrary, it is greater in the proportion of radius to the cosine of the same obliquity. The astronomical day is diminished in the first case, and lengthened in the second.

To have a mean astronomical day independent of these causes of inequality, astronomers have supposed a second i sun to move uniformly on the ecliptic, and to pass over' the extremities of the axis of the sun’s orbit at the same instant with the real sun. This removes the inequality arising from the inequality of the sun’s motion. To remove the inequality arising from the obliquity of the ecliptic, conceive a third sun to pass through the equinoxes at the same instant with the second sun, and to move along the equator in such a manner that the angular distances of the two suns at the vernal equinox shall be always equal. The interval between two consecutive returns of this third sun to the meridian forms the mean astronomical day. Mean time is measured by the number of the returns of this third sun to the meridian; and true time is measured by the number of returns of the real sun to the meridian. The arc of the equator, intercepted between two meridian circles drawn through the centres of the true sun and the imaginary third sun, when reduced to time, is what is called the Equation of Time. This will be rendered plainer by the following diagram.

Let Zɤ2^ (fig. 29) be the earth; ZFRz its axis; abcde, &c. the equator; ABCDE, &c. the northern half of the ecliptic from ɤ to =^=, on the side of the globe next the eye; and MNOP, &c. the southern half on the opposite side from i ≤⅛ to ɤ∙ Let the points at A, B, C, D, E, F, &c. mark off equal portions of the ecliptic, gone through in equal times by the real sun, and those at a, b, c, d, e, f, &c. equal portions of the equator described in equal times by the fictitious sun; and let Zɤz be the meridian.

As the real sun moves obliquely in the ecliptic, and the fictitious sun directly in the equator, any point between ɤ and F on the ecliptic must be nearer the meridian Z ɤ z i than the corresponding point on the equator from ɤ to f, that is to say, than the point whose distance from <p is expressed by the same number of degrees; and the [7:3:777]more so, as the obliquity is greater; and therefore the true sun comes sooner to the meridian every day whilst he is in the quadrant ɤF, than the fictitious sun does in the quadrant ɤ f ; for which reason the solar noon precedes noon by the clock, until the real sun comes to F, and the fictitious to f : which two points being equidistant from the meridian, both suns will come to it precisely at noon by the clock.

Whilst the real sun describes the second quadrant of the ecliptic FGHIKL from Cancer to =c⅛, he comes later to the meridian every day than the fictitious sun moving through the second quadrant of the equator from f to =o=; for the points at G, H, I, K, L, being farther from the meridian, their corresponding points at g, h, i, k, l, must come to it later; and as both suns come at the same moment to the point W, they come to the meridian at the moment of noon by the clock.

In departing from Libra through the third quadrant, the real sun going through MNOPQ towards >j> at R, and the fictitious sun through mnopq towards r, the former comes to the meridian every day sooner than the latter, until the real sun comes to R, and the fictitious to r, and then they come both to the meridian at the same time.

Lastly, as the real sun moves equably through STUVW, from R towards Υ>, and the fictitious sun through stuvw, from r towards ɤ, the former comes later every day to the meridian than the latter, until they both arrive at the point ɤ, and then they make iţ noon at the same time with the clock.

Having explained one cause of the difference between true and mean time, we now proceed to explain the other cause of this difference, namely, the inequality of the sun’s apparent motion, which is slowest in summer, when the sun is farthest from the earth, and swiftest in winter, when he is nearest to it.

If the sun’s motion were equable in the ecliptic, the whole difference between the equal time as shown by the clock, and the unequal time as shown by the sun, would arise from the obliquity of the ecliptic. But the sun’s motion sometimes exceeds a degree in 24 hours, though generally it is less; and when his motion is slowest, any particular meridian will revolve sooner to him than when his motion is quickest; for it will overtake him in less time when he advances a less space than when he moves through a larger.

Now, if there were two suns moving in the plane of the ecliptic, so as to go round it in a year, the one describing an equal arc every 24 hours, and the other describing in the same time sometimes a less arc, and at other times a larger, gaining at one time of the year what it lost at the opposite; it is evident that one of these suns would come sooner or later to the meridian than the other, as it happened to be behind or before the other; and when they were in conjunction, they would both come to the meridian at the same moment.

As the real sun moves unequably in the ecliptic, let us suppose a fictitious sun to move equably in a circle coincident with the plane of the ecliptic. Let ABCD (fig. 30) be the ecliptic or orbit in which the real sun moves, and the dotted circle abed the imaginary orbit of the fictitious sun; each going round in a year according to the order of letters, or from west to east. Let HIKL be the earth turning round its axis in the same direction every 24 hours; and suppose the two suns to start from A and a, at the same moment, A and a being in the same straight line, and in the plane of the meridian EH. Suppose also that the real sun, when at A, is at his greatest distance from the earth, where his motion is slowest. In the time that the meridian revolves from H to H again, according to the order of the letters HIKL, the real sun has moved from A to F; and the fictitious sun with a quicker motion from a to f, through a larger arc: therefore the meridian EH will revolve sooner from H to h under the real sun at F, than from HE to k under the fictitious sun at/; and consequently it will then be noon by the sun-dial sooner than by the clock.

As the real sun moves from A towards C, the velocity of his motion increases all the way to C, where it is at its maximum. But notwithstanding this, the fictitious sun gains so much upon the real, soon after his departing from A, that the increasing velocity of the real sun does not bring him up with the equally-moving fictitious sun till the former comes to C and the latter to c, when each has gone half round its respective orbit; and then being in conjunction, the meridian EH, revolving to EK, comes to both suns at the same time, and therefore it is noon by them both at the same moment.

But the increased velocity of the real sun now being at its maximum, carries him before the fictitious one; and therefore the same meridian will come to the fictitious sun sooner than to the real; for whilst the fictitious sun moves from c to g, the real sun moves through a greater arc from C to G; consequently the point K has its noon by the clock when it comes to k, but not its noon by the sun till it comes to l. And although the velocity of the real sun diminishes all the way from C to A, and the fictitious sun by an equable motion is still coming nearer to the real sun, yet they are not in conjunction till the one comes to A and the other to a, and then it is noon by them both at the same moment.

Mean solar time and sidereal time being both uniform, it is easy to compare the one with the other, and assign the number of degrees, minutes, &c. which the sun and a star will respectively describe in a given portion of sidereal or mean solar time. In a mean solar day the sun’s right ascension and mean longitude are increased by 59' 8"∙33; consequently 360° 59' 8"∙33 of the equator pass the meridian in 24 mean solar hours. The sidereal time corresponding to this period is 24 hours 3 min. 56∙555 sec.; therefore 24 mean solar hours are equal to 24 hours 3 min. 56∙555 sec. of sidereal time; and 24 hours of sidereal time are equal to 23 hours 56 min. 4∙0907 sec. of mean solar time, or to 24 hours minus 3 min. 55∙9093 sec. This difference of 3 min. 55∙9093 sec. is called the acceleration of the fixed stars in mean solar time; and the preceding excess of 3 min. 56∙555 sec. is the retardation of the sun in sidereal time. Hence the one species of time may be easily converted into the other; and the arc of the equator passed over by the meridian in a given mean time may be calculated. Thus,

≡∞≤2 = 15« 2-27-84708

24

is the arc described by a star in one hour of mean solar time.

But the process of converting sidereal or mean solar time into true or apparent time, or of computing from the instants of apparent time the corresponding mean solar and sidereal times, is attended with much greater difficulty. The reason is, that the interval between two successive transits of the sun over the meridian, which, in apparent time, measures the day, is a variable quantity; and hence there cannot exist any constant ratio between true and mean time, as there does between mean find sidereal time. The correction or equation by which apparent time is reduced to mean time, is technically called the Equation of Time, and is composed of the aggregate of the several variable terms which denote the inequalities of the sun’s motion in longitude. Besides the eccentricity, [7:3:778]I the obliquity of the ecliptic, and the variations of that; obliquity occasioned by the nutation of the earth’s axis, it is affected also by the small alterations of the sun’s right ascension, which result from the effect of the planetary perturbations on the earth; and hence the equation of time cannot be exactly computed without the aid of Physical Astronomy.

It will be evident, from what has preceded, that the equation of time expresses merely the difference between the true and mean right ascensions of the sun, reduced to time. Its different parts may be calculated numerically in the following manner. Let NA (fig. 31) represent the mean motion of the sun during a given interval, then NA is the sun’s mean longitude, N being the first point of Aries. Let NA = Μ, and from this mean longitude subtract the longitude of the apogee, the remainder will be the sun’s mean anomaly. From the mean anomaly let the equation of the centre be found and denoted by E. Take AB = E, then NB = Μ + E is the longitude corrected for the eccentricity.

Let us next suppose BC to be the small quantity by which the sun’s longitude is increased in consequence of the perturbations of the planets, and let BC = P; then NC = Μ + E+ P is the true and exact longitude of the sun. Through C let the arc CD be drawn perpendicular to the equator; the point D will be that point of the equator which passes the meridian at the same time with the sun. Let R = NC — ND = the reduction to the ecliptic; we have then the sun’s right ascension = Μ + E + P—R.

Let NF = NA, and F will be the place which the sun would occupy in the equator at the same instant that he occupies the point A in the ecliptic, if he moved uniformly in the former circle; for NF as well as NA will represent the mean diurnal motion of 59' 8"∙33 multiplied by the number of days elapsed in the interval between the equinox N and the time of the observation. The mean sun would therefore pass the meridian with the point F, whereas the true sun passes it with the point D; therefore, at the instant of true noon, when the sun C and the corresponding point D are on the meridian, the mean sun is at a distance from D expressed by the arc FD=M + E+P—R — M=E + P — R. Now, the arc of the equator FD measures the horary angle between the mean and the true sun, or the angle at the pole between the meridian and the hour-circle passing through D; it is therefore converted into time by the following proportion :—

360°: FD:: 24 mean solar hours: time from true noon; consequently the difference between mean and true noon, or the equation of time,

= ¾^ = 1 ⅛(E+ p -R)=<rt.

But this equation is not yet perfectly accurate: it requires to be corrected for the effects of nutation. Now it is known that the variation of the mean longitude of the sun, arising from the unequal precession of the equinoxes in consequence of the nutation occasioned by the inclination of the lunar orbit, is expressed by the formula 18" sin. (360° — moon’s node) = 18" sin. N. This variation, reduced to the direction of the equator, will therefore be 18" sin. N cos. e (e being the obliquity of the ecliptic). The difference between the two expressions is 18" sin. N (1 — cos. e) = 36" sin. 2 1 2 e sin. N = 1" ∙4887 sin. N, which, reduced to time, becomes 0∙09925 sec. × sin. N. This small correction, amounting to less than a tenth of a second, was long omitted in computing the equation of time. When it is included, that equation becomes í/T = f >θ (E + P — R) + 0∙09925 sec. × sin. N.

The quantities E, P, R, and N, are to be computed se- '. parately from the astronomical tables; and it must be ob-∙ i served, that the result will be expressed in mean solar' time.

The cosine of the obliquity, that is, cos. 23° 28', is equal to ∙9173 = ⅛ very nearly. Hence, since the equation of time is equal to the sun’s true right ascension, diminished by his mean longitude and the effects of nutation in right ascension (= ND — NA =f∑ 18" sin. N cos. e), it may, on denoting the true right ascension by A, be expressed as follows :—

cTĨ = A — Μ →= 18" sin. N × ⅛.

This is the form under which the equation of time was expressed by Dr Maskelyne.

The equation of time is at its maximum about the 3d of November, when it amounts to 16' 16"∙7, and is subtractive. At four different times of the year it vanishes, namely, about the 25th of December, the 16th of April, the 16th of June, and the 1st of September. These epochs, however, do not remain constant; for, on account of the change which the line of the apsides is constantly undergoing in reference to the line of the equinoxes, the difference between the true and mean right ascensions of the sun—in other words, the equation of time—varies continually in different years.

Sect. V.— Of the Spots of the Sun, his Rotation, and Constitution.

The sun, the great source of light, heat, and animation, when beheld with the naked eye, appears only as a luminous mass of uniform splendour and brightness; but when examined with the telescope, his surface is frequently observed to be mottled over with a number of dark spots, of irregular and ill-defined forms, and constantly varying in appearance, situation, and magnitude. These spots are occasionally of immense size, so as to be even visible without the aid of the telescope; and their number is frequently so great that they occupy a considerable portion of the sun’s surface. Dr Herschel observed one in 1779, the diameter of which exceeded 50,000 miles, more than six times the diameter of the earth; and Scheiner affirms that he has seen no less than 50 on the sun’s disk at once. Most of them have a deep black nucleus, surrounded by a fainter shade, or umbra, of which the inner part, nearest to the nucleus, is brighter than the exterior portion. The boundary between the nucleus and umbra is in general tolerably well defined; and beyond the umbra a stripe of light appears more vivid than the rest of the sun.

The discovery of the sun’s spots has been attributed to different astronomers. They appear to have been first taken notice of in a work of Fabricius, the friend of Kepler, which was published at Wittenberg in 1611, under the title of Joh. Fabricii Pl try sii de Maculis in Sole Observatis, et Apparente earum cum Sole Conversione Narratio. It contains, however, nothing more than a few vague conjectures respecting the spots, the phenomena of which he could not have observed with any degree of accuracy, inasmuch as he seems to have been unacquainted with any method of protecting the eye by intercepting a portion of the solar rays; for he recommends to those who should repeat his observations, to admit into the telescope only a small portion of the sun at once, till the eye should by degrees become able to support the full blaze of light. About the same time the discovery was warmly disputed by the illustrious Galileo, and Schciner, a German Jesuit, professor of mathematics at Ingolstadt. The whole circumstances connected with this dispute are narrated at great length by Galileo in his work entitled Istoria e 1A- [7:3:779]mostrazioni intorno alle Macchie Solari, e loro Accidenti, from which it appears certain that he observed the spots so early as April 1611. In a letter published by him in 1612, he remarks that the spots are situated on the surface of the sun, or that at least their distance from it is imperceptible; that the time of their continuance varies from 2 or 3 to 30 or 40 days; that their figures are irregular and variable; that some are seen to separate, and others to unite, even on the middle of the disk; that besides these peculiar motions, they have also a common motion, in virtue of which they traverse the disk in parallel lines. From this general motion he infers that the sun turns on an axis from west to east; and he adds as a curious remark, that the spots are confined within a zone extending only about 28 or 29 degrees to the north and south of the sun’s equator. Galileo illustrates all these positions by mathematical reasoning, and by drawings of the spots made on many successive days.

Scheiner’s observations were first announced in January 1612, in three letters addressed to his friend Marc Velser, a magistrate of Augsburg. In the first of these, the date of which is November 1611, he says that he had observed the spots seven months before, but that, having a different object in view, he had given little attention to them. He observed them again in the following October, and at that time imagined the appearance was owing to some imperfection of his telescope, till he was convinced by repeated observations that it was necessary to refer it to .the sun. From these remarks it is pretty clear that Scheiner had formed no accurate notions respecting the spots before October 1611, that is, six months after they had been observed by Galileo. Scheiner made the observation of the solar spots his whole occupation during the following eighteen years, in the course of which he discovered the position of the solar equator, and formed a theory much more complete than that of Galileo. The account of his observations was published in 1630, under the title of Rosa Ursina, sive Sol ex admirando Facularum et Macularum suarum Phaenomeno Varius, &c.

The discovery of the solar spots has also been claimed for our countryman Harriot. Amidst these conflicting pretensions it is perhaps impossible to arrive at the truth; but the matter is of little importance; the discovery is one which followed inevitably that of the telescope, and an accidental priority of observation can hardly be considered as establishing any claim to merit.

The solar spots furnish an extensive subject of curious speculation, but in an astronomical point of view they are chiefly interesting on account of their establishing the fact, and affording the means of determining the period, of the rotation of the sun. In order to obtain a precise idea of the position of a spot, and the path which it describes, it is necessary to project that path on the plane passing through the centre of the sun, and perpendicular to the visual ray drawn from the earth to the sun’s centre. Suppose the diameter of a circle ASB (fig. 32) to be divided into as many parts of unity as there are seconds in the apparent diameter of the sun. Let CP be taken equal to the number of seconds contained in the difference of the longitudes of the spot and the sun’s centre, and the perpendicular PM equal to the number of seconds in the latitude of the spot; then Μ will represent its position on the surface of the sun. By repeating the same operation a number of days consecutively, a series of points Μ Μ' Μ", &c. will be obtained in the apparent path of the spot on the sun’s disk, or rather in the projection of that path on the plane perpendicular to the visual ray. This projection is in general an oval slightly differing from an ellipse; and it is found that all the spots observed at the same time describe similar and parallel curves. They also return to the same relative positions in the same time, and their period is about 27⅛∙ days.

The paths described by the spots undergo very considerable changes, according to the season of the year at which they are observed. About the end of November and beginning of December they appear simply as straight lines Mm, M'm', M"m" (fig. 33), along which the spots move in the direction Mm, that is, they enter on the eastern and disappear on the western edge of the sun’s disk; and the points at which they disappear are more elevated, or nearer the north pole of the ecliptic, than those at which they enter. After a certain time the lines Mm begin to assume a curved appearance, and form ovals, as represented in fig. 34. During the winter and spring the convexity of the ovals is turned towards the north pole of the ecliptic; but their inclination, or rather the inclination of the straight lines joining their extreme points, to the plane of the ecliptic continues to diminish, and about the beginning of March disappears; so that the points at which they seem to enter and leave the sun’s disk are equally elevated, as in fig. 35. From this time the curvature of the ovals diminishes; they become narrower and narrower till about the end of May or beginning of June, when they again appear under the form of straight lines (fig. 36); but their inclinations to the ecliptic are now precisely in a contrary direction to what they were six months before. After this they begin again to expand, as in fig. 37, and their convexity is now turned towards the south pole. Their inclinations also vary at the same time, and about the commencement of September they are seen as represented in fig. 38; the points at which they enter and disappear being again equally elevated. After this period the ovals begin to contract and become inclined to the ecliptic, and by the beginning of December they have exactly the same direction and inclination as they had the previous year.

These phenomena are renewed every year in the same order, and the same phases are always exhibited at corresponding seasons. Hence it is evident that they depend on a uniform and regular cause, which is common to all of them, since the orbits described by the various spots are exactly parallel, and subject in all respects to the same variations. The simplest method of explaining the phenomena is to suppose with Galileo that the spots are adherent to the surface of the sun, and that the sun uniformly revolves round an axis inclined to the axis of the ecliptic. If the axis of revolution were perpendicular to the plane of the ecliptic, the spots, supposing them to adhere to the sun’s surface, would describe circles parallel to that plane, which, seen from the earth, would appear as so many parallel straight lines; but by supposing the axis to have a suitable inclination, all the phenomena become explicable in a very simple manner. While the sun is carried round in his orbit, his axis, constantly preserving its parallelism, will successively assume different positions relatively to the earth; and the planes of the circles described by the spots, which planes are always perpendicular to the axis, will consequently be presented to us under different inclinations; hence the variations of their apparent curvature. In two opposite points of the orbit the visual ray drawn from the earth to the centre of the sun is perpendicular to the axis of rotation. In these two positions the poles of the sun, or the points in which the axis meets the surface, are both visible at the same moment, and the spots appear to move in parallel straight lines. But as the axis retains this perpendicular position only for an instant, and declines from it very sensibly while the spot traverses the sun’s disk, the path of [7:3:780]the spots over the entire disk is neither a straight line nor an ellipse, one of which it would necessarily be if the sun, while revolving about his axis, did not change his place in his orbit. When the axis is not perpendicular to the visual ray, the path of the spots will appear to be a curve of which the concavity is turned towards that pole which is visible from the earth. This inclination of the solar axis to the plane of the ecliptic also explains the reason why the points of the disk at which the spots appear are more elevated during one half of the year, and more depressed during the other half, than those at which they disappear. It will also follow from the same hypothesis, that the curvature of the ovals must be the greatest possible when the straight lines joining their extremities are parallel to the ecliptic; and, on the contrary, least when the same straight lines are most inclined to the ecliptic; all which is exactly conformable to observation.

The various appearances which we have now described may be accurately represented by means of a common celestial or terrestrial globe. Let the wooden horizon of the globe, which is here supposed to represent the sun, be placed horizontally in the same plane with the eye of the spectator, and the pole be inclined about 7° from the zenith. The wooden horizon will now represent the ecliptic; and if the spectator walk round the globe, always keeping his eye in the plane of the wooden horizon, the circles of latitude will appear to him as ellipses of different inclinations and eccentricities: in two opposite points they will appear as straight lines, and, in short, exhibit in their various positions all the phenomena of the oval paths described by the spots of the sun.

The consequences deduced from the hypothesis of the rotatory motion of the sun are so perfectly conformable with observation, as to render the inference inevitable, that the sun revolves from west to east, on an axis inclined about 93° to the plane of the ecliptic. The plane which passes through the centre of the sun, perpendicular to the axis of rotation, is the Equator of the sun; the straight line joining the points in which it intersects the ecliptic is called the Line of the Nodes of the equator. The Nodes themselves are the two opposite points in which this straight line, produced indefinitely, meets the celestial sphere.

In order to determine the situation of the solar axis in space, it is necessary to find its inclination to the ecliptic, and the angle which the line of the nodes makes with any given line on the plane of the ecliptic, for example, with the line of the equinoxes. The requisite data for the solution of this problem are three different positions of the same spot, which must be obtained by observation. Let S (fig. 39) be the centre of the sun’s disk, AB a parallel to the terrestrial equator, and Μ the place of a spot, the co-ordinates of which, as referred to AB, are SX and MX. In this figure the earth is supposed to be situated in a straight line passing through S perpendicular to the plane of the paper, and it is to be recollected that SX and MX are in fact arcs of the solar globe, though so small when seen from the earth that they may be regarded as straight lines. By comparing the times of the transits of B, the border of the disk, and the spot Μ, we shall find BX, from which, as SB the semidiameter of the sun is known, we shall have SX the difference of the right ascensions of the spot and sun’s centre. The line CY, which is the difference of the declination of the border of the disk and that of the spot, is measured by the micrometer. This will give MX the declination of the spot.

Μ X c v

Now, tan. MSX = and SM = ⅛-∕ r , t ,v “therefore

SX cos. MSX

SM is also a known quantity. Let EE be the ecliptic, ß = obliquity of the ecliptic, and © = the longitude of' the sun. The angle BSE, which is technically called the 2 Angle of Position, is the complement of the angle made by ' the ecliptic with the circle of declination passing through the sun, and is therefore given by the formula tan. BSE = tan. e cos. ©: hence MSE (=MSX — BSE) is also given. On SE let fall the perpendicular Mm, then Mm = SM sin. MSE is the geocentric latitude of the spot, and Sm = SM cos. MSE is the difference between its geocentric longitude and that of the centre of the sun: the object is now to determine the angles subtended by these lines at the centre of the sun, that is, to convert the geocentric into heliocentric latitudes and longitudes.

Suppose two straight lines to be drawn from the earth, one to the centre of the sun, and the other to the centre of the spot Μ, and let ô be their inclination, which is measured by SM, and is consequently known, being the geocentric distance of the spot from S, the centre of the sun’s disk. Let R = distance of the sun’s centre from the earth, r = semidiameter of the sun, φ = the angle made at the centre of the sun by the straight lines drawn from it to the earth and the spot Μ, and let -ψ be the remaining angle of the triangle formed by the lines joining the earth, the centre of the sun, and the spot. We have then r : R:: sin. ô : sin. ∙ψ, whence sin. ∙ψ = — sin. ð. But R: r as radius to the sine of half the sun’s r 1 . . ,, . , sin. 0

true diameter; therefore sm. ∙ψ = —— i -— π .

sm. ⅛ sun s diameter

Now φ — 1δ0° — + — d, consequently we have the number of degrees, minutes, and seconds in <ţ>, the heliocentric distance of the spot from the straight line which joins the centres of the earth and sun. Having thus obtained the hypothenuse SM in parts of a great circle of the solar orb, the sides Sm and Mm will be obtained in similar parts from the common trigonometrical formulae for the resolution of a right-angled spherial triangle. These formulae give tan. Sm = tan. SM cos. MSE, sin. Mm = sin. SM sin. MSE.

It will be remarked that Mm is the heliocentric latitude of the spot, Sm the difference of its longitude and that of the earth; and as the longitude of the earth is equal to 180° + that of the sun, the heliocentric longitude of the spot will be 180° + © — SM if the spot is behind or to the east of the centre of the sun, and 180° + © + Sm if it precedes the centre.

In this manner the heliocentric longitudes and latitudes of the spots are deduced from their observed right as- 1 censions and declinations. The next step is to show how they are employed in determining the position of the solar axis. This problem is in practice somewhat laborious, although the principles on which its solution rests are sufficiently simple. The planes of the circles described by the spots are parallel to the sun’s equator: if, therefore, the position of one of them can be found, the position of the equator, and consequently of the axis, will be found at the same time. Now, the position of a plane is determined by three given points through which it is required to pass; consequently, by three observations of the same spot, we shall have three points in its plane, and thence the plane itself. The problem is therefore one of pure geometry, and may be-solved in various ways. The results of the most accurate observations make the inclination of the solar equator to the ecliptic amount to 7 o 19'23" or, 7° ⅜ very nearly; and the heliocentric longitude of the ascending node, that is, the point in which the equator of the sun intersects the ecliptic, in passing from [7:3:781]south to north, 80° 7' 4". The position of the node seems to undergo no variation, except such as may be supposed to arise from the precession of the equinoxes.

It has already been mentioned that the mean time in which a solar spot returns to the same position, relatively to the earth, is 27∙3 days. This, however, is not the time in which the sun makes a revolution about his axis. In the interval of 27∙3 days the sun describes in the ecliptic an arc equal to 26° ∙91 (for 365 d ∙25: 27 d ∙3:: 360°: 26° ∙91), and by virtue of this motion alone he exhibits every day different points of his surface, the whole of which would be successively shown to the earth in the course of a year, independently of the motion of rotation. Hence results an apparent annual rotation round an axis perpendicular to the ecliptic, and we must abstract the effects of this optical illusion in order to arrive at the time of a real rotation. This apparent rotatory motion will be easily understood by referring to fig. 40, in which S is the sun, E the earth, and C the point in which the visual ray ES intersects the surface of the sun. Suppose the sun to have advanced in his orbit from S to S', the visual ray drawn from the sun’s centre to the earth will now meet the surface in a different point C', and the angular distance between C and C' will be found by drawing S'E' parallel to SE; for the point c in which S'Ľ meets the surface will correspond to C, and the arc C'c, or the angle E'S'E, will be the apparent rotation, while the sun advances in his orbit from S to S'. Now, the angle E'S'E is equal to S'ES, or the apparent rotatory motion is equal to the angular motion of the sun in his orbit; hence, since the real rotation is in a contrary direction, it is obvious that, if the axis were perpendicular to the plane of the ecliptic, when a spot appears to have made one revolution, it has in reality passed over an arc equal to 360° + 26° ∙91, or 386 ’ 91 degrees. By a simple proportion, therefore, 386∙91: 360:: 27∙3: 25*4; and consequently 25∙4 days is the real time of the sun’s revolution. This result is, however, not quite accurate, on account of its having been supposed that the axis of rotation is perpendicular to the ecliptic, whereas it differs from a perpendicular to that plane by 7° ⅛∙; but the correction necessary on account of the difference is so small as to fall far within the limits of the errors to which the observations are liable; it is therefore unnecessary to have regard to it.

Observations of the spots, from which the elements of the sun’s rotation are deduced, are attended with a very considerable degree of uncertainty. The semidiameter of the sun, which at the surface of the earth subtends an angle of only 16', or 960", is equivalent to 90° at the centre of the sun. Hence the error of a single second (and it is impossible to answer for one or two seconds in observations of this sort) corresponds to about 338", or 5' 38", on the surface of the solar globe. An observer may, therefore, notwithstanding the greatest care, be mistaken to the extent of 10' with regard to the length of a heliocentric arc; and when to this we add, that the margins of the spots are ill defined, and even changeable, so that their centres, which it becomes necessary to observe, are not always the same, it will not appear surprising that a considerable discordance exists among the results of different observations. The uncertainty respecting the time of rotation amounts to no less than 10 or 12 hours. From a careful discussion of a numerous set of observations, Delambre found the time of rotation to be only 25Ό1154 days, instead of 25∙4 given above; and he remarks that, in order tourender the last-mentioned number admissible, it is necessary to suppose either an error of 2° 22', which is scarcely possible, or else that the spot has a proper motion in the same direction as the motion of rotation of the earth and all the planets.

From four different combinations of equations, derived from eleven observations of the same spot, Delambre computed the following table of the elements of rotation.
[table]

With regard to observations on the sun’s spots, Delambre remarks that he attaches little value to them, first, because it is impossible to make them well; and, secondly, because, even if they were sure, they only lead to results of little importance to astronomy. He discussed a hundred different spots, each observed at least three times by Messier, and deduced thirty different determinations of the elements of rotation. The more he multiplied his calculations, the more certain he became of the impossibility of a good solution; of which, indeed, there is no other chance than in a compensation of errors, little probable on account of their enormous magnitudes. These discrepancies render it probable that the spots, besides partaking of the general motion of the solar globe, have also proper motions either of displacement, or occasioned by a change of form, which may long prevent this part of astronomy from reaching a greater degree of exactness than it has already attained.

From the circumstance that the sun is incontestably endowed with a rotatory motion, Lalande concluded that, according to the rules of probability, he ought also to have a motion of translation in space. This idea was adopted by Herschel, and is now generally received as probable. It explains some, though not all, of the proper motions which have been supposed to be observed among the stars. If, indeed, the stars are so many suns, revolving like ours each on its own axis of rotation, it is extremely probable that they have also motions of translation; and thus, there being no fixed points in the heavens, the problem of their proper motions becomes so complicated as to be altogether insoluble.

The only interesting fact which has been deduced from the observations of the solar spots is the rotation of the sun. The curious appearances which they exhibit have, however, attracted great attention, and given rise to numerous theories respecting the constitution of that immense body which governs and vivifies the planetary system. We will now proceed to give a brief account of the appearances, and mention some of the theories that have been founded on them; premising, that after all that has been written on this subject, we are not yet, and probably never will be, in possession of any definite knowledge. The nature of the spots, and the physical constitution of the sun, afford fruitful subjects of harmless conjecture and speculation, but form no part of science.

The phenomena of the solar spots, as delivered by Scheiner and Hevelius, may be summed up in the following particulars. 1. Every spot which has a nucleus, or comparatively dark part, has also an umbra, or fainter shade surrounding it. 2. The boundary between the nucleus and umbra is always distinct and well defined. 3. The increase of a spot is gradual, the breadth of the nucleus and umbra dilating at the same time. 4. In like manner the decrease of a spot is gradual, the breadth of the nucleus and umbra contracting at the same time.
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5. The exterior boundary of the umbra never consists of sharp angles, but is always curvilinear, how irregular soever the outline of the nucleus may be. 6. The nucleus of a spot, whilst on the decrease, often changes its figure by the umbra encroaching irregularly upon it, insomuch that in a small space of time new encroachments are discernible, whereby the boundary between the nucleus and umbra is perpetually varying. 7. It often happens, by these encroachments, that the nucleus of a spot is divided into two or more nuclei. 8. The nuclei of the spots vanish sooner than the umbra. 9. Small umbrae are often seen without nuclei. 10. An umbra of any considerable size is seldom seen without a nucleus in the middle of it. 11. When a spot which consisted of a nucleus and umbra is about to disappear, if it is not succeeded by a facula or spot brighter than the rest of the disk, the place where it was is soon after not distinguishable from the rest.

In the Philosopliical Transactions, vol. lxiv. Dr Wilson, late professor of astronomy at Glasgow, has given a dissertation on the nature of the solar spots, in which he mentions the following appearances. 1. When the spot is about to disappear on the western edge of the sun’s limb, the eastern part of the umbra first contracts, then vanishes, the nucleus and western part of the umbra remaining; then the nucleus gradually contracts and vanishes, while the western part only of the umbra remains. At last this disappears also; and if the spot remains long enough to become again visible, the eastern part of the umbra first becomes visible, then the nucleus; and when the spot approaches the middle of the disk, the nucleus appears environed by the umbra on all sides, as already mentioned. 2. When two spots lie very near to one another, the umbra is deficient on that side which lies next to the other spot: and this will be the case, though a large spot should be contiguous to one much smaller; the umbra of the large spot will be totally wanting on that side next the small one. If there are little spots on each side of the large one, the umbra does not totally vanish, but appears flattened or pressed in towards the nucleus on each side. When the little spots disappear, the umbra of the large one extends itself as usual. This circumstance, he observes, may sometimes prevent the disappearance of the umbra in the manner above mentioned; so that the western umbra may disappear before the nucleus, if a small spot happens to break out on that side.

In the same volume, p. 337, the Rev. Dr Wollaston observes that the appearances mentioned by Dr Wilson are not uniform. He positively affirms that the faculae or bright spots on the sun are often converted into dark ones. “I have many times,” says he, “observed near the eastern limb a bright facula just come on, which has the next day shown itself as a spot, though I do not recollect to have seen such a facula near the western one after a spot’s disappearance. Yet, I believe, both these circumstances have been observed by others, and perhaps not only near the limbs. The circumstance of the faculae being converted into spots, I think I may be sure of. That there is generally, perhaps always, a mottled appearance over the face of the sun, when carefully attended to, I think I may be as certain. It is most visible towards the limbs, but I have undoubtedly seen it in the centre; yet I do not recollect to have observed this appearance, or indeed any spots, towards the poles. Once I saw, with a twelve-inch reflector, a spot burst to pieces while I was looking at it. I could not expect such an event, and therefore cannot be certain of the exact particulars; but the appearance, as it struck me at the time, was like that of a piece of ice when dashed on a frozen pond, which breaks to pieces and slides in various directions.” He also observes, that the nuclei of the spots are not always in the middle of the umbrae, and gives the figure of one seen on the 13th of November 1773, which is a remarkable instance' to the contrary.

The faculae, or bright spots, were observed with particular attention by Messier, who frequently saw them enter on the eastern limb of the sun, disappear as they approached the centre of the disk, re-appear on the opposite limb, and continue visible, as they had done at the time of their first appearance, for about three days, till they were carried off the disk by the rotation of the sun. Spots frequently broke out in these faculae, and when this did not happen, they were succeeded by spots which generally became visible on the following day; and from the regularity of this occurrence, he was enabled to predict the appearance of a spot 24 hours before it entered on the sun’s disk. He observed, also, that the magnitude of the spots was proportional to the brightness of the antecedent faculae. Like the spots, the faculae are generally confined to the equatorial regions of the sun; but they have been occasionally observed by Schroeter on every part of the disk.

To explain these singular appearances, numerous theories, more or less plausible, have been proposed, but all: resting on many gratuitous assumptions, and subject to great difficulties. Scheiner imagined that the spots do not belong to the sun, but supposed them to be inferior planets revolving at no great distance from the central luminary. Galileo, Hevelius, and others, have supposed them to be scoriae floating in the inflammable liquid matter of which they imagined the sun to be composed. This opinion, although it accounts for the appearance of the spots in the equatorial regions of the sun, to which such scoriae would be carried by the centrifugal force resulting from his rotation, cannot be reconciled with the regularity with which the spots frequently re-appear on the eastern limb of the sun. Dr Wilson, having observed that the spots situated near the edge of the disk are narrow, and without a penumbra on the side next the centre, and that only the central spots are completely surrounded by a penumbra (appearances which would be exactly represented by a conical gulf or cavity presented to us under different aspects by the revolution-of the sun), was led to adopt the opinion, that the appearances of the spots are occasioned by real excavations in the solar globe. He supposed the sun to consist of a dark nucleus, covered only to a certain depth by a luminous matter, not fluid, through which openings are occasionally made by volcanic or other energies, permitting the solid nucleus of the sun to be seen; and that the umbra which surrounds the spot is occasioned by a partial admission of the light upon the shelving sides of the precipice opposite to the observer. It is evident that, in proportion as these excavations are seen obliquely, their apparent dimensions will be diminished; one of the edges will disappear as it approaches the sun’s limb, or come more into view as it advances towards the middle of the disk; when the spot is about to leave the disk, the bottom of the excavation, or the nucleus seen through it, will first disappear, but a sort of faint or obscure spot will remain visible as long as the visual ray penetrates the cavity. These appearances are all conformable to the laws of perspective; but Dr Wilsou, wishing to give a still more palpable demonstration of the accuracy of his theory, fitted up a large globe, into which holes of the proper dimensions were inserted; and this machine being placed at a distance, and made to revolve, was found, when examined through a telescope, to exhibit in the course of its revolution all the phenomena of the solar spots.
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Dr Wilson’s theory was keenly combated by Lalande, who adduced several observations of his own, and some by Cassini, that could not be explained by means of it; and urged with reason, that an hypothesis, founded on the uniformity of appearances which in reality are exceedingly variable, was entitled to little consideration. Lalande himself supposed the spots to be scoriae which have settled or fixed themselves on the summits of the solar mountains; an opinion which he grounded on this circumstance, that some large spots which had disappeared for several years were observed to form themselves again at the identical points at which they had vanished.

The late Sir William Herschel, with a view to ascertain more accurately the nature of the sun, made frequent observations upon it from the year 1779 to the year 1794>∙ He imagined the dark spots on the sun to be mountains, which, considering the great attraction exerted by the sun upon bodies placed at its surface, and the slow revolution it has upon its axis, he thought might be more than 300 miles high, and yet stand very firmly. He says that in August 1792 he examined the sun with several powers from 90 to 500, when it evidently appeared that the dark spots are the opaque ground or body of the sun, and that the luminous part is an atmosphere, through which, when interrupted or broken, we obtain a view of the sun itself. Hence he concluded that the sun has a very extensive atmosphere, consisting of elastic fluids that are more or less lucid and transparent, and of which the lucid ones furnish us with light. This atmosphere, he thought, cannot be less than 1843, nor more than 2765 miles in height; and he supposed that the density of the luminous solar clouds needs not be much more than that of our aurora borealis, in order to produce the effects with which we are acquainted. The sun, then, if this hypothesis be admitted, is similar to the other globes of the solar system with regard to its solidity, its atmosphere, its surface diversified with mountains and valleys, its rotation on its axis, and the fall of heavy bodies on its surface; it therefore appears to be a very eminent, large, and lucid planet, the primary one in our system, disseminating its light and heat to all the bodies with which it is connected.

Herschel supposed that there are two regions or strata of solar clouds; that the inferior stratum is opaque, and probably not unlike our own atmosphere, while the superior is the repository of light, which it darts forth in vast quantities in all directions. The inferior clouds act as a curtain to screen the body of the sun from the intense brilliancy and heat of the superior regions, and, by reflecting back nearly one half of the rays which they receive from the luminous clouds, contribute also greatly to increase the quantity of light which the latter send forth into space, and thereby perform an important function in the economy of the solar system. The luminous clouds prevent us in general from seeing the solid nucleus of the sun; but in order to account for the spots, he supposes an empyreal elastic gas to be constantly forming at the surface, which, carried upwards by reason of its inferior density, forces its way through the planetary or lower clouds, and mixing itself with the gases which have their residence in the superior stratum, causes decompositions of the luminous matter, and gives rise to those appearances which he describes under the name of corrugations. Through the openings made by this accidental removal of the luminous clouds, the solid body of the sun becomes visible, which, not being lucid, gives the appearance of the dark spots or nuclei seen through the telescope. The length of the time during which the spots continue visible renders it evident that the luminous matter of the sun cannot be of a liquid or gaseous nature; for, in either case, the vacuity made by its accidental removal would instantly be filled up, and the uniformity of appearance invariably maintained. Herschel supposed •the luminous clouds to be phosphorescent.

It would be a needless waste of time to enter into any discussion of a theory so entirely vague and fanciful, and so destitute of all solid foundation. Till we are better acquainted with the nature of light,· fire, and heat, and have attained to the knowledge of every possible mode in which these elements can be produced and propagated, all hypotheses respecting the construction of the sun can only be gratuitous and conjectural. Some interesting questions, however, arise out of this subject. Whether the inferior stratum of solar clouds is sufficiently dense, as Herschel imagined, to protect the body of the sun from the scorching effects of the surrounding regions of light and heat, and render it a fit habitation for human beings, is a question of no importance to man, or to any thing pertaining to his planet; but it is interesting for him to know whether the light and heat dispensed by the sun are liable to any variation or secular diminution, either connected with the spots, or resulting from a decrease of the sun’s volume.

Those philosophers who adopt the hypothesis of the production of light and heat by the vibrations of an ethereal fluid, consider the mass of the sun to be invariable. Those, on the contrary, who attribute these effects to an emanation from the sun, think his mass and volume must be diminished by the incessant discharge of torrents of luminous particles from his surface. During the two thousand years which have elapsed since the first astronomical observations, no diminution of the sun’s volume has been perceived; but it must be remarked that such an effect may have taken place, though not yet sensible to our instruments. The sun’s diameter is nearly 2000"; and at the distance of 95,000,000 miles a second corresponds to 460 miles. Now, supposing the solar diameter to suffer a daily diminution of two feet, which may be considered as enormous, considering the vast magnitude of the sun, and the excessive rarity of light, the diminution would amount to 800 feet in a year, and to 460 miles, or 1", in 3000 years. Thus, after thirty centuries, the diminution would still be imperceptible, inasmuch as our instruments are not sufficiently accurate to enable us to appreciate, in an observation of this sort, so small a variation as one second.

Some astronomers, after Herschel, have imagined that the existence of the solar spots has an influence on the temperature of the seasons. In 1823 the summer was cold and wet; the thermometer at Paris rose only to 23° ∙7 of Reaumur, and the sun exhibited no spots; whereas in the summer of 1807 the heat was excessive, and the spots of vast magnitude. The relation, however, between the temperature and the appearance of the solar spots is not so uniform as to give much weight to this opinion. Warm summers, and winters of excessive rigour, have happened in the presence or absence of the spots. The year 1783 was remarkable for its fertility and the magnitude of the solar spots; a dry fog enveloped the greater part of Europe, and was followed by the earthquake of Calabria. Another opinion entertained by Herschel was, that one hemisphere of the sun emits less light than the other, so that when viewed at a great distance he will resemble some stars of which the brilliancy is subject to periodical variations.

By reason of the globular figure of the sun, his surface towards the border of the disk is seen obliquely, and therefore a much greater portion of it is comprehend-I[7:3:784] ed under a given visual angle, than when the ray proceeds from his centre. Now, as every point of the sun’s surface is supposed to emit an equal quantity of light in all directions, it follows that the light ought to be much more intense near the circumference of the disk, because if greater number of rays will proceed from the larger surface, which forms the oblique base of the luminous cone. Bouguer, deceived by some imperfect experiments, thought the light more intense at the centre of the disk than towards the limb; a circumstance which could only be explained by supposing the light to be diminished by some cause which acts most powerfully with regard to the borders of the disk. Such would be the effect of a dense atmosphere surrounding the sun; for in this case the rays which proceed from the border must traverse a much greater extent of the solar atmosphere, and consequently be absorbed in greater proportion than those which proceed from the central parts and traverse it directly; just as the atmosphere of the earth renders the light of the stars at the horizon much feebler than at the zenith. It has been found, however, that Bougucr’s experiments were inaccurate, and that the light is equally intense at the border and the centre. The existence of a solar atmosphere cannot therefore be demonstrated in this manner; but it is clearly indicated by the faint light which is observable round the sun’s limb during a total eclipse.

Another very curious phenomenon connected with the sun, is the faint nebulous aurora which accompanies him, known by the name of the Zodiacal Light. This phenomenon was first observed by Kepler, who described its appearance with sufficient accuracy, and supposed it to be the atmosphere of the sun. Dominic Cassini, however, to whom its discovery has been generally but erroneously attributed, was the first who observed it attentively, and gave it the name which it now bears. It is visible immediately before sunrise, or after sunset, in the place where the sun is aboűt to appear, or has just quitted, in the horizon. In total eclipses it is seen surrounding the sun’s disk, and resembling the beard of a comet. It has a flat lenticular form, and is placed obliquely on the horizon, as represented in fig. 41, the apex extending to a great distance in the heavens. Its direction is always in the plane of the sun’s equator, and for this reason it is scarcely visible in our latitudes, excepting at particular seasons,' when that plane is nearly perpendicular to the horizon. When its inclination is great, it is either concealed altogether under the horizon, or at least rises so little above it, that its splendour is effaced by the atmosphere of the earth. The most favourable time for observing it is about the beginning of March, or towards the vernal equinox. The line of the equinoxes is then situated in the horizon, and the arc of the ecliptic SS' S" (fig. 42) is more elevated than the equator SEQ by an angle of 23⅜ degrees; so that the solar equator, which is slightly inclined to the ecliptic, approaches nearer to the perpendicular to the horizon, and the pyramid of the zodiacal light is consequently directed to a point nearer the zenith, than at any other season of the year. For example, at the summer solstice, S"T, a tangent to the ecliptic, is parallel to EQ the tangent to the equator, and the luminous pyramid is in a plane less elevated by 23° ^∙ than at the time of the vernal equinox.

Numerous opinions have been entertained respecting the nature and cause of this singular phenomenon. Cassini thought it might be occasioned by the confused light of an innumerable multitude of little planets circulating round the sun, in the same manner as the milky way owes its appearance to the light of agglomerated myriads of stars. Its resemblance to the tails of comets has been noticed by Cassini, Fatio Duillier, and others; and Euler endeavoured to prove that they are both owing to similar causes. Mairan, like Kepler, ascribed it to the atmosphere of the sun; and this hypothesis was generally adopted, till it was shown by Laplace to be untenable for the following reasons. The atmosphere of any planet, endowed with a motion of rotation, cannot extend to an indefinite distance: it can only reach to such a height that the centrifugal force is exactly balanced by the force of gravity. Beyond this height the atmosphere would be dissipated by the superior energy of the centrifugal force. Now the height above the sun at which the two forces are equal is that at which a planet, if placed there, woul’d revolve about the sun in the same time in which the sun performs a revolution on his axis. But the orbit of such a planet would be greatly inferior to the orbit of Mercury; for the time in which Mercury makes a revolution in his orbit is eighty-eight days, while the sun revolves about his axis in twenty-five; it is therefore certain that the atmosphere of the sun cannot extend to the orbit of Mercury. Now, the greatest elongation of Mercury doesnot exceed 28°, and the zodiacal light has been observed to extend to above 100°, reckoning from the sun to the apex of the luminous pyramid. Hence the phenomenon cannot proceed from the sun’s atmosphere. Laplace further remarks, that the ratio of the equatorial and polar axes of the solar atmospherical spheroid cannot exceed that of three to two; whence its form would not correspond with the lenticular appearance of the zodiacal light. But to whatever cause this luminous matter is to be attributed, it is certain that it is of extreme rarity, inasmuch as it does not intercept the light of the smallest stars which are seen through it without any diminution of splendour.

On the subject of the solar spots and zodiacal light the following works may be consulted :—Galileo, Istoria e Dimostrazioni intoτηo alle Macchic Solari,. Rome, 1613; Scheiner, Rosa Ursina, Bracciani, 1630; Hevelius, Selenographia, Gedani, 1647; Reutschius, De Maculis et Faculis Solaribus, Wittemb. 1661, 4to; Cassini, Nouv. Obserυ. des Taches du Soleil; Hooke, Tractatus de Maculis Solari' bus, et Lumine Zodiacali, in Oper. Posth. Lond. 1705; Weidler, De Coloribus Mac. Sol. in his Observ. Meteorol. 1728-9, Wittemb. 1729; Boscovich, De Mac. Sol. Rom. 1736, 4to; De Lisle, Mémoires pour servira Ĩ Histoire de l’ Astr. Petersb. 1738; Bernoulli, Lettres Astronomiques, 1771; Wilson, Phil. Trans. 1774, vol. lxiv.; Ibid. 1783; Wollaston, Phil. Trans. 1774; Lalande, Phil. Trans. YΠ⅛, Mém. Acad. 1776, and Astronomie, tom. iii. p. 277; Herschel, Phil. Trans. 1795 and 1801; Woodward on the Substance of the Sun, Washington, 1801; Biot, Traité de l’ Astronomie Physique, tom. ii.; Bohn, Disput. Astrom. de Fascia Zodiacali; Cassini, Mém. Acad.Par. tom. vii. p. 119, and viii. p. 193; Mairan, Traité Physique et Historique de l’ Aurore Doreale, 1731; Lalande, Astronomie, tom. i. p. 276; Laplace, Exposition du Système du Monde', Delambre, Hist, de l' Astronomie Moderne, tom. ii. p. 742.

CHAP. III.

OF THE MOON.

Next to the sun, the moon is to mankind the most important and interesting of all the celestial bodies. Her conspicuous appearance in the heavens, the variety of her phases, and the rapidity with which she changes her place among the fixed stars, have rendered her at all times an object of admiration to the vulgar; while her proximity to the earth, her physical effects on the ocean, the intricacy of the theory of her motions, and the vast importance of that theory to navigation and -geography, have equally [7:3:785]claimed the attention of the observer and mathematician; nor is there any other department of astronomy in which their researches have been crowned with more triumphant success, or been rewarded with more brilliant discoveries.

Sect. I.— Of the Phases, Parallax, and Magnitude of the Moon.

The different appearances or phases of the moon were probably the first celestial phenomena observed with any degree of attention. When the moon, after having been for some days invisible, is again seen on the eastern side of the sun, and at a distance of 20 or 30 degrees from him, she appears as a curved thread of silvery light; and her form is that of a crescent, the horns of which are turned towards the east. The breadth of the crescent increases continually in proportion as she separates herself from the sun, till, having obtained a distance of 90°, she appears under the form of a semicircle. At this point she is said to be in her first quarter. Continuing her motion to the eastward, the line which terminates the eastern side of her disk assumes the curvature of an elliptic arc, and her visible portion continues to increase till she has attained the distance of 180° from the sun, when she appears perfectly round. She is then full, and is said to be in opposition, rising as the sun sets, and consoling us by her pale light for the absence of the great luminary. Having passed this point, she begins to approach the sun; her western side now takes the form of the elliptic arc, and her luminous portion diminishes exactly in the same proportion as it increased through the first half of her orbit. About seven days after the full she again appears as a semicircle, the diameter of which is turned towards the west, and she is now at her third quarter, and at a distance of 90° from the sun. The semicircle after this changes into a crescent, and she continues to approach the sun, till, having advanced to within 20° or 30° from him, she again disappears, being lost in the splendour of his rays. These phases regularly succeed each other, and the time in which they run through all their changes is about 291 days. When the moon passes the meridian at the same time with the sun, she is said to be in Conjunction. The two points of her orbit in which she is situated when in opposition or conjunction are called the Syzygies ; those which are 90° distant from the sun are called the Quadratures; and the intermediate points between the syzygies and quadratures are called the Octants.

A slight attention to the lunar phases during a single revolution will be sufficient to prove that they are occasioned by the reflection of the sun’s light from the opaque spherical surface of the moon. This fact, which was recognised in the earliest ages, will be made obvious by, the help of a diagram. If the moon is an opaque body we can only see that portion of her enlightened side which is towards the earth. Therefore, when she arrives at that point of her orbit A (fig. 43) where she is in conjunction with the sun S, her dark half is towards the earth, and she disappears, as at a, there being no light on that half to render it visible. When she comes to her first octant, at B, or has gone an eighth part of her orbit from her conjunction, a quarter of her enlightened side is towards the earth, and she appears horned, as at b. When she has gone a quarter of her orbit from her conjunction, to C, she shows us one half of her enlightened side, as at c, and we say she is a quarter old. At D she is in her second octant, and by showing us more of her enlightened side she appears gibbous, as at d. At E her whole enlightened side is towards the earth, and therefore she appears round, as at e, when we say it is full moon. In her third octant, at F, part of her dark side being towards the earth, she again appears gibbous, and is on the decrease, as at/. At G we see just one half of her enlightened side, and she appears as a semicircle, as at g. At H we only see a quarter of her enlightened side, being in her fourth octant, when she appears horned, as at h. And at A, having completed her course from the sun to the sun again, she disappears, and we say it is new moon. Thus, in going from A to E, the moon seems continually to increase; and in going from E to A, to decrease in the same proportion, exhibiting like phases at equal distances from A and E.

The magnitude of the visible portion of the moon’s disk thus depends on the situation of the moon relatively to the sun and the earth, and is easily determined geometrically from her elongation or angular distance from the sun. Let ADBC (fig. 44) be the projection of the lunar orb on the plane which passes through the centres of the sun, moon, and earth; let S be the place of the sun, E that of the earth, Μ the centre of the projection, and AB, CD two diameters perpendicular to SM and EM respectively, and let EM meet the circumference in G. It is evident that AGDB represents the hemisphere illuminated by the sun, and CAGD that which is visible from the earth; the whole portion of the visible disk, therefore, is represented by AGD, or by AG and DG. Now, if we conceive the moon’s surface to be projected on the plane which passes through her centre perpendicular to the visual ray, the illuminated portion of the disk DG will appear as a semicircle, while the part GA will appear as a semi-ellipse, the minor axis of which is to its major as MF to MC, AF being perpendicular to MC. The eye at E will therefore see the semicircle DMG, together with the semi-ellipse GMA, and the visible part will be to the entire disk as DF: DC, that is, as 1 + sin. AME: 2, or, as 1 + cos. φ : 2; φ being equal to EMS the elongation of the earth from the sun as seen from the moon. The geocentric elongation of the moon from the sun, or the angle MES = ψ, may be substituted instead of φ, when we know the ratio of the distances ES = τ τ and MS = R; for sin. P = ļĮ sin ∙ ψ; whence cos. <ţ> = Į Ž 2 '.

1 —— sin. 2 ∙ψ, and the above proportion becomes

DF: DC:: 1 + Jl — sin. 2 ·ψ: 2.

When the moon is in opposition the angle φ = 0, whence 1 + cos. φ = 2, and the whole disk is visible. At the conjunction φ = 180°, and 1 + cos, Φ = 0; the moon is consequently invisible.

The triangle EMS likewise furnishes the means of finding the distance of the moon when that of the sun is known; or reciprocally, the distance of the sun from that of the moon: but for this purpose it is necessary to observe accurately the limit of illumination, which is not easily done. Let ADBE (fig. 45) be the illuminated part of the moon ACBE. The distance between the cusps A and B, measured by the micrometer, gives the major axis of the ellipse ADB, and the semicojugate axis DM is found by measuring in the same manner the distance ED, and subtracting EM, half the transverse AB. This

DM

gives = cos. EMS (fig. 44); this gives the angle EMS or Μ; and from SEM, which is given by observation, we have ESM or S = 180° — E — Μ; whence Μ and S are given angles; and from the property of the triangle we have EM = ES ™ or ES = EM The cal-

sin. Μ sin. b culation would be greatly facilitated if the observation [7:3:786] were made at the exact instant when the moon is dichotomized, or in her first or third quarter, for the elliptic arc then becomes a straight line and the angle EMS=90°; so that sin. Μ = l. In this case therefore we have EM = ES EM

sin. S = ES cos. E, and ES = 7 ς, which gives the cos. E o ratio of the distances of the sun and moon. In this manner Aristarchus of Samos attempted to compare the distance of the two luminaries; and although his result was extremely erroneous, inasmuch as he found the distance of the sun to be only between eighteen and twenty times greater than that of the moon, whereas it is actually 380 times greater, yet the idea was an ingenious one, and the method perfectly accurate in theory. The error arose from the inaccuracy of the ancient instruments, and the impossibility of observing to within a few minutes the exact time of the dichotomy. Besides, as the angle at the sun is extremely small, a very slight error either with regard to the time or the angle has a great influence on the result.

? The moon’s absolute distance from the earth is obtained by means of her parallax, which, on account of her proximity, is very considerable, and can therefore be determined by the methods which were explained in chap. i. sect. 2. On comparing her parallaxes observed at different times, they are found to differ considerably in value. Thus, according to Lalande, the greatest horizontal parallax at Paris is 6P 29", the smallest 53' 51"; and it assumes every possible value between these two extremes. The horizontal parallax being the angle which the earth’s radius subtends at the moon, and consequently equal to the radius of the earth divided by the moon’s distance, the perigean and apogean distances corresponding to the extreme parallaxes are respectively 55∙916 and 63∙842 semidiameters of the earth. The ratio of these two numbers, or of the extreme values of the parallax, is 1∙1417, and denotes the greatest variations of the moon’s distance from the earth. The ratio of the corresponding quantities in the case of the sun is only 1∙034; hence the moon’s distance is subject to much greater variations than that of the sun.

These differences in the value of the parallax arise from the variations of the moon’s distance from the earth; but it is also observed to differ sensibly at different points of the earth’s surface, even at the same instant of time. If the earth were spherical, the horizontal parallax of the moon, supposing her distance to be invariable, would be the same at whatever part of the earth it might be observed. The case is, however, different; for, on account of the spheroidal figure of the earth, the section made by every vertical plane gives a different ellipse, no one of which even passes through the centre of the earth, excepting indeed the cases in which the place of the observer is on the equator or under the pole; for in all other cases the perpendicular to the surface of the earth does not pass through its centre. Hence it is necessary, in speaking of the horizontal parallax, to specify the place of the observation.

Since the parallax of the moon is subject to incessant variation, it is necessary to assume a certain mean value of it, about which the true and apparent values may be conceived to oscillate. This is denominated the constant parallax. It is evident that it cannot be found by taking the arithmetical mean of the extreme values; for by reason of the disturbing force of the sun, the eccentricity of the lunar orbit, and consequently the perigean and apogean distances, are constantly varying; and as the quantity by which one of these distances is increased is not equal to that by which the other is diminished, it follows that the mean of the perigean and apogean parallaxes will not be a constant quantity, but different in successive revolutions of the moon. If we abstract from all the inequalities of the lunar orbit, and suppose the moon to be at her mean distance and mean place, the constant parallax will be the angle under which a given semidiameter of the earth is seen by a spectator at the moon in such circumstances. According to Lalande, the following are the values of the constant parallax :—
[table]

The mean equatorial parallax being 57' 5", its double is l 0 54′ 10", which expresses the angle subtended by the diameter of the earth at the distance of the moon. The angle subtended by the moon at the same distance is 31' 26"; whence the diameter of the moon is to that of the earth as 31' 26" is to l o 54' 10"; that is, as ∙524 to 1∙899, or as 3 to 11 nearly. According to the tables of Burg, the accurate expression of the above ratio is 1: 0∙27293; hence the true diameter of the moon is 0∙27293 diameters of the terrestrial equator. The surface of the moon is consequently (0∙27293) 2 = 0∙0744908 = γy∙⅛τy of that of the earth, and its volume (0∙27293) 3 = 0∙0203308 = ψg∙ι⅛3-, or, in round numbers, ⅛th of the volume of the earth.

Sect. II.— Of the Elements of the Lunar Orbit.

1. Nodes and Inclination of the Lunar Orbit. — A few simple observations of the right ascensions and declinations of the moon suffice to show that her path is confined nearly to a plane, having only a small inclination to the plane of the ecliptic. If her path were rigorously confined to this plane, a single revolution would be sufficient to determine its inclination; but as numerous disturbing causes exist, which produce inequalities in all the elements of the lunar motion, it is only by taking the mean value of a great number of observations that these elements can be astronomically determined. In fixing the position of the plane of the orbit, the first object is to determine the straight line formed by its intersection with the plane of the ecliptic; that is to say, to determine the line of the Nodes. This would be at once accomplished if it were possible to observe the instant at which the moon’s latitude is nothing; for at this instant her centre is in the plane of the ecliptic, and consequently her longitude is the same as the longitude of the node. But as the accuracy of modern practice requires that all important observations be made in the plane of the meridian, and as it could only happen by a very rare coincidence that the latitude deduced from the meridional right ascensions and declinations would be exactly zero, it is necessary to have recourse to a process of calculation to find the precise place of the node.

Let LNL' (fig. 46) be a portion of the ecliptic, MNM‘I of the lunar orbit, N the place of the node, and ML = l, M‘L' = l' be two latitudes on opposite sides of the ecliptic. The two right-angled triangles give, according to Napier’s Rules, tan. l tan. ĩ tan ∙ in ~ sin. NM ~ sin. NM'^ ; therefore, sin. NM _ tan. l . sin. NM' - tan. I ’ whence, sin. NM — sin. NM' _ tan. l — tan. l’ sin. NM + sin. NM' - tan. I + tan. I * [7:3:787]I that is, by the trigonometrical formulae, • tan. ∣(NM — NM')_ sin. (l — l') tan. ⅜(NM + NM') - sin. (Z + Z') ’.

or tan. f (NM — NM') = tan. ⅛MM'.

Having therefore obtained an expression for the difference of NM and NM', and knowing also their sum MM', which is the difference of the longitudes of the moon at the time of the two observations, it is easy to deduce NM or NM'; consequently the longitude of the node is determined.

The Ascending Node of the lunar orbit is that point of the ecliptic through which the moon passes when she rises above the ecliptic towards the north pole: it is distinguished by the character Q. The Descending Node, Ĩ3, is the opposite point of the ecliptic, through which she passes when she descends below that plane towards the south, pole. The nodes of the lunar orbit were anciently called the head and tail of the Dragon.

The position of the nodes is not fixed in the heavens. They move in a retrograde direction, or contrary to the order of the signs; and their motion is so rapid that its effects become very apparent after one or two revolutions. Regulus, a star of the first magnitude, is sometimes eclipsed by the moon; and as the latitude of this star is only about 27' or 28', it is certain that when that phenomenon occurs the moon is very near her node. Suppose it to be the ascending node: the month following it will be observed that the moon passes to the north of Regulus; and every succeeding month she will pass farther to the north of the same star, till at the end of four or five years, when her meridional altitude will be about 5° greater than that of Regulus. Having attained this elevation, her latitude begins to diminish; and, after a period of about nine years, Regulus is again eclipsed by the moon in her descent to the southern side of the ecliptic. The moon after this passes to the south of the star for the following nine years; and at the end of 18į years the nodes have returned to their first position, after accomplishing a complete revolution. The same result is found from the observation of eclipses, the magnitude of which depends on the moon’s latitude; and therefore, if the nodes remained fixed, the eclipses would always be of the same magnitude in the same quarter of the heavens. This, however, is not the case; and it is only after about 18⅜ years that they begin to return in the same order.

The mean retrograde motion of the nodes is found by the comparison of observations made at distant epochs to amount to 19° 19' 42" , 316 in a mean solar year, and the time in which they make a complete revolution is consequently 6798 days, 4 hours, 17 min. 43∙18 sec. The longitude of the ascending node on the 1st of January 1801 was 13 9 53' 17"∙7; hence its position at any other epoch is easily deduced, since we know the rate of its mean motion.

The inclination of the lunar orbit may be determined either by observing the moon’s greatest latitudes, or it may be computed from the formula tan. N = ∙γj > when her latitude and distance from the node are known. The angle N, or the inclination, is observed to vary between 5° and 5° 17' 35". The mean inclination may therefore be taken at 5° 8' 48".

The retrograde motion of the moon’s nodes, as well as all the other inequalities of the lunar motion, is occasioned by the attracting power of the sun, which varies in intensity with the sun’s longitude, or distance from his perigee. The principal inequality to which the inclination of the lunar orbit is subject, is proportional to the cosine of double the sun’s distance from the moon’s ascending 1 node. In fact, the sun and earth being always in the plane of the ecliptic, when the moon is also in that plane the' action of the sun will have no tendency to increase or diminish her latitude; it will only affect her radius vector, or distance from the earth. But if the moon is not in the plane of the ecliptic, the sun’s attracting force will not only affect her distance from the earth, but also tend to bring her nearer to the ecliptic; and this action will be greater in proportion to her deviation from that plane. When the nodes are in the quarters, and the limits of latitude in the syzygies, the inclination is 5°, as was found by Ptolemy; but when the nodes are in the syzygies, and the limits in the quarters, the inclination is 5° 17' 35". It has therefore a mean value of 5° 8' 48", when the nodes and limits coincide with the octants. Denoting therefore the greatest deviation from the mean values by x, this deviation will pass through all its values between + x and — x while the sun’s distance from the ascending node varies from 0° to 90°. It may therefore be represented by a function which varies as the cosine of twice that distance, for the cosine of an angle runs through all its changes from + 1 to — 1 between 0 and 180°. Hence the equation which expresses the principal inequality of the moon’s latitude is

(8' 48") cos. 2 Q’s distance from ]) ’s 2∙

The co-efficient 8' 48" is called, in astronomy, the Co-efficient of the Argument, the argument itself being the function by which the inequality is represented.

This periodic inequality of the moon’s latitude was discovered by Tycho Brahe, from a comparison of the greatest latitudes at different epochs with the correponding positions of the moon with reference to her nodes. He observed that these latitudes oscillate about the mean value of 5° 8' 48"; and as the greatest latitudes give immediately the inclination of the moon’s orbit to the plane of the ecliptic, it necessarily followed that the inclination is subject to similar variations. The motion of the node of the lunar orbit is also subject to an inequality, depending on the same angle as the preceding, but proportional to its sine. Tycho explained these two inequalities by a very simple hypothesis, similar to that by which we have explained the oscillations of the terrestrial equator, namely, a slight nutation of the earth’s axis. In the present case it is only necessary to place the mean pole of the lunar orbit at the distance of 5° 8' 48" from the pole of the ecliptic, and to suppose the true pole to describe a small ellipse about the mean pole in the same time that the sun occupies in making a semi-revolution with regard to the nodes of the moon’s orbit, that is, in 173∙309 days.

There are various other inequalities which affect the latitude of the moon and the inclination of her orbit, to correct which, no fewer than twelve equations are given in the recent and most accurate tables. The greater part of them are, however, only known from theory, and are so small that even their accumulated effects are scarcely perceptible to observation.

The inclination of the lunar orbit to the plane of the terrestrial equator occasions considerable differences in the intervals between the moon’s rising or setting on successive days, and gives rise to the phenomenon of the Harvest Moon. As the daily motion of the moon is about 13 degrees from west to east, it follows that if she moved in a plane parallel to the equator, she would rise 50 minutes later every successive evening, because her orbit would then make the same angle with the horizon at all seasons of the year, and the intervals between her consecutive risings would be constant. For the sake of explanation, we may here suppose the moon to move in the plane of [7:3:788] 1 the ecliptic. Now, the time in which a given arc of the ecliptic rises above the horizon depends on its inclination to the horizon. In our latitudes the inclination of the ecliptic at different points to the horizon varies so much, that at the first point of Aries an arc of 13° becomes visible in the short space of 17 minutes, while at the 23d of Leo the same arc will only rise above the horizon in one hour and 17 minutes. Hence, when the moon is near the first point of Aries, the difference of the times of her rising on two successive evenings will be only about 17 minutes; and as this happens in the course of every revolution, she will rise for two or three nights every month at nearly the same hour. But the rising of the moon is a phenomenon which attracts no attention, excepting about the time when she is full, that is, when she rises at sunset. In this case she is in opposition to the sun, and consequently, if she is in Aries, the sun must be in Libra, which happens during the autumnal months. At this season of the year, therefore, the moon, when near the full, rises for some evenings at nearly the same hour. This circumstance affords important advantages to the husbandman, on which account the phenomenon attracts particular attention.

It is obvious, that as this phenomenon is occasioned by the oblique position of the lunar orbit with regard to the equator, the effect will be greater than what has just been described if the plane of that orbit makes a greater angle with the equator than the plane of the ecliptic, does. But we have seen that the plane of the moon’s orbit is inclined to the ecliptic in an angle exceeding 5°; consequently, when her ascending node is in Aries, the angle which her orbit makes with the horizon will be 5° less than that which the ecliptic makes with the horizon; and the difference of time between her risings on two successive evenings will be less than 17 minutes, as would have been the case had her orbit coincided with the ecliptic. On the contrary, when the descending node comes to Aries, the angle which her orbit makes with the horizon will be greater by 5°, and consequently the difference of the times of her successive risings will be greater than if she moved in the plane of the ecliptic. If when the full moon is in Pisces or Aries the ascending node of her orbit is also in one of those signs, the difference of the times of her rising will not exceed one hour and forty minutes during a whole week; but when her nodes are differently situated, the difference in the time of her rising in the same signs may amount to 3∣ hours in the space of a week. In the former case the harvest-moons arc the most beneficial, in the latter the least beneficial, to the husbandman. All the variations in the intervals between the consecutive risings or settings take place within the period in which the line of the nodes makes a complete revolution.

2. Dimensions, Eccentricity, and Apsides of the Lunar Orbit ; and the Inequalities of the Moons motion. — Since the moon moves in a plane orbit, the projection of her path on the surface of the celestial sphere will be a great circle. But the variations observable in the magnitude of her apparent diameter, and in the velocity of her motion, prove that if her orbit is a circle, the earth is at least not placed at its centre. The numerous perturbing causes which affect her motion render the exact determination of her orbit a matter of great difficulty; but observation shows that it deviates very little from an ellipse, of which the centre of the earth occupies one of the foci. It has also been found that the spaces passed over by the radius vector of the moon are very nearly proportional to the times of description; which is the distinctive character of the elliptic motion.

In order to obtain a correct idea of the figure and posi-Ί tion of the orbit, it is necessary to know its major axis, its-^ eccentricity, and the position of its apsides. The major v axis is equal to the sum of the greatest and least distances, and these have already been stated to be respectively 63∙842 and 55∙916 semidiameters of the earth; whence, supposing the earth’s radius to be 4000 miles, the major axis of the lunar orbit will amount in round numbers to 480,000 miles. It may be here remarked that the moon’s distance is an element which may be practically determined with great exactness. A variation of 1" in the parallax would occasion an error of only about 67 miles in the determination of the distance; therefore, since the parallax is certainly known to within 4", the greatest error in the distance deduced from it cannot exceed 280 miles out of about 240,000 miles.

An approximation to the eccentricity of the orbit of the moon is easily obtained from the variations of her apparent diameter. The moon’s apparent diameter is observed to vary between 29' 30" and 33' 30" very nearly. Now, let D denote the mean, D' the apogean, D" the perigean diameter, and e the eccentricity. We shall then.have

D' = τ ^-= 29' 30"

1+e

D"= r ^- = 33' 30", 1— e

D' l —e_29' 30"

D" - 1 + e - 33' 30" ’ whence we deduce

33'30" — 29' 30" 4'

e ~ 33' 30" + 29' 30" - 63' - °’ 0635 ·

This eccentricity corresponds to an equation of the centre amounting to 7° 16', and is much more considerable than that of the solar orbit, which, as we have before seen, amounts only to 0° ∙0168. The result just given is, however, only to be considered as approximative, and is, in fact, considerably too large. In the best and most recent tables the value assigned to the eccentricity of the lunar orbit is 0∙0548442, and the corresponding equation of the centre 6° 17' 12"∙7. (See Astronomical Tables and Formulae, by Francis Baily, Esq. London, 1827.)

The place of the apsides is likewise found from observations of the moon’s apparent diameter, because at these points of the orbit the apparent magnitude has its maximum and minimum values. But as the apparent magnitude varies very slowly at the apsides, it is preferable, as Lalande remarks, to choose the points of mean distance where its variations arc most rapid. Having selected two observations which give the same apparent diameter at two opposite points of the orbit, we shall have two points evidently at equal distances from the perigee, the longitude of which will therefore be given at the middle time between the two observations. In comparing the positions of the apsides thus determined at different epochs, it is found that they have a rapid progressive motion, that is, according to the order of the signs. In Mr Bally’s tables the motion of the perigee is stated to be 4069° ∙046278, or 11 revolutions + 109° 2' 46"∙6, in 36525 mean solar days. Its mean motion in a mean solar day is consequently 6' 41"; and in 365 mean solar days, 40° 39' 46"∙36. Hence the time in which it completes a sidereal revolution is 3232∙575343 mean solar days. The period of a tropical revolution of the apsides is 3231∙4751 mean solar days = 3231 days, 11 hours, 24 minutes, 0*8 seconds, or nearly 9 years. This mean motion of the lunar apsides is, however, subject to periodic inequalities of considerable magnitude, and therefore can only be determined accurately by means of observations separated from [7:3:789]each other by a long interval of time. In the tables just quoted, the epoch is the commencement of the present century, when the longitude of the perigee was in 266° Iff ,7"∙5. From this and the rate of the mean motion, the longitude of the perigee at any other epoch may be easily computed.

If the moon’s place in her orbit were not subject to the influence of any disturbing force, her true longitude, or distance from the apsis, would be found at any instant from the theory of the elliptic motion, supposing her mean motion, or the time of a return to the same perigee or apogee, to be accurately known. Thus, supposing E to denote the maximum elliptic equation, which, as we have already stated, amounts to 6° 7' 12"∙7, we should have the true longitude = mean longitude + E sin. A, A being the moon’s mean anomaly, and the longitudes counted from the perigee. This first equation or correction of the mean motion results solely from the circumstance that the moon moves in an elliptic orbit, according to the laws of Kepler, in the same manner as the sun and all the planets. Its argument is the mean anomaly A, and its period the anomalistic month. But the moon’s longitudes, as determined in this manner, are far from agreeing with observation; and several other corrections must be applied, the accurate determination of which forms the principal object of the lunar theory.

After the equation of the centre, the most considerable of the inequalities of the moon’s longitude is that to which Boulliaud gave the name of the Evection. It was noticed by Hipparchus; but it was Ptolemy who discovered its Iaw, and gave a construction which represents its general effects with great accuracy. These effects are to diminish the equation of the centre when the line of the apsides lies in syzygy, and to augment it when the same line lies in quadratures. Thus, supposing the apsides to lie in syzygy, and that it is sought to compute the moon’s true longitude about seven days after she has left the perigee, by adding the equation of the centre to the mean anomaly, the resulting longitude will be found to be above 80 , less than that which is given by observation. But if the line of the apsides lies in quadratures, the place of the moon at about the same distance, that is, 90° from the perigee, computed in the same manner, will be found to be before the observed place by above 80'; that is, the computed will be greater than the observed longitude, by more than 80 minutes. After a long series of observations, astronomers have found that the inequality in question is represented with great exactness, by supposing it proportional to the sine of twice the mean angular distance of the moon from the sun, minus the mean anomaly of the moon. In Baily’s tables, its maximum value is stated to be 1° 20' 29"∙9; whence, representing the angular distance of the sun and moon by î — Θ, and the mean anomaly as before by A, the correction due to the evection will be

(1° 20′ 29"∙9) sin. [2(D — Θ)-A].

A third inequality of the lunar motion, called the Variation, was discovered by Tycho Brahe, who found that the moon’s place, calculated from her mean motion, the equation of the centre, and the evection, does not always agree with the true place, and that the variations are greatest in the octants, or when the line of the apsides makes an angle of 45° with that of the syzygies and quadratures. Having observed the moon at different points of her orbit, he found that this correction has no dependence on the position of the apsides, but only on the moon’s elongation from the sun. Its maximum value is additive in the octants which come immediately after the syzygies, where the elongation is 45°, or I80° + 45°, and subtractive in the octants which precede the syzygies, where the ' elongation is 360° — 45°, and 180° — 45°. It vanishes al- - together in the syzygies and quadratures where the elonga-’ tion is 0°, 180°, 90°, or 270°; and on this account it was not perceived by the ancient astronomers, who only observed the moon in those positions. It will be readily perceived, from the limits between which it varies, that it is proportional to the sine of twice the angular distance between the sun and moon. Its maximum value, or the co-efficient of its argument, is 35' 41"∙9; hence the correction necessary on account of it is represented by the formula

(35' 41"∙9) sin. 2 ( D — ©).

When this equation is added to the two preceding ones, the differences between the computed and observed places of the moon are brought within narrower limits; but they do not yet disappear altogether except in the months of June and December, and about the time of the equinoxes they are found to amount to eleven or twelve minutes. This inequality was observed by Tycho and Kepler, though neither of them determined its magnitude. They supposed it to be an equation of time peculiar to the moon; and under this form Horrox inserted it in his tables, assigning its maximum value at 11' 16", which differs only by four seconds from that which is obtained from the most accurate modern observations. On account that it is regulated by the seasons, and depends not on the lunar orbit, but the anomaly of the sun, Kepler gave it the name of the Annual Equatioìi. By reason of this inequality the motion of the moon is slower than the mean motion during the winter months, when the motion of the sun is most rapid; and, on the contrary, is most rapid in summer, when the sun’s motion is slowest. Hence its argument is the same as that of the equation ‘of the centre of the sun, but with a contrary sign. It is therefore proportional to the sine of the sun’s mean anomaly; its period is the anomalistic year, and its maximum value is found by observation to be 11' 11"∙9; hence it is expressed by the formula

(11' 11"∙9) sin. O’s mean anomaly.

If we now collect these equations we shall obtain the following corrected expression of the moon’s true place: J) ’s true longitude = D’s mean longitude,

+ (6° 17' 12"∙7) sin. A

+ (1° 20' 29"∙9) sin.[2( 5 — ø)—A] —( 35' 41"∙9) sin. 2 ( » — ø)

—( 11' 11"∙9) sin. O’s mean anomaly.

For the physical cause of the three last-mentioned inequalities, viz. the evection, the variation, and the annual equation, see Physical Astronomy, sect. ii. par. 37, 38, and 39. They are the principal but not the only periodic inequalities which affect the longitude of the moon; for the profound investigations of modern science have detected many others, which, by reason of their small values, and the manner in which they are blended with one another, would have eluded all attempts to discover their period or their law by observation alone. Observation can make known only the joint effect of the independent equations, and thereby indicate their existence. In order to separate them and determine their respective values, it is indispensable to have recourse to the physical theory, from which alone the various circumstances of the lunar motion can be properly determined. Newton was the first who attempted to compute a priori the inequalities of the moon’s motion, and thereby led the way in a research which was prosecuted with infinite ingenuity and profound analytical skill by D’Alembert, Clairaut, Euler, and other illustrious mathematicians of the last century, and which was only brought to a successful termination by the labours of Laplace. The formulae given in the seventh [7:3:790]book of the Mécanique Celeste, for a great number of new inequalities derived from the theory of attraction, form the basis of the lunar tables of Burg and Burckhardt It is to these theoretical researches that we are indebted for the precision with which the moon’s motion is actually known, and the great advantages which thence result to geography and navigation. The recent tables contain no fewer than twenty-eight equations of longitude, all which may be regarded as corrections of the mean values of the four which we have explained, and which are themselves corrections of the mean motion. It is thus that astronomy approaches nearer and nearer, by every successive step, to the last term of a series which would represent the orbits and motions of the celestial bodies with absolute accuracy.

The lunar inequalities which we have as yet considered are all of a periodic nature, are compensated in the course of a comparatively small number of years, and have passed through many complete revolutions since the commencement of the history of astronomical observations. But there are, as in the case of the sun, others of a different kind, the periods of which are so long that, with reference to the duration of human life, they may be considered as permanently affecting the elements of the lunar orbit. These are the Secular Inequalities, the most remarkable of which is the acceleration of the moon’s mean motion.

On comparing the lunar observations made within the last two centuries with one another, there results a mean secular motion greater than that which is given by comparing them with those made by Ebn-Jounis, near Cairo, towards the end of the 10th century, and greater still than that which is given by comparing them with observations of eclipses made at Babylon in the years 719, 720, and 721 before our era, and preserved by Ptolemy in the Almagest. These comparisons, which have been made by different astronomers with the utmost care, prove incontestably the acceleration of the motion of the moon from the Chaldeans to the Arabians, and from the Arabians to the present times.

The acceleration of the moon’s mean motion was first remarked by Dr Halley, and mentioned in his Notes on the observations of Albategnius, inserted in the Philosophical Transactions for 1693. It was fully confirmed by Dunthorne, who was led by the discussion of a great number of ancient, observations of eclipses, to suppose that it proceeded uniformly at the rate of 10" in a hundred years; a supposition which consequently gave him a correction for the mean longitude of the moon proportional to the square of the time, or of 10" multiplied by the square of the number of years elapsed between 1700 and the epoch of the calculation. This was the first attempt to estimate the value of the secular equation, which had hitherto been confounded with the mean secular motion. By a similar discussion of ancient observations, Mayer was likewise led to a secular equation proportional to the square of the time, which, in his first Lunar Tables, published in 1753, he valued at 7" for the first century, counting from the year 1700, but which he advanced to 9" in his last tables, published in 1770. Lalande found it to amount to 9"∙886, and therefore agreed with Dunthorne in estimating the acceleration at 10" for the first century after 1700. Delambre subsequently undertook to determine accurately the actual motion of the moon in a century, from a comparison of the best modern observations. He found it to amount to 307° 53' 9" (rejecting the circumferences), while the most ancient observations agree in making it less by 3 or 4 minutes. The equation, which, in consequence of these comparisons, was empirically introduced into the tables, has been confirmed by the theory of gravitation; and the discovery of the phy- r sical cause, and of the law of the acceleration, due to La- * place, forms one of the most brilliant triumphs of modern science. In 1786 Laplace demonstrated that the acceleration is one of the effects of the attraction of the sun, and connected with the variations of the eccentricity of the earth’s orbit in such a manner that the moon will continue to be accelerated while the eccentricity diminishes, but that it will disappear when the eccentricity has reached its maximum value; and when that element begins to increase, the mean motion of the moon will be retarded.

In order to take into account the effects of the acceleration in the determination of the mean longitude, Μ. Damoiseau has given the following formula for the secular variation; 10"∙7232 n 2 +0"∙019361 n 3 , where n is the number of centuries from 1801; so that if L is the mean longitude of the moon on the 1st of January 1801, and m is the secular motion at that epoch, the mean longitude will be L + mn + 10"∙7232 n 2 + 0"∙019361 n? after n centuries. By means of this equation the tables satisfy the most ancient observations, and may be extended to at least a thousand years from the present epoch. It is only necessary to observe, that, in applying the formula, n must be taken negatively, if the epoch is anterior to 1801.

The same cause which gives rise to the acceleration of! the mean motion, namely, the diminution of the eccen- ’ tricity of the earth’s orbit, also occasions secular inequal- ’ ities in motion of the perigee and nodes of the orbit of i the moon. These two inequalities are, however, affected with opposite signs to that of the former; that is, while the mean motion of the moon is accelerated, the motion of her perigee and that of her nodes are retarded. They were deduced by Laplace from theory, and the equations by which they are expressed are connected with one another by a very simple ratio. If we take A to represent the secular acceleration of the mean motion, the secular variation of the perigee found by Laplace is — 4 ’ 00052 A, and that of the nodes — 0∙735452 A. From this Laplace concluded that the three motions of the moon, with respect to the sun, to her perigee, and to her nodes, are accelerated, and that their secular equations are in the ratio of the above numbers. By pushing the approximations to a great length, MM. Plana and Carlini, and Μ. Damoiseau, in Memoirs which obtained the prize of the Academy of Sciences for 1820, have found different numbers; those of Damoiseau are 1, 4∙702, áïid 0∙612, These important results of theory are all confirmed by observation.

The three secular inequalities which have been pointed out will obviously occasion others; for all quantities depending on the mean motion, the motion of the perigee, or of the nodes, must be in some degree modified by them. Thus the mean anomaly, which is the difference of the mean longitude of the moon and the mean longitude of the perigee, is subject to a secular equation equal to the difference of the secular equations affecting the longitudes of the moon and the perigee. The radius vector, the eccentricity and inclination of the orbit, are affected by the secular inequality of the mean motion, which, although too minute to have been hitherto appreciable, will acquire sensible values in the course of ages. The major axis of the ellipse is the only element exempted from inequalities of this sort. It is not probable, however, that the utmost efforts of science will ever make us acquainted with all the irregularities to which the moon’s motion is subject. They can only be developed by the complete integration of the differential equations of motion; an integration which is laboriously performed term by term, and which, when attempted to be carried beyond a certain point, transcends [7:3:791]the limits of human patience and industry. What is most essential is to select, among the multitude of terms, such as may possibly acquire considerable co-efficients by integration.

The following table exhibits in one view the different elements of the lunar orbit. The epoch is the commencement of the present century.
[table]

Sect. III.— Of the different Species of Lunar Months.

In treating of the sun, we took notice of three different species of revolutions or years, namely, the mean solar year, or the interval of time which the sun employs in performing a complete revolution with regard to the equinoxes; the sidereal year, or the time in which he returns to the same fixed star; and the anomalistic year, or the time in which he returns to the same point of his ellipse. In like manner, if we understand by the term month the time which the moon employs to make an entire revolution relatively to any given point, movable or fixed, we shall have as many different species of months as there are different motions with which that of the moon can be compared. For example, if we estimate her revolution relatively to the sun, the month will be the time which elapses between two consecutive conjunctions or oppositions. This is called the synodic month, lunar month, or lunation. If we consider her revolution as completed when she has gone through 360° of longitude counted from the movable equinox, we shall have the tropical or periodic month. The interval between two successive conjunctions with the same fixed star is the sidereal month. A revolution with regard to the apsides of her orbit, that is to say, the time in which she returns to her perigee or apogee, gives the anomalistic month: and, finally, the revolution with regard to the nodes is the nodical or draconitic month. Of these different periods the most important to mankind is the synodic month. It is also that which, by reason of the striking manner in which it is marked out by the phases of the moon, would first offer itself to the attention of the observer; and when its period is accurately determined, the other months may be deduced from it without difficulty, when the relative motions of the sun and moon are known with sufficient precision. The eclipses furnish a simple means of determining the synodic revolution with a great degree of accuracy. A few rude observations suffice to show that the period of a lunation is very nearly 29∣ days; and with this knowledge we are in a condition to compare two distant eclipses without running any risk of mistaking the number of revolutions that have taken place in the interval. The most ancient observation recorded by Ptolemy is an eclipse of the moon, which happened 720 years b.. c., on the 19th of March, at 6 hours 48 minutes mean time at Paris, according to Lalande. In order to make use of this observation directly for the purpose of determining the synodic revolution, it is necessary to compare it with another of the same kind in which the moon occupied the same point of her ellipse, or the same position relatively to her apsides; for as it is the true place of the moon which is observed, and the mean motion which we are in’ quest of, the equation of the centre ought to be the same in both observations. Now, a similar observation is furnished by an eclipse which happened in 1717, on the 9th of September, at 6 hours 2 minutes, the moon’s anomaly being very nearly the same as in the Babylonian observation. In the interval between the two observations the moon had therefore completed a whole number of revolutions, with regard to her mean as well as her true motion. The interval between the eclipses is 2437 years and 174 days minus 46 minutes, which expressed in days is 890287∙9680555... days. In this interval it is found, from the approximate value of 29∣ days, that 30148 synodic revolutions had happened; hence the mean length of . κ ,. , . 890287∙968055

the synodic month is = —— = 29 days 12

hours 44 min. 2 ’ 8497 sec.

To deduce the other revolutions, let N = the number of days in a synodic month, n the number of synodic revolutions in the interval between the two observations, m the mean motion of the sun, and T the time of a tropical revolution. We shall then have the proportion T: N:: 360°: n∙360° + m; whence

τ = Ν>36σ ° - n -

⅞∙360° + τ⅞ ~~ m m " h 360 o + n∙360°

N ∫ m f m ∖ 2 ì (1 ~τr360° + ∖τr360°∕ — &c -p

Let S = the time of a sidereal revolution. The value of S is immediately found from T; for let p = the precession of the equinoxes in the time T, we shall have S: T:: 360°: 360° — p ; whence .

c _ T∙360° _ T

b - 360° —p ~ 1 _ p 360° = τ { 1 + 3⅛^+(3⅛ 7 ) + &c ·}·

In like manner, if we represent by u the motion of the apside, and by v that of the node, in the time T, the times of the anomalistic and nodical revolutions will be respectively obtained by the substitution of u and — ν (the motion of the node being retrograde) in the last formula in place of p. Instead, however, of proceeding in this manner, it is better to assume as the basis of the calculation the mean motion of the moon in longitude, which is accurately known by the observations of 2000 years, and thence to deduce the time of the mean tropical revolution.

The mean motion of the moon in 100 Julian years, or 36525 days, is found to be 1336 circumferences + 307° 52' 43"∙5; or, by reducing the degrees, &c. to the fraction of a circumference, 1336∙85521875 circumferences. The

. 1 „ 35625 , t .

periodic month is therefore 133g.g5221875 da ^ s, ln order to reduce the denominator of this fraction to a smaller number of digits, we may divide its terms by 3, multiply them by 32, then divide them by 900, and there will re- AQ9∙ft0ft su It ————-+, the denominator of which contains only 7 15∙84421

digits in place of 12. On performing the division, we shall find the periodic month or tropical revolution = 27∙321582388 days = 27 days 7 hours 43 min. 4*7183 sec.
[7:3:792]

The mean motion of the moon in 36525 days is 1336° 307° 52'43"∙5

that of the sun 100 0 45 45

hence the relative motion is 1236° 307° 6'58"∙5

= 445267∙11625 degrees. The time of a synodic revolu-tion is therefore 445 9 67 . ll6 25 da J' s ’ or ’ multiplying ∏u- ; 1 . l onn 10519200000

merator and denominator by 800, = " ğðğ ğ ļġģgg 5 whence the synodie month is 29∙5305885391 days = 29 days 12 hours 44 min. 2∙849778 see.

In order to obtain the sidereal revolution, we subtract the secular motion of the equinoctial points = 5010" = 1° 23' 30" from the moon’s tropical revolution; the remainder, which gives the sidereal motion of the moon in 36525 days, is 1336° 306° 29' 13"∙5 = 481266∙4870833 degrees. Hence the time of a sidereal revolution is ⅜81¾66-487i⅞ 33 da ^ 0n mul ⅛ 1 y⅛ the ° f *⅛ _ ∙ 1 o∣AA∙ t ∙ 1 ,1. 31557600000 ... .

fraction by 2400, it is reduced to . 1 rrn⅛ αι - r . t √ > which gives the sidereal month = 27 days 7 hours 43 min. 11∙544375 see.

In like manner, by subtracting the motion of the perigee in 100 years from the secular motion of the moon, we shall find the anomalistic revolution to be 27 days 13 hours 18 min. 34∙9488 sec.; and by adding the retrograde secular motion of the node to the secular motion of the moon, the revolution in respect of the nodes is found to be performed in 27 days 5 hours 5 min. 35∙60769 see.

The following table exhibits the different kinds of lunar periods and motions :—
[table]

According to Ptolemy, the synodic month is 29 days 12 hours 44 min. 3⅜ see., which differs from the above Ί only by half a second. The same great astronomer madethe tropical month to consist of 27 days 7 hours 43 min. ’ 7į see. which exceeds the true time by 2∣ seconds; an error into which he was led by assigning too great a value to the mean motion of the sun.

The ancient astronomers paid great attention to these different revolutions, for the purpose of regulating their lunisolar calendar, and of avoiding the calculation of eclipses, which is attended with difficulties that to them must have proved almost insuperable. Their object was therefore to assign composite periods, after the revolution of which the eclipses would again return in the same order. Now, it is easy to see that a period which will bring back eclipses of the same magnitude and duration, on the same day of the year and at the same longitude, must be an exact multiple of the different lunar months. The return of the moon to the same distance from her node will give an eelipse , of the same magnitude; if she returns at the same time to the same point of her orbit, the eelipse will also be of the same duration; and if, in addition to these circumstances, she has also returned to the same longitude, the eelipse will take place on the same day of the year. But the numbers in the above table being incommensurable, it is impossible to find any period, however long, that will embrace all these conditions; the ancients therefore formed periods of different lengths, according as they aimed at satisfying the different conditions with a greater or less degree of precision. For an account of some of the most remarkable of the ancient lunisolar periods, see Calendar.

On account of the acceleration of the mean motion of the moon, the ratios of the different species of months are constantly undergoing alterations, and therefore the different eyeles, supposing them exact at the time of their formation, cannot continue so for an indefinite length of time. This circumstance is, however, little to be regretted; for, in the present state of astronomical science, they are not of any great use, inasmuch as we are in possession of surer methods of predicting eclipses, the calculation of which, from the ephemerides, is now a matter of comparative facility. They are, however, interesting in an historical point of view, and their formation was a principal object of the labours of the early astronomers. (On this subject see Lalande, Astronomie, tome ĩi. p. 185; Delambre, Astronomie Théorique et Pratique, tome ii. p.319; Sehubert, Traité d’ Astronomie Théorique, tome ii.; Woodhouse’s Astronomy, p. 665.)
[7:4:1]

THEORETICAL ASTRONOMY.

OF THE MOON.

WHEN we reflect on the immense addition that has been made to our knowledge of the constitution of the universe by means of the telescope, we may reasonably indulge an expectation that the further advances yet to be made will be closely connected with the extension of the power of that invaluable instrument.

Besides the power which the telescope gives of penetrating into remote space, it enables us to discover the nature and constitution of the bodies which are nearer to us. The admirable discoveries of Newton, as will be explained in the sequel, have taught us that the planets are not all composed of matter of the same density. But besides the property of weight, which belongs to all matter whatever, it has others which are peculiar to certain bodies, such as solidity, fluidity, &c. The Newtonian law of gravity reveals nothing of these; and it is only by the aid of the telescope that we can ever hope to acquire the slightest knowledge of the constitution of the planetary matter.

The structure of the more remote planets is probably beyond our knowledge, even when aided by the most improved telescopes; but it seems not unreasonable to indulge a hope of going some way in ascertaining the nature of the bodies nearer to us. Jupiter’s belts have with great probability been supposed to be masses of clouds in the region of his equator; and Herschel discovered, to the great delight of astronomers, that the polar regions of Mars assumed a whiteness of appearance when obliquely exposed to the light of the sun, just as if, like those of the earth, they were covered with snow. Above all, the moon presents to the astronomer an extensive field, not barely for conjecture, but for instructive observation. The solar spots have given occasion to much ingenious speculation; but the distance of the moon is only one four hundredth part of the distance of the sun from the earth, and in the same proportion our power of becoming acquainted with the constitution of the latter exceeds that of ever knowing intimately the constitution of the former. It is no wonder, then, that the attention of astronomers in all ages has been directed to the companion of our earth, whose structure may with great probability be supposed to resemble considerably the body round which it revolves, and of which it is not improbable that it may have once been a part.

CONTINUATION OF CHAP. III.

Sect. IV.— Of the Potation and Libration of the Moon.

The surface of the moon is diversified by dark patches, the number of which appears prodigiously increased in the telescope. These patches or dark spots present no changes of form like those of the sun, but permanently exhibit the same uniform appearances, and.retain their relative situations with regard to each other, and also, with some slight variations, to the apparent centre of the moon. The moon, therefore, at all times presents very nearly the same face to the earth. But if this were rigorously the case, it would follow that the moon revolves about an axis, perpendicular to the plane of her orbit, in the same time in which she completes a sidereal revolution about the earth, and that the angular velocities of the two motions are exactly equal. It is, however, proved by observation, that there are some variations in the apparent position of the spots on the lunar disk. Those which are situated very near the border of the disk alternately disappear and become visible, making stated periodical oscillations. But as they suffer no sensible changes in their respective positions, and always re-appear under the same form and magnitude when they return to the same position, it is inferred that they are permanently fixed to the surface of the moon; and their oscillations consequently seem to indicate a sort of vibratory motion of the lunar globe, which is known by the appellation of its libration. This motion has, however, no real existence. The phenomenon is the complicated result of several optical illusions, and does not depend in any degree on the rotation of the moon, which, relatively to us, is perfectly equable; or at least, if it be subject to irregularities, they are too minute to be appreciated.

In order to form a precise idea of the phenomenon of the libration, we must consider that the disk of the moon, seen from the centre of the earth, is terminated by the circumference of a great circle of the moon, the plane of which is perpendicular to a line drawn from the earth’s centre to that of the moon. The lunar hemisphere is projected on the plane of this circle turned towards the earth; and if the moon did not revolve round her axis, the projection would incessantly present different appearances to us, inas [7:4:2] much as the radius vector drawn from the centre of the earth by which the plane of projection is determined, would intersect the surface of the moon in a different point, at every new position in her orbit. But in consequence of her rotation, the radius vector is always directed to nearly the same point of the lunar surface, and would be always directed exactly to the same point if the angular velocity of rotation corresponded exactly with the angular velocity in the orbit. But the rotation of the moon is sensibly uniform; while the motion of revolution, being affected by the periodic inequalities, is sometimes slower and sometimes more rapid. The apparent rotation occasioned by the revolution of the moon round the earth is consequently in such cases not exactly counterbalanced by the real rotation, which remains constantly the same. Hence the different points of the lunar globe must appear to turn about her centre, sometimes in one direction, and sometimes in the contrary, and the same appearances be produced as would result from a small oscillation of the moon, in the plane of her orbit, about the radius vector drawn from her centre to the earth. The spots near the eastern or western edge of her disk disappear according as her motion in her orbit is more or less rapid than her mean motion. This is called the Libration in Longitude.

Further; the axis of rotation of the moon is not exactly perpendicular to the plane of her orbit. If we suppose the position of this axis fixed, during a revolution of the moon it inclines more or less to the radius vector, so that the angle formed by these two lines is acute during one part of her revolution, and obtuse during another part of it; hence the two poles of rotation and those parts of her surface which are near these poles are alternately visible from the earth. This is the Libration in Latitude.

Besides all this, the observer is not placed at the centre of the earth, but at its surfaced It is the radius drawn from his eye to the centre of the moon which determines the middle point of her visible hemisphere. But, in consequence of the lunar parallax, it is obvious that this radius must cut the surface of the moon in points sensibly different according to the height of that luminary above the horizon. An observer at the surface of the earth perceives points on the upper part of the moon’s disk, at the time of her rising, which could not be seen from the centre. In proportion as the moon acquires a greater elevation, these points approach the border of the disk, and finally disappear, while new ones become visible on the eastern part of the disk, which increase in number as the moon descends towards the horizon; so that in the course of a day she appears to oscillate about her radius vector in the direction of the earth’s rotation. This phenomenon constitutes what is called the Diurnal Libration, and is evidently the effect of the lunar parallax.

The libration in latitude and the diurnal libration were discovered by Galileo soon after the invention of the telescope. It was Hevelius who discovered the libration in longitude, and explained it by the equable rotatory motion of the moon combined with her unequal velocity in her orbit.

To the inhabitants of the moon, if such there be, the earth will appear as a species of moon, much larger than the moon appears to us, but visible only to that hemisphere which is turned towards the earth. At those places which are situated near the border of her visible disk, the earth will sometimes rise a few degrees above the horizon; and an inhabitant of the moon placed near the middle of the hemisphere presented to the earth will always see the earth near his zenith, making oscillations of only a few degrees in consequence of the libration. But an inhabitant of the other hemisphere will never see the earth at all; so that while one hemisphere of the moon is constantly enlightened, during her long night, by the light reflected from the earth, the other remains in constant darkness. In regard, therefore, to the distribution of light, one of the lunar hemispheres enjoys very great advantages over the other.

The elements of the rotation of the moon, that is to say, the position of her equator, the place of its nodes, and its inclination to the plane of the ecliptic, are found by the same methods which have been explained for determining the corresponding elements relatively to the sun. The geocentric positions of the spots are observed in the same manner; and in converting them into selenocentric latitudes and longitudes, the same formulae maybe employed, with a slight modification rendered necessary by the inclination of the lunar orbit to the ecliptic. One circumstance, not less remarkable than the coincidence which obtains between the times of rotation and sidereal revolution, is, that the nodes of the lunar equator coincide with those of the moon’s orbit, if not exactly, at least so nearly, that the differences are so small as to fall within the probable errors of observation and calculation. All the observations since the time of Hevelius agree in showing that the longitude of the descending node of the equator is very nearly equal to the mean longitude of the ascending node of the orbit; whence it follows that the nodes of the equator have a retrograde motion equal to that of the nodes of the orbit. With regard to the inclination of the lunar equator to the ecliptic, Mayer states it to be 1° 29', and Lalande 1° 43'. According to the latest computations made from the observations of Bouvard, the mean inclination of the lunar equator to the ecliptic is 1° 28' 42". Mr Baily makes it 1° 30' 10"∙8. Since the descending node of the equator coincides with the ascending node of the orbit, it is evident that its plane must be situated between the planes of the ecliptic and orbit, making an angle of about 1° 30' with the first, and of 3° 39' with the second.

Suppose three planes to pass through the centre of the moon, one of which represents her equator, the second the mean plane of her orbit, and the third parallel to the ecliptic. It is evident, from what precedes, that these three planes have a common section; that the first falls between the other two, making with them respectively the angles 3° 39' and 1° 30'. In the space of 6793 days, the time of a revolution of the nodes of the lunar orbit, the poles of the first two planes describe about the pole of the ecliptic, in a direction contrary to the order of the signs, two small circles parallel to the ecliptic, and of which the semidiameters are respectively the arcs 1° 30' and 5° 9'. Hence the difference between the longitudes of these two poles is constantly 180°, and the three poles are situated on the same great circle, that of the ecliptic being between the two others.

These results, which rank among the most curious discoveries of modern astronomy, were first obtained by Dominic Cassini: they were shown by Lagrange to be necessary consequences of the attraction which the earth exercises on the lunar spheroid.

The positions of the spots on the moon’s surface are determined by their distance from the lunar equator, and from a conventional meridian, that is, by their selenocentric latitudes and longitudes, after the manner in which the position of places is determined on the surface of the earth. The first meridian is assumed to be that which passes through the pole of the visible hemisphere when the true place of the moon in her orbit is equal to her mean place; hence the first meridian is always very near the middle of the face which the moon turns towards the earth, never deviating from it farther than by a quantity [7:4:3] equal to the equation of the moon, or her libration in longitude. The rotatory motion being equal to that of revolution, the selenocentric longitude of the first meridian, at any epoch, is found by adding 180° to the mean longitude of the moon; and this gives also the distance of the first meridian from the ascending node of the lunar equator. The position of the equator and first meridian being determined, the co-ordinates of a spot are computed without difficulty; and in this manner catalogues of the spots have been formed, and arranged according to their latitudes and longitudes.

Sect. V.— Of the Nature and Constitution of the Lunar Substance.

It has already been observed, that a slight attention to the different phases of the moon is sufficient to prove that she is an opaque spherical body, shining only by virtue of the light which she receives from the sun. The line bounding the visible part of her surface has exactly the form which would be produced by an illuminated hemisphere brought into different positions with respect to the eye; and the circular contour of the obscured portion of the sun during a solar eclipse could only be caused by the interposition of a spherical body. Besides, the parts of her hemisphere turned towards the earth, which the sun’s rays do not reach, are in some circumstances sufficiently discernible, and she has then the same circular appearance which she exhibits when at the full.

But if the light which comes to us from the moon is only that which she receives from the sun and reflects back to the earth, how does it happen, it may be asked, that the portion of her disk not directly exposed to the solar rays is distinctly visible for some days after the new moon? This phenomenon was ascribed by the ancients to the native light of the moon, to which, on account of its pale ashy hue, they gave the name of lumen incinero-sum. The explanation which is now generally given was first suggested by a celebrated painter, Leonard da Vinci. It consists in supposing that a portion of the light which is reflected from the illuminated hemisphere of the earth to the moon undergoes a second reflection at the lunar surface, and is transmitted back to the earth. The ancients were confirmed in their opinion respecting the native light of the moon, by observing that she is not altogether invisible in her eclipses. Plutarch, indeed, ingeniously ascribes her appearance under these circumstances to the light of the stars reflected from the moon; a cause, however, totally inadequate to produce the effect. This phenomenon is now generally ascribed to the scattered beams of the sun bent into the earth’s shadow by the refraction of its atmosphere.

The opinion that the moon’s light is chiefly, if not wholly, caused by the reflection of the sun’s rays at the lunar surface, has prevailed in all ages; and indeed no other explanation seems to have been thought of till it was suggested by Licetus, professor of philosophy at Bologna, as being more probable that the moon possesses a phosphorescent quality, and that the sun’s influence is only wanted to occasion the propulsion of the light which lies absorbed in her substance. This idea has been adopted by Professor Leslie, whose arguments in its support are at least extremely plausible and ingenious. So far as the appearances, and the explanation of the phases, are concerned, it is evidently matter of indifference whether we suppose that the solar rays are reflected from the surface of the moon, or that they exert an action in virtue of which the moon emits rays of her own. In either case it is that part of her surface only which is exposed to the impact of the solar rays that sends forth light to the earth.

The principal argument in favour of the phosphorescent nature of the moon is founded on the quantity of light which proceeds from her surface. It is evident that the moon does not act as a polished speculum, and reflect the whole of the incident rays; for in that case, as is known from the laws of Catoptrics, she would merely reflect an image of the sun, equally bright, varying in size according to her different positions relatively to the sun and the earth, and increasing till she arrives at her opposition, when her diameter would appear equal to about the 458th part of its real dimensions. Her phases could never be distinguished, and she would only appear to approach to or recede from the earth, in proportion as her diameter increased or diminished. It follows, therefore, that the moon’s surface must be irregular, or what is termed a mat surface, that is, of such a nature, that from every point of it the rays of light are reflected in all directions indifferently. According to the experiments ofBouguer, a white surface of this sort, for instance paper, or plaster of Paris, reflects only about the 150th part of the rays which fall upon it in a perpendicular direction; and the proportion is less as the angle of incidence becomes more oblique. Making allowance for the irregular surface and obscure spots of the moon, Mr Leslie computes that the solar light which she remits to the earth must be attenuated at least 105 million times; but Bouguer’s experiments show that the moon’s light is between the 250,000th and 300,000th part of the direct light of the sun, or about 350 times greater than the computed amount of reflected light. If every part of the moon’s surface reflected the light in the most perfect manner, it may be shown that only the 210,000th part of the rays which she receives from the sun would be thrown off in the direction of the earth; a quantity not much exceeding that which, according to Bouguer’s estimate, we actually receive from her. Mr Leslie states that he found the intensity of the moon’s light to approach the 150,000th part of the direct light of the sun; a result which, if admitted, must be entirely decisive of the question; for as the utmost possible quantity of reflected light cannot exceed the 210,000th part, it follows that the excess must be owing to the spontaneous light of the moon. Hence this ingenious philosopher concludes that the body of the moon is a phosphorescent substance, like the Eolognian Stone, which possesses the property of shining for some time when carried into a dark room, after having been exposed to the light of the sun. A fact first observed by the celebrated Arago seems to increase the probability of this opinion. All rays reflected from a surface not metallic acquire a peculiar modification, or become polarized ; but as the rays of the moon are not so modified, it is inferred that they have not undergone a reflection at her surface.

The secondary light of the moon, of which we have already made mention, affords arguments in favour of her native light precisely similar to the above. If the earth reflected, like a mirror, the whole of the incident rays, the illumination produced by the reflection would amount to about a 16,000th part of that which is caused by the sun; but as the sea reflects only about a 55th part of those rays, and the land a still smaller proportion, we may suppose that the reflected light of the earth does not exceed a millionth part of the direct light of the sun. It is extremely doubtful whether a light so greatly attenuated would suffice to render the moon visible. The lucid bow, or silvery thread of light, which, proceeding from the extremities of the lunar crescent, seems to embrace her unenlightened orb, is easily explicable on this hypothesis; [7:4:4]whereas it can hardly be satisfactorily accounted for by ascribing it to the secondary illumination from the earth. “I should rather refer it,” says Mr Leslie, “to the spontaneous light which the moon may continue to emit for some time after the phosphorescent substance has been excited by the action of the solar beams. The lunar disk is visible although completely covered by the shadow of the earth: nor can this fact be explained by the inflection of the sun’s rays in passing through our atmosphere; for why does the rim appear so brilliant? Any such inflection could only produce a diffuse light, obscurely tinging the boundaries of the lunar orb; and, in this case, the earth, presenting its dark side to the moon, would have no power to heighten the effect by reflection. But even when this reflection is greatest about the time of conjunction, its influence seems extremely feeble. The lucid bounding arc is occasioned by the narrow lunula, which, having recently felt the solar impression, still continues to shine; and from its extreme obliquity, glows with concentrated effect.” {Inquiry into the Nature and Propagation of Heat.')

Although these arguments go far to support the ancient opinion of the native light of the moon, they are not entirely conclusive; and indeed cannot be easily reconciled with some of the phenomena. If the moon shines in virtue of her native light, rays will be emitted in all directions from every point of her surface; whence, since a visual angle of a given magnitude includes a much larger portion of a spherical surface near the extremities of its apparent disk than towards the centre, and as the number of rays is proportional to the surface from which they proceed, it follows that the intensity of the moon’s light ought to be greater near the border than at the centre of her disk. The reason why this is not the case with regard to the sun is, that a greater proportion of the rays are absorbed in passing through a greater extent of the solar atmosphere; but the moon, having no atmosphere, ought to be sensibly most brilliant near the circumference of her orb. The contrary is, however, the case; her light is greatest at the centre, and less intense towards the circumference, exactly as it ought to be on the supposition of its being occasioned by the reflection of the solar rays. With regard to the ingenious argument of Arago, it cannot be held to be conclusive till we become more certainly acquainted with the nature of the lunar substance. The only property which we can safely ascribe to it as yet is density : whether in its physical properties it resembles the substances with which we are acquainted, is a question hardly within the bounds of legitimate investigation.

The spots of the moon, affording grounds for conjectures relative to her physical constitution and the nature of her surface, have been observed with great interest since the discovery of the telescope; and as they are of some service in the observation of eclipses, astronomers have been at much pains to determine their selenographic positions. On account of their number, it has been found necessary to distinguish them by particular names. Riccioli designated the most conspicuous of them by the names of astronomers, and other eminent men. Hevelius gave them the names belonging to countries, islands, seas, and regions on the earth, without reference to situation or figure. The nomenclature of Riccioli has, however, been deservedly preferred by Schroeter and o’thers who have particularly observed the phenomena of the lunar surface, and is now universally followed. Mayer gave a catalogue of 89 of the most remarkable of the spots, with their selenographic latitudes and longitudes referred to a first meridian, namely, that which passes through the centre of the moon’s apparent disk, perpendicular to the lunar equator, accompanied by an accurate map of her surface. Delineations of the lunar disk have also been given by Hevelius in his Selenographia ; by Cassini, Russel, Schroeter, Lohrmann, and others. The engraving (fig. 47) which accompanies this article gives a pretty accurate view of the appearance of the moon in her mean libration.

The following table contains the selenographic positions of some of the principal spots. The sign + indicates a northern, and — a southern latitude.
[table]

That there are prodigious inequalities on the surface of the moon, is proved by looking at her through a telescope at any other time than when she is full; for then there is no regular line bounding the dark and illuminated parts, but the confines of these parts appear as it were toothed and cut with innumerable notches and breaks; and even in the dark part, near the borders of the enlightened surface, there are seen some small spaces enlightened by the sun’s beams. Upon the fourth day after new moon there may be perceived some shining points, like rocks or small islands, within the dark body of the moon; but not far from the confines of light and darkness there are observed other little spaces which join to the enlightened surface, but run out into the dark side, which by degrees change their figure, till at last they come wholly within the illuminated face, and have no dark parts round them at all. Afterwards many more shining spaces are observed to arise by degrees, and to appear within the dark side of the moon, which, before they drew near to the illumined portion of the disk, were invisible, being totally immersed in the shadow. The contrary is observed in the decreasing phases, where the lucid spaces which [7:4:5]joined the illuminated surface recede gradually from it, and remain for some time visible after they are quite separated from the confines of light and darkness. Now it is impossible that this should be the case, unless these shining points were higher than the rest of the surface, so that the rays of the sun may illumine their summits before they reach their bases. Portions of considerable extent are also perceived on the lunar surface, which are never brilliant like the other parts, but remain constantly obscure. These are supposed to be deep valleys or cavities: they were formerly supposed to be seas, but, for reasons about to be given, this idea has been abandoned.

These phenomena render it certain that the surface of the moon is covered with mountains of a great height, with rocks or masses of unknown matter, but possessing the property of reflecting the sun’s light. The height of the lunar mountains may be determined in the following manner.

Let ABO (fig. 48) be the illuminated hemisphere of the moon, SO the tangential solar ray, and consequently 0 one of the points of the circle which separates the enlightened from the obscure hemisphere. All the part OD will be in darkness; but if this part contain a mountain öM so elevated that its summit M reaches the solar ray SO, the point M will be enlightened. Now, if the line OM can be determined by observation, it will be easy to deduce the arc Oa, and thence the height of the mountain βM, in terms of the moon’s radius. Let E be the place of an observer on the earth; draw the lines EM, EO, EC (C being the centre of the lunar orb), and Om perpendicular to EM. The distance of the moon from the earth being known, we have the distance EO: the angle Oem is measured by the micrometer; therefore Om, which is the projected distance of OM, is a given

Oτ∕⅛ quantity. Now OM = ———; and since Oem is a n j cos. MO»í very small angle, EOw may be considered a right angle, consequently MOffl = MOE — 90°; therefore OM

Qm Om Orn i .

“cos. (MOE — 90°) = sin. MOE ~ sin. EOS ' that 1S ’ the distance between the summit of the mountain and the illuminated part of the moon’s disk is equal to the projected distance measured by the micrometer, divided by the sine of the moon’s elongation from the sun. Suppose this distance OM = n ∙ CO; we shall then have CM = CO √1 + n 2 , and the height of the mountain - γΑ aM. = CO(√^1 + n 2 — 1) = — CO (neglecting in the development of the radical the powers of n which are higher than the square). According to the observations of Hevelius, the greatest value of n is τ L, which gives the height of the mountain equal to ∙ 3 + τ of the semidiameter of the moon. Schroeter, who estimated the heights of the lunar mountains by measuring the projections of their shadows at the time the sun was near their horizon, makes it in some cases -,f v . The highest mountains on the earth do not reach an elevation greater than T ðg ö °f the terrestrial radius; the lunar mountains, therefore, in proportion to the diameters of the earth and moon, are nearly five times higher than those of our globe. Their absolute height is above five English miles. It is easy to see that these determinations are susceptible of very little accuracy.

Not only is the moon’s surface rendered irregular by high and precipitous mountains, but numerous cavities appear in every part of her surface, some of which, according to Schroeter, are upwards of four English miles in depth, and forty in circumference at the orifice. An insulated mountain is frequently observed to rise in the centre of these enormous pits or caverns; and they are surrounded by high annular ridges, the masses of which would exactly fill the inclosed cavities. From this circumstance, it is probable that the elevations and hollows which abound on the surface of the moon have been produced by volcanic eruptions. Herschel imagined that he even observed volcanoes in activity. At the time of the new moon he perceived on different parts of her obscure disk three luminous points, resembling pieces of burning charcoal, covered with a thin coat of white ashes, one of which, by a comparison with the third satellite of Jupiter, appeared to be upwards of three miles in diameter. Their brilliancy continued during several days, undergoing variations altogether independent of the increase of the moon’s apparent magnitude, and at the end of that time they appeared to become extinct.

The existence of a lunar atmosphere has been a fertile subject of controversy among astronomers. It has been urged that, as the brightness of the moon is sensibly equal at all times when she is not obscured by clouds in the terrestrial atmosphere, she cannot be surrounded by an atmosphere similar to that of our earth, so variable in its density, and so liable to be obscured by clouds and vapours. The assumption of an equable brightness in the appearance of the moon has not, however, been allowed to pass uncontroverted. Hevelius relates that he has several times found in skies perfectly clear, when even stars of the sixth and seventh magnitude were visible, that, at the same altitude of the moon, at the same elongation from the earth, and with the same telescope, the moon and her maculae do not appear equally lucid and clear at all times, but are much brighter and more distinct at some times than at others. Several other observers have noticed that the moon is not always equally conspicuous during total eclipses; a fact which is held to be indicative of accidental variations in the state of the lunar atmosphere.

Another objection against the existence of a lunar atmosphere is derived from the circumstance that, if it existed, its influence would be perceptible in the occultations of the planets, or fixed stars, by the moon. When the moon approaches so near to a star that part of her atmosphere (supposing she has one) is interposed between the star and the eye of the observer, the star, it is contended, ought to suffer a change in its colour in consequence of the absorption of some of its light in traversing the denser parts of the moon’s atmosphere. But when we consider that the region of the vapours and clouds in the terrestrial atmosphere does not exceed the height of four miles, or the 1980th part of the earth’s diameter, and that consequently the obscure part of the lunar atmosphere, supposing it to be similarly constituted, would not subtend an angle of one second (the mean apparent diameter of the moon being 1889 seconds), and that this space is passed over by the moon in less than two seconds of time, it will scarcely be expected that observation will be able to decide whether the supposed obscuration takes place or not. Some observers have, however, remarked instances of stars, when about to be occulted, presenting an evident diminution of light. (See a Paper by Mr Ramage of Aberdeen, in the 2d vol. of the Memoirs of the Astronomical Society.')

The existence of a lunar atmosphere cannot increase or diminish the apparent diameter of the moon, because its effect on the rays of light when they enter it will be exactly counteracted at the time of their emergence. Itmightseem, therefore, that in order to determine whether the light of a star or planet is inflected when it passes very near the [7:4:6]border of the lunar disk, it would be sufficient to compare; the observed duration of the occultations of the stars, or eclipses of the sun, with the time, calculated from the theory of the moon’s motion, which she consumes in passing through a space equal to her apparent diameter. But if, as is generally supposed, a certain irradiation is produced around luminous objects, by which their image is dilated, the apparent diameter of the moon will thereby be augmented when she is projected on the dark sky. On the contrary, when the disk of the moon is projected on that of the sun in annular eclipses, the irradiation, by dilating the luminous ring which seems to surround the moon, will cause her apparent diameter to appear smaller than it actually is. On this account it becomes very difficult to ascertain by direct measurement the amount of the inflection of light at the border of the lunar disk, or even to be certain of its existence. An indirect method of arriving at the desired object was, however, imagined by Dionis du Séjour, who regarded the irradiation and inflection as two unknown quantities to be determined simultaneously from the observations of the phases of eclipses, upon which their effects are different. The observations best adapted for this purpose arc those of the magnitude of the luminous crescent, and its successive increase in annular eclipses. Du Séjour calculated with great care the annular eclipse of the sun which took place in the year 1764, and was visible over all Europe; and which, by reason of the extent and variety of its phases, presented a great many points of comparison. He particularly employed the observations made in different places of the instants of the formation and rupture of the ring. He likewise confirmed these observations by measurements made by Mr Short, at London, of the distance between the horns of the crescent at divers instants. It resulted, from the comparison of an immense number of observations, that the measurements could not be reconciled without supposing an irradiation of 3" on the semidiamcter of the sun, and an inflection of nearly the same amount round the disk of the moon, produced by her atmosphere. According to this result, the horizontal refraction at the surface of the moon amounts to 1''∙5. The mean horizontal refraction observed at the surface of the earth is 0° ∙585, and consequently about 1400 times greater than that at the moon. Hence, supposing the moon’s atmosphere of the same nature as that of the earth, its density must be 1400 times less, and consequently rarer than the most perfect vacuum which can be produced by the best pneumatic machines.

The existence of a lunar atmosphere, though of small extent, is indicated with considerable probability by the observations of Schroeter, who perceived that some ranges of lunar mountains, when in the dark hemisphere, are illuminated more feebly in proportion as they recede from the boundary of light and darkness; an effect which would be produced by the partial absorption of the rays of light which pass near the moon’s surface. He also remarked the same circumstance with regard to the cusps, and perceived other indications of an atmosphere in some changes of tint, which induced him to think that a twilight might be discernible towards the cusps. With a view to prove the accuracy of this conjecture, he examined the moon with great care under the most favourable circumstances, and at length descried a faint glimmering which he took to be crepuscular light, extending from the cusps into the dark body of the moon.' Its greatest breadth was 2"; and it extended 1' 20" from i the cusps, along the circumference of the lunar disk? From these data he computed that the height of the lunar atmosphere, to the limit where it ceases to inflect the rays of light or diminish the brightness of a star, does not exceed 5742 English feet. A ring of this breadth, at the distance of the moon, will subtend an angle of only 0"∙94; hence the almost imperceptible influence of the lunar atmosphere.

Some observations, related by Mr Ramage of Aberdeen, in a memoir already alluded to, of occultations of Jupiter and his satellites, with an excellent reflecting instrument, also tend to confirm the existence of a small lunar atmosphere. On the approach of the satellites no diminution of their light was perceptible. On coming into contact with the moon’s limb they did not disappear instantly, like fixed stars, but formed an indentation or notch in the limb, as if imbedded in it, but at the same time separated from it by a fine line of light. The indentation continued visible till about half their diameters were immersed, when it disappeared. (Memoirs of the Astronomical Society of London·, vol. ii. p. 87.)

The dark spots on the moon’s surface were formerly supposed to be water; but as elevations and cavities are distinctly perceptible in them, that hypothesis is evidently erroneous. Besides, the extreme tenuity of her atmosphere is inconsistent with the existence of water at her surface. It is only by the weight of the terrestrial atmosphere that the liquids at the surface of the earth are prevented from being dissipated in vapours. If the present atmosphere were removed, every liquid would continue to be dissipated in this manner till a new atmosphere was formed, to which each would contribute in proportion to its elastic force; and the evaporation would only cease when the tension of the vapour of each liquid was equal to its elastic force in a vacuum at the same temperature. But if the vapours were removed as they arose, by any absorbing cause, the evaporation would continue till the liquids entirely disappeared. Now we may suppose this to have been the case with respect to the moon, and that at one time she may have had an atmosphere, which the attractive force of the earth, aided by some accidental circumstance, may have swept away and united with our own. (Biot, Asir. Phys, tome ii. p. 413.)

Under these circumstances it is evident, that no animal, similarly constituted to those which inhabit the earth, could respire at the surface of the moon.^[3. It has always been a favourite opinion with mankind, that the celestial bodies, resembling the earth in some respects, are also, like it, peopled with rational beings. The author of the verses attributed to Orpheus ascribes to the moon many mountains, and cities and palaces: fl ∙πολλ nifi ιχli, 7rολλ , versa, πcλλa μιλatξx, ] Every thing there appears solid, desolate, and unfit for the production and support of organized substances; and the excessive cold which certainly prevails must be sufficient to destroy every source of animal or vegetable life. May it not then be supposed that the moon is a planet which has not yet reached a state of maturity—a maturity to be prepared by successive volcanic eruptions; or that, having fulfilled its destiny, it is now in a state of decay?

On the subject of this section, see Hevelius, Selenogra-phia; Sehroeter, Selenotopographische Fragmente; Hooke’s Micrographia, 1665; Lemonnier, Selenographie ; Mayer, Cosmographische Nachrichten, 1748; Boscovich, De Lunce Atmosphoera, 1753; Dunn, Pliil. Trans., 1762; Herschel, Phil. Trans., 1780 and 1787, p. 229; Ferguson’s Astronomy, by Brewster; Ε> ∖ οl, Astronomie, tome ii. p. 413; Schubert, Traité d ’ Astronomie Théorique, tome ii. p. 364.
[7:4:7]

Sect. VI.— Of Eclipses and Occultations.

In describing these interesting phenomena, we will first consider the eclipses of the moon, which, for anygiven place on the earth, are much more frequent than those of the sun, and, by reason of certain circumstances about to be explained, can also be computed with much greater facility.

1.— Eclipses of the Moon.

The earth being an opaque, round body, much smaller than the sun, must project behind it in space a conical shadow, limited by straight lines drawn from the extremities of the sun’s disk to touch the surface of the earth. When the moon enters this shadow, and a portion of her disk is still enlightened by the sun, the enlightened part will necessarily have the form of a luminous crescent, the concavity of which is turned to the conical shadow of the earth; and this appearance will likewise be exhibited when the moon begins to emerge from the shadow. As she approaches towards the shadow, her light is not suddenly eclipsed, but passes insensibly through all the successive gradations of obscurity, till the darkness attains its greatest intensity. The reason of this will be easily comprehended by considering that when an opaque body is placed between an object and the sun, so as to conceal only a part of his disk, the object is then less enlightened than when none of the solar rays are intercepted; and this in proportion as more or less of the sun is concealed. Between full illumination and total obscurity there are consequently intermediate tints and gradations of light, which are denominated the Penumbra, in contradistinction to the Umbra which covers those places to which the sun’s rays are completely intercepted. Let S and E (fig. 49) be the centres of the sun and earth, and ABB'A' any plane whatever passing through the axis SE, in which, Iet AB, A'B' be tangents to the sun and earth on opposite sides of the axis SE; these tangents will meet the prolongation of the axis in a point C. Let also AB ʹ, A ʹ B touch the sun and earth on opposite sides of SE, and let C ʹ be their intersection. If we now suppose the plane to turn round the axis SC, a conical shadow or umbra, of which the apex is C, will be formed behind BB'; and if the moon is situated within any part of the cone BCB ʹ, the sun will be entirely invisible. As soon as the moon emerges from this cone, a part at least of the sun’s rays will fall upon her, and she will be in the penumbra. If situated, for example, at M, and MN be drawn to touch the earth and meet the solar disk in N, then the part of that disk between A and N will be visible at M. The extent of the penumbra is therefore determined by the angle DBC. When M is situated in the straight line BD, the whole of the sun’s disk is visible; when M is in BC the sun entirely vanishes, and his visible portion diminishes from the instant M passes BD till it reaches BC. The intensity of the penumbra, therefore, goes on increasing from the first of these limits to the second, where it is confounded with the total darkness. This explains the progressive obscuration of the moon’s disk in her eclipses.

If we now attempt to determine the circumstances under which an eclipse of the moon can take place, it is obviously necessary to inquire, in the first place, into the length of the conical shadow BCB ʹ; for an eclipse can only happen on the supposition of its extending beyond the orbit of the moon. For this purpose let L’F (fig. 50) be drawn .parallel to AB, and meeting SA in F. We have then sin. T'Γ'Γ> ∙ CΓΓ SF SA— EB ∙ τ> ∙ τ> LCB = sm. SEF = ≈= = = sm. R — sm. ®, R

SE SE - r being the apparent semidiameter of the sun, and∕> his horizontal parallax. By reason of the smallness of these angles the arcs may be substituted for the sines; hence the angle EB

ECB = R — p. But CE = —- τ √∕⅛, therefore making' sin. LCB o

L’B, the radius of the earth, = a, CE = —— ..

sin. (R — p) From this expression it is evident that the value of CE depends on the horizontal parallax, and therefore varies with the sun’s distance from the earth. On calculating, by means of the values of R, a, and p, given in the preceding sections, the values of CE at the perigean, mean, and apogean distances, the following results will be obtained:—

Lengths of CE.

Sun in perigee 212∙896 terrestrial radii.

at mean distance 216 ’ 531

in apogee 220∙238

Now the greatest distance of the moon from the earth is less than 64 terrestrial radii (sect, ii.), consequently the shadow of the earth is projected into space between three and four times farther than the distance of the moon. Hence it appears, that if the moon moved in the ecliptic she would traverse the earth’s shadow and be eclipsed every revolution. On account of the inclination of her orbit to that plane, the eclipses can only happen when she is in or near her nodes. The greatest distances from the nodes at which they can take place are called the Lunar Ecliptic Limits.

In order to determine these limits, it is necessary first of all to know the apparent diameter of a section of the earth’s shadow, at the place where it is traversed by the orbit of the moon. Let ZMZ ʹ (fig. 50) be a part of the moon’s orbit, intersecting the lines AC and A ʹ C in m and m'; then mm' is the geocentric diameter of the shadow, the half of which is measured by the angle m EC. Now the angle mEC is the difference between the angles EmA and ECA, the first of which, namely, EmA, is the apparent semidiameter of the earth seen from the moon, or, ’ in other words, the moon’s horizontal parallax. Let this angle, therefore, be denoted by P. It has already been shown that EC A = R— p; consequently we have w?EC = P + p — R in all cases; that is to say, the semidiameter of the shadow is equal to the sum of the horizontdl parallaxes of the sun and moon, diminished by the apparent semidiameter of the sun. On calculating the amount of this expression from the values of P and p given above, the following table will be obtained, exhibiting the magnitude of the apparent diameters of the earth’s shadow for different distances of the sun and moon.

Apparent diameters of earth’s shadow.

Ç J in apogee 1° 15' 24 ″ ∙30

Sun in perigee, < at mean distance 1 23 2 ∙31

( in perigee 1 30 40 ∙31

ŕ J) in apogee 1 15 56 ∙86

Sun at mean distance, < at mean distance 1 23 34 ∙87

in perigee 1 31 12 ∙87

Íj) in apogee 1 16 28 ∙29

at mean distance 1 24 6 ∙30

in perigee 1 31 44*30

The greatest apparent diameter of the moon being only 33′ 31 ″ ∙07, which is about a third part of the diameter of the earth’s shadow, it follows that the moon may not only be completely enveloped in the shadow, but, since she passes over a , space nearly equal to her own breadth in an hour, that she may continue to be totally eclipsed during a space of about two hours.

The above determination refers only to the umbra or cone of total darkness; but the diameter of the penumbra is obtained in a manner exactly similar. In the same [7:4:8]diagram the semidiameter of the penumbra is measured •by the visual angle ZEC; but ZEC = E∕C' + EC'ζ and EC? = EC? = C'EA + C'AE; therefore ZEC = EZC'+ C'E A + C'AE; or, retaining the same denominations as above, ŻEC = P+ p + R. Hence, the semidiameter of the penumbra is equal to the horizontal parallaxes of the moon and sun, augmented by the apparent semidiameter of the sun.

All the different numerical values of the semidiameter of the. penumbra corresponding to particular positions of the sun and moon may be computed exactly in the same manner as in the case of the umbra.

The two expressions for the semidiameters of the umbra and penumbra, viz. P + p — R, and P + p + R, give immediately the distance of the moon’s centre from the axis of the cone when her disk comes into contact with the shadow. Representing the apparent semidiameter of the moon by r, her disk will just touch the umbra when the distance of her centre from the axis SE is equal to P+∕>—R + ^; and it will touch the penumbra when the distance is P+, p + R∙ + r. But on account of the very feeble obscurity of the penumbra towards its extreme border, it is impossible to observe with any degree of precision the time at which the moon enters it. In computing the ecliptic limits, therefore, it is only necessary to have regard to the umbra.

In the table given above, the extreme values of P + p — R, the. apparent semidiameter of the earth’s shadow, are ⅜ (1° 15' 24"∙30) = 37' 42''∙15, and ⅜ (1° 31' 44"∙30) = 45' 52"·15; and the least and greatest values of r, the apparent semidiameter of the moon, are respectively 14' 45" and 16' 45" (sect, ii.); therefore the least distance of the moon’s centre from the axis of the shadow at the time of her immergence or emergence is 37' 42"∙15 + 14' 45" = 52' 27"∙15, and the greatest 45' 52"∙15 + 16' 45'' = 62' 37"∙15. The first is the limit within which an eclipse must necessarily happen; the last that beyond which it cannot happen.

It is now easy to ascertain the limits of the moon’s distance from her node, within which the eclipses take place. Let NC (fig. 51) be a portion of the ecliptic, NM part of tfie moon’s orbit, N its node, C the centre of a section of the earth’s shadow, Μ the centre of the moon; the verge of the lunar disk touching, but not penetrating, the shadow at a. It is evident that, if the moon be at a greater distance than CM from NC, there can be no eclipse. The greatest value of CM, as we have just seen, is 62' 37"∙15, from which the corresponding ecliptic limit NC is easily computed by means of the formula rad. × sin. CM = sin. NC × sin. CNM. Supposing the angle CNM, that is, the inclination of the moon’s orbit to the ecliptic, to be 5°, which is its minimum value, the value of NC is found from the logarithmic tables to be 12° 2' nearly. Hence an eclipse of the moon can only happen when she is within about 12° 2' ofher node. Under the most favourable circumstances, however, the limits may extend to 13° 21'. A lunar eclipse will certainly take place if the moon’s distance from her node, at the time of her mean opposition, is not greater than 7° 4'.

When the moon’s disk only comes into contact with the shadow, as in fig. 51, the phenomenon is called an appulse; when the disk only enters into the shadow in part, the eclipse is said to be partial; it is called total if the moon entirely disappears, and central when her centre coincides with the axis of the cone, or if at the time of the eclipse the moon is exactly in her node.

In the preceding determinations the shadow has been supposed to be conical, whereas, on account of the compression of the earth at the poles, it is not exactly a cone, but a conoid on. an elliptic base; and it varies at every instant by reason of the earth’s rotation. To determine rigorously the figure of the cone at every instant, and the diameter of its section where the moon enters and leaves it, would require calculations of great complication and prolixity. Such precision is, however, unnecessary; and indeed it is impossible to attain to absolute accuracy so long as the figure of the earth is not exactly known. It is usual and natural to employ the largest diameter which the shadow can have at the distance of the moon.

It has also been supposed that the shadow is terminated by tangents to the sun and the earth; that is to say, that all the rays of light which are not obstructed by the globe of the earth pass in straight lines from the sun to the moon. But the earth being surrounded by an atmosphere which near the surface exerts a powerful action on the solar rays, it is to be presumed that those rays which, if unobstructed, would glance by the surface of the earth, are absorbed by the lower strata of the atmosphere. The effect of this will be to enlarge the diameter of the shadow; and the requisite correction may be regarded as an augmentation of the earth’s radius, by adding to it a part of the atmosphere; or, which amounts to the same thing, an augmentation of the lunar parallax. Now, it is found by experience that such a correction is necessary; and, according to Mayer, the lunar parallax must be augmented by a 60th part in order to satisfy the observations.

Another effect of the action of the earth’s atmosphere on the solar rays is to render the moon dimly visible even when she is totally eclipsed,—a circumstance to which allusion was made in the preceding section. Let the circle f g lι i (fig. 52) concentric to the earth include that part of the atmosphere which is sufficiently dense to produce a sensible refraction of the rays of light. All those rays which do not fall within that circle, such as N ∖ fw, Niv, proceed in their direct course without suffering any refraction; but those which enter the atmosphere between f and h, and between i and l, on opposite sides of the earth, are gradually more bent inward as they go through a greater portion of the atmosphere, until the rays W7eand V/, touching the earth at m and n, are bent so much as to meet at q, a little short of the moon; and therefore the dark shadow of the earth is contained in the space mopqm, where none of the sun’s rays can enter; all the rest, R, R, being mixed by the scattered rays which are refracted as above, is in some measure enlightened by them; and some of those rays falling on the moon, give her the colour of tarnished copper, or of iron almost red-hot; so that if the earth had no atmosphere, the moon would be as invisible in total eclipses as she is when new. If the moon were so near the earth as to go into its dark shadow, suppose about p or q, she would be invisible during her stay in it, but visible before and after in the fainter shadow R R.

After having pointed out the general phenomena of the lunar eclipses, and the limits within which they take place, it only remains to show in what manner the time of their commencement, end, and duration, and also their magnitude, may be determined by computation.

Let the line NE (fig. 53) represent the ecliptic, NO the orbit of the moon, C the centre of the terrestrial· shadow, and Μ the centre of the moon at the instant of the opposition; then CM will be the circle of latitude on which the opposition takes place. The centre of the shadow C being always in opposition with the sun, moves along the ecliptic from west to east, or from N towards E, with the same velocity as the sun. The moon also, at the same time, moves in her orbit from west to east, or from N towards 0. Now the velocities of these two motions are given by the astronomical tables, and the question is to determine the instant of time at which the circles [7:4:9]representing sections of the moon and the earth s shadow • meet each other either before or after the opposition.

' At the time of an eclipse the apparent distance of the centre of the shadow from the moon is very small, consequently CM, and also the differences of the respective longitudes and latitudes of C and Μ, may be regarded as straight lines. During the short iuterval between the commencement and end of an eclipse, the motion of the sun, and consequently that of the centre of the shadow, may likewise be regarded as uniform. By these suppositions, sufficiently accurate for our present purpose, the problem is considerably simplified.

Suppose now that C' and Μ' are two simultaneous positions of the shadow and moon at any instant before or after the opposition. Let Μ' P be perpendicular, and MQ parallel, to NE. The velocities of the moon and terrestrial shadow being known from the tables, the lines C' P and Q M', which represent the mutions of the centre of the moon relatively to that of the shadow in longitude and latitude, are known also; whence C' P and P Μ' are given, and consequently C' Μ' the distance of the centres. Let us assume

λ = CM or ]) ’s latitude when in opposition, s = ø’s motion in longitude, m = J’s horary motion in longitude, n = D’s motion in latitude, t = time from Μ to M', c = C'M' the distance of the centres.

Now, since we suppose that CP and QM' are the 5 ’s motion in longitude and latitude respectively in the time t, it is evident that CP = m t, and QM' = n t. But CC' is the sun’s motion, or the motion of the terrestrial shadow in longitude during the same time; therefore CC'= st. We have then C'P = m t — st, and PM' = ∖ A∙n t; consequently c 2 — (mt — s i) 2 +(λ+ n tf.

In this quadratic equation, if t is regarded as the unknown quantity, the only arbitrary quantity contained in it will be c, the distance of the centres, the others being all determined from the tables. On assigning, therefore, any arbitrary value to c, the resolution of the equation will give the corresponding value of t, and consequently the circumstances or different phases of the eclipse which we may wish to determine.

On arranging the terms of the above equation so as to obtain the resolution relatively to t, we have [fm — sf + w 2 ĵ f 2 + 2 λ ’ η t = c 2 — λ 2 , which may be still simplified by introducing an auxiliary

2Z

angle d, such that tan. ð = ; for by this substitution m — s j there will result nft ft + 2 λ∙ η t sin. 2 Ô — ((ft — λ 2 ) sin. 2 ∂, which gives the two following values of t : t = -( — λ sin. 2 Ô z±z sin. ô V eft — λ 2 cos. 2 ô ∖ n∖ J

The first of these denotes the time at which the moon enters, and the second that at which she quits, the umbra or penumbra.

The time at which the different phases of the eclipse happen, are calculated directly from this equation. If, for example, we wish to determine the time at which the moon’s disk begins to enter the shadow, we make c = P + p — R + r (neglecting the small augmentation of the shadow occasioned by the refraction of the atmosphere). In the case of the penumbra we must take c = PA-p-Ļ- R + r; and it is evident that, if in either case λ is of such a magnitude that c is less than λ cos. β, the value of t will be impossible; in other words, no eclipse can take place.

If we suppose P+77 — R+r = λ cos. ô, the two values ' of t will be equal, and the duration of the phase will only 2 be for an instant, as in the case of the appulse, in which the κ moon’s limb just touches the shadow without entering it. In general, the portion of the diameter of the eclipsed part is P +77—R∙ + r — λ cos. ô ; and consequently the diameter of the part not eclipsed is equal to the diameter of the moon, or 2r, minus this quantity, that is, equal to λ cos. ∂ — P —77 + R + r. When this expression is equal to nothing the eclipse is just a total one; when negative, the upper boundary of the moon’s limb will be under the upper boundary of the section of the shadow, and the total eclipse will continue for some time.

The instant at which the middle of the eclipse happens will evidently be that at which the two values of t are equal, or when the radical disappears, that is, when c = λ cos. ô. In this case ŕ =— ~ λ sin. 2 Ô, and the instant is n called that of the greatest phase. It is usual to express the quantity of the eclipse in digits, or twelfths of the lunar diameter; so that the eclipsed part is represented

12

by 2Ĩ ( P +P— P + r ~ λ cos. ô). Thus, taking the moon’s apparent diameter at 33' 18", and supposing the eclipsed part to be 24' 52", this part expressed in digits 24' 52"

will be ⅛-pp × 12 = 8∙96 digits.

The obscurity of the penumbra renders observations of the commencement and termination of the lunar eclipses extremely uncertain. To obviate in some degree this inconvenience, care is taken to observe as accurately as possible the instants at which the shadow arrives at or passes different known spots on the moon’s disk; so that the same eclipse offers in fact a great number of different observations, the mean of which may be regarded as more certain than any individual one. But after all the precautions that can be taken, the eclipses of the moon are far from affording results equally precise and certain as those of the sun. They were formerly of much greater importance than they are in the present state of astronomy; for the ancients had no other means of determining the geographical longitudes of places on the earth. In fact, as the eclipse is occasioned by the moon’s being deprived of her light, the different phases of the eclipse happen at exactly the same physical instant of time to all observers to whom the moon is visible. The difference of the time reckoned by two observers at the instant of the phenomenon will therefore give the difference of the horary angles, or of their meridians; but supposing each to have made a mistake of 4 minutes of time in an opposite sense (and the ancients could scarcely guarantee a greater degree of accuracy), the resulting error in the difference of longitude would amount to 2°. The geographical tables of Ptolemy contain errors of still greater magnitude.

2.— Eclipses of the Sun.

The eclipses of the sun are caused by the interposition of the moon between the sun and the earth, and their general phenomena may be explained in the same manner as those of the moon. When the conical shadow which the moon projects behind her in space reaches the earth, those points of the earth’s surface on which it falls are completely deprived of the light of the sun, and involved in total darkness. Those parts of the earth which aré covered by the penumbra arc only partially deprived of the sun’s light, because the moon does not conceal the whole, but only a part of the solar disk.
[7:4:10]

In order to appreciate the different circumstances of a solar eclipse, the procedure to be adopted is in many respects the same as that which has been explained in regard to the eclipses of the moon. The length of the moon’s shadow, the first object of inquiry, is found exactly in the same manner as that of the earth; and it is only necessary to substitute in the formulae already given the values of the apparent diameter and parallax of the sun which they would have at the surface of the moon. Now, these values are easily found, for the diameter of the sun as seen from the moon is equal to his diameter as seen from the earth, increased in the ratio of the distances of the moon and earth from the sun. In the same manner the parallax of the sun relatively to the moon is equal to his parallax relatively to the earth, augmented in the ratio of the distances, and diminished in the ratio of the diameters, of the moon and earth. Thus, let D represent the distance of the earth from the sun, d the moon’s distance from the sun, m the moon’s true semidiameter, and a the semidiameter of the earth; the sun’s apparent semidiameter as seen from the moon will be R ∙ (R being his apparent semidiameter as seen from the earth), and his horizontal parallax will b<ψ ∙ ∙ ~∙ The formula then which expresses the length of the terrestrial shadow CE (fig. 50), namely, ——-∕ r, adapted to the case of the moon, becomes sm. (R— p) r

. , — . , and expresses the distance be-

M(1 →∙≡)⅞}

tween the centre of the moon and the apex of her shadow. By means of this formula the following results, which refer to the extreme cases in which the length of the shadow is a maximum and minimum comparatively with the moon’s distance from the earth, may be computed.

Length of Moon’s shadow. distance.

Sun in apogee, Moon in perigee, 59∙730 55∙902

Sun in perigee, Moon in apogee, 57*760 63∙862

In the first case the shadow of the moon will reach beyond the centre of the earth; in the second it will not reach even to the surface. It follows, therefore, that even if the orbit of the moon coincided with the ecliptic, she would not produce a total obscurity every time she comes between the sun and the earth. At her greatest distances, where the shadow does not reach the earth, the effect of her interposition would be to conceal only a part of the sun’s disk.

By introducing into the other formulae modifications similar to the above, we shall find the apparent diameter of the shadow and the solar ecliptic limits. The apparent semidiameter of the earth’s shadow has been shown to be equal to P + p — R; consequently the semidiameter of the lunar shadow at the distance of the earth, as seen by an observer placed on the moon, is equal to the parallax of the earth, plus the parallax of the sun relatively to the moon, minus the apparent semidiameter of the sun seen from the moon. The parallax of the earth means simply the apparent semidiameter of the moon seen from the earth; and if we neglect the parallax of the sun, which cannot influence the result to the extent of half a second, we shall have the following theorem: The semidiameter of the lunar shadow is equal to the excess of the apparent semidiameter of the moon above the apparent semidiameter of the sum Hence, denoting the moon’s apparent semidiameter by r r, the semidiameter of the lunar shadow will be expressed by τ — R ∙ ~; or,

... . , 11 . , ∞ D

taking into account the sun s parallax, r + p

R∙-5. The ratio of the distances in this and the pre- d d ’ ceding formula, may be expressed in terms of the parallaxes; for since p = ĝ-, and P = py~~~ p p therefore p τtι> τ

— . Since also — = -≈ τ , according to the nature

P— p a P

. , ,. 1 m D r P

of parallax, we have likewise τ + p ·—·— = p--~^> therefore, by substituting these values, the expression for the semidiameter of the lunar shadow becomes

P

(r —R)∙-L-.

v z P — p

If to the apparent semidiameter of the shadow at the point where it is ouched by the earth, we add the apparent semidiameter of the earth as seen from the moon, that is to say, the moon’s horizontal parallax (P), the distance between the centres of the moon’s shadow and of the earth will be p

P + (r — R)p— , x P — p from which expression the solar ecliptic limits may be readily computed. The result of the computation is, that a solar eclipse may take place if the moon’s distance from her node, at the time of her mean conjunction with the sun, does not exceed 19° 44'. If her distance from the node is less than 13° 33', the sun will certainly be eclipsed in some part of the world.

The solar eclipses present a great variety of appearances, depending on the relative positions of the sun, the moon, and the spectator. If the apparent diameter of the moon happens to surpass that of the sun, the eclipse will be total ; but if the moon’s diameter be the smaller, the observer will see a luminous ring, formed by that part of the sun’s disk which exceeds that of the moon, and the eclipse will in that case be annular. If the centre of the moon is not in the same straight line which joins the observer and the centre of the sun, the eclipse can only be partial, as the moon can only conceal a part of the sun’s disk. When the moon merely touches without penetrating the solar disk, the phenomenon is called an appulse ; and the eclipse is central if the observer is placed at the centre of the shadow, on the straight line joining the centres of the sun and moon.

When the change happens within 17 degrees of the node, and the moon is at her mean distance from the earth, the point of her shadow just touches the earth, and she eclipses the sun totally to that small spot whereon her shadow falls; but the darkness is not of a moment’s continuance.

The moon’s apparent diameter, when largest, exceeds the sun’s, when least, only two minutes of a degree; so that in the greatest eclipse of the sun that can happen at any time and place, the total darkness continues no longer than whilst the moon passes over two minutes in her orbit, that is, about 3 minutes and 56 seconds of an hour.

The moon’s shadow covers only a spot on the earth’s surface about 180 English miles broad, when her diameter appears largest, and the sun’s least; and the total darkness can extend no farther than the limits of the dark shadow. Yet the partial shadow or penumbra may then cover a circular space 4900 miles in diameter, within all which the sun is more or less eclipsed, as the places are less or more distant from the centre of the penumbra. When the moon changes exactly in the node, the penumbra is circular on the earth at the middle of the general eclipse, because at [7:4:11]that time it falls perpendicularly on the earth’s surface; but at every other moment it falls obliquely, and will therefore be elliptical; and the more so as the time is longer before or after the middle of the general eclipse; and then much greater portions of the earth’s surface are involved in the penumbra.

To make several of the above and other phenomena plainer, let S (fig. 52) be the sun, E the earth, M the moon, and AMP the moon’s orbit. Draw the straight line W e from the western side of the sun at W, touching the western side of the moon at c, and the earth at e: draw also the straight line V d from the eastern side of the sun at V, touching the eastern side of the moon at d, and the earth at e : the dark space c e d included between these lines is the moon’s shadow, ending in a point at e, on the surface of the earth, because in this figure the moon is supposed to be at her mean distance from the earth. Had the moon been in her perigee, the shadow would have covered a space on the surface of the earth of about 180 miles in diameter, to all places within which space the eclipse would have been total. Had she been in her apogee, the shadow would have terminated in a point above e, and to an observer at e the sun would have been eclipsed annularly. Draw the straight lines WXtZΛ and VXcy, touching the contrary sides of the sun and moon, and ending on the earth at a and b ; draw also the straight line SXM from the centre of the sun’s disk, through the moon’s centre to the earth, and suppose the two former lines WXiZΛ and VX¢^ to revolve on the line SXM as an axis, and the points a and b will describe the limits of the penumbra TT on the earth’s surface, including the large space aba, within which the sun appears more or less eclipsed, according as the places are more or less distant from the verge of the penumbra a b.

Draw the right line y 12 across the sun’s disk, perpendicular to SXM the axis of the penumbra; then divide the line y 12 into twelve equal parts, as in the figure, for the twelve digits or equal parts of the sun’s diameter; and at equal distances from the centre of the penumbra at e (on the earth’s surface YY) to its edge ab, draw twelve concentric circles.

To an observer on the earth at b, the eastern limb of the moon at d seems to touch the western limb of the sun at W when the moon is at M, and the sun’s eclipse begins at b ; but at the same moment of absolute time, to an observer at a, the western edge of the moon at c leaves the eastern edge of the sun at V, and the eclipse ends. At the very same instant, to all those who live on the circle next to ab, the moon cuts off or darkens a twelfth part of the sun, and eclipses him one digit; to those who live on the next interior circle, the moon cuts off two twelfth parts of the sun; to those on the following circle, three parts; and so on to the centre at e, where the sun is centrally eclipsed. The different appearances of the eclipse, as seen by spectators in these different situations, with regard to the centre of the shadow, are represented in fig. 54, under which figure there is a scale of hours and minutes, to show at a mean state how long it is from the beginning to the end of a central eclipse of the sun on the parallel of London, and how many digits are eclipsed at any particular time from the beginning at A to the middle at B or the end at C. Thus, in 16 minutes from the beginning, the sun is two digits eclipsed; in an hour and five minutes, eight digits; and in an hour and 37 minutes, 12 digits.

Having determined the diameter of the moon’s shadow at the earth, and the limits within which eclipses of the sun can take place, the next object is to determine the time of their commencement and termination. If the position of the observer were on the moon instead of the ' surface of the earth, our solar eclipses would appear to 2 him as eclipses of the earth, and they would commence at the instant when the earth’s disk began to penetrate the lunar shadow. A spectator so situated might therefore compute all the circumstances of a terrestrial eclipse in exactly the same manner in which we compute those of the moon; the same formulae would suffice, with the slight modifications which have already been made. But relatively to an observer placed on the earth the case is altogether different. To him the eclipse does not begin when the moon’s shadow comes into contact with the earth’s disk, but when it begins to obscure his station. This, therefore, is one circumstance which renders the computation of solar more complicated and difficult than that of lunar eclipses; for it is necessary not only to determine generally what portion of the terrestrial disk is covered by the shadow, but also its position relatively to the equator and to a given meridian, and likewise the path described by the centre and contour of the umbra and penumbra on the surface of the earth. There is another circumstance which still further augments the difficulty of the computation, namely, the position of the observer, on account of which it is necessary to introduce the particular conditions which depend on the parallax. In the case of a lunar eclipse, it is only necessary that the moon’s disk enter the earth’s shadow in order that the eclipse may be visible to any part of the terrestrial hemisphere opposite to the moon; but a solar eclipse may happen in some parts of the earth without being visible at others—a circumstance which is occasioned entirely by parallax.

When we abstract from the effects of parallax, or suppose the observer to be placed at the centre of the earth, the problem of determining the different circumstances of a solar eclipse is exactly the same as that relative to an eclipse of the moon. But when the spectator is supposed to be placed at the surface, the latitudes and longitudes corresponding to his situation are different from the geocentric latitudes and longitudes; and in order to adapt the formula from which the time is given to these new circumstances, it is necessary, as a preliminary step, to compute the corrections which must be applied in consequence of the effects of parallax in longitude and latitude. The chief circumstance, therefore, in which the calculation of solar eclipses differs from that of lunar, consists in its being necessary to compute the effect of parallax in the direction of the angular distances which form the data of the problem, in order to apply the requisite correction to the values of those distances furnished by the tables. The development of these computations belongs to Practical Astronomy.

The sun’s ecliptic limits exceeding 17° 21', while those of the moon are only 11° 26', it follows that the eclipses of the sun must be much more frequent than those of the moon. Yet the lunar eclipses being visible to every part of the terrestrial hemisphere opposite to the sun, and those of the sun visible only to the small portion of the hemisphere on which the moon’s shadow falls, it happens that for any particular place on the earth the lunar eclipses are much more frequently visible.

In any year the number of eclipses of both luminaries: cannot be less than two, nor more than seven: the most 1 usual number is four, and it is very rare to have more than 1 six; for the sun passes through both the nodes but once a year, unless he passes through one of them in the beginning of the year; and if he does, he will pass through the same node again a little before the year is finished; because, as these points move 19^ degrees backwards every year, the sun will come to either of them 173 days after the other; [7:4:12]and. when either node is within 17 degrees of the sun, at the time of new moon, the sun will be eclipsed. At the subsequent opposition, the moon will be eclipsed in the other node, and come round to the next conjunction again ere the former node be 17 degrees past the sun, and will therefore eclipse him a second time. When three eclipses take place about either node, the like number generally happens about the opposite, as the sun comes to it in 173 days afterwards; and six lunations contain but four days more. Thus, there may be two eclipses of the sun and one. of the moon about each of her nodes. But when the moon changes in one of the nodes, she cannot be near enough the other node at the next full to be eclipsed; and in six lunar months afterwards she will change nearer the other node: in these cases there can be but two eclipses in a year, and they are both of the sun.

The eclipses of the sun being of great importance for the determination of geographical longitudes, it is of consequence to be in possession of some easy method of assigning the time at which they may be expected to occur, in order to avoid the necessity of long and tedious calculations. This may be done in a very simple manner, by considering that if a time can be assigned after which the sun and moon occupy exactly or nearly the same positions with regard to the nodes of the lunar orbit, their motions after that interval will recommence under the same circumstances, and the eclipses be reproduced in the same order. Now, it has been shown (chap. iii. sect. 2) that the nodes of the lunar orbit retrograde at the rate of 19° ∙3286 in a year, consequently the time in which the sun returns to the moon’s node is that which he requires to describe an arc of 360° — 19° ∙3286, or, as is found by a simple proportion, 346*619851 days. On comparing this with 29∙5305887 days, the time of a lunation, it will be observed that these, numbers are nearly in the ratio of 223 to 19, so that after 223 synodic revolutions the moon has returned 19 times to the same position relatively to the sun. But 223 synodic revolutions are completed in 18 mean solar years and 10 or 11 days, consequently after that interval all the eclipses, whether of the sun or the moon, return again in nearly the same order; which gives a very simple means of predicting them, since only 18 years of observation are required. This period was known to the astronomers of the remotest ages, and is generally supposed to be that which the Chaldeans distinguished by the name of Saros. (See Part I. of this article.)

But the ratio of 223 to 19 is not exact, and it is besides subject to variation from the secular inequalities of the sun and moon, by reason of which the rates of their mean motions are sensibly changed. Discordancies will hence arise; and in the course of time the order of eclipses observed in one of these periods will require correction. But the variations are slow and gradual; the lunisolar periods may, therefore, continue to be employed when approximative results only are required. When rigorous accuracy is wanted, recourse must be had to computation from the astronomical tables.

Occultations of planets and stars by the moon are phenomena of which the calculation depends on exactly the same principles, and is even made by the help of the same formulae,’as the eclipses of the sun. Let E (fig. 55) denote the centre of the earth, Μ that of the moon, and S a star or planet concealed by the moon; the straight line SM will represent the axis of the lunar shadow, that is to say, the portion of space which the rays proceeding from the star cannot reach in consequence of their being intercepted by the moon; and the angle EMO (EO being perpendicular to SM) will be the apparent distance of the centre of the .shadow from the centre of the earth.

The expression for this angle m a function of the time is obtained in exactly the same manner as in solar eclipses; and by equating it with the different values of the angle which correspond to the different phases of the occultation, and regarding the time as the unknown quantity, the epochs will be obtained at which the phases take place. There is only one circumstance which renders a slight modification of the formulae necessary. In computing the angle SEM, or the apparent distance of the centres of the star and moon seen from the centre of the earth, that distance may be regarded in solar eclipses as the hypothenuse of a right-angled triangle, the sides of which are respectively the latitude of the moon, and the difference between the longitudes>of the moon and sun. But, in occultations of the planets or stars, the star may be. out of the ecliptic, and consequently its latitude not zero; so that the sides of the right-angled .triangle, of which the apparent distance of the centres is the hypothenuse, are the difference of latitude of the star and moon, and the difference of longitude reduced to the moon’s place, that is to say, multiplied by the cosine of the moon’s latitude. It is evident that any of the planets may suffer an. occultation by the moon; but with regard to the fixed stars, it is only those which are situated at a distance from the, ecliptic not greater than the moon’s extreme latitude, that can ever be hid by the interposition of the lunar disk.

The following is a list of all the solar eclipses that will be visible in this country during the present century. The time of the commencement of the eclipse, and the number of digits eclipsed, are computed for the middle of England. (Sec Baily’s Tables, &c. p. 52.)
[table]
[7:4:13]

CHAP. IV.

' Ο L THF ELaNETS. Sect. I.— General Phenomena of the Planetary Motions.

Having now described the motions and explained the phenomena of the sun and moon, our attention will be next occupied by the planets, those no less interesting bodies, whose remarkable peculiarities of apparent motion have attracted the curiosity, and formed a principal object of the labours, of astronomers in all ages. The sun and moon move among the stars always in the same direction, and with velocities nearly uniform; but the planets, though their apparent motions are most frequently from west to east, sometimes appear to have no proper motions, or to remain stationary among the fixed stars; at other times they appear to move in a contrary direction, or to retrograde; and hence the earth cannot be the centre of the planetary orbits. The determination of that centre, and the order of distance in which the orbits of the different planets are placed around it, is comparatively an easy task since the telescope and micrometer have made us acquainted with the phases and variations of the apparent diameters of the planets; but the ancients, who were guided by the apparent motions alone, found greater difficulty in extricating the elements of their theories from observations, and in framing hypotheses by which the phenomena could be represented with tolerable accuracy. The different hypotheses which have been proposed for this purpose are called, with sufficient impropriety, Systems of the World.

In order to obtain a general notion of the path traced by a planet in the heavens, it is necessary to attend closely to the various phenomena which it exhibits. As an example, we may take Venus, the most brilliant and remarkable of all the planets.

A slight attention to the position of Venus, continued a few days, suffices to show that she changes her place with considerable rapidity among the fixed stars. If we observe her in the evening, we shall soon find that her greatest distance from the sun never exceeds an arc of about 47°; that after attaining this distance she begins again to approach the sun, the time which she continues above the horizon after sunset gradually diminishing, till at last she sets simultaneously with the “un, and is lost in the effulgence of his rays. From the circumstance of her appearing in the evening, and not remaining visible more than about three hours after the sun has descended below the horizon, Venus has obtained the name of ’ E<mgoc, Hesperus, or the evening star; sometimes also she is called the shepherd’s star.

A few days after the evening star has disappeared, a brilliant star is observed in the morning preceding the sun in the east, which was not seen while Venus followed him in the west. At first it rises only a few minutes before the sun, but every succeeding morning somewhat earlier, till its distance from him is between 45 and 47 degrees. It then begins gradually to fall back; its elongation or distance from the sun becomes less and less, till it approaches so near to him as to be again lost in his rays. This has been called Φωff⅛sogoj, Lucifer, or the morning star.

Hesperus and Lucifer were long regarded as different stars. It could not fail, however, to be remarked, that during the time the first continues to shine in the evening, the other is invisible in the morning, and as soon as the bright harbinger of day makes its appearance, the evening star ceases to be visible. It was, besides, observed that the one star disappears within a very short distance from the sun, and that the other, wnen first seen after its periodical disappearance, is equally near to him on the opposite side. The distance which each recedes from the sun was likewise remarked to be the same; and the times during which they are alternately visible were found to be equal. These phenomena led some bolder genius to affirm that Hesperus and Lucifer are the same star, which is alternately visible in the morning and evening, according as it precedes or falls behind, the sun. Obvious as this conclusion is, it was not arrived at till after many r ages of reflection and experience.

Since Venus never appears at a greater distance from the sun than about 47°, it is evident that the earth is neither the centre of her orbit, nor included within it; for in either case she would sometimes, like the moon, be seen in opposition to the sun. From the appearances which have as yet been described, it can only be inferred that Venus is a satellite of the sun, and that her orbit is carried along with him in his annual revolution in the ecliptic; but the phenomena which she presents when seen through the telescope afford the means of deducing more definite conclusions.

When Venus, after having been for some time visible in the evening, begins to approach the sun, she appears through the telescope as a fine luminous crescent, the horns of which are turned towards the east, and which becomes narrower as her distance from the sun diminishes. After she has passed the sun, and begins to appear in the morning, the horns of the crescent are turned towards the west, and its breadth gradually enlarges in proportion as the planet recedes from the sun, till she has gained her greatest elongation, when her disk becomes a semicircle. After this she begins to approach the sun with an accelerated motion, and her disk becomes gibbous, the illuminated or visible part being greater than a semicircle; and when she overtakes the sun, her disk has attained the dimensions of a full orb. Having passed the sun, the orb begins again to contract, and passes through the same gradation of changes on the eastern side of the sun, till the planet comes again into conjunction, when it vanishes entirely. All these phases, which nearly resemble those of the moon, are illustrated∙by fig. 56. When Venus is in the superior conjunction at A, or in the same straight line with the sun, she presents the full orb, because the hemisphere enlightened by the sun is turned directly towards the earth at E∙ Arrived at B, the illuminated hemisphere is not turned exactly towards the earth, and consequently one side of her orb must appear elliptical, the major axis being to the minor as radius to the cosine of the inclination of the planes of illumination and vision, as was shown in explaining the phases of the moon. (Chap. iii. sect. 1.) At C, where the straight lines drawn from the planet to the earth and sun form a right angle, the minor axis of the ellipse vanishes, and we have the half-illuminated orb. At D only a small portion of the enlightened hemisphere is visible from E; and when the planet arrives at F, the inferior conjunction, her dark side is wholly turned towards the earth, and she is invisible.

Mercury exhibits phenomena exactly analogous to those of Venus. Like Venus, he oscillates on opposite sides of the sun; but his oscillations are much quicker, and performed in a much smaller arc. His greatest elongation or distance from the sun does not exceed 28° 20'; so that he never appears above the horizon longer than an hour and 50 minutes after sunset or before sunrise. He emits a very vivid white light; but, by reason of his proximity to the sun, he is seldom visible to the naked eye. His phases resemble those of Venus, and he is frequently seen as a dark spot passing over the sun’s disk.
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From these phenomena several important conclusions may be drawn. In the first place, Mercury and Venus are opaque bodies, which are only visible in consequence of the sun’s rays reflected from their surfaces. In the second place, their orbits are described about the sun, and do not embrace the earth, because both planets pass between the sun and the earth, and their digression from the sun never exceeds a certain limit. For this reason they are called Inferior Planets. In the third place, since the digressions of Venus are much more considerable than those of Mercury, it is obvious that her orbit includes that of Mercury. This last fact is established by other phenomena. Their angular velocities may be compared by means of the times which they respectively employ in returning to their conjunctions; and in this way it is found that the angular velocity of Mercury is nearly three times greater than that of Venus—a circumstance of itself sufficiently indicative of his greater proximity to the sun. But the occultation of one of these planets by the other furnishes a decisive evidence of the disposition of their orbits. On the 17th of May 1737 Mercury was observed to be eclipsed by Venus near their inferior conjunction; whence it follows that Venus is nearer the earth, and consequently at a greater distance from the sun. This fact was known to the Egyptians, and the name of the Egyptian System given to that theory according to which Mercury and Venus were regarded as satellites of the sun; but as the ancient astronomers were unacquainted with one of its strongest proofs, namely, the transits, and could only form inferences from the digressions, it was not generally adopted by them.

The revolution of Venus and Mercury about the sun may also be inferred from the variations of their apparent diameters, although, on account of the unequal distances of the sun from the earth, these variations are subject to considerable irregularities. When Venus approaches nearest to the earth, her apparent diameter subtends an angle of about 61", while it does not amount to lö" when she is at her greatest distance. The apparent diameter of Mercury varies from 12" to 5", indicating corresponding variations of distance. These variations of his apparent diameter are not sensible to the naked eye, on account of the irradiation which surrounds the disks of the planets, and which renders it impossible to form any correct judgment respecting the magnitudes of small luminous objects seen from so great a distance.

The phenomena of the other planets differ in some respects from those of Mercury and Venus. Instead of remaining constantly within a certain distance from the sun, their angles of elongation assume all possible values, and they are frequently seen in opposition, and consequently more distant from him than the earth is. Their orbits, therefore, embrace the earth; and as they are never observed in their conjunctions to pass, like the inferior planets, over the sun’s disk, even when the direction of their motion traverses the sun, it follows that the sun is also included within their orbits. This fact is rendered certain by the appearance of their disks at the time when they are in conjunction. If they were then placed between the sun and the earth, their disks would appear cornicular, like those of Mercury and Venus; but they uniformly present a full orb at the time of their conjunction, and consequently the same hemisphere is presented to the earth and the sun, at least if, as is certain, they derive their light from him. They must therefore be situated beyond the sun. From the circumstance that their orbits, include both the earth and the sun, they are called Superior Planets. In order to determine whether the sun or the earth is the centre of their motions, it will be convenient to have recourse to observations of their apparent diameters, which, if their orbits are circular, and have the earth in their centre, will always be of the same magnitude. But the apparent diameter of Mars gradually increases from his conjunction to his opposition; and therefore, since the distance of the planet is directly proportional to the magnitude of its apparent diameter, Mars is nearer to the earth at his opposition than at his conjunction. The variations of the apparent diameter of this planet are very considerable, the limits being 18"∙28 and 3"∙6; so that Mars is five times farther from the earth at his greatest distance than at his least. His orbit, therefore, cannot be a circle described about the earth. We might indeed suppose it to be an ellipse, or other elongated curve; but the enormous eccentricity which it would be necessary to assign to it renders this supposition extremely improbable. The analogy of Venus and Mercury will rather lead us to infer that the sun is the centre of his motion; and of this we have a geometrical proof in the circumstance that the difference of his greatest and least distances is equal to the diameter of the earth’s orbit. Jupiter, Saturn, and Uranus, present exactly similar phenomena. We therefore conclude in the same manner that they are superior planets, circulating about the sun.

The order of distance in which the superior planets are disposed about the sun and the earth may be inferred either from the rate of their motion when they are in opposition with the sun (it being natural to suppose that their velocities will diminish in proportion to their distances or the magnitude of the orbit which they describe), or it may be determined from the variation of their apparent diameters. Now, the apparent diameters at the conjunctions and oppositions are nearly in the following ratio: those of Mars as 1 to 5, of Jupiter as 10 to 15, of Saturn as 1 to 1∙23; hence the diameter of the terrestrial orbit is to the diameters of the orbits of Mars, Jupiter, and Saturn, as the difference of the preceding numbers is to their sum, or as unity to the numbers 1∣, 5, and 9ļ. In this manner we find that Jupiter is at a much greater distance from the earth than Mars, and Saturn than Jupiter.

It has already been remarked that the apparent motions of the planets are not always in the same direction. Through the most considerable part of their orbits they move from west to east, according to the order of the signs; and their motion is most rapid when they are at the greatest distance from the earth. It gradually relaxes until the planet has reached its greatest eastern digression if it is an inferior planet, or its eastern quadrature if a superior one; after which its proper motion is slower than the sun’s motion in the ecliptic. In the course of a short time the planet seems stationary among the stars for some days, its right ascension undergoing scarcely any variation. In the course of a few days more, however, it begins again sensibly to change its place, and now moves in an opposite direction, or retrogrades. This retrograde motion continues to be accelerated as the planet approaches its inferior conjunction, or its opposition, at which point it attains its maximum. After this it begins to be retarded; the planet becomes a second time stationary, and then assumes its direct motion, to pass through another series of similar changes. The arc and time of retrogradation are different for each of the planets, being greatest in the case of those which are nearest the earth, and least for those which are at the greatest distance.

. These phenomena, which are called the Stations and Petrogradations of the planets, were observed with great attention by the ancients and the astronomers of the middle ages, to whom their explanation gave much embarrassment on account of their being incompatible with [7:4:15] the supposition that the earth is the centre of the celestial motions. In fact, if the earth is supposed immovable, the path described by the planets is a curve so extremely complicated and irregular, that Aristotle, and even Riccioli, who lived in an age when the celestial motions were much better known, were reduced to the necessity of supposing a genius or angel to reside in each of the planets, directing its motions as the mind of man directs the motions of his body. Suppose the earth to be at rest in E (fig. 57), the orbit of a superior planet will resemble the curve abcdef, &c. When the planet arrives at b, before it comes into opposition with the sun at A it becomes stationary. From b to c it retrogrades, and is again stationary at c. Its motion then becomes direct, and its distance from the earth continues to increase while it runs through the arc defgh, and till it arrives at h, and is in conjunction with the sun at B. It is then at its greatest distance, and in passing through the arc him continues to approach the earth till it arrives at m, in opposition with the sun at C. Here it exhibits the same phenomena as at the former opposition, becoming stationary, retrograding, &c.; and at every succeeding opposition describes a sort of node or loop, similar to lrnn. Its path is thus made up of an infinity of nodes, and presents a sufficiently striking resemblance to an epicycloid. Fig. 58, which has been copied from Cassini into most of the elementary treatises of astronomy, represents the apparent motions of Saturn, Jupiter, and Mars, in respect of the earth.

Such are the general phenomena presented by the motions of the planets. We must next endeavour to determine the nature of the curve described by each of them, and the law according to which it is described, in order to arrive at the solution of the principal problem of astronomy, viz. to express the position of the heavenly bodies in terms of the time reckoned from a given instant.

The first successful attempt to frame a system by means of which the motions of the planets might be numerically calculated, was made by Apollonius of Perga. Apollonius supposed that a planet, instead of describing a circle about the earth, moves in the circumference of a second circle, the centre of which is carried round the circumference of the first. Let us suppose two unequal circles, situated in the same plane, the greater of which ACB (fig. 59), which is called the deferent, carries on its circumference the centre of the smaller PRP'S, which is called the epicycle; and let us also suppose that each of these circles turns uniformly about its centre, according to the order of the signs, that is, from west to east. The earth is situated at E the centre of the deferent,, and the planet whose motions we consider is placed on the circumference of the epicycle. The phenomena resulting from this disposition will be different according to the ratio of the velocities of the two circles. In the first place, suppose the velocities to be equal, and that at the first instant the planet is placed at P, the extremity of the straight line which joins the centres of the deferent and epicycle: the planet is then at its apogee, and it is evident that its apparent velocity is the greatest possible, being the sum of the velocities in the two circles. It is evident also that the apparent velocity of the planet will diminish as it approaches the lower extremity P' of its epicycle, or its perigee, and that in this point it will vanish altogether, because the perigee is carried backwards in the direction RP'S by the motion of the epicycle with a velocity equal to that by which it is carried forward by the motion of the deferent. From this point the apparent motion will receive a gradual augmentation of velocity, till the planet arrives again at its apogee, where it will recommence a course perfectly similar to the preceding. In this case, therefore, the apparent motion of the planet, though alternately accele- l rated and retarded, will always be direct.

Let us next suppose that the velocity of the epicycle is greater than that of the deferent, which is the hypothesis adopted by Apollonius and Ptolemy. From P to R the motion of the planet is direct. At the point R its path coincides with the tangent RE, and it would be stationary if the epicycle were immovable; but as the epicycle advances according to the order of the signs, the planet will continue to move directly till it arrives at the point -τ, where its retrograde motion in the epicycle, in the direction <ĩĩ P', is equal to the direct motion in the deferent. At the perigee P' the planefr i will retrograde, because the directions of the two motions are diametrically opposite, and the retrograde motion in the epicycle is greater than the direct motion in the deferent. At π' it becomes a second time stationary, after which it resumes its direct motion, precisely according to the actual phenomena.

Lastly, if we suppose the velocity of the deferent to be greater than that of the epicycle, there will be neither station nor retrogradation, and the planet will always advance in the same direction, but contrary to that of the motion of the deferent. This, in the Copernican system, would be the motion of the moon with reference to a spectator placed on the sun, and regarding the orbit of •the earth about the sun as the deferent, and that of the moon about the earth as the epicycle; for then the velocity of the earth about the sun would be about thirty times greater than that of the moon about the earth.

This method of representing the geocentric motions of the planets was adopted and fully developed by Ptolemy, who assigned the ratios of the radii of the epicycles and deferents of each of the planets, and disposed the orbits in the manner most conformable to the apparent motions. It may be remarked that the absolute lengths of the radii are immaterial; it is only their relative lengths to which it is necessary to have regard. As observations were multiplied, and new inequalities in the motions of the planets detected, the system of simple epicycles was in most instances found insufficient to explain the phenomena. Double and triple systems were therefore introduced, in which the first epicycle was regarded as a second deferent carrying its own epicycle, the second epicycle as a third deferent, and so on, till every irregularity of motion was explained. It is easy to conceive all this mechanism, and even to reduce it to general sormulae; but although it affords considerable facility for calculation, it is much too complicated to have place in nature. It is now well known that the orbits of the planets are not epicycloids, but ellipses: nevertheless, at a time when the circle and straight line alone were admitted in the solution of geometrical problems, it was extremely natural to inquire whether any construction could be found, without employing other curves, to represent the planetary motions; and the system of epicycles which resulted from this inquiry will remain a perpetual monument of the ingenuity of its authors, Apollonius and Ptolemy.

In arranging the planetary system, Ptolemy placed the earth at the centre of the universe, and nearest to it the moon, whose synodic revolution is the shortest of all, being performed in 29⅜ days. Next to the moon he placed Mercury, who returns to his conjunctions in 116 days. After Mercury followed Venus, whose periodic time is 584 days. Beyond Venus he placed the sun, then Mars, next Jupiter, and lastly Saturn, beyond which is the sphere of the fixed stars. (See fig. 60.) Plato and some other philosophers had placed the orbit of the sun immediately after that of the moon, and Mercury and Venus beyond the sun, on account that these planets [7:4:16]lwere never seen on the solar disk. Ptolemy, however, • remarked that this reason was inconclusive, because the planets might easily be supposed to pass between the sun and the earth, without appearing exactly on the sun’s disk, in the same manner as the new moons do not always cause a solar eclipse. In the system of Ptolemy, it is, however, a matter of absolute indifference whether the orbits of these planets are placed above or below that of the sun, inasmuch as the phenomena are exactly the same in both dispositions. This fact doubtless furnishes a strong objection to his system, and might have led him, we are apt to suppose, to adopt the system of the ancient Egyptians with regard to Mercury and Venus, and place the sun at the common centre of their orbits. But it must be recollected that Ptolemy had no means of measuring the diameters of the planets, or of forming any accurate notions of their distances; he was unacquainted with the phases of Venus, which demonstrate the revolution of that planet about the sun; and, in short, knew of no phenomenon which could not be reconciled with his system. His object was solely to represent the apparent motions by a geometrical construction, and by such means as geometry at that time could legitimately employ; and this object he fully accomplished. He never regarded his system in any other light than as a mere hypothesis, by means of which the celestial phenomena could be reduced to calculation.

But while thus much must be conceded in favour of the system of Ptolemy, it must be confessed that scarcely any thing could well be imagined more complicated and cumbersome; more at variance with the simplicity which pervades the economy of nature, or, in a physical point of view, more absurd. Yet so prone is the human mind to cling to the ideas which have been first presented to it, and with so much difficulty are errors which have once obtained a firm footing eradicated, that, till the beginning of the 16th century, it was implicitly followed by astronomers of all countries. The glory of bursting the fetters of prejudice and authority, and of building the true system of the world on the ruins of a fabric which had been erected with so much address and labour, was reserved for Copernicus. This great man was led, by a profound meditation on the different hypotheses which had been imagined to account for the apparent motions of the heavenly bodies, to adopt the ideas of some of the ancients, and remove the earth from the centre of the world, ascribing to it a double motion of rotation about its own axis and of revolution about the sun. From some scattered hints contained in the writings of the ancient philosophers, Copernicus composed the system which retains his name; a system of the truth of which the complete development of the planetary theory has furnished the most convincing and satisfactory proofs. According to this system the sun is the common centre of the orbits of all the planets, which revolve around him in the following order :—Mercury, Venus, the Earth, Mars, Jupiter, and Saturn. Far beyond the orbit of Saturn he supposed the fixed stars to be placed, which formed the boundaries of the visible creation.

Although the great simplicity and beauty of the Copernican system soon recommended it to the adoption of the most philosophical of the astronomers of that period, yet for a long time it met with considerable opposition. The most eminent of its opponents was the celebrated Tycho Brahe, who could never bring himself to adopt the supposition of the motion of the earth. The principal objections which he urged against it were the immense distance at which it is necessary to suppose the fixed stars to be placed, in order to account for the smallness, or rather the entire absence, of the annual parallax; the improbability that a heavy mass like the earth should have so rapid a motion; and some passages of Scripture which seem to suppose the motion of the sun and the immobility of the earth. Tycho, therefore, proposed another system, in which he endeavoured to retain the most essential advantages of the Copernican theory, and at the same time preserve the earth’s stability. In the system of Tycho the earth is supposed to be the centre of the solar and lunar orbits, anti the sun to be the centre of all the planetary orbits, the orbits of the two inferior planets being smaller, and those of the superior larger, than the orbit of the sun; a distinction which is necessary for the explanation of the conjunctions and oppositions. This disposition of the orbits will be understood by referring to fig. 61. The system of Tycho is far less philosophical than that of Copernicus, and he has therefore exposed himself to the charge of having made a retrograde movement in science; but it must be confessed that it affords a satisfactory explanation of all the phenomena, and that the objections which can be urged against it are not of an astronomical, but of a physical or mechanical nature. In fact, if the planets are supposed to revolve about the sun, it is absolutely indifferent, so far as regards the phenomena, whether the annual motion is ascribed to the sun or the earth; but it is a physical absurdity to suppose that the sun, with its whole train of attendant planets, revolves about the earth, which, in comparison of them, is a mere atom.

Taking the truth of the Copernican theory for granted, let us consider what effect the motion of the earth has upon the apparent motions of the other planets. Were the earth to stand still in any part of its orbit, the places of conjunction both in the superior and inferior semicircle, as also of the greatest elongation, and consequently the places of direct and retrograde motion, and of the stations of an inferior planet, would always be in the same part of the heavens; whereas, on account of the earth’s motion, the places where these appearances happen are continually advancing forward in the ecliptic, according to the order of the signs. In fig. 62 let ABCD be the orbit of the earth; efgh that of Mercury; © the sun; GFKI an arc of the ecliptic extended to the fixed stars. When the earth is at A, the sun’s geocentric place is at F; and Mercury, in order to be in conjunction, must be in the line AF; that is, in his orbit he must be at f or h. Suppose him to be at/in his inferior semicircle: if the earth stood still at A, his next conjunction would be when he is in his superior semicircle at h; the places of his greatest elongation also would be at e and g, and in the ecliptic at E and G. But supposing the earth to go on in its orbit from A to B; the sun’s geocentric place is now at K; and Mercury, in order to be in conjunction, ought to be in the line BK at m. As by the motion of the earth the places of Mercury’s conjunctions are thus continually carried round in the ecliptic according to the order of the signs, so the places of his greatest elongations must also be carried forward in the same direction. Thus, when the earth is at A, the places of his greatest elongation from the sun are in the ecliptic E and G; the motion of the earth from A to B advances them forward from G to L and from E to I. But the geocentric motion of Mercury will best be seen in fig. 63. Here we have part of the extended ecliptic marked c γ>, b , ∏, &c., in the centre of which S represents the sun, and round him are the orbits of Mercury and the Earth. The orbit of Mercury is divided into 11 equal parts, such as he goes through once in eight days; and the divisions are marked by numeral figures, I, 2, 3, &c. Part of the orbit of the earth is like [7:4:17]wise divided into 22 equal arcs, each are being as much •as the earth goes through in eight days. The points of division are marked with the letters α, b, c, d, e,f, &c. and show as many several stations from whence Mercury may be viewed from the earth. Suppose then the planet to be at 1 and the earth at a ; draw a line from a to 1, and it shows Mercury’s geocentric place at A. In eight days he will have advanced to 2, and the earth to b ; draw a line from 2 to b, and it shows his geocentric place at B. In other eight days he will have proceeded to 3, and the earth to c; a line drawn from 3 to c will show his geocentric place at C. In this manner, going through the figure, and drawing lines from the earth at d, c,fg i &c. through 4, 5, 6, 7, &c. we shall find his geocentric places successively at the points D, E, F, G, &c. where we may observe, that from A to B, and from B to C, the motion is direct; from C to D, and from D to E, retrograde. In this figure 22 stations are marked in the earth’s orbit from whence the planet may be viewed, corresponding to which there ought to be as many in the orbit of Mercury; but as the periodic time of that planet does not include so many intervals of eight days, his place is marked at the end of every eight days for two of his periodical revolutions; and to denote this, two numeral figures are placed at each division.

The geocentric motion of Venus may be explained in a similar manner; only, as the motion of Venus is much slower than that of Mercury, her conjunctions, oppositions, elongations, and stations, all return much less frequently than those of Mercury.

To explain the stationary appearances of the planets, it must be remembered, that the diameter of the earth’s orbit, and even that of Saturn, are but mere points in comparison of the distance of the fixed stars; and therefore any two lines, though absolutely parallel, drawn at the distance of the diameter of Saturn’s orbit from each other, would, if continued to the fixed stars, appear to us to terminate in the same point. Let, then, the two circles fig. 64 represent the orbits of Venus and of the Earth; let the lines AE, BF, CG, DH, be parallel to SP, we may nevertheless affirm, that if continued to the distance of the fixed stars, they would all terminate in the same point with the line SP. Suppose, then, Venus at E while the earth is at A, the visual ray by which she is seen is the line AE. Suppose again, that while Venus goes from E to F, the earth goes from A to B, the visual ray by which Venus is now seen is BF parallel to AE; and therefore Venus will be all that time stationary, appearing in that point of the heaven where SP, if extended, would terminate: this station is at her changing from direct to retrograde. Again, suppose, when the earth is at C, Venus is at G, and the visual line CG; if, while the earth goes from C to D, Venus goes from G to H, so that she is seen in the line DH parallel to CG, she will be all that time stationary, appearing in the point of the celestial sphere determined by the prolongation of SP. This station is at her changing from retrograde to direct; and both are in her inferior semicircle.

As the superior planets move in larger orbits than the earth, they can only be in conjunction with the sun when they are on the side opposite to the earth; as, on the other hand, they are in opposition to him when the earth is between the sun and them. They are in quadrature when their geocentric places are 90° distant from that of the sun. In order to understand their apparent motions, we shall suppose them to stand still in some part of their orbit while the earth makes a complete revolution; in which case any superior planet would have the following appearances;—1. While the earth is in its most distant semicircle, the motion of the planet will be direct. 2. While the earth is in its nearest semicircle, the planet will be retrograde. 3. While the earth is near those places of its orbit where a line drawn from the planet would be a tangent, it would appear to be stationary. Thus, in fig. 65, let abed represent the orbit of the; earth, S the Sun, EFG an arc of the orbit of Jupiter, ABC an arc of the ecliptic projected on the sphere of the fixed stars. Suppose Jupiter to continue at F, while the earth goes round in its orbit according to the order of the letters abed. While the earth is in the semicircle most distant from Jupiter, going from a to b and from b to c, his motion in the heavens would appear direct, or from A to B and from B to C; but while the earth is in its nearest semicircle cde, the motion of Jupiter would appear retrograde from C to B and from B to A; for α, b, c, d, may be considered as so many different stations from whence an inhabitant of the earth would view Jupiter at different seasons of the year; and a straight line drawn from each of these stations through F, the place of Jupiter, and continued to the ecliptic, would show his apparent place there to be successively at A, B, C, B, A. While the earth is near the points of contact a and c, Jupiter would appear stationary, because the visual ray drawn through both planets does not sensibly differ from the tangent Fα or Fc. When the earth is at b, a line drawn from b through S and F to the ecliptic shows Jupiter to be in conjunction with the sun at B. When the earth is at d, a line drawn from d through S, continued to the ecliptic, would terminate in a point opposite to B; which shows Jupiter then to be in opposition to the sun; and thus it appears that his motion is direct when he is in conjunction, but retrograde when he is in opposition with the sun.

The direct motion of a superior planet is more rapid the nearer it is to a conjunction, and slower as it approaches to a quadrature with the sun. Thus, in fig. 66, let Θ be the sun, the little circle round it the orbit of the earth, of which abcdefg is the most distant semicircle, OPQ an arc of the orbit of Jupiter, and ABCDEFG an arc of the ecliptic in the sphere of the fixed stars. If we suppose Jupiter to stand still at P, by the earth’s motion from a to g he would appear to move directly from A to G, describing the unequal arcs AB, BC, CD, DE, EF, FG, in equal times. When the earth isat√, Jupiter is in conjunction with the sun at D, and there his direct motion is swiftest. When the earth is in that part of its orbit where a line drawn from Jupiter would touch it, as in the points e or g i Jupiter is nearly in quadrature with the sun; and the nearer the earth is to any of these points, the slower is the geocentric motion of Jupiter; for the arcs CD and DE arc greater than BC or EF, and the arcs BC and L’F are greater than AB or FG.

The retrograde motion of a superior planet is more rapid the nearer it is to an opposition, and slower as it approaches to a quadrature with the sun. Thus, let Θ,fig. 67, be the sun, the little circle round it the orbit of the earth, whereof ghihlmn is the nearest semicircle, OPQ an arc of the orbit of Jupiter, NKG an arc of the ecliptic. If we suppose Jupiter to stand still at P, by the earth’s motion from g to n he would appear to move from G to N, describing the unequal arcs GH, HI, IK, KL, LM, MN, in equal times. When the earth is at h, Jupiter appears at K in opposition with the sun, and there his retrograde motion is swiftest. When the earth is either at g or n (the points of contact of the tangents Pg and Pn), Jupiter is nearly in quadrature with the sun: and the nearer he is to either of these points the slower is his retrogradation; for the arcs IK and KL are greater than HI or LM, and the arcs HI and LM are greater than GH [7:4:18]lor MN. Since the direct motion is swiftest when the earth is at d (fig. 66), and continues diminishing till it changes to retrograde, it must be insensible near the time of change; and, in like manner, the retrograde motion being swiftest when the earth is in k (fig. 67), and diminishing gradually till it changes to direct, must also at the time of that change be insensible; for any motion gradually decreasing till it changes into a contrary one gradually increasing, must at the time of the change be altogether insensible.

The same changes in the apparent motions of this planet will also take place if we suppose him to advance slowly in his orbit; only they will happen successively when the earth is in different parts of its orbit, and consequently at different times of the year. Thus (fig. 65), let us suppose that while the earth goes round its orbit, Jupiter goes from F to G; the points of the earth’s orbit from which Jupiter will now appear to be stationary will be x and y; and consequently his stations must be at a time of the year different from the former. Moreover, the conjunction of Jupiter with the sun will now be when the earth is at f, and his opposition when it is at e ; for which reason these also will happen at times of the year different from those of the preceding opposition and conjunction. The motion of Saturn is so slow, that it occasions but little alteration either in the times or places of his conjunction or opposition; and the same will take place in a more eminent degree in Uranus; but the motion of Mars is so much swifter even than that of Jupiter, that both the times and places of his conjunctions and oppositions are thereby very much altered.

Fig. 68 exemplifies the geocentric motion of Jupiter in a very intelligible manner. In this figure Θ represents the sun; the circle 1, 2, 3, 4, the orbit of the earth, divided into twelve equal arcs for the twelve months of the year; PQ, an arc of the orbit of Jupiter, equal to that which he describes in a year, and divided in like manner into twelve equal parts, each representing the arc he describes in a month. Now suppose the earth to be at 1 when Jupiter is at a, a line drawn through 1 and a shows Jupiter’s place in the celestial ecliptic to be at A. In a month’s time the earth will have moved from 1 to 2, Jupiter from a to b ; and a line drawn from 2 to b will show his geocentric place to be in B. In another month the earth will be in 3, and Jupiter at c, and consequently his geocentric place will be at C; and in like manner his place may be found for the other months at D, E, F, &c. It is likewise easy to observe that his geocentric motion is direct in the arcs AB, BC, CD, DE; retrograde in EF, FG, GH, HI; and direct again in IK, KL, LM, MN. The inequality of his geocentric motion is likewise apparent from the figure.

Supposing the orbits of the planets to be circular, the points of station, and the extent and duration of the retrogradations, may be geometrically determined with great facility; but if we attempt to take into consideration all the inequalities of the orbits, the problem becomes one of extreme complication and difficulty. This, however, is the less to be regretted, on account that it is more a problem of curiosity than of any real importance, and the mean values which result from the supposition of circular orbits are more than sufficiently accurate for all the uses which are now made of these phenomena.

Let S, fig. 69, be the sun, E and V two planets (which we may suppose to be the Earth and Venus) revolving in their respective orbits in the times T and t : the problem is, to determine those points at which one of them, as seen from the other, will become stationary, and the extent of the arc through which it will appear to retrograde. Suppose the spectator to be placed at E, and V to be the point at which Venus is stationary. Also let the semidiame- ' ters of their orbits SE and SV be R and r respectively, the ∙ angle of digression SEV=p, and the angle SVE=∙ψ. During a short time, at the station, the lines EV and cv (Ee and Nν being very small arcs) may be considered as parallel; therefore <p=Suv= Sey + ESe=<p+ δφ + ESe, whence δφ= — ESe. In like manner ∙ψ=Sire=Sve—VS-υ=ψ + δ∙ψ—VSv; whence δ∙ψ=VSv. But as the orbits are described uniformly, T: t :: VSw: ESe, and consequently T: t :: δ∙ψ: δ<p. Now, the triangle SEV gives the equa tion sin. ∙ψ=- sin. p,from which, by differentiating, we get

R

δ∙ψ cos. ∙ψ = — δ<p cos. φ, and consequently ð-ψ: δ<p ::

R cos. <p : r cos. ∙ψ. Comparing this with the above proportion, we have T: t :: R cos. φ : τ cos. ψ, whence r cos. ∙ψ

=-ψR cos. <p. Taking the squares, and adding r 2 sin. 2 ∙ψ =R 2 sin. 2 φ, there results r 2 T 2 =R 2 (< 2 cos. 2 ^>+T 2 sin. s p), whence we derive

1 p∙ 2 T 2 -R¾ 2 ∖⅜ sm '^ - R∖ T 2 - t i ) ’

0 T t or, if we suppose -jξ=r', and ^ψ=G

—-Ofe§}'·

From this formula, all the circumstances connected with the stations and retrogradations are easily computed, when the diameter of the orbit and the periodic time of the planet are known. Thus, relatively to Venus, the quantity ?, that is, the ratio of the diameter of her orbit to that of the earth, is, as will be shown in next section, equal to 0*723, and

224∙7 ∕1∙338×∙108

r =36S⅛5 = 0 βl62 ' whence sm ∙ y = √ l⅛19×∙88S =

0∙48208; and <p=28° 49'. The angle -ψ is obtained from the equation sin. ∙ψ=-^' -, which gives >ψ=138° IP. The angle VSE at the sun, or the heliocentric distance of Venus and the Earth, which we shall designate by χ, is 180° — (p + +)=13°. From this angle we likewise obtain the time from the station to the conjunction; for since the mean daily motion of Venus is 1° 36' 7"∙8, and that of the earth 59’ 8"∙3, the relative daily motion is 36' 59"∙5, 13°

therefore the time from conjunction will be √ qzz .√> or uD O*z o about 21 days, and consequently the retrograde motion of Venus will continue about 42 days.

ļ ∕∕ ι ^ r P^ —— R⅜2∖

If in the equation sin. φ — R∖T 5 f Γ^) we su PP ose rT = Rí, the angle <p vanishes; consequently there will be no regression, and the station will only be momentary at the inferior conjunction. In this case r : R:: t : T, or the distances of the planets are proportional to their periodic times. If Rí were greater than rT, <p would become imaginary, and the motion of the planet would always be direct. The stations and retrogradations are therefore consequences of that law of the system according to which the distances are to one another in a greater ratio than the periodic times.

The following table, given by Delambre {Astronomic, tςme iii. p. 9), exhibits the elements of the stations and retrogradations of the planets, calculated from formulae equivalent to the above.
[7:4:19]
[table]

Sect. H.— Of the Orbits of the Planets.

In determining the elements of the orbits of the sun and moon, the labour of the astronomer is facilitated by the circumstance that the earth, at which his observations are made, either is, or may be regarded as, the centre of motion. But in the case of the planets, the sun, and not the earth, is situated in the centre of their orbits, and consequently the elements of those orbits must be determined from the measurement of certain linear and angular distances from the sun: it is therefore necessary, as a preliminary step, to convert the geocentric into heliocentric observations, that is, to deduce the true place of a planet as seen from the sun, from its apparent place as seen from the earth. On this account the determination of the planetary orbits is attended with somewhat greater difficulties than those which present themselves in the cases of the suñ and moon.

1. Nodes and Inclinations of the Planetary Orbits. — Conformably with the plan which has been followed in the two preceding chapters, we will first consider those elements which determine the position of the plane of the orbit of a planet; and then those which regard the orbit itself, that is to say, the elements and position of the ellipse which the planet describes. When these are known, and the astronomer is also acquainted with the mean motion, or the time of the periodic revolution of the planet, and the instant at which it occupied any given point of its orbit, he is in a condition to assign the epoch at which it will again occupy the same or any other given point; and this comprehends the complete solution of the problem which he proposes to himself. It is proper, however, to remark here, that although this method of considering the elements separately is extremely convenient, perhaps indispensable, for the purpose of illustration, it is neither that which the practical astronomer follows, nor that by which the various truths which compose the actual system of theoretical astronomy were discovered. No discovery in astronomy has perhaps ever been made by a direct process. The elements first obtained from the observations are imperfect and inexact, and it is only by successive and frequently laborious approximations that they are advanced to accuracy. Hypotheses are first framed to account for or classify the phenomena; the results of these arbitrary suppositions are computed and compared with new observations; and the differences which are found to exist between the computed and observed quantities serve to verify or correct the assumption, or suggest other approximations still nearer the truth. In astronomy, as Woodhouse remarks, scarcely one element is presented simple and unmixed with others. Its value, when first disengaged, must partake of the uncertainty to which the other elements are subject, and can be supposed to be settled to a tolerable degree of exactness only after multiplied observations and many revisions. There are no simple theorems for determining at once the parallax of the sun, or the heliocentric latitude of a planet.

To give a general idea of the method of reducing geocentric to heliocentric longitudes and latitudes, let S (fig. 70) be the sun, E the earth, P the place of a planet in its orbit NP, NA the ecliptic, and N the ascending node, or that point of the ecliptic through which the planet passes when it comes to the north of that plane. Let Sɤ, Eɤ, be drawn from S and E to the first point of Aries; and because the parallax of the fixed stars is insensible, their distance being infinitely great in comparison of the distance of the earth from the sun, S ɤ and E ɤ are considered as parallel to one another. Let also PL be drawn perpendicular to the ecliptic, and the other lines as in the figure. It is easy to see that as NPC, NLA are regarded as the intersections of the planes of the orbits of the earth and the planet with the celestial sphere, the centre of which is occupied by the sun, they are portions of great circles; and consequently every question relative to the position of the planet may be resolved by the formulae of spherical trigonometry. In fact, PL being a circle of latitude, the arc <y>NL is the Heliocentric Longitude of the planet, ɤ N the longitude of the ascending node, and the spherical angle PNL the inclination of the orbit. In order that the longitude of the planet in its orbit may not differ greatly from its longitude referred to the ecliptic, the former is not reckoned from the node N, but from another point equally distant from the node with the vernal point <v; so that the longitude of the planet is obtained by adding NP to the longitude of the node. The latitude also depends immediately on the distance NP of the planet from the ascending node; hence NP is called the Argument of the Latitude ; and the difference between the longitudes in the orbit and ecliptic, that is, NP—NL, is called the Reduction to the Ecliptic. The angle PEL, or the planet’s apparent distance from the ecliptic to an observer at E, is the Geocentric Latitude ; PSL, its distance from the ecliptic when viewed from the sun, is its Heliocentric Latitude. It is evident from the mere inspection of the diagram, that ɤ EL and ɤSL are the geocentric and heliocentric longitudes respectively. The angles of the triangle ESL, from which the reductions are computed, being of very frequent use in practical astronomy, have received certain technical denominations with which it is convenient to be acquainted. The angle LES, which is the difference between ɤEL and <γΕS, the geocentric longitudes of the planet and the sun, is called the angle of Elongation ; ESL, the angle at the sun, is called the angle of Commutation ; and SLE, which is the difference between the heliocentric and geocentric longitudes, is called the Annual Parallax, or the Prosthaphoeresis of the Orbit. In the case of an inferior planet, the annual parallax SLE may have any value between zero and 360°; but if the planet be a superior one, the greatest value of the annual parallax has place when the straight line drawn from the planet to the [7:4:20] earth is a tangent to the earth’s orbit. In any other situation of the planet the sine of the annual parallax will be to the sine of the elongation as SE to SL.

In order to determine the position of the orbit of a planet relatively to the ecliptic, it is necessary to discover in the first place the place of the nodes. The longitude of the node is the heliocentric longitude of the planet at the instant when it is in the ecliptic, and its geocentric and heliocentric latitudes are consequently both equal to zero. If then, at the same instant, the planet should happen to be in opposition with the sun, or (if an inferior one) in its inferior conjunction, its geocentric longitude, which is equal to the longitude of the sun + 180°, would be the heliocentric longitude of the node. But it can happen very rarely that the latitude of a planet is zero at the time of its opposition or conjunction, especially with regard to those of which the periodic times are considerable. Mercury and Venus, indeed, by passing oftener through their nodes within a given time, afford more chances of the occurrence of the phenomenon; and it does occur occasionally under very favourable circumstances when those planets pass over the disk of the sun, for at such times the latitude must be very small, that is, the planet must be very near its node. But this simple method of determining the heliocentric longitude of the node, although it may frequently be had recourse to in the case of Mercury, can seldom be practised with regard to Venus, and is of no use whatever in the cases of the other planets, because, on account of the greater length of their times of revolution, the oppositions and passages through the node take place simultaneously only after very long intervals.

Leaving, therefore, out of consideration the circumstance of. opposition, let us inquire what other means are in our possession for determining the longitude of the node. It would be by an extraordinary chance that a planet should happen to be observed at the precise instant of its passage through its node; but if observed when very near it, the diurnal variations of latitude afford data for determining that instant by interpolation. Suppose, then, the planet to be in its node at Ń: the angle ɤ EN is obtained from observation, being the geocentric longitude of the node; and ɤES, the longitude of the sun, is known from the solar tables. Hence, in the triangle ENS the angle SEN = ɤEN—ɤES is known, as is also the side SE, which is the sun’s distance from the earth. If, therefore, the angle at the planet, that is ENS, can be found, every part of the triangle will be determined, and we shall then have the heliocentric longitude ɤSN (equal to 180° —ESN + ɤES), and also the radius vector SN, or the distance of the planet from the sun. Now, the sides of a triangle being proportional to . 1 ∙ f n . . i . SN sin. SEN

the sines of their opposite angles, rrr; = -∑ =

11 δ ’ SE sin. ENS

sin. (ɤEN — ɤES) , „ , .

- sin. (ɤ SN ɤEN) ’ but from t US e 9 uatl0n n θhing can be deduced, inasmuch as it contains two unknown quantities, SN and ɤ SN. Suppose, however, that the planet is observed a second time in its passage through the same node, and that the earth is then in another part of its orbit at e : this second observation will furnish another

. . SN sin. SeN sin.(ɤeN—<weS)

1 Se sin. eNS sm.(ɤSN—ɤeN) , e act ly similar to the former, and in which Se is equal to SE. If, then, we suppose the place of the node has not sensibly changed in the interval between the two observations, this last equation will furnish a second relation between the two unknown quantities SN and ɤSN, by combining which with the former, the values of both those ' quantities may be determined. By repeating the same; observation, it will be seen whether the place of the node ' is fixed, or is subject to any other variation than that which arises from the precession of the equinoxes. Its variation, if it does vary, can only be determined by observations made at distant epochs.

When the longitude of the node has been determined in this manner, it will be easy to deduce the inclination of the orbit. For this purpose the planet may be observed at the time when the sun’s longitude is equal to that of the node, and the earth is consequently situated in the line of the nodes. By this observation the elongation of the planet, which is then PEN (fig. 71), is given, as also its geocentric latitude PEL. Now, by reason of the right-angled triangles PEL and PSL, we have PL = EL tan. PEL = SL tan. PSL, consequently tan. PSL

EL

- ±± tan . PEL; but EL: SL:: sin. LSN: sin. LES, therefore tan. PSL = S ? n ‘ tan. PEL. Now, by Na-sin. LES j pier’s rules for circular parts, the right-angled spherical triangle PNL gives also sin. NL = cot. PNL tan. PL; whence tan. PL = sin. NL tan. PNL; or, since tan. PL = tan. PSL, and sin. NL = sin. LSN, therefore tan. PSL = sin. LSN tan. PNL. By equating these two values of tan. PSL, we have

• -r (^iλt τwτ-r Sin. LSN τ -, τ sm. LSN tan. PNL = —-.— τ -^,- c ,∙ tan. PEL; sin. LES therefore, ultimately, ττ tan. PEL tan. PNL = τ 1 , c ,-; sin. LES

that is, the inclination of the orbit is given in terms of the geocentric latitude and the longitude of the node.

The instant at which the sun is in the node of a planet’s orbit cannot be easily seized; but the sun’s mean motion being known, if his longitude is observed when it is nearly equal to that of the node, the time at which he passes through it may be determined with all the necessary accuracy by a simple proportion. It requires also to be remarked that the above method of determining the inclination takes it for granted that the position of the node is exactly known; but even should some uncertainty remain regarding this element, the resulting inclination would scarcely be affected in any sensible degree by a slight error, especially if at the time of the observation the planet is not very distant from its quadratures. In fact, if we make PNL = I, PEL = λ, and LES = p, the above equation becomes tan. I =^L^, which being differentiated

.. , . τ ı - ∙ i τ Ŵcos.øtan.X

with respect to 1 and 0, gives d tan. 1 = -— r -- .

sm. 2 p

But d tan. I = an d fr°m the equation itself we

Sl∏∙^ 0

derive cos. 2 1 = -?—≈ r ~ t -—2?; therefore, by substituting sin. Ç) -Į-tan. λ i ∕ Į _ P c0s, P tan ∙ sin. 2 φ + tan. 2 λ

From this it is evident that the error of inclination resulting from an erroneous position of the node will be smaller in proportion as φ is greater, and will disappear altogether when φ (that is, LES or the geocentric distance of the planet from its node) is 90°, that is to say, at the quadratures.

The process now explained for determining the position of the nodes gives at the same time the length of [7:4:21]the radius vector SN; and, in consequence of certain rela- ∙tions which will be pointed out in the present section, all 'the other elements of the orbit may be deduced from simple observations of the passages of a planet through its nodes. But when approximate values of these elements are known, the heliocentric longitudes and latitudes may be found from geocentric observations of various kinds. If, therefore, the planet is observed several times before and after its passage through the node, the diurnal variations of longitude and latitude (which we at present suppose to be known very nearly) will give by interpolation the instant at which the latitude was nothing, and the heliocentric longitude, computed for that instant, will give the place of the node. By means of this more accurate determination the other elements may be corrected; and on repeating the same process with the corrected values, the longitude of the node will be obtained with still greater precision. These indirect methods of correcting the elements by one another were first employed by Kepler, and are of extensive application in every department of astronomy. They lead to more exact values of the elements of the planetary orbits than could be obtained from any direct method whatever.

As the passage of a planet through either of its nodes takes place only once during each periodic revolution, a remote planet, like Uranus, which moves very slowly, can very seldom be observed in those positions. Astronomers have therefore sedulously employed themselves in devising other methods by which all the elements of a planet’s orbit may be discovered when a very small portion of it has been made known by observation. But these methods are the results of profound mathematical theory, based on the principle of Universal Gravitation, and could not therefore be properly explained in this place. Their application belongs to Practical Astronomy.

When the longitudes of the nodes of the planetary orbits are deduced by the methods explained above from ancient andmodern observations,it is found that the nodes are not altogether fixed, but that, in respect of the fixed stars, they retrograde, or move from east to west; buttheir motion is so slow, that, although it is in reality, like every motion of the system, alternately accelerated and retarded, it may be regarded as uniform during a very long period of time. The inclinations of the orbits also undergo small variations, scarcely sensible to observation. The retrograde motion of the nodes of the planets, exactly analogous to that of the nodes of the moon, is a necessary consequence of the mutual gravitation of all the planetary bodies. The same is the case with the variations of inclination, of which the period and limits can only be calculated from the same theory to which we owe the knowledge of their existence. It is only indeed by the aid of the theories of Physical Astronomy that the greater part of these smalI secular variations can be disengaged from the periodic inequalities.

The following table exhibits the inclinations of the orbits of the planets, the positions of their nodes, and the variations of those elements. For the old planets the epoch is the commencement of the present century, that is to say, midnight preceding the 1st of January 1801. For Vesta, Juno, Ceres, and Pallas, the epoch is 1820. The annual increment of the longitude of the node is referred of course to the equinoctial point of Aries: to find its motion in respect of the fixed stars, it is only necessary to subtract the precession of the equinoxes, or 50"∙1, from the numbers given in the table.
[table]

2. Of the Figures of the Orbits. — After the position of the plane of a planet’s orbit in space has been determined, it remains to trace its path on that plane, or to determine the figure and elements of the orbit itself. This, it is evident, may be accomplished, if we are in possession of the means of assigning, at any instant, the planet’s distance from a fixed point in the plane of its orbit, and likewise the angle formed by the radius vector with a straight line given by position on the same plane. With these data a series of points may be laid down, representing the positions successively occupied by the planet; and the curve formed by joining them together will represent the orbit. Now, when the place of the nodes and the inclination are known, the radius vector and elongation may be computed by the rules of spherical trigonometry from a single observation of the planet’s geocentric latitude and longitude at any epoch whatever; but by making the observations when the planet occupies certain situations of its orbit with respect to the earth, the difficulties attending the computation may be in a great measure eluded. If a planet, for example, is observed at the time of its opposition or conjunction, there will be no occasion for a previous deduction of the heliocentric from the geocentric longitude; because, in either of those cases, the radius vector of the planet, and the straight line EP (fig. 72) drawn from it to the observer, are projected on the plane of the ecliptic in the same straight line ESL; and consequently the planet, whether seen from the earth or the sun, is referred to the same point of the ecliptic, excepting indeed the inferior planets, which in their inferior conjunctions are referred to points diametrically opposite. Let P, therefore, be the place of a planet in opposition; then, in the spherical triangle PNL, right-angled at L, the side NL, being the longitude of the planet minus that of the node, is given by observation (the longitude of the node being supposed to be previously determined); the inclination PNL is also known; therefore, by Napier’s rules, τ , xτ tan. NL tan. PN = τ -- τ ; cos. PNL

that is, the elongation of the planet is given in terms of its longitude and the inclination of its orbit. The same [7:4:22]triangle PNL also affords data for determining the radius vector; for tan. PL = tan. PNL sin. NL; but PL is the measure of the angle PSL, and consequently PSL, the heliocentric latitude, also becomes known. Now, in the triangle PES the angle PES is given by observation, therefore EPS (= PSL — PES) is also given; but

PS: SE:: sin. PES: sin. EPS, whence PS = SE

sin. EPS or the radius vector is found in terms of the sun’s distance from the earth and given angles.

In consequence of the incommensurable relation subsisting between the times of the periodic revolutions of the planets and that of the earth, the successive oppositions and conjunctions never take place at the same points of the orbits; therefore, by making a number of similar observations, as many different angles of elongation and radii vectores will be found, which, unless the orbits are circular, will all have different values. By computing, therefore, the values of these radii vectores in terms of the greater axis of the earth’s orbit, and laying them down in their true positions round the sun, we shall obtain an approximation to the curve described by the planet on the plane of its orbit. A few observations of this sort will suffice to show that the orbit is eccentric; and as we have already seen that the solar orbit is an ellipse, analogy will immediately suggest the probability that that of the planet is also an ellipse, having the sun in one of its foci. It is easy to determine whether this be the case or otherwise. Three points given by position on a plane completely determine an ellipse of which a focus is known: having therefore computed an ellipse which satisfies three observations of a planet, its periphery, if the hypothesis of elliptic motion is correct, will comprehend all the places of the same planet computed from any other observations. Now, it is found that this is what takes place with regard to every one of the planets; hence results the first of the three laws discovered by Kepler which form the basis of the whole system of modern astronomy, namely, that the orbits of the planets are ellipses, of which the sun occupies one of the foci.

By a comparison of the sectors formed by two contiguous radii vectores and the arcs included between them, with the time consumed by the planet in describing those arcs, Kepler was led to the discovery of the second great law of the planetary motions, namely, that the areas described by the radius vector of a planet are proportional to the times employed in describing them. The data from which this important conclusion was deduced were not rigorously exact; but the fact itself has been confirmed by an infinity of observations since the time of Kepler; and Newton, by the application of geometry to dynamics, demonstrated that it is necessarily true of all motions regulated by a central force, whatever the nature of that force may be.

Kepler having, by means of the most laborious computations, established the existence of these two laws with regard to each of the planets separately, next undertook to discover whether any analogy could be found regulating their mean distances from the sun. As there were only six planets known in his time, he commenced his investigation by comparing the intervals between their respective orbits with the five regular geometrical solids. Having failed in this speculation, and in various others suggested by a mind equally fertile and persevering, he at length, by a happy inspiration of genius, thought of comparing the mean distances of the planets with their respective periods of revolution, and soon perceived that the numbers which represent the periodic times of the planets, beginning with Mercury and ending with Saturn, increase in a much greater ratio than those representing their mean distances from the sun. Jupiter, for example, vs four times more distant from the sun than the earth is; but his period is twelve times greater than that of the earth. Kepler then tried the various powers of the distances and periods. Unfortunately an error of computation for some time concealed the discovery from him; but having resumed the subject, he at length found, on the 15th of May 1618 (and few days, as Mr Playfair has remarked, are more memorable in the annals of science), with all the delight which a great and important discovery gives rise to, that the squares of the numbers which ' express the times of revolution are to one another as the cubes 1 of those which express the mean distances of the planets from the sum

These three general laws, which are necessary consequences of the law of gravitation, directly as the masses, and inversely as the squares of the distances, greatly facilitate the investigation of the orbit of a newly discovered planet, and may even be regarded as more accurate than the results obtained from any moderate number of observations, however exact. Thus, instead of having recourse to observation for the determination of the radius vector, which cannot be found by direct observation but with considerable difficulty, it is preferable to deduce it by means of the third law from the planet’s periodic time, which can always be determined with great precision, and with much greater facility. They also afford a strong analogical proof of the annual motion of the earth. In fact, if we compute from the third law the periodic time of a body placed at the same distance from the sun as the earth is known to be, the result will give exactly the sidereal year. It is therefore certain that the earth obeys the same laws as the planets; and, when classed among them, the most perfect analogy pervades the whole system.

The periodic time of a planet may be determined by observing the interval which elapses between two consecutive passages through the same node. The retrograde movement of the nodes being, as we have seen, inconsiderable, the planet, in returning to its node, also returns to the same position very nearly with reference to the fixed stars, and therefore has in the interval completed a sidereal revolution. Hence the mean sidereal motion is known, and consequently, from the third law of Kepler, the planet’s mean distance from the sun, the double of which is the transverse axis of the orbit. But the observation of the passages through the nodes not only gives the time of a sidereal revolution, but also approximate values of the eccentricity and the position of the line of the apsides; for, unless the orbit is circular, or the line of the apsides coincides with that of the nodes, the time the planet consumes in passing from the ascending to the descending node, or from ♯ to ö, will be different from that in which it passes from S to $2; and, by a comparison of this difference with the theory of elliptic motion, it is easy to deduce the greatest difference between the true and mean anomalies, that is, the greatest equation of the centre, and thence the eccentricity. The method is exactly similar to that which has already been described in chap. ii. sect. 2.

But as these methods lead to results of no great accuracy, and as the passages through the nodes in the case of the distant planets occur only after considerable intervals of time, the astronomer requires some more accurate and expeditious means of determining the elements of an orbit; and three geocentric observations of the planet in any part of its orbit whatever are sufficient to determine these elements, when the nodes and inclination, mean motion and mean distance, are known. Having already [7:4:23]explained in last section the manner in which the geocentric are converted into heliocentric longitudes, and reduced to the plane of the orbit, let us assume v, d, v", to represent three heliocentric longitudes so reduced, and ¶ the longitude of the perihelion, also reduced to the plane of the orbit. The true anomalies corresponding to the three observations will consequently be v — ¶, d — if — ¶. Now, if, as in the series given in chap. ii. sect. 2, for the true anomaly in terms of the mean, we express the mean motion by n, and the time elapsed since the passage through the perihelion by t, the three corresponding mean anomalies will be respectively nt, nt’, n t’ ; and it will be observed, that although t is unknown, yet as t , — t a ∏d t'' — t (the intervals between the first and the other two observations) are known, t' and t" are both given in terms of t.

Omitting the square, and all the higher powers, of the eccentricity e, the series just referred to gives the three following equations, nt —v —ff—2esin. (v — ιr) η t’ =v t — ¶ — 2 e sin. (v r — ff) n t’'=v” — ¶ — 2esin. (v" — ff) from which we must deduce the three unknown quantities t, e, π, that is to say, the epoch of the planet’s passage through the perihelion, the eccentricity, and the longitude of the perihelion. By subtracting the first of these equations from each of the other two, we obtain n(t' — t)=v ’ — v — 2e [sin. (√ —ff)— sin. (v — ff)] n(t ’ ' — t)-v" — v — 2 e [sin. (v" — V) — sin. (v — ff)ĵ; and if we assume n(t' — i) — (ν t — v')z∑a n(f' — t) — (v" — v)=b (a and b being thus known quantities), these two equations give us the two following, a— — 2 e [sin. (v’ — π) — sin. (v — ff)]

δ= — 2 e [sin. (√' — ff) — sin. (v — ∙κ) J; whence, by division, a sin. (√—ff)— sin. (υ — ff) b~ sin. (vl' — ff)— sin. (v — ff) Now, the numerator of this fraction

. . fsin.(√-π) ì

= sin. Cv — ¶) V ——y < — 1 f v tsm. (v — ¶·) J

. , x ( sin. d cos. ¶—cos. d sin. ff ì

= sin. (v — ff) < - 1?

Ism. “cos. w—cos. v sm. ff j

. , . ssin.î/— sin.v — tan.fffcos.î/—cos.v)1

= sm.(ι ι —ff) λ 7 i 2 ∙

t sin. v — cos. v tan. ff 1

and, similarly, the denominator

. z λ fsin.√'—sin.∙υ—tan.ff(cos.∙zf—cos.v)i = sin. (v — π) 4 7 ï L V ∙

I sin. v — cos. v tan. w J therefore

π sin, vl — sin, v — tan, ff (cos, r d > — cos. v) b sin. v" — sin. v — tan. ¶ (cos. v" — cos. f) ’ whence

. ι ~ 0 (sin. vt' — sin. ν) — ó(sin. v’ — sin. v') «(cos. v" — cos. v') — Z>(cos. v' cos. îD ’

an equation which gives ff, the longitude of the perihelion reduced to the orbit, the angles v, v ’ , v", being given by observation.

Having thus found ff, and consequently v' — <κ, v" — ff, it is easy to determine the eccentricity e. From the equa-

α ~ — 2 e [sin. (v’ — ff) — sin. (v — ff)ļ we obtain 2e = sin. (v’ — ff) — sin. (v — ff) ’ a formula which may be rendered better adapted for logarithmic calculation by being put under the form a ¢ —» -- - - - - _ - -

. . v' — v — (v' + v ∖ , 4 sin. —- cos. I q — — ff y

By means of the values of e and v — <ιr, which have now been found, we obtain t, the epoch of the passage of the planet through its perihelion, immediately from the equation n t — v — ff — 2 e sin. (v — ff).

The only element which now remains undetermined is the radius vector, and this is given by the polar equation of the ellipse, viz.

_ fl (1- c 2)

1 + e cos. (v — ff)’ in which a represents the semi-axis major, or mean distance found by the third law of Kepler.

In this manner approximate values of the different elements are obtained, but no method can be proposed which will give them, without repeated corrections, so accurately as to accord with the precise observations of the present day; and the great object of the practical astronomer is to advance them gradually nearer and nearer to the truth. From the laws of Kepler applied to a few observations, formulae arc constructed which represent the mean values of the elliptic elements nearly; every successive observation gives a geocentric latitude and longitude; and by comparing these with the corresponding latitude and longitude computed from the formulae, equations of condition are obtained, by means of which the formulae are corrected and rendered still more exact. New observations give new equations of condition, to be joined with the former; and after a great number of such equations have been obtained, geometry teaches us how to combine them so that each may have its just influence in the determination of the final result. Thus the tables are gradually approximated to perfection, or to such a state that they differ only insensibly from the mean of a great number of observations.

But the great difficulty of obtaining an exact conformity between observation and the results of computation arises from the circumstance that the different elements of the orbits have no fixed values, but are incessantly though slowly varying, in consequence of the mutual disturbances which the planets occasion to the motions of each other. If the planets had no mutual attraction, and obeyed only the central force of the sun, the problem of determining the different circumstances of their motion would be one of easy solution, and each of them would accurately describe an ellipse on the plane of its orbit, according to the laws of Kepler. In consequence, however, of their mutual attraction, every planet is compelled to deviate more or less from its ellipse; so that when the matter is viewed with mathematical precision, the laws of Kepler belong only to an ideal system, and have no actual existence in nature. It is in the determination of the amount of these causes of perturbation, and the secular variations they give rise to, that the most profound and intricate theories of physical astronomy find their application; for it is from theory alone that we can derive any knowledge of the laws and periods of changes which are completed only after many hundreds or thousands of years. The planets whose motions are most disturbed by their mutual attractions are Jupiter and Saturn; and even with respect to them, observation, although it makes [7:4:24]Known the existence of inequalities, can neither detect their laws nor assign their periods. The only elements which are exempted from secular changes occasionedby the perturbing forces are the greater axes of the orbits, and the mean motions depending on them according to the third law of Kepler. The eccentricities of the orbits vary slowly. With regard to Mercury, Mars, and Jupiter, those elements are at the present time increasing; in the cases of Venus, the Earth, Satura, and Uranus, they are diminishing. The perihelia are also gradually shifting their places on the planes of the orbits. These motions are direct in the case of all the planets excepting Venus, the perihelion of whose orbit, when referred to the fixed stars, moves in a direction contrary to that of the signs.

The following table exhibits the mean motions of the planets, and the elements and positions of their ellipses on the planes of their orbits, at the commencement of the year 1801. For the four new planets the epoch is 1820.
[table]

3. Of the real Dimensions of the Planetary Orbits, and the Transits of Venus and Mercury over the Suns Dish. —In what has hitherto been said respecting the mean distances of the planets from the sun, those distances have been estimated in parts of the semi-axis major of the earth’s orbit; a convenient scale, which enables us to form a very precise idea of the relative dimensions of the several orbits, and likewise serves to express their absolute dimensions, provided we can determine the distance between the sun and the earth in terms of any measure with which we are familiar. It is evident, indeed, in consequence of the relation that subsists between the mean distances and mean motions, that it is only necessary to determine the mean distance of any one of the planets from the sun, in order to determine the mean distances of all the others, and assign the dimensions of the whole solar system. Now there are various ways of determining the sun’s distance from the earth in terms of the earth’s semidiameter. The distance of a planet from the sun may likewise be. obtained if we can find the means of measuring its distance from the earth at any epoch; for the geocentric positions of the sun and the planet being known from the theory of their motions, the radius vector of the orbit, or planet’s distance from the sun at that epoch, may be found by a simple trigonometrical computation. To determine the distance of a planet from the earth, it might seem only necessary to determine its horizontal parallax; but in general the parallaxes of the planets are quantities by far too small to be directly observed. That of Mars, however, becomes very appreciable in particular circumstances, that is to say, when Mars is in opposition with the sun, and at the same time near the perihelion of his orbit. Thus, in the year 1751, on the 6th of October, that planet, being near his opposition, was observed at the same instant of time by Lacaille at the Cape of Good Hope, and by Wargentin at Stockholm; and the horizontal parallax deduced from the two observations, in the manner explained in chap. i. sect. 2, was found to amount to 24"∙6. Now the distance of the planet being equal to unity divided by the sine of the horizontal parallax, the distance of Mars from the earth at the time of the observation was consequently

24"∙6 × 3∙1416 648000 o ol 1 ÷ 180° ×60×60 = ^7L28^ = 8381 terrestnal radii ∙

But the distance of Mars from the earth at that time, as computed from the theory of his elliptic motion, was 0 ’ 435 parts of the radius of the earth’s orbit; consequent-

1 × 8381

ly the whole length of that radius is — 7r -. -- r -- = 19226 ; o 0∙435

semidiameters of the earth. Hence the distance of Mars from the sun = 19226 + 8381 = 27607 semidiameters of the earth. It will be remarked that these numbers are only approximative, our present object being to explain the method, and not to determine the exact quantities.

But besides the inconvenience attending the determination of a planet’s distance by this method, which requires observations to be made simultaneously on opposite sides of the earth, the method is in itself liable to great uncertainty. The error of a resùlt is always in a certain proportion to the error of observation, and in the present case a very large quantity is to be determined from a very small one; hence a very slight error of observation will occasion a very erroneous result in the computation of the mean distance. It would be difficult in a single observation of this nature to answer for an error of 2"; but here 2" is a twelfth part of the whole parallax; consequently an uncertainty amounting to a twelfth part affects the mean distance.

A much more accurate method of determining the sun’s distance, and thence the dimensions of the planetary orbits, is afforded, though rarely, by the transits of Venus over the sun’s disk. When Venus is at her inferior conjunction, and at the same time very near her node, her body will be projected on the disk of the sun; and through the effect of her proper motion, combined with that of the earth, she will appear as a dark spot passing over the disk, and describing a chord which will be seen under different aspects by spectators placed at different points on the earth, because, by reason of the parallax, they refer the planet, to different points on the solar disk. The position of the spectator not only occasions a difference in the apparent path described by the planet, but has also a very sensible influence on the duration of the transit, in consequence of which the parallaxes both of Venus and the sun can be determined with great exactness. In order to illustrate this, let E (fig. 73) represent the earth, V Venus, and S the sun. An observer placed at E, the centre of the earth, would see Venus in the direction[7:4:25] of the visual ray EV: she would consequently; appear to him projected on the sun’s disk at S, and in her successive positions would appear to describe the line DS. Other observers placed at O' and 0" on the earth’s surface would see the planet at V' and V": to the first she would appear to describe the chord D'V', and to the second D''V". This is a necessary result of the difference of the parallaxes of Venus and the sun; and as the chords D'V', D''V" differ in length according as they are more or less remote from the centre of the disk, the duration of the transit will be longer or shorter according to the situation of the observer and the geocentric latitude of the planet. If by reason of the relative parallax the time of a transit is longer than the true time in one hemisphere, it will be shorter in the opposite; and hence the difference of the times (which may be observed with great accuracy) at places having very different latitudes may serve to determine the relative parallax, or the difference between the parallax of Venus and that of the sun. But the parallaxes are reciprocally proportional to the distances; and the ratio of the distances being known, therefore the ratio of the parallaxes is also known; and having thus the ratio and the difference of the two parallaxes, it is easy to compute the separate amount of each.

This particular application of the transits of Venus to the determination of the sun’s distance was first pointed out by Dr Halley, when he announced the transits of 1761 and 1769. Kepler had before announced the occurrence of a transit, but he regarded it only as a curious, and till that time unobserved, phenomenon.

The transit of Venus which occurred in 1769 was anxiously expected by astronomers, and observed in many different parts of the world. The result of the whole of the observations renders it extremely probable that the parallax of the sun is included within the limits of 8"∙5 and 8"∙7. The mean 8"∙6 has been adopted by Delambre and Lalande. From the following table, computed from the different observations, and published by Delambre in the second volume of his Astronomie, p. 505, an idea may be formed of its probable accuracy. Delambre, indeed, remarks that the sun’s parallax is now sufficiently well known for all the practical purposes of astronomy.
[table]

Here the mean of the first 5 results is nearly...8"∙59

of the next 5 8 '41

of the next 4 8 '75

of all 8 ∙57

Having once obtained the value of the solar parallax, it is easy to deduce the sun’s distance, and consequently the dimensions of all the planetary orbits. For this purpose we have sin. 8"∙6: 1:: radius of earth: sun’s distance; that is, on reducing the radius of a circle to seconds, , ,. 360° ×60×60 oone . i . 1 ...

sun s distance =—.— λ - Ί . ιc ,,= 23984 terrestrial radii.

8∙6 × 2 × 3∙ 14159

Now, if we assume the semidiameter of the earth to be 4000 miles in round numbers, the distance of the sun, or radius of the earth’s orbit, will consequently be 23984 X 4000=95936000 miles.

By means of this value the mean distances of the planets from the sun, which in the table given above were expressed in terms of the mean distance of the earth, may be converted into miles. The following are the results in round numbers.
[table]

The transits of Venus being phenomena of great importance, in consequence of their practical application to the problem of the sun’s distance, it becomes interesting to determine the periods at which they successively occur. It is evident that, by reason of the inclination of the orbit, they can only take place when Venus is very near one of her nodes. Two conditions must therefore be satisfied: Venus must be within a short distance of her node, and at the same time in her inferior conjunction. Now, the interval between two successive conjunctions, that is, the period of a synodic revolution, is easily deduced from the sidereal revolutions of the planet and the earth. Thus, generally, let A and B be two planets, T and t the times of their sidereal revolutions respectively, and suppose T to be greater than t. In the time T, A describes a complete circumference, therefore T: t :: 1: -^ r = part of a circumference described by A in the time t. But during the same time t, B describes a whole circumference; therefore 1—-ψ-is what B gains on A in the time t. But the successive conjunctions will always take place when B has gained a whole circumference; therefore, denoting by S the interval between two successive t Tí

conjunctions, we have 1 ψ: 1:: t : S, whence S = ψ—-,

From this simple formula the synodic revolution of any of the planets is found by substituting for T and t the times of the sidereal revolutions of the earth and planet. Suppose, for example, we wish to find the time of a synodic revolution of Mercury. In this case T=365∙256 days, and t = 87∙969 days; consequently T— t — 277∙287, and „ 365∙256 × 87∙969 _. c or , f , 1 .. . 1 _

S = ry ⅛⅛' = 115∙877 days, which, therefore, tĉ í t £όΊ is the time of a synodic revolution of Mercury.

In the case of Venus we have t = 224∙700, whence T — t — 365∙256 — 224*700 = 140*556 s and consequently c , 365∙256 × 224∙700 coo __,,. 1 . t , . , S = . = 583∙92 days; which is the period

140∙556 ; r of her synodic revolution.
[7:4:26]

Let us next attend to the other condition which must be satisfied before a transit can take place, namely, that the planet has returned to its node as well as to its inferior conjunction. If we represent by rothe number of revolutions of the earth in the required period, and by n the number of synodic revolutions of the planet in the same time, it is evident that we shall have mA = ¶ŗ—-, whence — = ∙ In the case of Mercury, therefore, — —

η T— t η

87969 , . _ or , nn n ∙ . 1 c

2 77287 ’ t ^ lat 1S t ° sa y , a ^ t θ r ð ' θθ 9 years, in the course of which Mercury will have been 277287 times in conjunction, the earth and Mercury will be again in conjunction, occupying the same points of their orbits as at the commencement of the period, supposing the nodes fixed. But periods of such enormous length are of no practical use: it is necessary to find an approximating ratio expressed by smaller numbers. For this purpose it is convenient to have recourse to the method of continued fractions, from which the following series is obtained :—

87969 _1 1

277287 - 3+p . 1 1 6+ īo+-LjËĨ

i ^ r 1079’

Here the first approximating fraction is ⅛, which denotes that in one year, during which there will happen three synodic periods, Mercury will not be very far from his conjunction, nor from the same point of his orbit in which he was at the commencement of that time. The

Į_ i θ

next approximation is 3 + θ = τθ, showing that after six years, during which there will have been 19 conjunctions, Mercury will be again nearly in conjunction at the same point of his orbit. By continuing the process we obtain the following series of fractions, each approaching nearer

87969 t0 277287’ " amely,

1 6 7 13 33 46

3’ 19’ 22’ 41’ ĪŌ4’ 145’ &C '

of which the numerators express the number of years, and the denominators the corresponding number of synodic revolutions.

Approximative fractions might be found in the same manner to express the ratio of the number of revolutions of Mercury to those of the earth, or of ∙^ r ; but it is unnecessary to have recourse again to division, inasmuch as they are easily obtained from the above. For example, since the fraction ∣∙ denotes that Mercury has gained three revolutions on the earth in one year, it is evident that he must have completed 3+1=4 revolutions. In the same way, in six years Mercury gains 19 on the earth, or completes 19 + 6 = 25. Hence, in the series required the numerators of the fractions will continue the same as in the series above, while the denominators will be the sums of the terms of the corresponding fractions. The new series will therefore be

1 6 7 13 33 46

4’ 25’ 29’ 54’ Ī37’ Ī9Ï’ &C ’

the numerators being the number of years, and the denominators the corresponding periods of Mercury.

Applying to the case of Venus and the earth the for- 1 m t , . mula — = τp—-, we shall have t = 224∙7008, T — t — n 1— t ’ m 2247008 365-2563—224-7008 = 140-5555; therefore —= ———.

η 1405555

The series of fractions approximating to this ratio, obtained in the manner indicated above, is

1 2 3 8 227 235 243 1’ 1’ 2’ 5’ 142’ 147’ 152’ &C ’ the numerators of which, as before, express the years, and the denominators the synodic periods. Taking the fourth fraction of the series, it appears that after eight years, in which there are five synodic periods, Venus will again occupy nearly the same position with respect to the earth and the nodes of her orbit. This will take place more nearly after 227 years, and more nearly still after 235; consequently, 235 years after a transit has taken place, the occurrence of another may be expected with great probability. The alteration, however, which takes place in the position of the line of the nodes, which in the preceding computations we have regarded as fixed, renders the numbers a little uncertain.

Since Venus returns to her conjunction at nearly the same point of her orbit after eight years, it may happen, and sometimes indeed actually does happen, that a transit will take place in about eight years after the occurrence of a former one. But in that time the latitude of Venus, in consequence of the inclination of her orbit, undergoes a variation amounting to 20' or 24 , ; in sixteen years the change of latitude increases to 40 ’ or 48', which is considerably greater than the diameter of the sun. It cannot happen, therefore, that three transits will take place within sixteen years. From the above series of fractions we might infer that another could not take place before 227 years; but that series was obtained on the supposition that the transits only happen when Venus returns to the same node; and it is evident that they may equally occur when the planet is near the other node, and consequently after an interval of half the length of the former, or 113 years. If, at the occurrence of the first transit, Venus has passed her node, the next will happen eight years sooner; or, if she has not reached the node, eight years later. Hence, after two transits have occurred within eight years, another cannot be expected before 105, 113, or 121 years, that is, 113 z±z 8 years. But these periods sometimes fail: that of 235 brings about the phenomenon with greater certainty, and 243 (which is the double of 121) is the surest of all. The periods of 235 and 251 are that of 243 diminished or augmented by 8. The whole calculation, therefore, reduces itself to periods of 121 and =±∑ 8 years. The last transits took place in 1761 and 1769; the next will not happen till the years 1874 and 1882; and thus the infrequency of these phenomena adds to the interest they derive from their real importance.

Delambre has given a list of all the transits of Venus for a period of 2000 years, from which the following is extracted. (Astronomie, tome ii. p. 473.)
[table]
[7:4:27]

Sect. III.— Of the Physical Constitution of the Planets, their Magnitude, Rotation, and other remarkable objects. Mercury.

Mercury is a small star, but emits a very bright white light, though, by reason of his always keeping near the sun, he is seldom to be seen; and when he does make his appearance, his return to the sun is so rapid, that he can only be discerned for a short space of time. Delambre was able to observe him only twice with the naked eye.

Mercury is about 3140 English miles in diameter, and his mean distance from the sun about 37 millions of miles. On account of his smallness and brilliancy it is extremely difficult to find any spot on his disk so distinctly marked as to afford the means of determining his rotation. Besides, by reason of his proximity to the sun, an observation of a spot, if made in the evening, can scarcely be well begun before the planet sets; or, if in the morning, before the increasing twilight renders the spot invisible. Hence it is only possible to observe daily a very small arc of a small circle; and if the spot re-appears on the succeeding day, it is doubtful whether the arc which it has passed over exhibits the whole motion, or if one or more circumferences ought to be added. By an attentive observation of the variations of the phases of Mercury, Schroeter has, however, remarked that he revolves about his axis in the space of 24 hours 5 minutes 30 seconds. Μ. Harding discovered in 1801 an obscure streak on the southern hemisphere of the planet, the observations of which, together with those of a spot discovered by Schroeter, gave the same period of rotation. The results of Schroeter’s researches on Mercury may be summed up as follows: Isŕ, The apparent diameter of Mercury at his mean distance is 6"∙02; 2ćZ, His form is spherical, exhibiting no sensible compression; 3J, His equator is very considerably inclined to his orbit, and the differences of his days and seasons must consequently be very great; Mh, There are mountains on his surface which cast very long shadows, and of which the height bears a greater proportion to the diameter of the planet than those of the Earth, the Moon, or even of Venus. The height of Chimboraço is - 1 ∙ 5 1 17 - of the radius of the earth; one of the mountains in the moon has been estimated at of her radius; the highest in Venus at ψ⅛i and one in Mercury at τ +y. The highest mountains are in the southern hemisphere, which is also the case in respect of the Earth and Venus. There are no observations to prove decisively whether Mercury is surrounded by an atmosphere.

Venus.

Venus, the most beautiful star in the heavens, is about 7700 English miles in diameter, and placed at the distance of 68 millions of miles from the sun. Although the oscillations of this planet are considerably greater than those of Mercury, and she is seldom invisible, yet on account of the uniform brilliancy of her disk, it is extremely difficult to ascertain the period of her rotation. Dominic Cassini, after having long fruitlessly attempted to discover any object on her surface so well defined as to enable him to follow its motions, at length, in 1667, perceived a bright part, distant from the southern horn a little more than a fourth part of the diameter of the disk, and near the eastern edge. By continuing his observations on this spot, Cassini concluded the rotation of Venus to be performed in about 23 hours; but he does not seem to have considered this conclusion as deserving of much confidence. In the year 1726 Bianchini, an Italian astronomer, made a number of similar observations for the same purpose, from which he inferred that the rotation of the planet is performed in 24 days 8 hours. The younger Cassini has shown, however, that the observations of Bianchini, as well as those of his father, could be explained by a rotation of 23 hours and 21 or 22 minutes, whereas the rotation of 24 days 8 hours cannot be reconciled with the appearances observed by the elder Cassini. The determination of Cassini was regarded by astronomers as the more probable of the two, particularly as Bianchini was not able to make his observations in a connected manner, on account that a neighbouring building intercepted his view of the planet, and obliged him to transport his telescope to a different situation. The question of the rotation of Venus was finally settled by Schroeter, who found it to be performed in 23 hours 21 minutes 19 seconds. Each of the three observers found the inclination of the axis of rotation to the axis of the ecliptic to be about 75°. Some doubt, however, still exists with respect to the value of this element.

Schroeter’s observations on this planet were principally directed to a mountain situated near the southern horn. The line which joins the extremities of the horns is always a diameter; and the horns of the crescent of a perfect sphere ought to be sharp and pointed. Schroeter remarked that this was not always the case with regard to the horns of Mercury and Venus. The northern horn of the latter always preserved the pointed form, but the southern occasionally appeared rounded or obtuse,—a circumstance which indicated that the shadow of a mountain covered the part Bo (fig. 74), so that the line joining the extremities of the horns appeared to be Ao, and not AB (perpendicular to CD, the axis of the ecliptic); but at d, beyond oB, he remarked a luminous point, which he supposed to be the summit of another mountain, illuminated by the sun after he had ceased to be visible to the rest of that hemisphere. Now, in order that the horn of the crescent may appear obtuse in consequence of the shadow of a mountain falling upon it, and another mountain d present a luminous point, the two mountains must be at the same time both at the edge of the disk and on the line separating the dark from the enlightened part of the planet. But this position cannot be of long continuance; for the rotation will cause d to rise into the enlightened part, or sink into the dark hemisphere, and in either case the mountain will cease to be visible. If, however, the rotation is completed in 23 hours 21 min., the mountain d will appear 39 min. sooner than it did on the previous day; for in the course of a day the boundary of light and darkness will hardly have shifted its position on the surface of the planet through the effects of the orbital motion. Hence it is possible to obtain several consecutive observations, from which an approximate value of the period may be found; and this being once obtained, it may be rendered still more exact by observations separated from each other by a longer interval. Thus Schroeter found that an interval of 20 days 11 hours 15 min. between two apparitions of the mountain being divided by 23 hours 21 min., gave 21 ’ 005 revolutions. That intervals of 121 days 14 hours 25 min., 142 days 1 hour 40 min., 155 days 18 hours 11 min., divided each by 23 hours 21 min., gave 125∙01, 146Ό2, 165*09 revolutions respectively. All these comparisons prove that the revolution of 23 hours 21 min. is somewhat too short. They ought to have given 21, 125, 146, and 165 revolutions exactly, supposing the observations to have been perfectly accurate. On dividing the intervals by 21, 125, 146, and 165 respectively, the quotients will be each the time of a revolution very nearly; and by taking a mean among the whole, the most pro [7:4:28]bable result at least will be obtained. In this manner •Schroeter found the period of rotation already stated, namely, 23 hours 21 min. 19 sec.

Since the time of rotation of Mercury and Venus is nearly equal to that of the earth, the compression of these planets at the poles, which results from the centrifugal force, ought also to be nearly in the same proportion. But at the distance of the earth the compression must be imperceptible even in the case of Venus; for, supposing it to amount to the difference between the radius of her poles and that of her equator would only amount to a tenth of a second as seen from the earth.

During the transits of Venus over the sun’s disk in 1761 and 1769 a sort of penumbral light was observed round the planet by several astronomers, which was occasioned, without doubt, by the refractive powers of her atmosphere. Wargentin remarked that the limb of Venus which had gone off the sun showed itself with a faint light during almost the whole time of emersion. Bergman, who observed the transit of 1761 at Upsal, says that at the ingress the part which had not come upon the sun was visible, though dark, and surrounded by a crescent of faint light, as in fig. 75; but this appearance was much more remarkable at the egress; for as soon as any part of the planet had disengaged itself from the sun’s disk, that part was visible with a like crescent, but brighter (fig. 76). As more of the planet’s disk disengaged itself from that of the sun, the part of the crescent farthest from the sun grew fainter, and vanished, until at last only the horns could be seen, as in fig.77. The total immersion and emersion were not instantaneous; but as two drops of water, when about to separate, form a ligament between them, so there was a dark shade stretched out between Venus and the sun, as in fig. 78; and when this ligament broke, the planet seemed to have got about an eighth part of her diameter from the limb of the sun (fig. 79). The numerous accounts of the two transits which have been published abound with analogous observations, indicating the existence of an atmosphere of considerable height and density. Schroeter calculated that its horizontal refraction must amount to 3 O' 34'', differing little from that of the terrestrial atmosphere. A twilight which he perceived on the cusps afforded him the data from which he deduced this conclusion.

Cassini and Montaigne imagined that they had observed a satellite accompanying Venus; but this appears to have been an optical illusion arising from the strong light of the planet reflected back from the convex surface of the eye upon the eye-glass of the telescope, and thence reflected a second time back to the eye. This hypothesis at least will explain the appearances which they have described; and although astronomers have sought tor this pretended satellite with great care, they have neither observed it on the sun during the transits of Venus in 1761 and 1769, nor in any other part of her orbit.

Mars.

After Venus, Mars is the planet whose orbit is nearest to the earth. His diameter is about one half, and his volume only about one fifth part of that of our globe. He is of a dusky reddish colour, by reason of which he is easily recognised in the heavens. His mean distance from the sun is about 142 millions of miles.

The rotation of Mars was suspected before the year 1643 by Fontana, a Neapolitan astronomer; but it was reserved for Cassini to demonstrate its existence and assign its period. Cassini began to observe the spots on the surface of Mars at Bologna in 1666; and after having continued his observations for a month, he found they returned to the same situation in 24 hours and 40 min. The planet was observed by some astronomers at Rome with longer telescopes; but they assigned to it a rotation of 13 hours only. This, however, was afterwards shown by Cassini to have arisen from their not distinguishing between the opposite sides of the planet, which, it seems, have spots pretty much alike. He made further observations on the spots of this planet in 1670; which confirmed his former conclusion respecting the time of rotation. The spots were again observed in subsequent oppositions, particularly for several days in 1704 by Maraldi, who took notice that they were not always well defined, and that they not only changed their shape frequently in the interval between two oppositions, but even in the space of a month. Some of them, however, continued of the same form long enough to allow the time of the planet’s rotation to be determined. Among these there appeared that year an oblong spot, resembling one of the belts of Jupiter when broken. It did not reach quite round the body of the planet; but had, not far from the middle of it, a small protuberance towards the north, so well defined that Maraldi was thereby enabled to fix the period of its revolution at 24 hours 39 min., only one minute less than what Cassini had determined it to be.

The near approach of Mars to the earth in 1719 afforded an excellent opportunity of observing him, as he was then within 2ģ-° of his perihelion, and at the same time in opposition to the sun. His apparent magnitude and brightness were thus so much increased, that he was by the vulgar taken for a new star. His appearance at that time, as seen by Maraldi through a telescope of 34 feet long, is represented in fig. 80. There was then a long belt that reached half-way round, to the end of which another shorter belt was joined, forming an obtuse angle with the former, as in fig. 81. This angular point was observed on the 19th and 20th of August, a little to the east of the middle of the disk; and 37 days after, on the 25th and 26th of September, it returned to the same situation. This interval, divided by 36, the number of revolutions contained in it, gives 24 hours 40 minutes for the period of one revolution; a result which was verified by another spot of a triangular shape, one angle whereof was towards the north pole, and the base towards the south, and which on the 5th and 6th of August appeared as in fig. 82. After 72 revolutions it returned to the same situation on the 16th and 17th of October. Some of the belts of this planet are said to be parallel to his equator; but that seen by Maraldi was very much inclined to it.

Besides these dark spots on the surface of Mars, astronomers had noticed that a segment of his globe about the south pole exceeded the rest of his disk so much in brightness, that it appeared to project as if it were the segment of a larger globe. Maraldi informs us that this bright spot had been taken notice of for 60 years, and was more permanent than the other spots on the planet. One part of it is brighter than the rest, and the least bright part is subject to great changes, and has sometimes disappeared.

A similar though less remarkable brightness about the north pole of Mars was also sometimes observed, the existence of which has been confirmed by Sir W. Herschel, who examined the planet with telescopes of much greater power than any former astronomer ever was in possession of. A very full account of Herschel’s observa-] tions on this planet is given in the 74th volume of the∑ Philosophical Transactions. Some of the remarkable ap- t pearances there described are represented in fig. 83—88. The magnifying powers he used were sometimes as high [7:4:29]as 932; and with this the south polar spot was found to •be 4P" in diameter. Fig. 96 shows the connection of the other figures marked 89, 90, 91, 92, 93, 94, 95, which Complete the whole equatorial succession of spots on the disk of the planet. “The centre of the circle,” Herschel observes, “marked 90, is placed on the circumference of the inner circle, by making its distance from the circle, marked 92, answer to the interval of time between the two observations, properly calculated and reduced to sidereal measure. The same is done with regard to the circles marked 91, 92, &c.; and it will be found by placing any one of these connected circles in such a manner as to have its contents in a similar situation with the figures in the single representation, which are marked with the same number, that there is a sufficient resemblance between them; though some allowance must be made for the distortions occasioned by this kind of projection.”

From these observations Herschel concluded that the diurnal rotation of Mars is accomplished in 24 hours 39 minutes 21-∣ seconds; that his equatoris inclined to his orbit in an angle of 28° 42', and his axis of rotation to the axis of the ecliptic in an angle of 30° 18'. Hence the time of rotation and the seasons of this planet are little different from those of the earth.

The bright appearance so remarkable about the poles of Mars is ascribed by Herschel to the reflection of light from mountains of ice and snow accumulated in those regions. “The analogy between Mars and the earth,” says he, “is perhaps by far the greatest in the whole solar system. Their diurnal motion is nearly the same, the obliquity of their respective ecliptics not very different; of all the superior planets, the distance of Mars from the sun is by far the nearest alike to that of the earth; nor will the length of the Martial year appear very different from what we enjoy, when compared to the surprising duration of the years of Jupiter, Saturn, and the Georgium Sidus. If we then find that the globe we inhabit has its polar region frozen and covered with mountains of ice and snow, that only partly melt when alternately exposed to the sun, I may well be permitted to surmise, that the same causes may probably have the same effect on the globe of Mars; that the bright polar spots are owing to the vivid reflection of light from frozen regions; and that the reduction of those spots is to be ascribed to their being exposed to the sun.”

Since the discovery of the flattened form of the earth, it was to be presumed that the rotation of the other planets would produce a similar effect on their figures, and this supposition has been fully confirmed by observation. The time of the rotation of Mars is nearly equal to that of the earth, but his diameter being only about half that of the earth, the velocity of a point on his equator is consequently only half as great as that of a point on the earth’s equator: hence we might expect that the deviation of his figure from a perfect sphere would be much less considerable. The contrary, however, appears to be the case; and his compression seems to be much greater than that of the earth. According to Herschel, the ratio of his equatorial and polar axes is 103 to 98. Schrocter estimates the same ratio to be that of 81 to 80. This remarkable compression at the poles of Mars arises in all probability from considerable variations of density in the different parts of his globé

it has been commonly related by astronomers, that the atmosphere of this planet is possessed of such strong refractive powers as to render invisible the small fixed stars near which it passes. Dr Smith relates an observation of Cassini, in which a star in the water of Aquarius, at the distance of six minutes from the disk of Mars, became so faint before its occultation, that it could not be seen b y the naked eye, nor even with a three feet telescope. This would indicate an atmosphere of a very extraordinary size and density; but the following observations of Herschel seem to show that it is of much smaller dimensions. “1783, Oct. 26th. There are two small stars preceding Mars, of different sizes; with 460 they appear both dusky red, and are pretty unequal; with 218 they appear considerably unequal. The distance from Mars of the nearest, which is also the largest, with 227, measured 3' 26" 20'". Some time after, the same evening, the distance was 3' 8" 55"', Mars being retrograde. Both of them were seen very distinctly. They were viewed with a new 20 feet reflector, and appeared very bright. October 27th: the small star is not quite sobright in proportion to the large one as it was last night, being a good deal nearer to Mars, which is now on the side of the small star; but when the planet was drawn aside, or out of view, it appeared as plainly as usual. The distance of the small star was 2' 5" 25'". The largest of the two stars,” adds he, “on which the above observations were made, cannot exceed the 12th, and the smallest the 13th or 14th magnitude; and I have no reason to suppose that they were any otherwise affected by the approach of Mars, than what the brightness of its superior light may account for. From other phenomena it appears, however, that this planet is not without a considerable atmosphere; for, besides the permanent spots on its surface, I have often noticed occasional changes of partial bright belts, and also once a darkish one in a pretty high latitude; and these alterations we can hardly ascribe to any other cause than the variable disposition of clouds and vapours floating in the atmosphere of the planet.”

Ceres, Pallas, Juno, and Vesta.

The commencement of the present century was rendered remarkable in the annals of astronomy by the discovery of four new planets circulating between Mars and Jupiter. Kepler, from some analogy which he found to subsist among the distances of the planets from the sun, had long before suspected the existence of one at this distance; and his conjecture was rendered more probable by the discovery of Uranus, with regard to which the analogy of the other planets is observed. So strongly, indeed, were astronomers impressed with the idea that a planet would be found between Mars and Jupiter, that, in the hope of discovering it, Baron Zach formed an association of 24 observers, who divided the sky into as many zones, and undertook each to explore one carefully. A fortunate accident anticipated a discovery which might have required years of toil. An error in the catalogue of Wollaston, who had laid down a star in a position in which it is not to be found, engaged Piazzi, the superintendent of the observatory at Palermo, to observe for several successive days all the small stars in the neighbourhood of the place indicated. On the first day of the present century, the 1st of January 1801, he observed a small star in Taurus, which, on the day following, appeared to have changed its place. On the 3d he repeated his observation, and was then satisfied that it had a diurnal motion of about 4' in right ascension, and 3'⅜ in declination towards the north pole. He continued to observe it till the 23d, when he communicated his discovery to MM. Bode and Oriani, giving them the positions of the star on the 1st and 23d, and only adding, that between the 11th and 13th its motion had changed from retrograde to direct. Before the communication reached them, however, the planet was lost in the sun’s rays; and, owing to its extreme[7:4:30] smallness, the difficulty of finding it after its emergence was so great, that it was not again seen till the 31st of the following December, when it was detected by Zach. In this was recognised the planet which Kepler bad suspected to circulate between Mars and Jupiter. Piazzi, in honour of Sicily, gave it the name of the tutelary goddess of that country, Ceres ; and her emblem, the sickle, , has been adopted as its appropriate symbol.

Ceres is of a reddish colour, and appears to be about the size of a star of the eighth magnitude. The eccentricity of her orbit is somewhat greater than that of Mercury; and its inclination to the ecliptic greater than that of any of the old planets. The distance of Ceres from the sun is about 3∙2 times that of the earth, or nearly 270 millions of miles. Schroeter found her apparent diameter to be 2", corresponding to about 1624 miles; but Herschel reduced this measurement to 0"∙5, which would indicate a diameter of about 160 miles. The nebulosity which surrounds the planet renders it almost impossible to distinguish the true disk; and hence arises the great discrepancy between the above estimates of its magnitude. From a great number of observations, Schroeter inferred that Ceres has a dense atmosphere, rising to the height of no less than 675 English miles above the planet, and subject to numerous changes. On this account he conceives that there is little chance of discovering the period of its rotation.

The difficulty of finding Ceres induced Dr Olbers of Bremen to examine with particular care the configurations of all the small stars situated near her geocentric path. On the 28th of March 1802 he observed a star of the seventh magnitude, which formed an equilateral triangle with the stars 20 and 191 of Virgo in Bode’s catalogue. He was certain that he had never seen a star in that place before, and at first imagined it might be one of those which are subject to periodical changes of brilliancy; but after examining it for two hours, he remarked that its right ascension was diminishing, while its northern declination continued to augment nearly in the same manner as had been the case with Ceres when that planet was first seen by him in almost the same position. On the following day he found its right ascension had diminished 10', while its northern declination had increased 20'. From observations continued during a month, Μ. Gauss calculated an elliptic orbit, the eccentricity of which amounted to ∙24764, much greater than that of any of the other planets. He also found its inclination to be 34° 39', exceeding the aggregate inclinations of all the other planetary orbits; and its mean distance 2*770552, almost the same as that of Ceres. On account of these three circumstances, the new planet, otherwise of little importance, became the most singular in the whole system. One planet had been suspected to exist between Mars and Jupiter, and two were now discovered. The great inclination of the last rendered it necessary to enlarge the boundaries of the zodiac; but the extent of the zodiac is entirely arbitrary, and had been limited by the extreme latitudes of Venus. There is no reason, as Delambre remarks, why it may not be extended even to the poles. Dr Olbers gave the new planet the name of Pallas, choosing for its symbol the lance, γ , the attribute of Minerva.

The most surprising circumstance connected with the discovery of Pallas was the existence of two planets at nearly the same distance from the sun, and apparently having a common node. On account of this singularity Dr Olbers was led to conjecture that Ceres and Pallas are only fragments of a larger planet, which had formerly circulated at the same distance, and been shattered by some internal convulsion. Lagrange made this hypothesis the subject of an ingenious memoir, in which he determined the explosive force necessary to detach a fragment from a planet with a velocity that would cause it to describe the orbit of a comet. He found that a fragment detached from the earth in this manner, with a velocity equal to 121 times that of a cannon-ball, would become a direct comet; and if with a velocity equal to 156 times that of a cannon-ball, its motion would be retrograde. For

121 or 156

other planets the velocity must be —;—— ·. , and

√mean distance consequently less as the mean distance of the planet from the sun is greater. A smaller velocity would be required to cause the detached fragment to move in an elliptic orbit; and with regard to the four small planets we are now considering, an explosive force less than twenty times that of a cannon-ball would have sufficed to detach them from a primitive planet, and cause them to describe ellipses similar to their actual orbits. This hypothesis served also to explain the great eccentricities and inclinations by which these planets are distinguished from the others belonging to the system; for it is evident that the explosive force must have projected the different fragments in all directions, and with different velocities. It followed also, that other fragments of the original planet might probably exist, revolving in orbits which, however they might differ in respect of inclination and eccentricity, would still intersect each other in the same points, or have common nodes, in which the several fragments would necessarily be found at each revolution. Dr Olbers therefore proposed to examine carefully every month the two opposite parts of the heavens in which the orbits of Ceres and Pallas intersect each other, with a view to the discovery of other planets, which might be sought for in those parts with greater chance of success than in a wider zone embracing the whole limits of their orbits. Subsequent discoveries demonstrated the soundness of his conjecture, and gave a degree of probability to his hypothesis.

According to Herschel, the diameter of Pallas is only about 80 English miles, or about one half of that which he assigned to Ceres, while Schroeter estimates it at 2099 miles, Schroeter also found the atmosphere of Pallas to be about two thirds of the height of that of Ceres, or about 450 miles. The light of the planet undergoes considerable variations, the cause of which is uncertain.

While Μ. Harding, of the observatory of Lilienthal, near Bremen, was engaged in forming a complete zodiac of the small telescopic stars near the orbits of Ceres and Pallas, with which these planets were likely to be confounded, he determined, on the 22d of September 1804, the position of a small star, by comparing it with the two stars marked 93 and 98 of Pisces in Bode’s catalogue. These two stars are situated very near the equator, and at a small distance from one of the nodes of Ceres and Pallas, and exactly in that sort of defile where, according to Dr Olbers, an observer would be certain of detecting in their passage the other fragments of the original planet of which Ceres and Pallas are parts. On the 4th the star was no longer in the same position, but had moved a little to the south-west. On the 5th and 6th Μ. Harding, by means of a circular micrometer, determined the rate of its motion to be 12' 42" in declination to the south, and 7' 30" in right ascension, retrograde, the interval between the observations being 24 hours 14 min. 12 sec. From this it was evident that the body belonged to the planetary system. It had then the appearance of a star between the eighth and ninth magnitudes. It was without any nebulosity, and of a whitish colour. A few days [7:4:31]afterwards the elements of its orbit were computed by Gauss. This planet has received the name of Juno, and for its symbol ÿ, the starry sceptre of the queen of Olympus.

Juno is distinguished from the other planets by the great eccentricity of her orbit, which is so considerable, that she describes that half of it which is bisected by the perihelion in about half the time which she employs to describe the other half. This planet is somewhat smaller than Ceres and Pallas, and, though free from nebulosity, must have, according to Schroeter’s observations, an atmosphere of greater density than that of any of the old planets.

The success of Μ. Harding encouraged Dr Olbers to renew the plan of research which he had pointed out on the discovery of Pallas; and on the 29th of March 1807 he perceived, in the constellation of Virgo, a star of the fifth or sixth magnitude, which he suspected from the first observation to be a new planet. A few subsequent observations rendered this conjecture certain. Dr Olbers left to Gauss the care of giving a name to the new planet, and of determining the elements of its orbit. Gauss named it Vesta, and chose for its symbol Jį, an altar surmounted with a censer holding the sacred fire. Vesta is the smallest of all the celestial bodies known to us. Her volume is only about a fifteen thousandth part of that of the earth, and her surface is about equal to that of the kingdom of Spain. She is distinguished by the vivacity of her light, and the luminous atmosphere with which she is surrounded.

Jupiter.

Jupiter is by far the largest planet in the system. His diameter is about 11 times, and his volume 1281 times, greater than that of the earth. His distance from the sun is 5Í times the radius of the ecliptic, or nearly 125,000 terrestrial semidiameters, and consequently above 490 millions of miles. His apparent diameter, which, at his mean distance, is 36"∙7, and varies between 45"∙8 and 30", would subtend an angle of 3' 17" if seen at the same distance as the sun. From Jupiter the sun will appear under an angle of & at most; the sun’s disk will appear to be 27 times smaller than when seen from the earth, consequently the light and heat which Jupiter receives from the sun will be only the 27th part of what is received by our globe. His density is ∙99239, that of the sun being considered as unity, or is about one fourth of the density of the earth; and a body which weighs one pound at the equator of the earth, would weigh 2∙444 pounds if removed to the equator of Jupiter.

Jupiter has the same general appearance with Mars, only the belts on his surface are much larger and more permanent. Their usual appearance, as described by Dr Long, is represented fig. 97-100; but they are not to be seen but by an excellent telescope. They are said to have been first discovered by Fontana and two other Italians, but Cassini was the first who gave a good account of them. Their number is very variable, as sometimes only one, but seldom more than three, may be perceived. Messier at one time saw so great a number that the whole disk seemed to be covered by them. They are generally parallel to one another, but not always so, and their breadth is likewise variable, one belt having been observed to grow narrow, while another in its neighbourhood has increased in breadth, as if the one had flowed into the other: and in this case a part of an oblique belt lay between them, as if to form a communication for this purpose. The time of their continuance is very uncertain: sometimes they remain unchanged for three months, at other times new belts have been formed in an hour or two. In some of these belts large black spots have appeared, which moved swiftly over the disk from east to west, and returned in a short time to the same place; whence the rotation of this planet about its axis has been determined. On the 9th of May 1664, Dr Hooke, with a twelve feet telescope, observed a small spot in the broadest of the three obscure belts of Jupiter; and observing it from time to time, found that in two hours it had moved from east to west about half the visible diameter of the planet. In 1665 Cassini observed a spot near the largest belt of Jupiter, which is most frequently seen. It appeared round, and moved with the greatest velocity when in the middle, but appeared narrower, and moved slower, the nearer it was to the circumference; showing that the spot adhered to the body of Jupiter, and was carried round upon it. This principal, or ancient spot as it is called, is the largest and the most permanent of any hitherto known; it appeared and vanished no fewer than eight times between the years 1665 and 1708: from the year last mentioned it was invisible till 1713. The longest time of its continuing to be visible was three years, and the longest time of its disappearing was from 1708 to 1713. It seems to have some connection with the principal southern belt; for the spot has never been seen when that disappeared, though the belt itself has often been visible without the spot. Besides this ancient spot, Cassini, in the year 1699, saw one of less stability, that did not continue of the same shape or dimensions, but broke into several small ones, whereof the revolution was but 9 hours 51 min.; and two other spots that revolved in 9 hours 52ļ min. The changes in the appearance of the spots, and the difference in the time of their rotation, make it probable that they do not adhere to Jupiter, but are clouds transported by the winds, with different velocities, in an atmosphere subject to violent agitations.

By means of the spots, which can be easily observed, the rotation of Jupiter has been determined with considerable precision. The time of rotation, according to Cassini, Maraldi, and others, is 9 hours and between 55 and 56 minutes: Schroeter makes it 9 hours 55 min. 33 see. The inclination of his equator to his orbit is only 3° 5' 30", so that the variations of his seasons must be almost insensible.

The radius of Jupiter being nearly 11 times (10*86) that of our earth, and his rotation being 2*4 times more rapid, it follows that the space passed over by a point on his equator is 26 times greater than that passed over by a point of the terrestrial equator in the same time. Hence the centrifugal force is 26 times greater; and if the spheroidal form of the earth is occasioned by the diurnal motion, we may expect to find the same effects on a much larger scale exhibited in the form of Jupiter. And this is in fact observed to be the case; for the compression of Jupiter is about ∙⅛th of his radius, the diameter of his equator being to that of his poles as 15 to 14 nearly, while that of the earth is only κ ⅛ 3 -th. According to Struve, the equatorial diameter at tĥe mean distance subtends an angle of 38"∙327, the polar 35"∙538; and the ellipticity is 0Ό728 = ļ ĝsÿļ·

The annual parallax of Jupiter is less than 12°, consequently the earth, as seen from Jupiter, will never appear at a greater distance than 11° or 12° from the sun. The digressions of Mars would be 17° 2', those of Venus 8°, and those of Mercury only 4° 16'. An inhabitant of Jupiter must therefore be probably ignorant of the existence of Mercury, which will be almost constantly plunged in [7:4:32] the sun’s rays, and likewise greatly diminished in splendour, on account of his great distance. From this we may infer the possibility of the existence of planets inferior to Mercury, and invisible to us, for similar reasons.

On observing Jupiter through the telescope, he is seen accompanied by four little stars, which oscillate on both sides of him, and follow him in his orbit as the moon follows the earth. On this account they are called satellites or attendants. They were first noticed by Galileo within a year after the discovery of the telescope; and it was soon perceived that they revolve around Jupiter in narrow circles, the planes of which deviate little from that of the equator of the planet. They are distinguished from one another by the denomination of first, second, third, and fourth, according to their relative distances from Jupiter, the first being that which is nearest to him. Their apparent motion is oscillatory, like that of a pendulum, going alternately from their greatest elongation on one side to their greatest elongation on the other, sometimes in a straight line and sometimes in an elliptic curve, according to the different points of view in which we observe them from the earth. They have also their stations and retrogradations, and exhibit in miniature all the phenomena of the planetary system.

Since the satellites revolve in orbits about the huge orb of Jupiter, it is evident that occultations of them must frequently happen, by their going behind their primary, or by coming in between us and it; in the former case when they proceed towards the middle of their upper semicircle, and in the latter when they pass through the same part of their inferior semicircle. Occultations of the former kind happen to the first and second satellites at every revolution; the third very rarely escapes an occultation; but the fourth more frequently, by reason of its greater distance. It is seldom that a satellite can be discovered upon the disk of Jupiter, even by the best telescopes, excepting at its first entrance, when, by reason of its being more directly illuminated by the rays of the sun than the planet itself, it appears like a lucid spot upon it. Sometimes, however, a satellite in passing over the disk appears like a dark spot, and can be easily distinguished. This is supposed to be owing to spots on the body of these secondary planets; and it is remarkable that the same satellite has been known to pass over the disk at one time as a dark spot, and at another appearing so luminous that it could not be distinguished from Jupiter himself, except at its coming on and going off. To account for this diversity of appearance, we must suppose either that the spots are subject to change, or, if they be permanent, like those of our moon, that the different portions of the surfaces of the satellites are not equally luminous, and that at different times they turn different parts of their globes towards us. Possibly both these causes may contribute to produce the phenomena just mentioned. By reason of the spots, also, both the light and apparent magnitude of the satellites are variable; for the fewer spots there are upon that side which is turned towards us, the brighter it will appear; and as the bright parts only can be seen, a satellite must appear larger the more of its bright side it turns towards the earth, and smaller the more it happens to be covered with spots. The fourth satellite, though generally the smallest, sometimes appears larger than any of the rest. The third sometimes seems least, though usually the largest; nay, a satellite may be so covered with spots as to appear less than its shadow passing over the disk of the primary, though we are certain that the shadow must be smaller than the body from which it is projected. To a spectator placed on the surface of Jupiter, each of these satellites would put on the phases of the moon; but as the distance of any of them from Jupiter is but small when compared with the distance of that planet from the sun, the satellites are illuminated by the sun very nearly in the same manner with the primary itself; hence they appear to us always round, having constantly the greater part of their enlightened half turned towards the earth; and, indeed, on account of their small size, their phases can scarcely be discerned even through the best telescopes. Their spots, or rather the observed variations of their brilliancy at different times, have afforded the means of determining the fact and the period of their rotation; and it is a very remarkable circumstance that they all, like the moon, constantly turn the same face towards their primary, or complete a rotation about their respective axes in the same time in which they perform a revolution in their orbits.

When the satellites pass through their inferior semicircles, they may cast a shadow upon their primary, and thus cause an eclipse of the sun; and in some situations this shadow may be observed going before or following the satellite. On the other hand, in passing through their superior semicircles, the satellites may be eclipsed in the same manner as our moon, by passing through the shadow of Jupiter. And this is actually the case with the first, second, and third of these bodies; but the fourth, by reason of the greater magnitude and inclination of its orbit, passes sometimes above or below the shadow, as is the case with the moon. The beginnings and endings of these eclipses are easily seen through the telescope, when Jupiter is at a sufficient distance from the sun. The same satellite disappears at different distances from the planet, according to the relative situations of Jupiter, the sun, and the earth; but always on that side of the disk where the shadow of the planet is known from computation to be. With regard to the first and second satellites, the immersions only are visible while Jupiter is passing from his conjunction to his opposition with the sun, and the emersions while he passes from his opposition to his conjunction. The third and fourth sometimes disappear, and again appear on the same side of the disk; and the time during which the satellite continues invisible is exactly that in which, according to computation, it would pass through the planet’s shadow. When Jupiter is near his opposition, the eclipses take place when the satellites are close to his disk; because the eye of the spectator is then nearly in the axis of the dark cone formed by his shadow.

These various phenomena will be better understood by referring to fig. 101, where A, B, C, D, represent the earth in different parts of its orbit; J Jupiter in his orbit MN, surrounded by his four satellites, the orbits of which are marked 1, 2, 3, 4. At a the first satellite enters the shadow of the planet; at b it emerges from it, and advances to its greatest eastern elongation at c. It appears to pass over the disk of Jupiter like a dark spot at d, and attains its greatest western elongation at e. Similar phenomena take place with respect to the other satellites. Now, since the shadow of Jupiter is always directed away from the sun, the immersions only will be visible to the earth when the earth passes from the position C to the position A; for the eastern limit of Jupiter conceals the satellite at the time of emersion, as is evident by drawing f 9 in the direction of the visual ray. For the same reason the emersions only are visible while the earth is passing from A to C, or when Jupiter advances from his opposition to his conjunction. This, however, is only strictly true of the first satellite; for the third and. fourth, as we have already remarked, and sometimes even the second, owing to their greater distances from Jupiter, occasionally disappear and re-appear on the same side of the disk.

The disks of the satellites having no sensible magnitude[7:4:33] except in the very best telescopes, their diameters have only recently been determined by direct measurement. Schroeter and Harding attempted to measure them by observing the time which the satellite takes to pass over the disk of Jupiter; but such observations are liable to great uncertainty, by reason of differences in the magnifying power of the telescopes, the sight of the observer, the state of the atmosphere, the distance of the satellites from the primary, their altitude above the horizon, and even on account that, by reason of their rotation, they do not always present to us the same hemisphere. Schroeter estimates their diameters relatively to Jupiter as follows :—That of the first =∙⅛, of the second =⅛, of the third —⅛ t ^ ιe f° urt b = ⅛∙ The following are the results of a series of micrometrical measurements made by Professor Struve at the Dorpat observatory, with the great refractor of Fraunhofer :—

Diameter of the first 1"∙015

Diameter of the second 0 ∙911

Diameter of the third 1 *488

Diameter of the fourth 1 ∙273

These dimensions are adapted to the mean distance of Jupiter, namely 5∙20279. Compared with the earth, the diameters of the satellites are approximately as follows :— I. = ⅜, II. = į, III. = ⅜, IV. = + As seen from Jupiter, the apparent magnitude of the first will be nearly equal to that of our moon seen from the earth; the second and third somewhat greater than half; and the fourth nearly equal to a quarter of that of the moon. These four moons must present to the inhabitants of Jupiter a spectacle of endless variety, on account of the rapid rotation of the planet, the short period of their revolutions, and their eclipses, which happen almost daily.

Saturn.

Saturn, the remotest of the planets known to the ancient astronomers, circulates round the sun at a distance equal to about 9∣ times the semidiameter of the terrestrial orbit, or nearly 900 millions of miles. His apparent diameter at his mean distance is only about 16"∙2, yet his true diameter is nearly 10 times, and his volume about 995 times, that of our globe. The area of the sun’s disk, as seen from Saturn, is only of its apparent magnitude as seen from the earth; consequently the light and heat which any point on his surface receives from the great luminary is 80 times less than that which we enjoy. His density, compared with that of the sun considered as unity, is supposed to be ∙55, or about J of the density of the earth; and a body which weighs one pound at the equator of the earth, would weigh about P01 pound if transferred to the equator of Saturn.

This planet, in consequence of a luminous double ring with which he is surrounded, presents one of the most curious phenomena in the heavens. This singular appendage was first noticed by Galileo, to whom the planet presented a triple appearance, the large orb being situated between two small bodies or ansoe. Sometimes the ansae are so enlarged as to present the appearance of a continuous ring; at other times they entirely disappear, and Saturn appears round like the rest of the planets. After a certain time they again become visible, and gradually increase in magnitude; and they evidently do not adhere to the surface of the planet, inasmuch as a vacant space between them is distinctly perceived even in ordinary telescopes.

These curious appearances were shown by Huygens to be occasioned by an opaque, thin, circular ring, surrounding the equator of Saturn, and at a considerable distance from the planet. Saturn moving in the plane of his orbit carries the ring along with him, which, presenting itself to the earth under different inclinations, occasions all the phenomena which have been described. The ring being only luminous in consequence of its reflecting the solar light, it is evident that it can be visible only when the sun and the earth are both on the same side of it: if they are on opposite sides it will be invisible. It will likewise be invisible in two other cases, namely, 1sZ, when its plane produced passes through the centre of the earth, for then none of the light reflected from it can reach us; and, 2d, when its plane passes through the sun, because its edge is then only enlightened; and being very thin, the whole quantity of reflected light will scarcely be sufficient to render it visible. It is, however, evident that in these two cases the effect will be modified in some degree by the power of the telescope. In ordinary telescopes the ring disappears sometimes before its plane comes into either of the situations mentioned; but Herschel never lost sight of it, either when its plane passed through the earth or the sun. In the last case the edge of the ring appeared as a luminous line on the round disk of the planet, measuring scarcely a second in breadth; but at the distance of Saturn a second corresponds to 4000 miles, which is equal to the semidiameter of the terrestrial globe. The reason of the ring’s disappearance will be easily understood by referring to fig. 102, where the circle abed represents the orbit of the earth, A B C D that of Saturn 9 3 times more distant from the sun. When Saturn is at A, the earth and sun are both in the plane of the ring; its edge is consequently turned towards us, and it will be invisible unless telescopes of very high power are used. As Saturn advances from A to B the ring gradually opens, and it attains its greatest breadth at C, where its face is turned more directly towards us, or a straight line perpendicular to its plane makes a more acute angle with the visual ray than in any other situation. As the planet advances towards D, the plane of the ring becomes more oblique to the visual ray; the breadth of the ring consequently contracts, and it again disappears at Ľ. From E to F, G, and A, the same phenomena will be repeated, only in this case it is the southern side of the ring which is visible to the earth, whereas, while Saturn was in the other half of his orbit, it was the northern side.

The successive disappearances of the ring form a period of about 15 years, with some variations arising from the different positions of the earth in its orbit. At present (1830) the south side is presented to us; it will be invisible in 1833; its northern side will become visible in 1838; it will again disappear in 1847, and show its southern side in 1855.

Sir W. Herschel’s observations have added greatly to our knowledge of Saturn’s ring. According to him, the ring is separated into two annular portions by a dark belt or zone, which he has constantly found on the north side. As this dark belt is subject to no change whatever, it is probably owing to some permanent construction of the surface of the ring; and it is evidently contained between two concentric circles, for all the phenomena correspond with the projection of a circular zone. The matter of the ring Herschel thinks no less solid than that of Saturn, and it is observed to cast a strong shadow upon the planet. The light of the ring is also generally brighter than that of the planet; for it appears sufficiently luminous when the telescope affords scarcely light enough for Saturn, It is remarkable that the outer ring is much less brilliant than the inner. Herschel concludes that the edge of the ring is not flat, but spherical or spheroidal. The dimensions of the ring, or of the two rings with the space between them, he gives as follows :—
[7:4:34]

Ailles.

Inner diameter of smaller ring 146345

Outside diameter of ditto 184393

Inner diameter of larger ring 190248

Outside diameter of ditto 204883

Breadth of the inner ring 20000

Breadth of the outer ring 7200

Breadth of the vacant space, or dark zone 2839

The following measures were taken by Professor Struve, at Dorpat in Russia, in 1828, with a repeating wire-micrometer attached to the large refracting telescope of Fraunhofer, belonging to the observatory at that place, and may be regarded as decidedly the most accurate of any that we possess. (See Memoirs of the Astronomical Society, vol. iii. p. 301.)

Outer diameter of the outer ring 40"∙095

Inner diameter of the outer ring 35 ∙289

Outer diameter of the inner ring 34 ∙475

Inner diameter of the inner ring 26 - 668

Breadth of the outer ring 2 ∙403

Breadth of the division between the rings 0 ∙408

Breadth of the inner ring 3 ∙903

Distance of the ring from the ball 4 ∙339

Equatorial diameter of Saturn 17 ∙991

These dimensions are adapted to the mean distance of Saturn, 9∙53877. According to the same excellent astronomer, the inclination of the plane of the ring to the ecliptic is 28° 5'∙9.

In observing the ring with very powerful telescopes, some astronomers have remarked, not one only, but several dark concentric lines on its surface, which divide it into as many distinct circumferences. In common telescopes these are not perceptible; for the irradiation, by enlarging the space occupied by each ring, causes the intervals between them to disappear, and the whole seems blended together in one belt of uniform appearance. (See fig. 103.) Struve, however, noticed no trace of the division of the ring into many parts.

By means of some spots observed on the surface of the ring, Herschel found that it revolves in its own plane in 10 hours 32 minutes 15 ’ 4 seconds; and Laplace arrived at the same result from theory. It is particularly worthy of remark, that this is the period in which a satellite, having for its orbit the mean circumference of the ring, would complete its revolution according to the third law of Kepler. This circumstance furnishes a physical explanation of the reason why the ring is able to maintain itself about the planet without touching it; or at least brings the fact within the general law by which the planets are sustained in their orbits. The centrifugal force resulting from its rotation, and the attraction of the planet, suffice to maintain its equilibrium.

From observations of some obscure belts, and a very conspicuous spot on the surface of Saturn, Herschel concluded that his rotation is performed in 10 hours 16 minutes, on an axis perpendicular to the belts and to the plane of the ring; so that the planes of the planet’s equator and ring coincide. According to the same astronomer, the ratio of the equatorial and polar diameters of Saturn is 2281 to 2061, or nearly 11 to 10. But it is very remarkable that the globe of Saturn appears to be flattened at the equator as well as at the poles. The polar compression extends to a great distance over the surface of the planet, and the greatest diameter is that of the parallel of 43° of latitude, where, consequently, the curvature of the meridians is also the greatest. The disk of Saturn, therefore, resembles a square of which the four corners have been rounded off. According to the latest observations of Herschel, the axis of rotation, the diameter of the equator, and the greatest diameter under the parallel of 43°, are to one another as the numbers 32, 35, 36. (See the Philosophical Transactions for 1806, Part II.)

Saturn is attended by seven satellites, but so small that they can only be seen by the help of powerful telescopes. Huygens first discovered one of these satellites in 1655. It is the sixth in the order of distance, and is the largest of them all. Four others were discovered about twenty years afterwards by Dominic Cassini; and, lastly, SirW. Herschel, in 1789, discovered two new satellites, at a time when the ring was visible only in a telescope of forty feet. The orbits of these are interior to those of the five satellites formerly discovered, but exterior to the ring, though so near to it that it is only when the ring disappears that they can be seen. All the satellites appear to revolve in the plane of the ring, with the exception of the two last. The inclinations of their orbits are, however, not known with much certainty.

Uranus.

Uranus is the remotest planet belonging to the system, and is scarcely visible excepting through the telescope. His distance from the sun is nineteen times the radius of the ecliptic, or about 1800 millions of miles; and his sidereal revolution is performed in 83 years 150 days and 18 hours. His diameter is about 35,112 English miles, or nearly six and a half times that of the earth, and, seen from the earth, subtends an angle of only 4", even at the time of his opposition. The apparent diameter of the sun, seen from this planet, is 1' 40"; consequently the surface of the sun will there appear 400 times less than it does to us, and the light and heat which is received will be less in the same proportion. Analogy leads us to infer that Uranus is opaque and revolves on his axis, but of this there is no direct proof. Laplace has concluded from theory, that the time of his diurnal rotation cannot be much less than that of Jupiter and Saturn, and that the inclination of his equator to the ecliptic is very inconsiderable. His density is supposed to exceed somewhat that of the earth. Schroeter thinks that certain variations in the appearances of his disk indicate that great changes are going on in his atmosphere.

This planet was discovered by Sir W. Herschel atBath on the 13th of March 1781. His attention was attracted to it by the largeness of its disk in the telescope, which exceeded that of stars of the first magnitude, while to the naked eye it appeared only as a star of the seventh magnitude. In the course of a few days its proper motion became sensible, consequently it could not be a fixed star. Herschel at first took it for a comet, but it was soon perceived that it described a path which, instead of resembling the eccentric orbits of the comets, was almost circular, like that of the planets. It was then recognised to be one of the principal planets of the solar system; and the observations of the last fifty years have not only confirmed this fact, but afforded data from which the elements of its orbit have been determined with great precision. Herschel gave it the name of the Georgium Sidus, in honour of his royal patron George III. Foreigners for some time generally called it the Herschel, after its discoverer; but the mythological name of Uranus, suggested by the late Professor Bode of Berlin, is now generally adopted. In 1787 Herschel discovered that it was attended by two satellites: he subsequently discovered four others, so that Uranus is accompanied by six satellites; and the whole number of satellites now known to belong to the system is consequently eighteen.

Before concluding this section, it will be proper to take notice of the following curious relation of the numbers [7:4:35]which express, approximately at least, the distances of the planets from the sun. It was first pointed out by Bode, and, though purely empirical, and not even very accurate, served to confirm the German astronomers in their anticipations of the discovery of a new planet between the orbits of Mars and Jupiter.

Let the number 10 be assumed to represent the semidiameter of the earth’s orbit; then the semidiameters of the orbits of the other planets may be expressed in round numbers as follows :—

Mercury...4

Venus 7 = 4 + 3∙2°

Earth 10 = 4 + 3∙2 1

Mars 16 = 4 + 3∙2 2

Ceres 28 = 4 + 3∙2 3

Jupiter...52 = 4 + 3∙2 4 Saturn ...100 = 4 + 3∙2 5 Uranus..196 = 4 + 3∙2 g

It will be remarked that every succeeding term of this series of numbers, after the second, is the double of the preceding, minus 4; the general term being 4 + 3∙2" -2 , commencing with Venus, and n indicating the rank of the planet.

A view of the proportional magnitudes of the orbits and disks of the planets, as also of the comparative magnitudes of the sun seen from each planet, is given in fig. 104.

On the subjects contained in this section the following works may be consulted: Galileo, Nuntius Sidereus ; Simon Marius, Mundus Jovialis anno 1609 detectus, &c. 1614; Cassini, Martis circa proprium axem revolubilis Observationes Bononienses, Bonon. 1666; Idem, Disceptatio Apologetica de Maculis Jovis et Martis, Bonon. 1667; Idem, Nouvelles Découvertes dans le Globe de Jupiter, Paris, 1690; Bianchini, Hesperi et Phosphori Nova Phaenomena, 1728; Cassini, Elemens d’ Astronomie, 1740; Schroeter, Aphro-ditographische Fragmente, Helmstadt, 1796; Idem, Lilien-thalische Beobachtungen der neu entdeckten planeten, Göttingen, 1805; Gauss, Theoria Motus Corporum Caelestium, Hamburg!, 1801; Idem, Journal of Gotha, 1811; Laplace, Mécanique Céleste, tome iv. p. 135; Idem, Système du Monde ; Delambre, Astronomic, tome ii. chap, xxvii.; Schubert, Traité d’ Astronomie Théorique, tome ii. Petersburg, 1822; and the numerous papers of Sir W. Herschel in the Philosophical Transactions. Sect. IV.— Of the Orbits of the Satellites.

In order to establish a theory of the motions of the satellites, the first thing necessary is to ascertain the directions in which they move with reference to the primary planet. Now, it is observed that their motion is sometimes towards the east, and at other times towards the west; but that the satellites are never eclipsed except in passing from the west to the east of the planet. When an eclipse takes place the satellite is always moving eastward; on the other hand, the satellite is always moving westward when it appears on the planet’s disk. From this it results that the true motions of the satellites around the planets are from west to east, according to the order of the signs, or in the same directions as the motions of the planets about the sun. This fact, which holds true of the moon, and the satellites of Jupiter and Saturn, is one of the most remarkable in the planetary system. With regard to the satellites of Uranus, their motions are performed in orbits almost perpendicular to the ecliptic; they cannot therefore with propriety be said to be either direct or retrograde.

The eclipses of the satellites of Jupiter present an easy method of determining their mean motions and periodic ' times; and, by reason of the small inclinations of the or- j bits, these are of very frequent occurrence; for the first three satellites traverse the shadow of Jupiter in every revolution, and the fourth only passes by it sometimes, in consequence of the greater inclination of its orbit. If the instants can be observed at which a satellite enters and emerges from the shadow, the middle point of time between these two instants will be that of the heliocentric conjunction of the satellite with its primary. The interval between two central eclipses gives the synodic period of the satellite; whence, since the motion of the primary is known, the sidereal period of the satellite, and its mean angular motion with regard to the straight line joining the centres of the sun and the planet, are easily deduced. Instead of two successive eclipses, it is preferable to compare two that are separated from each other by a long interval of time; the interval divided by the number of sidereal revolutions will give the mean time, unaffected by any periodic inequalities which may exist in consequence of the mutual action of the satellites on one another. In order to render the result as accurate as possible, those eclipses are chosen which take place when the planet is nearly in opposition.

The distances of the satellites from their primary are ascertained by measurement with the micrometer, at the time of the greatest elongations. On comparing the distances with the times of revolution, the beautiful law of Kepler is found to prevail; and, as in the system of the planets, so in the various systems of the satellites, the squares of the periodic times vary as the cubes of the mean distances from the central body. The distances of the satellites of Jupiter and Saturn, compared with the diameters of their respective primaries, are represented in fig. 105.

The inclinations of the orbits of the satellites of Jupiter, and the positions of their nodes, together with the other elements of their elliptic motion, are determined by means of their eclipses. The plane of the orbit of the first satellite nearly coincides with the plane of the equator of Jupiter, the inclination of which to the plane of his orbit is 3° 5' 30". The inclination of the orbit of the second to the plane of the planet’s equator is 27' 49"∙2. Its nodes have a retrograde motion on that plane, and go through an entire circuit in the space of 30 years. The orbit of the third is inclined to the equator of Jupiter in an angle of 12' 20"; and the line of the nodes retrogrades through a whole circumference in 142 tropical years. Hence tĥe inclinations of the orbits of these two satellites to the orbit of Jupiter are variable; that of the second varying between 3° 19' 24"∙6 and 2° 51' 35"∙4, and that of the third between 3° 17' 50'' and 2° 53' 10". The inclination of the orbit of the fourth satellite to the orbit of Jupiter is also variable. Its nodes have a retrograde motion, and complete a revolution in 531 years. Since the middle of the last century the inclination of this satellite has been observed to increase, and the motion of its nodes to diminish. (Laplace, Mécanique Céleste, tome iv. livre viii.)

The orbits of the four satellites are doubtless elliptical, but those of the first and second are so small that it has been found impossible to determine their eccentricity. The eccentricity of the third is perceptible, that of the fourth much more so. According to Laplace, the greatest equation of the centre of the third, at its maximum in 1682, amounted to 13' 16"∙4, and at its minimum in 1777, only to 5' 7"∙5. The eccentricity of the orbit of the fourth is still greater, and also subject to considerable variations. The line of the apsides has a direct motion, amounting to 42' 58"∙7 annually. The great in [7:4:36] fluence of the compressed figure of Jupiter on these elements gives each of the orbits an eccentricity peculiar to it; but each also participates in the eccentricities of the others. The mutual perturbations of the satellites greatly affect their motions, and render their analytical theory exceedingly complicated and difficult.

On comparing the mean longitudes with the mean motions of the first three satellites of Jupiter, Laplace discovered the two following relations, which, by reason of their remarkable simplicity, may perhaps be regarded as among the most curious discoveries cver made in astronomy. Denoting by m', m", nι"', the mean motions of the three satellites respectively, and their mean longitudes by t, l”, l'", these laws are expressed by the formulae, ml + 2m'" — 3m'' — 0

l' +2Z w — 3Z" =180°; that is to say, the mean sidereal motion of the first, added to twice that of the third, is equal to three times that of the second; and the mean longitude of the first satellite, plus twice that of the third, minus three times that of the second, is always equal to a semicircumference. The first of these relations is true of the synodical as well as of the sidereal revolutions; and it follows from the second, that the first three satellites can never be eclipsed at the same time, because in that case their longitudes would be equal, or Z' + 2Z'"— 3Z''=0. These results of theory agree so nearly with observation, that we are tempted to regard them as rigorously exact, and to ascribe the slight differences that may be perceptible to the unavoidable errors of observation, and to the periodic inequalities in the motions of the satellites, by reason of which their true motions are alternately greater and smaller than their mean motions. We must therefore infer that these relations depend on a physical cause, by which they will be preserved for ever, or at least during a long series of ages, notwithstanding the small oscillations to which, from various sources of perturbation, the mean longitudes of the satellites are subject.

From observations of the eclipses of Jupiter’s satellites, Roemer was led to the very important discovery of the successive propagation of light. The times at which these eclipses happen are found to differ from the times computed from the sidereal revolutions of the satellites, being sometimes earlier and sometimes later, according to the position of Jupiter relatively to the sun and the earth. When Jupiter is in opposition with the sun, and his distance from the earth consequently less than his distance from the sun by the whole radius of the earth’s orbit, the satellites are eclipsed sooner than they ought to be according to computation. On the contrary, when Jupiter is in conjunction, and his distance from the earth greater than his distance from the sun by the same quantity, the eclipses happen later. These differences, which are exactly the same for all the satellites, cannot be ascribed either to the eccentricity of the orbit of Jupiter, or to inequalities in their motion; for the oppositions and conjunctions of the planet correspond successively to all the different points of his orbit, and the eclipses also happen when the satellite is at different points of its own orbit. The simplest and most natural way of explaining the phenomenon is to suppose that the light reflected from the satellites is not transmitted to the earth instantaneously, but occupies a sensible portion of time in traversing the diameter of the terrestrial orbit. When Jupiter is near his conjunction, the eclipses are observed to happen about 16 minutes 26 seconds later than when he is near his opposition; the difference between his distances from the earth in these two positions is equal to the diameter of the earth’s orbit, supposing the orbits to be circular: it follows, therefore, that light employs 16 minutes 26 seconds in traversing the terrestrial orbit, and consequently the half of that time, or 8 minutes 13 seconds, in coming from the sun to the earth. The exact agreement of this hypothesis with observation renders its truth unquestionable. The fact of the successive transmission of light led to another discovery of the utmost importance in astronomy, namely, the aberration.

The eclipses of Jupiter’s satellites are useful in determining the longitude of places on the earth, and on this account the theory of the motions of these bodies has been cultivated with the most laborious care. The epochs at which the eclipses take place are calculated in advance, and inserted in the Ephemerides. On comparing these epochs, computed for a given meridian, with the immediate results of observation made in another place at a given hour, the difference of time is obtained, whence the difference of longitude is immediately deduced. The method is the same as for the eclipses of the moon. Unfortunately, by reason of the magnifying power required to render the satellites visible, it cannot be employed at sea, the instability of the vessel rendering the telescopes unserviceable. The tables of Delambrc, which were computed from the theory of Laplace, and the comparison of an immense number of observations, give the places of the satellites with all the precision which it is perhaps possible to obtain.

The satellites of Saturn have not the practical utility of those of Jupiter, because, by reason of their great distance, their eclipses are invisible; and indeed some of them cannot be perceived at all, excepting through telescopes of extraordinary power. Their periods, mean motions, and some of the other elements of their orbits, are determined by the micrometrical measurement of their greatest digressions from their primary. At the time of their greatest digressions they are always situated in the same straight line with the greater axis of the ring, and their distances from Saturn are then equal to the semitransverse axes of their apparent orbits. In their conjunctions the minor axes of their apparent ellipses seem only half as great as their transverse axes, whence it is inferred that the sine of their inclination is one half, and consequently that the inclination itself is about 30°, which is nearly the inclination of the ring. Hence the satellites seem to move in the plane of the ring. The only one which deviates considerably from that plane is the seventh. From certain observations made by Bernard in 1787, Lalande makes the inclination of this satellite 22° 42' to the orbit of Saturn, or 24° 45' to the ecliptic. There exists, however, considerable uncertainty with regard to the inclination of the orbits of the satellites, as well as with regard to that of the ring.

According to the theory of Laplace, the spheroidal figure of Saturn must maintain the ring, and the orbits of the interior satellites, in the plane of his equator. But the amount of the compression is unknown: all that is certain is, that the inclination of the orbits of the first five satellites to the equator of the planet is insensible at the distance of the earth; consequently all the satellites, excepting the seventh, and probably also the sixth, will appear, as well as the ring, to move in the plane of the equates. The orbit of the seventh satellite preserves the same mean inclination to the plane of the equator of the planet; and the line of its nodes has a retrograde motion nearly uniform.

The orbit of the sixth satellite is elliptical. For the meridian of Paris its longitude in 1800 was 67° 25' 47"; that of its inferior apside 203° 35' 7". Its mean motion in 36525 days is 2290 revolutions + 202° 12'; in one day [7:4:37] 22° 34' 37"∙186. The eccentricity is Ό4887; and its greatest equation 5° 36' 8". By particular observations on this satellite, Bessel found its inclination to the ecliptic to be 24° 30', or 25° 55'; differing very sensibly from that of the ring or the equator of the planet. It is, however, not improbable that these two planes may have some inclination to each other. (Delambre, Astronomie Théorique et Pratique, tome iii. p. 510.)

The satellites of Uranus can be perceived with still greater difficulty than those of Saturn. The orbits of these satellites are almost perpendicular to the ecliptic. The elements of the second and fourth have been determined by actual measurement; the periods of the two others have been theoretically deduced from the third law of Kepler. The inclination of the fourth is 89° 30', or 90° 30', and the ascending node 171° or 249°, according as it is conceived to be direct or retrograde.

The following table exhibits the mean distances and sidereal revolutions of the satellites of Jupiter, Saturn, and Uranus.
[table]

CHAP. V.

OF COMETS.

The comets form a class of bodies belonging to the solar system, distinguished from the planets by their physical appearances and the great eccentricity of their orbits. By reason of the smallness of their diameters, and a nebulosity which renders them ill adapted to reflect the rays of light, the greater part of them are only visible in the telescope, and continue to be so only during a short period of time; for as they advance to and recede from the sun almost in straight lines, and with prodigious velocities, they are soon carried far beyond the limits of vision. They have received the name of comets {coma, hair) from the bearded appearance which they frequently exhibit.

Sect. I.— Of the Orbits of Comets.

The comets are not more remarkably contrasted with the planets in the singularity of their physical appearances, than in the directions of the paths which they follow in space. While the orbits of all the planets are confined within a narrow zone, or to planes not greatly inclined to the ecliptic, those of the comets are inclined in all possible angles, and some of them are even observed to be perpendicular to the ecliptic. Nor is the contrast less striking with regard to the figures of the orbits, which, instead of being nearly circular, like those of the planets, have the appearance of being almost rectilinear. Kepler was of opinion that the cometary orbits are straight lines; Cassini supposed them to be very eccentric circles; and Tycho was for some time of the same opinion, but he afterwards found that the hypothesis of the eccentric circle would not satisfy the observations of the comet of 1577. Hevelius seems to have been the first who discovered, by means of a geometrical construction, that the orbits might be represented by parabolas; and Dörfel first calculated their elements on this hypothesis. After it was known, however, that certain comets return to the sun in the same orbits, it became necessary to adopt an opinion, already probable from analogy, that, in conformity with the laws of Kepler, the cometary orbits are ellipses having the sun in one of the foci. This hypothesis is now universally admitted; but as the ellipses are in general extremely elongated, and the comets are only visible while they describe a small portion of their orbits on either side of their perihelia, their paths during the time of their appearance differ very little from parabolas; whence it is usual, on account of the facility of computation, to assume that they really move in parabolic curves. Newton employed the hypothesis of an elliptic motion to compute the orbit of the famous comet of 1680. Since that time the orbits of more than a hundred and fifty different comets have been computed on the elliptic hypothesis, and their elements determined so as to satisfy all the observations. It is possible that the orbits of some comets may be in reality parabolic; but in this case the comet, after having passed its perihelion, would recede to an infinite distance from the sun, and never again visit our system. Burckhardt imagined that the observations of the comet of 1771 were best represented by supposing the orbit to be an hyperbola. In fact, it is demonstrated in the Principia, and every treatise on Physical Astronomy, that the species of curve which one body describes about another, in virtue of an attractive force varying inversely as the square of the distance, depends only on its velocity of projection. The curve must necessarily be a conic section; but it may be an ellipse, a parabola, or an hyperbola, according as the primitive impulsive force falls within or exceeds certain assignable limits.

If the comets moved in parabolas or hyperbolas, and had consequently only a temporary connection with the solar system, the determination of their orbits would be a matter of mere curiosity, and of no consequence whatever to astronomy. But it is only by the accurate determination of the elements of their orbits that it can be discovered whether those bodies ever revisit the system; for the appearances which they exhibit, depending on the situation of the earth in its orbit with relation to them at the time they are visible, are far too variable and uncertain to afford any sure means of recognising them. The comet of 1811 was scarcely visible in the months of April and May; it was subsequently lost in the sun’s rays, and, having passed its perihelion, re-appeared in August with a splendour and magnificence that rendered it an object of admiration. It is therefore only by observing that a comet follows the same orbit in its successive returns to the sun that we can be assured of its identity. But even the determination of the orbit is not always sufficient to lead to the detection of a comet in a [7:4:38]1 subsequent revolution; for if, in the course of the previous one, it came within the sphere of attraction of Jupiter, or any of the larger planets, the elements of its orbit may have been greatly or entirely changed. The orbit of the comet of 1770 was calculated by Lexell, and subsequently by Burckhardt, and both these astronomers found that the observations could only be represented by an ellipse in which the time of revolution was five years and a half; yet the comet has never been seen since, or at least seen moving in the same orbit. Hence it is concluded with certainty that the attraction of Jupiter, near which planet it approached, was so great as to compel it to move in a totally different ellipse. Other causes also conspire to render the chances of the discovery of periodic comets extremely small. In the first place, it is only within a very small portion of their orbits that they are visible; and this, on account of its proximity to the sun, is passed over with inconceivable rapidity. But in proportion as they recede from their perihelia the solar action diminishes; and, towards the aphelion of its orbit, a comet may be almost motionless, and for this reason not return for thousands of years. In the second place, it may happen, that during the greater part of the time the comet continues in the visible portion of its orbit, it may rise above the horizon only during the day, in which'case it will be invisible, and may consequently pass through our system without being observed. The comet of 1818 was present in all its splendour long before it became visible, but in full day. Seneca relates a very curious instance of one having been seen during a total eclipse of the sun, in the year 60 before our era. A third cause of uncertainty consists in the difficulty of observing their true places with sufficient precision to enable the elements of their orbits to be exactly determined. The small comets are only nebulous points, which can be distinguished with difficulty; the larger ones are surrounded with a variable, ill-defined, and indistinct nebulosity. The comet of 1729 continued visible during six months: its orbit was computed by three different astronomers, whose results were far from coinciding. The same uncertainty exists with regard to the orbits of several other comets. When, therefore, all these circumstances are taken into consideration, it will not appear surprising that, although the elements of above 150 comets have been computed, there are only three which are certainly known to have been observed in their successive revolutions.

The first of the three periodic comets with which we are yet acquainted was made known to astronomers by Dr Halley. That active and indefatigable genius, having perceived that in 1682 the elements of its orbit were nearly the same as those of two comets which had respectively appeared in 1531 and 1607, concluded that the three orbits belonged to the same identical comet, of which the periodic time was about 76 years. After a vague estimate of the perturbations it must sustain from the attraction of the planets, Dr Halley predicted its return for 1757,—a bold prediction at that time, but justified by the event, for the comet again made its appearance as was expected, though it did not pass through its perihelion till the month of March 1759, the attraction of Jupiter and Saturn having caused, as was computed by Clairaut previously to its return, a retardation of 618 days. This comet had been observed in 1006; and the accounts which have been preserved represent it as having then appeared to be four times the size of Venus, and to have shone with a light equal to a fourth of that of the moon. History is silent respecting it from that time till the year 1456, when it passed very near to the earth: its tail then extended over 60° of the licavens, and had the form of a sabre. An object so striking and so terrific could not fail, in a superstitious age, to excite universal dismay, and be regarded as portentous of the greatest calamities to the human race, if not of the destruction of the world. Accordingly Pope Calixtus ordered public prayers to be said over Christendom, in which he exorcised the comet, and the Turks, who had at that time made themselves masters of Constantinople, and overthrown the eastern empire. Dr Halley’s comet will return to its perihelion in 1835; but whether on this occasion it will present any resemblance to its former appearances, or whether it will even be visible in Europe, cannot be certainly determined. The following table of its elements in 1835 is given by Pontécoulant (Théorie Analytique du Systeme du Monde, tome ii. p. 147):

Instant of the passage through the perihelion in October 1835 31 d ∙2

Semiaxis major 17 ∙98355

Ratio of the eccentricity to the semiaxis major..0 ∙967453 Place of the perihelion on the orbit 304° 34' 19"

Longitude of the ascending node 55 6 59

Inclination 17 46 50.

The two other comets whose periodic returns have been verified by observation have received the names of Encke and Biela, the astronomers who first computed their orbits, or recognised them as having been observed in their previous revolutions. The first returns to its perihelion in 1208 days, and the second in 2440 days. Encke’s comet, although its identity was not discovered till 1818, has been frequently observed, as in 1789, 1795, 1801, and 1805, and on these occasions it exhibited very different appearances, having been seen with and without a nucleus, with and without a tail,—circumstances which account for its having so long escaped being recognised as a regular attendant on the sun. In its returns to its perihelion in 1808, 1812, and 1815, it escaped detection; but it re-appeared in 1818, and it was from the observations of this year that Encke computed the elliptic elements of its orbit. On its next return, in 1822, it was invisible in Europe; but it was observed at Paramatta, in New South Wales, during the whole month of June, and the time of its perihelion passage was found to differ only by about three hours from that previously computed by Encke. On its returns in 1825 and 1828 its observed and computed places agreed equally well. The following, according to Pontécoulant, are its elements for 1829-30, computed from the observations at Paramatta:

Passage through the perihelion, 1829, January 10 d ∙573 Mean diurnal motion 1069 ∙5570

Eccentricity 0 ∙8446862

Place of perihelion 157° 18' 35"

Longitude of ascending node 334 24 15

Inclination 13 22 34.

Encke’s comet presents in some respects a considerable analogy with the planet Ceres, the inclination and greater axis of its orbit being the same, while its sidereal revolution is only 46 days shorter than that of Vesta. The orbit is, however, greatly more elongated, for its perihelion falls within the orbit of Mercury, and its aphelion is situated between Jupiter and the new planets. The perturbations it sustains are chiefly occasioned by the attraction of Jupiter, that of the earth and Venus being extremely small, while the action of Mercury is insensible.

The third periodic comet with which our knowledge of the solar system has recently been enriched receives its name from Biela, by whom it was first perceived in Bohemia, on the 25th of February 1825. The parabolic elements computed from the first observations presented a striking resemblance with those of two comets [7:4:39]which had been observed in 1772 and 1806, which induced MM. Clausen and Gainbart, the first at Marseilles and the second at Altona, to compute the elements of the three comets on the hypothesis of elliptic orbits; and, after some attempts, each found an ellipse which represented all the observations so accurately as to leave no doubt of the identity of the comet. Its period is six years and about nine months, and it will return to its perihelion in November 1832, about the same time with Encke’s. The following table of its elements has been computed from the observations of 1826, and the theory of the perturbations (Pontécoulant, tome ii. p. 158):

Passage through the perihelion 1832, November 27⅞4808 Eccentricity 0∙75l 4481

Place of the perihelion 109° 56' 45"

Longitude of ascending node 248 12 24

Inclination 13 13 13

Semiaxis major 3∙53683.

Such is the present state of astronomy with respect to periodic comets. There are two others besides, of which the orbits are supposed to be known, though their returns to their perihelia have not yet been verified. The first is that which appeared in 1680, and of which Newton computed the period to be 575 years. It may therefore be identical with those which are recorded in history to have appeared in 1106,∙ 531, 34 b. c. and 619 b. c. The second is that which appeared in 1556, and is supposed to have made a former visit in 1264. This comet is expected to appear again in 1848.

The following table, taken from one given by Delambre in the third volume of his Astronomie, shows the comparatively small distances within which the greater part of the comets hitherto observed approach to the sun, and the apparently fortuitous inclinations of the planes of their orbits to that of the ecliptic. Supposing the sun’s distance from the earth to be unity, then, of 120 comets, there are

5 whose perihelion distance is less than...0∙1

3 between 0∙1 and 0 ’ 2

6 0∙2 and 0∙3

11 0∙3 and 0 ’ 4

10 0∙4 and 0∙5

22 0∙5 and 0∙6

12 0∙6 and 0∙7

11 0∙7 and 0∙8

8 0∙8 and 0∙9

9 0∙9 and 1

21 . 1 and 2

1 equal to 2∙293

1 4∙069.

Of the same number there are

4 whose inclination is between 1° and 5°

3 5 and 10

4 10 and 15

3 15 and 20

2 20 and 25

2....∙. 25 and 30

7 30 and 35

2 35 and 40

4 40 and 45

] 45 and 50

3 50 and 55

4 55 and 60

7 60 and 65

3 65 and 70

3 .70 and 75

3 75 and 80

4 80 and 85

2 85 and 90.

The motions of the above 61 are direct; those of the remaining 59 are retrograde, and their orbits are distributed over the whole quadrant in the same random manner. This circumstance sufficiently indicates that the mechanical causes, whatever they were, which gave the same direction to the two motions of translation and rotation of all the planets and satellites, exercised no influence on the comets. Hence many astronomers have entertained the idea that these bodies have only a casual or transient connection with the planetary system. , Sect. II.— Of the Appearances and Physical Constitution of Comets.

Of all the celestial phenomena, those of the comets are the most striking, and the most calculated to impress the ignorant with the idea of supernatural agency. Appearing suddenly in the heavens, and under aspects the most uncommon and terrific, they have been almost universally regarded as visible demonstrations of the wrath, and harbingers of the vengeance, of offended deities. These superstitious terrors, arising from thatγain propensity of the mind of man to regard the universe as created for himself alone, have only been dissipated by the progress of sound philosophy, and a more extended acquaintance with the riches of nature, and the endless variety of her productions.

The appearances exhibited by the comets are exceedingly diversified, and sometimes extremely remarkable. That which appeared in the year 130 b. c., at the birth of Mithridates, is said to have had a disk equal in magnitude to that of the sun. Ten years before this, one was seen, which, according to Justin, occupied a fourth part of the sky, that is, extended over 45°, and surpassed the sun in splendour. Another, equally remarkable, appeared in the year 117 of our era; and in 479 there was one of which the disk, according to Freret, was of such magnitude that it might have occasioned the extraordinary eclipse of the sun which took place about that time. In 400 one was observed, which is said, on the authority of Gainas, to have resembled a sword, and to have extended from the zenith to the horizon. That of 531 was of greater magnitude still, and its appearance more terrific. Those which appeared in 1066 and 1505 exhibited disks larger than that of the moon. It is, however, highly probable that all these accounts have been greatly exaggerated, through the ignorance and credulity of the historians by whom they are related; for, since comets have been observed by astronomers, no instances have occurred in which their magnitudes and appearances have been so extraordinary. The most remarkable among those of which we possess accurate accounts appeared in the years 1456, 1618, 1680, 1744, 1759, 1769, 1807, and 1811.

Fig. 106 is a representation of the celebrated comet of 1680, taken from Lemonnicr’s Histoire Celeste. It exhibits the nucleus or disk with its surrounding atmosphere. Above is a sort of ring, wider at the summit, and narrower towards the sides. A coma or beard succeeds the ring; and lastly, an immense train of luminous matter, somewhat less vivid than the nucleus. This luminous train, or tail as it is called, is by far the most singular and striking feature presented by the comets. That of the comet of 1744 was one of the most remarkable. It was divided into six branches, all diverging, but curved in the same direction; and between the branches the stars were visible. It is represented in fig. 107. The tail of the comet of 1811 was composed of two diverging parts inclined to each other in an angle which varied from 90 to 15 or 20 degrees. These branches were curved in opposite directions, and descended from the nucleus like a veil: between the [7:4:40] branches, and surrounding the nucleus, was a space comparatively obscure.

The nucleus, which is the densest and most luminous part, may be said to form the true body of the comet. It is, however, so far from having the dense and solid appearance of the planets, that some astronomers have imagined it to be diaphanous, and even supposed that they have observed stars through it. But supposing such an observation certain, it may be accounted for with much greater probability by the effects of refraction; and it is besides extremely difficult to distinguish the nucleus from the surrounding nebulosity. If the nucleus were an opaque globular body, it would exhibit phases like Venus or Mars, according to its different positions with relation to the sun and the earth; and such were observed, or at least were supposed to be observed, in the case of the comet of 1682, by Hevelius, Picard, and Lahire. But the nebulosity renders the phases exceedingly obscure and indistinct, and prevents the true-body of the comet from being seen; in the same way as a globe of roughened glass prevents us from distinguishing the form of the flame of an inclosed lamp. The real nucleus has probably never been observed by any astronomer; and, from the appearances, we are led to infer that a comet, at least near its perihelion, is only an agglomerated mass of vapours. As it recedes from the perihelion, the vapours may be condensed by cold into a solid substance. This hypothesis is also favoured by the extreme smallness of the density of the comets, which is known certainly from the circumstance that they produce no' appreciable effect on the motions of the planets. The comet of 1770 traversed the system of Jupiter’s satellites without causing any sensible perturbation of those small bodies. This comet also passed very near the earth; and Laplace calculated, that if its mass had been equal in density to that of the earth, the effect of its attraction would have increased the length of the sidereal year by two hours and twenty-eight minutes. But since its influence was altogether insensible, it is certain that its mass was not equal to the five thousandth part of that of the earth, and probably much inferior even to this quantity.

If the nuclei of comets are solid, the matter of which they are composed must be extremely fixed in order to enable them to resist the intense heat they necessarily experience in their approaches to the sun. According to the computation of Newton, the great comet of 1680, at its perihelion, was only distant from the sun by the 163d part of the semidiameter of the earth’s orbit, where it would be exposed to a heat above 2000 times greater than that of red-hot iron,—a temperature of which we can form no conception, and which would instantly dissipate any substance with which we are acquainted.

In order to explain the singular phenomena of the train of light which frequently attends the comets, the following theory was proposed by Newton. The comets move in very eccentric orbits, and consequently, towards their perihelia, approach very near to the sun. The excessive degree of heat they sustain near this point of their orbits must convert into vapour every substance capable of vaporization; and hence the prodigious extent of their atmosphere in comparison of the smallness of their nuclei. When this atmosphere has acquired all the volatility of which it is susceptible, the impulsion its vapours receive from the solar rays, however feeble that force may be conceived to be, is sufficient to put them in motion, and drive them off in a direction opposite to the sun. Thus it is remarked, that the tail becomes most conspicuous after the comet has passed the perihelion, and that its direction, as was first observed by Appian, is the straight line joining the centres of the sun and comet. The Slight curvature which is generally observed may be accounted for by combining the motion given to the vapours by the impulsion of the sun’s rays with the motion of the comet in its orbit; for the detached vapours arc driven by the impact of the luminous particles beyond the sphere of the comet’s attraction, and consequently cease to follow the direction of the nucleus. Hence the curvature of the tail must be greatest towards its extremity; and this is observed to be actually the case. It may be remarked, however, that although this hypothesis serves to explain the phenomena when all the branches of the tail are bent in the same direction, it is inapplicable when the directions of the curvature are opposite, as was the case with regard to the comet of 1811. In fact, no ultimate reasons for several of the phenomena exhibited by comets have yet been given which can be considered as entirely satisfactory.

The discovery of the periodic returns of certain comets necessarily put an end to the apprehensions and terrors which their unusual appearances were well calculated to excite, and proved them to be permanent bodies belonging to the same system, and acted on by the same laws, as the planets. But this very discovery gave rise to apprehensions of another kind, more natural, though, when closely examined, hardly more reasonable. Since the comets are so numerous, and their orbits traverse the planetary system in all directions, and come within the orbit of the earth, is there not a probability that some of them may come into contact with our globe, and destroy it by the direct collision; or at least approach so near as to produce the most disastrous effects by their attraction? Halley found that the comet of 1680 had approached its perihelion about the time of the universal deluge, and thought it probable that that great catastrophe might have been immediately occasioned by the earth’s being enveloped in the aqueous vapours of its tail,—an idea which was afterwards more fully developed by Whiston. Lalande and Maupertuis have minutely detailed the terrible effects which might be produced by the shock of a comet, or even by its approach to our earth. The vapours brought by the tail would mingle with the atmosphere, and render it less respirable. The attraction of the nucleus would destroy the equilibrium of the ocean, and cause extraordinary inundations, which might sweep off the greater part of the human species. The direct shock might change the position of the earth’s axis, or even cause the earth to leave its present orbit. It might then become a satellite of the comet, and be carried away with it to the extreme limits of the sun’s attractive influence: or, as the mass of the comet would probably be inferior to that of the earth, the earth would carry the comet along with it in its orbit, and thus acquire a second moon; and it has even been surmised that the moon we actually enjoy may owe its origin to an accidental occurrence of this kind. But all these reveries have disappeared before the calculus, by which it is demonstrated that the orbit of the moon can never at any time have been greatly different from what it now is. The collision of a comet with the earth is not an impossible event; though so infinitely little probable, that it can never excite any just cause of alarm. The conjunctions of the planets anciently caused terrors still more unreasonable; and the eclipses, which now scarcely attract the notice of the vulgar, long rivalled the comets in the terrors which they occasioned to the inhabitants of the earth.

Various opinions have been entertained respecting the nature, and formation, and uses of the comets. Newton supposed that, as some of them pass so near to the sun ⅛ [7:4:41]to be involved in all probability within his atmosphere, the resistance they must consequently experience will cause them to approach nearer and nearer to the great luminary at every successive revolution, till at last they are precipitated into his substance; hence their use in the system may possibly be to repair the losses which the sun sustains from the constant emission of light. Whatever destinies they may be appointed to fulfil, the recent discovery of two comets of short periods must be regarded as of great importance to astronomy, inasmuch as the frequency of their appearance will enable observers to take notice of any great changes with which either their masses or orbits may be affected.

With regard to the actual number of comets belonging to the solar system, there are no data from which we can form any probable conjecture. Those only (with two or three exceptions) which come within the orbit of the earth are visible to us, and upwards of 150 have been observed since the discovery of the telescope. If, then, we suppose them to be equally distributed throughout the whole system, it would follow that the number of comets coming within the sphere of Uranus, the radius of which is twenty times that of the earth’s orbit, amounts to 1,200,000 (the cube of 20 multiplied into 150), assuming the average period of their revolutions to be that in which 150 have come within the sphere of the earth. Such computations, however, are scarcely deserving of notice.

On the subject of comets, the reader may consult Aristotle, Meteorol. lib. i. cap. vi.; Seneca, Quoest. Natural. vii.; Hevelii Cometographia ; Newton, De Mundi Systemate, and Princip. lib. iii. prop. 42; Halley, Synopsis Astronomiae Cometicae, and Phil. Trans, vol. xxiv.; Euler, Mem. Acad. Berlin, 1756; Dionis du Séjour, Essai sur les Comètes, Paris, 1775; Lexell, Pliil. Trans. 1779; Clairaut, Mem. Acad. Paris, 1760; Lambert, Lettres Cosmologiques, and ANm. Acad. Berlin, 1771; Bode, ibid. 1786, 1787; Sir H. Englefield on the Orbits of Comets, 4to, London; Pingré, Cometographie, 2 vols. 4to, Paris, 1784; Laplace, Mécanique Celeste, tome iv. p. 193, and Systeme du Monde, 4to, p. 127; Delambre, Astronomic Théorique et Pratique, tome iii. chap, xxxiii.; also the Connaissance des Terns, Memoirs of the Astronomical Society, and the various scientific Journals.

CHAP. VI.

OF THE FIXED STARS.

After having treated of the different classes of bodies which compose the solar system, it only remains for us to inquire what observation has been able to discover respecting that innumerable host which “studs the galaxy,”—that multitude of brilliant points which, on account of their always sensibly retaining the same relative positions, have received the name of Fixed Stars. Sect. I.— Of the Arrangement of the Fixed Stars.

The great multitude of stars visible even to the naked eye renders it impossible to distinguish each by a particular name: astronomers have accordingly, for the sake of reference, formed them into groups, to which they give the name of Constellations or Asterisms. To the different constellations the early astronomers gave the names of men, and animals, and other familiar objects, from some fancied resemblances or analogies, which, for the most part, are not easily traced. These denominations, consecrated by ancient usage, are preserved in modern catalogues; and the practice of delineating the object itself on celestial globes and charts has been only recently, if it has yet altogether, been abandoned. The ancients likewise distinguished some of the brightest stars in the different constellations by particular names; but when it was wished to include others less conspicuous, it became necessary to have recourse to a different mode of proceeding. According to the usual method, first introduced by Bayer in his Uranometria, each of the stars in every constellation is marked by a letter of the Greek alphabet, commencing with the most brilliant, which is designated by α, the next most conspicuous is called /3, the third γ, and so on. When the Greek letters are exhausted, recourse is had to the Roman or Italic; but even with the help of these the nomenclature cannot be extended far, and the simplest and most comprehensive method is undoubtedly to employ the ordinal numbers to particularize the stars belonging to each constellation. It requires to be remarked, that the order of the letters indicates only the relative brilliancy of the stars in the same constellation, without any reference to those in other parts of the heavens. Thus α Aquarii is a star of the same order of brightness as y Virginis. The stars of the first order of brightness are likewise denominated stars of the first magnitude ; those of a degree inferior in brightness are said to be of the second magnitude ; and so on with the third, fourth, &c. Below the sixth the same denominations an© continued; but the stars of the seventh and inferior magnitudes are no longer visible to the naked eye, and are therefore called telescopic stars. It is obvious that, in conferring these denominations, it has been assumed that the brilliancy of the stars is proportional to their magnitudes,—an hypothesis at least extremely doubtful. The terms are however only used for the sake of distinction, and no exact ideas can be attached to the numbers. Observers are even at variance on the subject, certain stars being regarded by some of them as being of the first, which are considered by others as being only of the second magnitude.

A few stars have preserved the names conferred on them by the Greek or Arabian astronomers. Some of those names, belonging to stars of the first magnitude, are the following :— Sirius, in the right shoulder of Orion; Bigel, in his left foot; Aldebaran, or the eye of the Bull; Capella; Lyra; Arcturus; Antares; Spica Virginis; Begulus, or the heart of the Lion; Canopus ; Fomalhaut ; Acharnar, &c.

The number of constellations given by Ptolemy is 48. They do not comprehend all the stars in his catalogue, and those not included in the figures are called by him unformed stars (αμοξ<pωτοι, informes), and given at the end of that constellation to which they are nearest. The following table includes Ptolemy’s constellations, and those which have been added by the moderns.

ptolemy’s constellations. North of the Zodiac.

1. Ursa Minor, The Little Bear.

2. Ursa Major, The Great Bear.

3. Draco, The Dragon.

4. Cepheus, Cepheus.

5. Bootes, Arctophilax.

6. Corona Borealis, The Northern Crown.

7. Hercules, Engonasin, Hercules kneeling.

8. Lyra, The Harp.

9. Cygnus, Gallina, The Swan.

10. Cassiopeia, The Lady in her Chair.

11. Perseus, Perseus.

12. Auriga, The Waggoner.
[7:4:42]

13. Serpentarius, Ophiuchus, Serpentarius.

14. Serpens, The Serpent.

15. Sagitta, The Arrow.

. fi ĵAquila, Vultur, et [The Eagle and

Į Antinous, ∣ Antinous.

17. Delphinus, The Dolphin.

18. Equulus, Equi Sectio, The Horse’s Head.

19. Pegasus, Equus, The Flying Horse.

20. Andromeda, Andromeda.

21. Triangulum, The Triangle.

In the Zodiac.

22. Aries, The Ram.

23. Taurus, The Bull.

24. Gemini, The Twins.

25. Cancer, The Crab.

5 The Lion, to which he joined some stars of Berenice’s Hair.

27. Virgo, The Virgin.

28. Libra, Cheloe, The Scales.

29. Scorpio, The Scorpion.

30. Sagittarius, The Archer.

31. Capriconus, The Goat.

32. Aquarius, The Water-bearer.

33. Pisces, The Fishes.

Southern Constellations.

34. Cetus, The Whale.

35. Orion, Orion.

36. Eridanus, Pluvius, Eridanus, the River.

37. Lepus, The Hare.

38. Canis Major, The Great Dog.

39. Canis Minor, The Little Dog.

40. Argo Navis, The Ship.

4L Hydra, The Hydra.

42. Crater, The Cup.

43. Corvus, The Crow.

44. Centaurus, The Centaur.

45. Lupus, The Wolf.

46. Ara, The Altar.

47. Corona Australis, The Southern Crown.

48. Piscis Australis, The Southern Fish.

The constellations added by Hevelius are the following _

1. Antinous, Antinous.

2. Mons Menelai, Mount Menelaus.

3. Asterion et Chara, The Greyhounds.

4. Camelopardalus, The Giraffe.

5. Cerbefus, Cerberus.

6. Coma Berenices, Berenice’s Hair.

7. Lacerta, The Lizard.

8. Lynx, The Lynx.

9. Scutum Sobieski, Sobieski’s Shield.

10. Sextans, The Sextant.

11. Triangulum, * The Triangle.

12. Leo Minor, The Little Lion.

The constellations added by Halley in the southern hemisphere are,—

1. Columba Noachi, Noah’s Dove.

2. Robur Carolinum, The Royal Oak.

3. Grus, The Crane.

4. Phoenix, The Phoenix.

5. Pave, The Peacock.

6. Apus, Avis Indica, The Bird of Paradise.

7. Apis, Musea, The Bee or Fly.

8. Chamaeleon, The Chameleon.

One of the most important objects of practical astronomy is the formation of catalogues of the fixed stars, in which their positions are determined for a given epoch for it is only by means of registered observations that the l state of the heavens can be compared at different times/ and any changes which take place be detected. The apparent place of a star is easily determined by observation; but in order to render such observations available for the purposes of comparison, the mean place of the observed star must be computed and reduced to a given epoch; and this reduction, which involves a knowledge of the precession, nutation, aberration, and in general of all the motions which affect the star’s apparent place, is only accomplished by a laborious process of calculation.

The principal catalogues-of the stars which we possess are the following :—

Ptolemy’s catalogue, which contains 1022 stars. The positions are referred to the ecliptic, and the longitudes are for the year 137 of our era. It is supposed that the greater part of the observations on which it is founded were made and computed by Hipparchus 267 years before, and that Ptolemy merely reduced them to his epoch by adding to each of the longitudes 2° 40', which, according to him, was the amount of the precession of the equinoxes in that interval. This catalogue forms part of the Almagest.

The catalogue of Ulugh Beigh, containing 1017 stars.

Tycho’s catalogue, which contains only 777 stars, in 45 constellations.

Riccioli’s catalogue, which contains 1468 stars. Part of it, however, was merely copied from more ancient catalogues.

Bayer’s catalogue, containing 1762 stars, in 72 constellations. It was published in his Uranometria in 1603. The third edition of this work appeared at Ulm in 1661.

The catalogue of Hevelius, which contains 1888 stars, of which it gives the latitudes, longitudes, right ascensions, and declinations, for the year 1661. Published in his Prodromus Astronomiae in 1690.’

Flamsteed’s catalogue, containing 2884 stars. Published in the Historia Caelestis Britannica in 1725. A less perfect edition was given by Halley in 1712.

Catalogues of Lacaille. — The first of these, published in his Astronomiae Fundamenta, contains 397 stars; the second, which is given in his Coelum Australe Stelliferum, contains 1942 of the stars in the southern hemisphere; and the third, which was reduced from his observations by the celebrated Bailly, contains the places of 515 zodiacal stars.

Mayer’s catalogue, containing 998 zodiacal stars. It appeared in his Opera Inedita, Göttingen, 1775, and was reprinted in the Connoissance des Tems for 1778.

Bradley’s catalogue, containing 587 stars. This was published in the first volume of his observations, edited by Hornsby, in 1798. The positions of 389 stars, calculated from Bradley’s observations, had been given by Mason in the Nautical Almanack in 1773. Bradley’s observations extended to 3000 stars, but the greater part of them remained useless to astronomy till they were reduced and made the subject of discussion by Bessel in his Fundamenta Astronomiae, Regiomonti, 1818.

Maskelyne’s catalogue of 36 stars.

Cagnoli’s catalogue, containing 501 stars. Published in the Memoirs of the Italian Society.

Bode’s catalogue, which contains 17,240 stars, reduced from the observations of various astronomers.

Piazzi’s catalogue, which contains 6748 stars, reduced to the year 1800. In 1814 Piazzi published a new catalogue, comprising 7646 stars.

Zach’s catalogue, inserted in his Tabuloe speciales Aberrationis et Nutationis, Gothae, 1806.
[7:4:43]

Catalogue of the Astronomical Society of London, containing 2881 stars, published in the second volume of their Memoirs. Almost all the stars comprised in this catalogue are to be found in the catalogues of Bradley or Piazzi, from which they have been reduced to the year 1830.

In addition to the above, the records of great masses of observations may be found in the Philosophical Transactions, the Connoissance des Terns, and the various astronomical and scientific Journals. Lalande has registered in the Memoirs of the Academy of Paris, and his Histoire Céleste, the positions of no fewer than 50,000; and Bessel, of the Königsberg Observatory, who continues to explore the heavens with unabated zeal, has already examined an equal number. Astronomers are now aware of the importance of extending their researches to the most minute sidereal objects.

Sect. II.— Of the Parallax, Distance, Magnitude, and Number of the Fixed Stars.

The fixed stars being the points of departure from which all the celestial motions are estimated, one of the first objects in astronomy is to determine the amount and law of all the minute variations of position, real or apparent, to which they are subject. One of the most obvious consequences of the hypothesis of the annual motion of the earth is the existence of an annual parallax of the stars; but on account of the enormous distances of these bodies, this effect of the earth’s motion is so small that it cannot be easily measured; and it is even doubtful, after all the attempts that have been made to detect it, whether it is at all sensible to the best instruments. The longest line which nature has furnished us with the means of actually measuring, is the circumference of our own globe. From this geometry teaches us how to find its diameter; and the diameter we employ as a scale with which to compare the distances of the sun and moon, and the other bodies of the solar system. But experience shows us that this scale, large as it is in our conceptions, is only an insensible point in comparison of the distances of the fixed stars. Astronomy has furnished us with another base, about 24,000 times longer than the former, or above 190 millions of miles. This is the diameter of the earth’s orbit, which is most conveniently used for expressing the distances of the planets and comets from the sun. Yet even this immense line is insensible when compared with the distances of the stars; for, on observing the same star from its two extremities, at the end of six months, no variation whatever is perceptible in the star’s position, after the proper corrections have been made for the small effects produced by different and known causes. The limits of the errors of modern observations cannot well be supposed to exceed 1". It follows, therefore, that, seen from the distance of the fixed stars, the diameter of the ecliptic, which exceeds 190 millions of miles, subtends an angle of less than 1". Had the annual parallax exceeded this small quantity, it could scarcely have escaped the multiplied efforts that have been made to detect it, not only by Bradley, whose observations, undertaken for the express purpose of determining the parallax of the stars, conducted him to the grand discoveries of the aberration and nutation, but also by other observers furnished with the more delicate instruments of the present day; and particularly the observations made with the splendid circle of the Greenwich Observatory, the chef d’oeuvre of Troughton. Astronomers are, however, not entirely agreed as to the fact that the parallax is altogether insensible; and Dr Brinkley of the Dublin Observatory has perceived indications of its existence with regard to several stars. Dr Brinkley could discover no parallax in his observations on the circumpolar stars, with regard to which the changes of declination resulting from the precession are most appreciable. He did not even observe it in the case of the stars of the constellation ĉ Cygni, some of which, having a very sensible proper motion, may, with great probability, be considered as being ata less distance from the earth; but in Wega (α Lyrae) he found a parallax of 1"∙13, and one of 1"∙42 in the star Athair, in the Eagle. Bradley supposed the parallax of Sirius to amount to 1". These facts are, however, disputed by other eminent observers. In a series of 14 stars Struve found the parallax to be negative; that is, the small change of position which the observations seemed to indicate was in a direction contrary to what it would have been if it had arisen from the annual motion of the earth. Mr Pond, the Astronomer Royal, thinks the probable value of the parallax cannot exceed 0"∙018, a quantity so extremely minute as to be altogether lost in the uncertainties of instrumental errors, and the errors of refraction, which are at least 20 times greater.

Let us concede for a moment the disputed parallax of 1", and inquire what must be the corresponding distance of the star. The semidiamcter of the terrestrial orbit being taken at 95 millions of miles, the distance of a star whose parallax is P'will be expressed by ∙∙τ , × 95000000=206264 ×

r j sin. 1"

95000000 = 19,595,080,000,000, or about 20 billions of English miles. To assist the imagination in forming some idea of this almost inconceivable distance, we may calculate that a ray of light which darts from the sun to the earth in the space of 8 min. 7 sec., would require 206264 X 8 min. 7 sec., or 3 years and 216 days, to reach us from the star. A spider’s thread before the eye of a spectator placed at the same distance would suffice to conceal the orbit of the earth; and the breadth of a hair would blot out the whole planetary system. But a star having a parallax of 1" is at a moderate distance in comparison of innumerable others, in which no parallactic motion whatever can be distinguished. Supposing the distance of one of these to be only a thousand times greater, a ray of light darted from it would travel between 3000 and 4000 years before it reached the earth; and if the star were annihilated by any sudden convulsion, it would appear to shine in its proper place during that immense period, after it had been extinguished from the face of the heavens. Pursuing speculations of this kind, we may conceive, with Huygens, that it is not impossible that there may exist stars placed at such enormous distances, that their light has not yet reached the earth since their creation.

When viewed with the naked eye, the magnitudes of the stars appear to be very different,—a circumstance which may be attributed either to a real diversity of magnitude, or, which is more probable, to the great differences in their distances. The sensible magnitudes which they exhibit when viewed in this manner are owing only to the numerous reflections of the rays of light from the aerial particles surrounding the eye; as is proved by looking at them through a long tube, which prevents any rays from reaching the eye excepting those which come directly from the star. In the telescope their dimensions are entirely inappreciable: the greater the power with which they are viewed, the smaller are their apparent diameters, because they are then more completely divested of the effects of irradiation. A star having a diameter of 1", and an annual parallax of 1", would be more than a million of times larger than the sun. Nevertheless, Sir W. Herschel assures us, that, by means of the great powers which his telescopes carried, he had seen the disks of some stars perfectly round, and had even succeeded in measuring [7:4:44]their apparent diameters. He found the diameter of ’ Wega to be f, that of Aldebaran 1"∙5, and that of Capella 2"∙5. Supposing the measurement accurate, and the annual parallax of this last not to exceed 1", its volume would be equal to 20 million times that of the sun.

With regard to the number of the stars, it is altogether impossible to form any satisfactory conjecture. Of those which are visible to the naked eye, the number does not, probably, at any time, exceed 1000, although, from the effect of their twinkling, and the confused manner in which they are seen, one is apt to suppose them to be much more numerous; but in the telescope they are prodigiously multiplied. Within the limits of a space extending 15° by 2°, Herschel counted no fewer than 50,000. In the single constellation of the Pleiades, instead of 6 or 7, which can be distinguished by the unassisted vision, Hooke, with a telescope of 12 feet, counted 78; and in telescopes of greater power the number appears to be vastly larger. Although, by reason of their very unequal distribution, no accurate estimate can be formed of the number contained in the whole sphere by the examination of any small portion of it, yet there is some reason to conclude that there cannot be less than 75 millions of stars altogether visible in a good telescope. Baron Zach estimates that there may be at least a thousand millions of stars in the entire heavens, without reckoning (what may probably exist) opaque bodies which cannot be perceived, and stars whose light has been extinguished. It is, however, evident that all estimates of this sort have no other limit than such as is imposed by the imagination. If an observer could be transported to the remotest star visible in his telescope, he would probably see extending before him in the same direction, a firmament equally rich and splendid as that which he beholds from our own insignificant planet.

Sect. III.— Of the Proper Motions of the Stars.

In Chap. II. four different causes of apparent motion with regard to the stars have been explained; namely, the precession of the equinoxes, the nutation of the earth’s axis, the secular diminution of the obliquity of the ecliptic, and the aberration of light; all which are occasioned by the various motions proper to the earth, and for that reason arc called apparent motions. But if, after due allowance has been made for the effects produced by them, it should happen that the observed place of any star, or number of stars, does not correspond with former observations, a new and peculiar motion would be indicated, which must be explained by a new hypothesis. Now there are obviously two ways of accounting for such a phenomenon, namely, a parallactic motion and a proper motion, from either of which the observed variation of position may result. It has already been shown that the annual parallax, or that apparent alteration in the places of the fixed stars which might be expected from the motion of the earth in its orbit, is insensible; but we may suppose, and indeed observation has even rendered it probable, that the sun, accompanied by his whole train of planets and comets, is in motion in space; and if this is the case, the stars must appear to change their relative situations as soon as the sun has described a space bearing a sensible ratio to their distance. The resulting motion would thus partake of the nature of parallax, and is hence termed parallactic. The other method of accounting for any observed alteration in the mutual position of the stars, is to ascribe to them a proper motion, instead of supposing them to be absolutely fixed. On this supposition the motion is real; yet it may happen that the observed changes of position result from the combined effects of a real and parallactic motion. The effects of these motions are in their nature sufficiently distinct, although, on account of the extremely minute quantities to be determined, and the uncertainty that still exists as to the exact amount of some of the apparent motions with which they are blended, it may for centuries to come be found impossible to separate them. If all the stars forming a group, or situated in the same quarter of the heavens, appear to recede from or approach to each other, their motions may with reason be ascribed to the translation of the solar system in space, and consequently be parallactic: but if, on the contrary, some appear stationary, while others appear to move in different directions, the phenomena will indicate a real change in the positions of the stars.

From the analogy of our own system we arc naturally induced to extend the principle of gravity to the sphere of the stars, and to suppose, as a necessary consequence, that none of the celestial bodies are in a state of absolute repose. Their mutual attraction must communicate to them a motion which would end in uniting them all in the same mass unless it were counteracted by a centrifugal force: hence the stars are supposed to move about distant centres in orbits analogous to those of the planets. But if the stars are not absolutely at rest in space, their motions must be extremely slow, inasmuch as the actual state of the heavens corresponds entirely with the descriptions that have been given of it by Hipparchus and Ptolemy. Ptolemy has transmitted to us a great number of observations on the relative situations of the fixed stars, made by Hipparchus, whose method was to observe those which are situated in an arc of the same circle, or which can be intersected by the same straight line; and he assures us that he himself, after an interval of 260 years, could perceive no alteration. In order to furnish posterity with the means of pursuing similar inquiries, Ptolemy added the positions of many other stars determined by his own observations; and we owe to his labours the certain knowledge that the relative positions of the stars, notwithstanding the numerous displacements of the ecliptic and equator, are at present nearly the same as they were 2000 years ago. The ancient observations were, however, of too rude a nature to admit of any satisfactory conclusion being deduced from them respecting the minute quantities in question. The accurate instruments of the present day have enabled observers to remark some changes of position too decided to admit of doubt; hence astronomers have inferred a proper motion in several stars, as well as a translation of the sun and the planets in space. From the comparison of a great number of observations, Sir W. Herschel was of opinion that many of the proper motions might be explained by supposing the solar system to have a motion directed towards the star λ in the constellation Hercules. This conjecture has not, however, been confirmed, or rather it has been entirely overthrown, by subsequent observations. Bessel, having subjected to a rigorous comparison a much greater number of the proper motions indicated by comparing the catalogues of Bradley, Mayer, Piazzi, &c. arrived at the conclusion that many points might be assigned in the sphere, some of them even diametrically opposite to each other, situated in the direction of those motions; but that, in whatever direction the sun is supposed to move, so many proper motionswill remain unaccounted for, that there is no reason for preferring one point to another. The proper motions of the stars which have hitherto been remarked are evidently subject to no one assignable law, and are directed to many different points in space. It is therefore infinitely probable that they are due, in part at least, to a real dis [7:4:45]placement of the stars, and not to a general translation of the solar system.

Some recent observers have supposed they have detected proper motions in a great number of stars, the rates of which they have inserted in their catalogues along with those of the precession; but there is great discordancy in the determinations of different astronomers. Baron Zach, comparing Maskelyne’s observations of the right ascensions of the Greenwich stars, reduced to 1802, with those of Bradley, reduced to 1760, found results which differ from those of Maskelyne himself, not only in amount, but in several instances even in the direction of the supposed proper motions. For example, the proper motions (in right ascension) of γ Pegasi, α Ceti, Rigel, Sirius, Spica, γ and β Aquilae, α Cygni, a Aquarii, and a Pegasi, are all positive according to Zach, while Maskelyne considers them as being all negative. Such is the uncertainty respecting the proper motions even of the Greenwich stars, which, by reason of the frequent observations they have undergone, and the rigorous scrutinies to which they have been subjected, are probably those whose places are the most accurately determined of any in the whole heavens. But the motions are so slow that they must remain, for a considerable number of years at least, blended with 1 the errors of observation. The double star 61 Cygni is indeed supposed by Bessel to have an annual proper motion' of + 5"O6 in right ascension, and 3"∙34 in declination; but in general the rates are confined within much narrower limits, and appear to amount only to a fraction of a second. The number of stars in which this proper motion has been supposed to be observed is, as we have already noticed, very considerable. Μ. Bessel, by a comparison of 2959 stars out of Bradley’s catalogue, with the same stars in the catalogue of Piazzĩ, found that 425 of them had an annual proper motion, amounting to more than 0"∙2, in the arc of a great circle. The following table, published by Mr Baily in the. second volume of the Memoirs of the Astronomical Society, contains a list of all those stars observed by Bradley and Mayer, whose annual proper motions, according to Piazzi, as given in his catalogue, amount to 0"∙5 either in right ascension or declination. The positive sign, in the column of declinations, denotes a motion towards the north, the negative a motion towards the south. The numbers prefixed are those of Flamsteed, unless when inclosed within a parenthesis, in which case they are those of Piazzi.
[table]
[table]

The proper motions of the stars are not susceptible of direct observation or measurement: they are only indicated by the minute differences which remain after applying to the observed position of any given star the corrections due to all the apparent motions with which we are acquainted. Thus, let Μ and Μ' be the mean places of any given star [7:4:46]at two different epochs, the interval between which is t •years; P the annual precession of the equinoxes, and x ' the annual motion of the star in right ascension. It is obvious that x will be given by the formula

Μ—Μ' ŋ x = 1; t so that the value of x, or the amount of the proper motion, is dependent not only on the accuracy of the observations, and on the computations from which Μ and M', the mean places of the star, have been deduced, but also on the quantity P, which is calculated by a method of approximation not entirely exact. For all these reasons it is easy to see that the determination of the proper motions must be attended with considerable uncertainty, and that hundreds, probably thousands, of years will be required to develope their rates and directions, and assign the distant centres round which they are performed.

Sect. IV.— Of Variable and Double Stars, Nebulce, and the Milky Way.

The periodical variations of brilliancy to which some of the fixed stars are subject, may be reckoned among the most remarkable of the phenomena exhibited by those bodies. Several stars, formerly distinguished by their splendour, have entirely disappeared; others are now conspicuous which do not seem to have been visible to the ancient observers; and there are some which alternately appear and disappear, or of which the light at least undergoes great periodic variations. Some seem to become gradually more obscure, as δ in the Great Bear ; others, like β in the Whale, to be increasing in brilliancy. Some stars have all at once blazed forth with great splendour, and, after a gradual diminution of their light, have again become extinct. The most remarkable instance of this sort is that of the star which appeared in 1572, in the time of Tycho. It suddenly shone forth in the constellation Cassiopeia with a splendour exceeding that of stars of the first magnitude, even of Jupiter and Venus at their least distances from the earth, and could be seen with the naked eye on the meridian in full day. Its brilliancy gradually diminished from the time of its first appearance, and at the end of sixteen months it entirely disappeared, and has never been seen since. During the whole time of its apparition its place in the heavens remained unaltered, and it had no annual parallax; its distance was consequently of the same order as that of the fixed stars. Its colour, however, underwent considerable variations. Tycho describes it as having been at first of a bright white; afterwards of a reddish yellow, like Mars or Aldebaran; and, lastly, of a leaden white, like Saturn. Another instance of the same kind was observed in 1604, when a star of the first magnitude suddenly appeared in the right foot of Ophiuchus; it presented phenomena analogous to the former, and disappeared in like manner after some months. Kepler wrote a book on this singular apparition. These instances sufficiently prove that the stars are subject to great physical revolutions.

A great number of stars have been observed whose light seems to undergo a regular periodic increase and diminution, and these are properly called variable stars. One in the Whale has a period of 334 days, and is remarkable for the magnitude of its variations. From being a star of the second magnitude, it becomes so dim as to be seen with difficulty through powerful telescopes. Some are remarkable for the shortness of the period of their variation. Algol has a period of between two and three days, ð Cephei one of 5ģ days, β Lyrce one of 6ŷ days, μ Antinoi one of 7 days. The regular succession of these variations precludes the supposition of their being occasioned by a real or permanent destruction of the stars; neither can they be supposed to arise from a change of distance; for as the stars invariably retain their apparent places, it would be necessary to suppose that they approach to and recede from the earth in straight lines,—an hypothesis which is at least extremely improbable. The most probable supposition is, that the stars revolve, like the sun and planets, about an axis; and that the surfaces of the variable stars are unequally covered with dark spots, or unequally fitted to emit light; whence their dark sides will be turned towards us after certain intervals by the effect of rotation. In this way Newton accounted for the phenomena. Other astronomers have devised different explanations. Maupertuis supposed that the figure of the stars is not globular, but flat, and that the variations of brilliancy depend on the angle which their flat sides make with the visual ray,—an angle which will be constantly varying if the stars are endowed with a rotatory motion. Others, again, have imagined that the partial obscurations of the stars may be occasioned by their being eclipsed by opaque bodies or planets revolving round them.

On examining the stars with telescopes of considerable power, many of them are found to be composed of two or more stars placed contiguous to each other, or of which the distance subtends a very minute angle. This appearance is probably in many cases owing solely to the optical effect of their position relative to the spectator; for it is evident that two stars will appear contiguous if they are placed nearly in the same line of vision, although their real distance may be immeasurably great. There are, however, many instances in which the angle of position of the two stars varies in such a manner as to indicate a motion of revolution about a common centre; and in this case the two stars form a binary system, performing to each other the office of sun and planet, and connected together by gravity or some equivalent principle. The recent observations of Messrs Herschel and South have placed this singular fact beyond doubt. Motions have been detected so rapid as to become measurable within very short periods of time; and at certain epochs the satellite or feebler star has been observed to disappear, either on passing behind or before its primary, or by approaching so near to it that its light has been absorbed by that of the other. The most remarkable instance of a regular revolution of this sort is that of the double star ξ Ursoe Majoris, in which the angular velocity is 6° ∙4 annually, so that the two stars complete a revolution about one another in the space of 60 years; and above three fourths of a circuit have been already described since its discovery in 1781. The double star p Ophiuchi presents a similar phenomenon, and the satellite has a motion in its orbit still more rapid, α Castoris, γ Virginis, ξ Cancri, ξ Bootis, δ Serpentis, and that remarkable double star 61 Cygni, together with several others, exhibit similar variations in their respective angles of position.

The examination of double stars was first undertaken by Sir W. Herschel, with a view to the question of parallax; for it is evident, and indeed had been remarked by Galileo, that the apparent distance of two stars which are very near each other will vary with the position of the earth in its orbit, unless they are both so remote that, in comparison of their distance from the earth, the diameter of the terrestrial orbit is insensible. His attention was, however, soon arrested by the new and unexpected phenomena which these bodies presented. Sir W. Herschel observed in all 2400 of them. Messrs South and Herschel have given a catalogue of 380 in the Transactions of the [7:4:47]Royal Society for 1824, and South added 458 to the list •in the volume for 1826. Mr Herschel has published three series of observations in the Memoirs of the Astronomical Society, containing altogether 1000 double stars; and the catalogue of Μ. Struve of Dorpat contains 3063 of the most remarkable. The object of these catalogues is not merely to fix the place of the star within such limits as will enable it to be easily discovered at any future time, but also to record a description of the appearance, position, and mutual distances, of the individual stars composing the system, in order that subsequent observers may have the means of detecting their connected motions, or any changes with which they may be affected. Μ. Struve has also taken notice of 52 triple stars, among which No. 11 of the Unicon, ξ of Cancer, and ξ of the Balance, appear to be ternary systems in motion. Quadruple and quintuple stars have likewise been observed, which also appear to revolve about a common centre of gravity. Every region of the heavens furnishes examples of these curious phenomena, especially those which abound in stars; though Μ. Struve remarks that some parts of the milky way contain very few, while others present them in great abundance.

Some of the double stars present curious instances of contrasted colours, and generally assume the complementary tints,—a circumstance which, Mr Herschel thinks, may be owing in some degree, at least in cases where red and green, yellow and blue stars are combined in a double star, to the influence of optical deception. “When I first observed,” says that ingenious philosopher, “the double star No. 881, AE. 19 hours 8 minutes 56 seconds, P. D. 95° 45' 33", I remarked it as a case of contrasted colour, the large star being ruddy, and the small one blue. But on closer attention I perceived the small star itself to be double; yet each of the two very minute stars of which it consists appeared equally blue while the eye continued under the influence of the large one; but when this was withdrawn from the field, they appeared of no particular colour, but just like other small stars in the neighbourhood. It may be remarked further, too, that yellow stars are, generally speaking, accompanied by blue small ones, blue being the complementary tint of yellow; but when the large star has an excess of red rays, the blue verges to green, as it ought on the hypothesis of contrast. A most remarkable instance is that of No. 895 (Struve’s catalogue), AE. 6 hours 12 minutes, P. D. 84° 12', in which the large star is of a full ruby red, and the smaller one of a fine green, but which colour it also loses when the large star is concealed behind the cross of the wires.” (Memoirs of the Astronomical Society, vol. iii. p. 186.)

The Nebulae, so called from their dim cloudy appearance, form another class of objects which furnish matter for curious speculation and conjecture respecting the formation and structure of the sidereal heavens. When examined with a telescope of moderate powers, the greater part of the nebulae are distinctly perceived to be composed of clusters of little stars, imperceptible to the naked eye, because, on account of their apparent proximity, the rays of light proceeding from each are blended together, through the effects of irradiation, in such a manner as to produce only a confused luminous appearance. In others, however, no individual stars can be perceived, even through the best telescopes; and the nebula exhibits only the appearance of a self-luminous or phosporescent patch of matter in a highly dilated or gaseous state; though it is possible that even in this case the appearance may be owing to a congeries of stars so minute, or so distant, as not to afford singly sufficient light to make an impression on the eye. In some instances the nebula presents the appearance of a faint luminous atmosphere, of a circular form, and of large extent, surrounding a star of considerable brilliancy. One of the most remarkable nebulae is that which is situated in the sword-handle of Orion. It was discovered by Huygens in 1656, and described and figured by him in his Systema Saturnium. Since that time it has been examined and described by various observers, particularly Fouchy, Mairan, Le Gentil, and Messier, who have given engravings of it; and if any trust can be placed in their descriptions of so indistinct and difficult an object, it must have undergone great changes in its form and physical appearances. Unfortunately, however, no satisfactory inference can be drawn from the comparison of the different descriptions; for it is found that the same nebula, viewed on the same night with different telescopes, presents appearances so different as to be scarcely recognizable as the same object. The effects of atmospherical variations also cause great differences in its appearance, even when it is viewed through the same telescope at different times; so that it is scarcely possible that any two observers will be found to agree in their delineations of its outline. Mr Herschel, in the second volume of the Memoirs of the Astronomical Society, has given a detailed description of this nebula as it appeared in his twenty feet reflector in 1824, together with a drawing which, on account of the superiority of his telescope, is probably a much more correct representation of the object than any which previously existed. Fig. 108 is copied from that drawing. Of that portion of the nebula which he calls the Huygenian region Mr Herschel gives the following account: “I know not how to describe it better than by comparing it to a curdling liquid, or a surface strewed over with flocks of wool, or to the breaking up of a macharel shy when the clouds of which it consists begin to assume a cirrous appearance. It is not very unlike the mottling of the sun’s disk, only (if I may so express myself) the grain is much coarser, and the intervals darker; and the flocculi, instead of being generally round, are drawn out into little wisps. They present, however, no appearance of being composed of small stars, and their aspect is altogether different from resolvable nebulae. In the latter we fancy that we see stars, or that, could we strain our sight a little more, we should see them; but the former suggests no idea of stars, but rather of something quite distinct from them.”

Another very remarkable nebula is that in the girdle of Andromeda, which, on account of its being visible to the naked eye, has been known since the earliest ages of astronomy. It was re-discovered in 1612 by Simon Marius, who describes it as having the appearance of a candle seen through horn, that is, a diluted light, increasing in density towards a centre. Le Gentil mentions that its figure had appeared to him for many years round, but that in 1757 it had become oval. He also remarks that its light was perfectly uniform in all parts,—a fact which is quite at variance with its present appearance, and which, if true, argues that the nebulous matter is in a rapid state of condensation. “At present,” says Mr Herschel, in the volume above referred to, “it has not, indeed, a star or any well-defined disk in its centre; but the brightness, which increases by regular gradations from the circumference, suddenly acquires a great accession, so as to offer the appearance of a nipple, as it were, in the middle, of very small diameter (10" or 12"), but totally devoid of any distinct outline, so that it is impossible to say precisely where the nucleus ends and the nebula begins. Its nebulosity is of the most perfect milky, absolutely irresolvable kind, without the slightest tendency to that separation [7:4:48] into flocculi above described in the nebula of Orion; nor; is there any sort of appearance of the smallest star in the centre of the nipple. This nebula is oval, very bright, and of great magnitude, and altogether a most magnificent object.”

Sir W. Herschel, who devoted himself to the examination of every uncommon appearance in the sidereal heavens, has given catalogues of 2000 nebulae and clusters of stars discovered by him, and has shown that the nebulous matter is distributed through the immensity of space in quantities inconceivably great, and in separate parcels of all shapes and sizes, and of all degrees of brightness between a mere milky appearance and the condensed light of a fixed star. Finding that the gradations between the two extremes were tolerably regular, he thought it probable that the nebulae form the materials out of which nature elaborates suns and systems, and conceived that, in virtue of a central gravitation, each parcel of nebulous matter becomes more and more condensed, and assumes a rounder form; that from the eccentricity of its shape, and the effects of the mutual gravitation of its particles, it acquires gradually a rotatory motion; that the condensation goes on increasing till the mass acquires consistency and solidity, and all the other characters of a comet or planet; that by a still further process of condensation the body becomes a real star, self-shining; and that thus the waste of the celestial bodies, by the perpetual diffusion of their light, is continually compensated and restored by new formations of such bodies, to replenish for ever the universe with planets and stars. (See the Philosophical Transactions for 1811.) These hypotheses or conjectures give no doubt a mechanical reason for the formation of stars, but· the answer to them is exceedingly obvious. Has any instance as yet been observed of a nebula being succeeded by a star, or cluster of stars, or even of becoming so much more condensed as absolutely to change its form? Till a change of this sort has been observed, the inferences are drawn from analogies too slender to entitle them to be regarded as any thing more than mere fancies and speculations. Even if every link in the chain were perfect, and the gradation distinctly traced from the most diffuse nebula to the most compact star, the facts would still be insufficient to warrant the conclusion that the celestial' matter had actually undergone a transition from the nebulous to the stellar state, or that any star or nebula in the heavens ever existed in a state different from its present. Though there is little reason to hope that we shall ever obtain a full knowledge of the mysterious processes which the great Architect of the universe has employed in the formation of the celestial bodies, yet a long series of observations with such instruments as are now constructed may lead to the detection of changes sufficiently indicative of the nature of the forces by which they are produced. These cosmological speculations, however, it may be well to remark, are not of the slightest value to astronomy.

That great luminous tract which encompasses the sky. like a girdle, and is called the Galaxy or Milhy-Way, is supposed by Sir W. Herschel to be a nebula of which the sun forms one of the component stars; and hence it appears immensely greater than other nebulae only in consequence of our situation with respect to it, and of its greater proximity to our system. On examining any part of it with a good telescope, we perceive a prodigious multitude of small stars, whose blended light occasions the whitish appearance which forms so remarkable an object in the heavens. Yet notwithstanding the apparent contiguity of the stars which crowd the galaxy, it is certain that their mutual distances cannot be less than a hundred thousand times the radius of the terrestrial orbit. From this we may attempt to form some notion of the inconceivable distances of the other nebulae, some of which, probably not inferior in magnitude to the milky-way, appear only as small luminous patches in the telescope.

See Tycho de Nova Stella anni 1572; Hevelius, Historiola mirae Stellae in Collo Ceti, anno 1660; Maupertuis, Figure des Astres, Oeuvres, tome i. Lyon, 1756; Michel, Phil. Trans. 1784; Pigott, Phil. Trans. 1785, 1797; Lemonnier, Mem. Acad. Par. 1789; Lalande, Astronomie (885); Herschel, Phil. Trans, passim. (s.)

PART III.^[4. This portion of the present treatise is reprinted from the article on Physical Astronomy contributed by the late Professor Playfair to the Supplement to the fourth, fifth, and sixth editions of this work. ]

PHYSICAL ASTRONOMY.

The whole science of astronomy may be reduced to two general problems. The first is to express the position of all the heavenly bodies in terms of the time reckoned from a given instant, either in the past or the future duration of the world. The same may be otherwise stated by saying, that the thing required is, to express the position of any one of the heavenly bodies in a function of the time, the time being considered as the only variable quantity, though combined with other known quantities, which enter into the function as the co-efficients of the different terms. This is the most general view of that which is usually called descriptive, or sometimes geometrical astronomy. The solution of this problem enables us to determine for any time the places of the heavenly bodies, relatively to one another, and Relatively to any point on the earth’s surface. It contains under it an endless variety of subordinate problems, embracing a long series of successive generalizations, from the first observations to the determination of the orbits of the heavenly bodies, and the final reduction of all that concerns their motions into the form of astronomical tables. The second problem is, to compare the laws of motion in the heavens, as discovered from the preceding investigations, with the laws of motion as already known on the surface of the earth, in order to find out whether or not they are the same; and, if not, in what their difference consists. The solution of this problem constitutes what is called Physical Astronomy: it is the same with inquiring into the causes of the celestial motions; for by causes we mean the general facts concerning the motion of bodies which are observed to take place on the surface of the earth.

Though the first of these two problems goes necessarily before the second, for the solution of which it affords the data, yet, after this solution is obtained, it affords great assistance to many of the researches involved in the first, and exemplifies, in a most remarkable manner, the use of theory in the investigation of facts, and the re-action, as it were, of the second problem on the first.

Taking for granted the solution of the first problem, as given under the other Parts of this article, we are now to [7:4:49]consider the second, and to explain the manner in which it has been resolved by Newton and the philosophers who have come after him.

The history of the first of these two problems is long and interesting, beginning from the remotest period to which the records or the traditions of mankind have ventured to ascend, and coming down to the present time; and, in the ages to come, it is never likely to know any limit but the movable instant which separates the past from the future,—as long, at least, as science and civilization are inhabitants of the earth.

The history of the second comes within small compass; because, between the first rude effort and the last refined investigation there is hardly any intermediate step but one.

The concentric orbs of the ancient philosophers were an attempt at an explanation of the physical causes of the celestial motions, or at an assimilation of those motions to such as we are accustomed to see on the surface of the earth. The great phenomenon to be explained was the diurnal motion of the heavens, by which so many bodies, very distant from one another, all describe circles round the earth, keeping time so precisely with one another, that the revolutions, whether great or small, arc accomplished in the same interval. This could not be, unless a connection subsisted between those bodies; and the most simple idea of that connection was, that the bodies were fixed in the surface of a sphere which revolved on an axis, and carried them along with it.

If the whole of astronomy had been confined to the single fact of the diurnal revolution of the fixed stars, the hypothesis just mentioned would have been quite satisfactory. But as some of the heavenly bodies, such as the sun and planets, did not revolve precisely in the same time with the rest, it was necessary to assign to them particular spheres of their own. Those spheres, therefore, must be transparent: light must find an easy passage through them, and hence they must be crystalline. By degrees, as more accurate knowledge was obtained of the motion of the planets, it was found necessary so to increase the number of the spheres, that the complication of the structure was burdensome to the imagination; the hypothesis did not answer the very first object of a theory, that of connecting the facts together; and it was so unlike any process of nature with which we are acquainted, that it was highly improbable. The hypothesis of the homocen-tric orbs therefore fell into discredit, and, after the discovery of the earth’s motion, was entirely abandoned.

The next attempt to explain the whole system of the celestial motions was that of Descartes, by means of vortices of subtile matter, and the pressure which, by the centrifugal force of those vortices, was produced on the grosser bodies of the stars. But as a taste for accurate knowledge increased, and as men reflected more on the true objects of philosophic theory, the system of vortices appeared more and more defective, and at length ceased to 'ιave any followers.

Newton succeeded, who, rejecting all the cumbersome machinery, both solid and fluid, of his predecessors, adopted a plan far more philosophical in the design, and far more difficult in the execution, than any thing yet known in the physical or mathematical sciences. Assuming as true the three general facts concerning the planetary system known by the name of the laws of Kepler, he proceeded to inquire by what sort of action on one another the planets could be made to describe orbits having the properties indicated by these three general facts. The general facts to which we now refer, are,

I. That every planet moves so, that the line drawn from it to the sun describes about the sun areas proportional to the times.

II. That the planets describe ellipses, each of which has one of its foci in the same point, viz. the centre of the sun.

III. That the squares of the times of the revolutions of the planets are as the cubes of their mean distances from the sun. Sect. I.— Of the Forces which retain the Planets in their Orbits.

1. If a body gravitating to a fixed centre have a motion communicated to it in a direction not passing through that centre, it will move in a curve, and the straight line drown from the body to the centre will describe areas proportional to the times.

Let S (fig. 109) be the centre to which the body A gravitates, at the same time that a motion is communicated to it in the direction AB. And first,· let the gravitating or centripetal force be supposed to act, not continually, but at intervals, producing instantaneously, at the beginning of each interval, the same velocity that it would have produced by acting continually during the whole of that time: let AC be the space which the body would describe by the action of this force alone; also let AB be the space which it would describe in the same time by the projectile force acting on it alone. It will therefore describe the line AD, the diagonal of the parallelogram contained by AB and AC, and at the end of the first interval will be in D. If, then, no new impulse of gravity were to act on it, it would in the second interval of time go on in the direction AD, and describe DF equal to AD. But if, at the beginning of the second interval, an impulse of the centripetal force be instantaneously impressed, sufficient to carry the body in that time from D to E in the line DS, it will describe the line DG, the diagonal of the parallelogram contained by DE and DF. The same is true of the third interval, in which the body will go from G to L, and of every subsequent interval. Join SB, SF, SK, &c. The areas of the triangles ABS, ADS are equal, the triangles being on the same base AS, and between the same parallels AS and BD. For the same reason, the triangles DGS, DFS are equal, and DFS is equal to ADS, because they have equal bases and the .same altitude. For the same reason, the triangle SGL is equal to SDG, or to ADS; and the same is true of all the other triangles that are described in the equal intervals of time by the line drawn from the body to the centre S. This holds, however short the intervals may be, and however great their number; and therefore it is true when the intervals are infinitely small, and their number infinitely great, that is, when the action of the centripetal force is continued.

But when the intervals of time become infinitely small, the rectilineal figure ADGL passes into a curve. For when these intervals diminish, the lines AB, DF, &c. the lengths of the parallelograms, diminish in the same proportion; but the lines AC, DE, &c. the breadths, diminish in a greater proportion, viz. in that of the squares of these intervals. Hence, the angles which AD, DG, GL, the diagonals, make with the sides AB, DF, GK, continually diminish; and therefore the angles ADG, DGL, or the angle which each diagonal makes with that which is contiguous, increases without limit, so that, as the diagonals diminish in length, the angles they make with one another become greater than any finite rectilineal angles, and therefore the figure becomes a curve line.

That the lines AC, &c. or the supposed effect of the centrifugal force, diminish as the squares of the times, is evident from the laws of the descent of heavy bodies, as explained under the head of Dynamics.

2. Hence, conversely, if a body move in a curve, so that the line drawn from it to a fixed point describe areas proportional to the times, the body gravitates to that point, or tends continually to descend to it.
[7:4:50]

For, since it does not move in a straight line, it must be • continually acted on by a deflecting force; and the direction of the deflecting force must always pass through the same point, otherwise the areas described about that point would not be proportional to the time.

3. Corollary. The velocities of a body in different points of the curve which it describes about a centre of force, are inversely as the perpendiculars drawn from the centre to the tangents of the curve at these points. Let ACA', fig. 110, be the curve which a body describes about the centre S. Let Aα and A'«' be two arches of the curve, described in the same indefinitely small portion of time. Join Sα, Sα', then the areas of the triangles AS«, A'Sα' are equal by this proposition. At A and A', draw the tangents AB' A'B', and from S let fall on them the perpendiculars SB and SB'. Because the areas of the triangles AS«, A'S«', are equal, A« × SB = A'«'× SB', or A«: A'«':: SB': SB; but A« is to A'«' as the velocity of the body describing the curve at A to its velocity at A, therefore these velocities are inversely as the perpendiculars SB, SB'.

The straight line AB (fig. 109), according to which the projectile motion was impressed on the body, is a tangent to the curve at the point A.

4. On comparing the first and second of these propositions with the first of Kepler’s laws, as just enumerated, it is evident that the primary planets all gravitate to the sun, and that the secondary planets gravitate every one to its primary. The next thing, therefore, is to discover the law observed by this force, or the function of the distance to which it is proportional; and also, whether, in that function, other variable quantities are not involved beside the distance. The general fact that the orbits, or curves described by the planets round the sun, are ellipses, may assist in this investigation, and in expressing the velocity of a planet, in terms of the radius vector, or its distance from the sun.

5. Let ADBE (fig. IlD be the orbit of a planet, S the focus in which the sun is placed, AB the transverse, and DE the conjugate axis, C the centre, and F the superior focus. Let the planet be anywhere at P; draw a tangent to the orbit in P, on which from the foci let fall the perpendiculars SG, FH. Draw also DK touching the orbit in D, and let SK be perpendicular to it. Let the velocity of the planet, when at the mean distance, or at D, be = c, and when at P = v. Join SP, FP. Then, by the corollary to the last proposition, the velocity at D is to the velocity at P as SG to SK, that is, c : v :: SG: DC, DC

0r ν “c ’ SG’

But because the triangles SGP, FHP, are equiangular, having right angles at G and H, and also the angles SPG, FPH equal, from the nature of the ellipse, SP: PF:: SG: FH, and therefore also SP: PF: SG 2 : SG × FIT. But SG × FH=CD 2 , therefore SP: PF:: SG 2 : CD 2 ,

, CD 2 _ PF CD , CD 2 am ⅛~sp∙ Now l , = c .-,° r ≈≈ = c ≈.-, and

PF therefore r 2 = c 2 ∙ —.

⅛Jl Hence, as the distance of a planet from the sun, at any point in its orbit to its distance from the superior focus, so the square of its velocity at its mean distance from the sun to the square of its velocity at the point just mentioned.

6. If SL be taken in the greater axis equal to SP, and FN = PF, so that SN = the transverse axis AB, 2 _ „ NL SN—SP

' — c ~ ∙ 1^∖i -d i ģp ∙ Then as SN is a given line, v is expressed in terms where SP, the distance from the sun, is the only variable quantity.

If, from the velocity of the revolving body thus expressed in terms of the distance, a transition can be made to that of a body descending in a straight line, the law of the centripetal force will be easily investigated. This will be facilitated by the following proposition:

An equal approach to the centre of force produces an equal increase of the square of the velocity, whether the body revolve in a curve about the centre, or descend to it in a straight line. In like manner, equal recesses from the centre of force produce equal diminutions of the square of the velocities, in whatever lines the bodies move.

Let ABC (fig. 112) be a curve which a body describes about a centre, S, to which it gravitates, while another body descends in a straight line AS, to that centre. Let BC be any arch of the curve ABC, and let BD, CH, be arches of circles described from the centre S, intersecting the line AS in D and H; the square of the velocity of the body, which describes the arch BC, will be as much increased as the square of the velocity of that which falls through DH.

From the centre S describe the arch bd, indefinitely near to BD, and draw Ęf perpendicular to the arch B6. Also let the centripetal force at B or D be called G. Now, the part of this force which is in the direction BZ>, and which is employed in accelerating the body moving ŋr in that line, is G × ŋy,; and the increment of the space

B/' × Bδ

being B6, therefore 2G × "jÿğ— ls the momentary increment of the square of the velocity of the body at B. But Bf × Bó = BE 2 , because BEž> is a right-angled triangle, and E∕ ’ the perpendicular on the hypothenuse. Therefore 2 G × ∖2< = 2G × = 2G × BE =

BE BE

2G × Ðd. But 2G × Dd is the momentary increment of the square of the velocity of the body at D, or the increment of that square while the body falls from D to d. These momentary increments therefore are equal; and as the same may be shown for the next and every subsequent instant, the whole increase of the square of the velocities of the bodies in moving over BC and DH are equal.

If the bodies moved in the opposite directions, the one from C to B, and the other from H to D, it would be proved, in the same manner, that the squares of their velocities would be equally diminished.

7. Hence it is evident, that, if the velocities of the revolving and of the falling body are equal in any one instance when they are equally distant’ from the centre, their velocities will always be equal when they are equally distant from that point; for equal quantities receiving equal increments continue equal.

8. Suppose now that a planet revolves in the elliptical orbit APB (fig. IlD, it will have at A, the higher ap-

√AF

Âg, or (if AN in the axis pro-

∕AN

duced be taken equal to AF) ≡c× -r~rp Let a body a * A a at A begin to descend towards S with the same velocity; then if SL = SP, the velocity of the planet at P will be the same with that of the falling body at L. But the ve-

√PF ∕NL

ījū — c × ⅛√tt’ therefore, a body descending from A, and falling directly to the sun under the action of the same centripetal force which urges the planet, would at any point L in its fall have its velocity = c × a ∕⅛⅛. Hence, at the point N [7:4:51] its velocity would be equal to 0, or the body must begin • to fall from N, in order that its velocity may be every- ' where equal to that which the planet has in its orbit, when at the same distance from the sun.

The law, therefore, according to which the planets gravitate is such, that any body under the influence of the same force, and falling direct to the sun, will have its velocity at any point equal to a certain velocity, multiplied into the square root of the distance it has fallen through, divided by the square root of the distance between it and the suns centre.

This is a fact with respect to the law of gravity in the solar system, of which, though there be no direct example, yet is it no less certain than the ellipticity of the planetary orbits, of which it is a necessary consequence.

9. From the law thus found to regulate the velocity of bodies falling in straight lines to the sun, the law of the force by which that velocity is produced may be derived by help of reasoning which is quite elementary.

Let C (fig. 113) be the centre to which the falling body gravitates, A the point from which it begins to fall, and let its velocity at any point B, be to its velocity in the √AB

— - to 1; it is required to find the law of the force with which the body gravitates to C.

Let DEF be a curve, such, that if AD be an ordinate or a perpendicular to AC, meeting the curve in D, and BE any other ordinate, AD is to BE as the force at A to the force at B, then will twice the area ABED be equal to the square of the velocity which the body has acquired in B. If, therefore, the velocity at B be v, that at the √AB

-pp, by hypothesis, and J5 L√

AB therefore 2ABED = c 2 ∙ 777,; and since AB = AC — JoC

Af> up /AC ∖

BC, 2ABED = c 2 . 1 , r , = e 2 (⅜r — 1 ). For the jdC ∕

(AC ∖

-r 7 z 1), and therefore the

6C ∕

difference of these areas, or 2BZ>eE, that is, 2EB × Bδ

„/AC AC∖ AC.Bδ w , , = c A30 — BĞ/ & ’ BC 2 ^ ' ’ Wherefore ’ elvidιn g AC AG

by B6, 2EB = c 2 . or EB = c 2 .; now c 2 and j BC 2 BC 2 AG arc given, therefore EB is inversely as BC 2 , that is, the centripetal force at B is inversely as the square of BC, the distance from the centre of force. in the planetary system, therefore, the force with 'which any planet gravitates to the sun varies in the inverse ratio of the square of the distance of the planet from the suns centre.

10. The line CG is the same with the mean distance of the planet, in an orbit of which AC is the length of the transverse axis, and if the gravitation at that distance • ∙ CL (?

— F, and the mean distance itself = a, F = c i , — r = — , a i a or «F = c 2 .

Let it next be required, the elliptic orbit of a planet being given, to find the time in which the planet will revolve round the sun.

If a be the mean distance, or the semi transverse axis, b the semicojugate, then ¢raò = the area of the orbit. But as c is the velocity at the mean distance, or the elliptic arch which the planet moves over in a second when it is at D, the vertex of the conjugate axis, therefore fbc is the area described in that second by the radius vector; and since this area is the same for every second of the planet’s revolution, therefore the area of the orbit divided by ⅜δc will give the number of seconds in which the revolution is completed, which is therefore = = ffa θ r

⅜oc c since c 2 = a F, the time of a revolution = - α = 2τr f ^ r .

√ i zF v F

11. Hence it is easy to compare the times of the revolutions of any two planets of which the mean distances are known. Let t and t' be the times of revolution for two different planets, of which the mean distances are a and and the gravitation at those distances F and F ’,

JL X

2 ,2

ĊZ (İ and, by what has just been shown, t : t , :: — - : —-, or

F 2 f 2 /7 ∩*

&: f 2 :; -p-: . But F: F':: a t2 : « 2 , by what is already shown (Art. 9), therefore i 2 : f 2 : ~, οrfi∙.t r2 ∙.∙. a 3 ; d’ 3 , that is, the squares of the times of revolution of any two planets are as the cubes of their mean distances from the sum Thus the third law of Kepler is explained by the conclusions deduced from the other two.

12. The share which this third law has in establishing the principle of universal gravitation does not seem to have been always clearly apprehended. From the elliptical orbit of a planet, it is fairly inferred that, over all the circumference of that orbit, gravitation is inversely as the square of the distance from the centre of the sun. That force is shown to be (x being the distance from the centre of force), and the same is true of every individual planet; but whether c 2 α was a constant quantity, or one which retained the same value through the whole planetary system, could not be known without comparing the periods of different planets with their distances from the sun. It was indeed highly probable that e 2 α was a given quantity, or the same for every part of our system; but it could not be considered as a thing demonstrated till the evidence of the third law was introduced.

13. These laws hold of the secondary planets relatively to their primary, just as with the primary planets relatively to the sun. Each system of secondary planets, however, has a different numerator to the fraction which expresses gravity; that is, the quantity c 2 α is the same for all the satellites of Jupiter, but it is a different quantity from that which belongs to the satellites of Saturn, and different from that which belongs to the primary planets. The quantity c 2 α seems therefore to depend on the central body of each system of planets, and the precise nature of this connection requires to be farther examined into.

14. Let the centripetal force tending to a given centre S be inversely as the squares of the distances, and let the intensity of that force at any given distance from the centre be also given; then, if a body be projected from a given point, with a given velocity, and in a given direction, it is required to determine the conic section which it will describe.

Let the semitransverse, or the mean distance to be found, = a, the semicojugate = b, the velocity at the distance a = c, and at the given distance d let the centripetal force = f ; and first let the direction of the initial motion be at right angles to the radius vector, so that the point of projection is either the higher or the lower apsis. Let the velocity of the projection = v, and the radius vector at the point of projection = r.

Because the areas described in equal times are equal, [7:4:52]be — rv ; and if F denote the centripetal force at the discs d 2 f tance a, (ft = a F, and F = —. But F = —⅛, therefore a a- ft dftf If

— = -j, and c = dJ-. Hence, by substituting for c, CL Cl λ *∙ Cl bd ft J — = rv, and lftdftf = aftvft. But lft = AS × SB =

τ (2 a — r), wherefore r (2 a — r) dftf = ar 2 v 2 , and a = γ(ffi ^i* φ

-- τ , ~-—-,. Thus a, the semitransverse axis, and there- 2dftf — rv 2 , fore the transverse axis itself, is found; and thence with the focus S and the apsis A, the conic section may be described.

15. The conic section will be a circle, when a — r, that is, d 2 f when 2 d 2 f — v 2 r = dftf, or when d 2 f = ιftr, and v 2 = —. 2d 2 f

16. But if 2d 2 f = rft, or v 2 = the denominator vanishes, and a becomes infinite, so that the trajectory is a parabola, of which the focus is S, the vertex A, and the parameter 4r. The square of the velocity which determines the trajectory to be a parabola is, therefore, double of the square of the velocity which determines it to be a circle.

17. When 2d 2 ff>ftr, the value of a is affirmative and the conic section is an ellipsis, and this ellipsis has its d 2 f d 2 f higher apsis at A, if ⅛∙ 2 <≤~+ but when ^ 3 ≥~, and less 2d 2 f . . , 7 7

—-, A is the lower apsis.

18. When v 2 goes beyond this last limit, or when 2d 2 f vft ≥ — f-, the value of a is negative, and the trajectory becomes an hyperbola.

19. Next, let the body be projected from B (fig.114) with the velocity v, in the direction BD, oblique to BS. Find the distance from which a body must fall to acquire at B the velocity v, and let OB, taken in SB produced, be equal to this distance; then is SO equal to the transverse axis. Let BE be drawn, making with BD the same angle that SB makes with BG, and let BE = BO, then is E the higher focus. Produce SE to N, so that SN = SO, and bisect EN in A, then is A the higher apsis; and if SP be made equal to EA, P is the lower apsis, and AP the transverse axis; and therefore the foci and the transverse axis being given, the elliptic orbit may be described.

20. From what has been shown at Art. 9, it is evident that the primary planets gravitate to the sun with forces that are inversely as the squares of the distances, and that the secondary gravitate toward the primary, according to the same law. This inference, however, does not apply exactly to the moon, which, being a single satellite, does not by comparison with any other afford a proof that, in bodies revolving round the earth, the squares of the periodic times are as the cubes of the mean distances. The centripetal force at the moon, however, from our knowledge of her periodic time, may be compared with the force of gravity at the earth’s surface, and will determine whether that force decreases as we recede from the earth in the inverse ratio of the squares of the distances.

Let a be the distance of the moon from the centre of the earth, r the radius of the earth, g the velocity acquired by a heavy body at the earth’s surface by falling during one second; let t be the period of the moon’s revolution in seconds, and c the velocity of her motion. Then, by Art. 14, αc 2 = ftg, and therefore c = r 1 J-∙

Now, the circumference of the circle described by the moon is 2πα, and this, divided by c, gives the periodic

„ 1 . 1 2<rα ∕α , , time of the moon in seconds, or — × /— = t, so that,

4√ 2 α 3 r 2 gft a 3 gft a ' ft = — 5 —, and a i = >. Hence —7 = + “and — =

r 2 g 4ff- r 3 4w 2 r r ( qp ∖ ~

—j~) 3 ∙ Hence, as g, r, and t are known, we may find

—, or the ratio of the moon’s distance to the radius of the r earth, which, if it come out the same that it is known to be from observations of the moon’s parallax, will prove that the force which retains the moon in her orbit is the same that causes bodies to fall at the surface of the earth, but diminished in the same ratio that the square of the moon’s distance is greater than the square of the radius of the earth.

Now g = 32∙166 feet, r = 3481279∙4 fathoms or

20887676∙4 feet, and t = 2360591∙5 seconds. Hence - r

= 60∙218.

Now the mean equatorial parallax of the moon is found by observation = 57' 0"∙9, from which the mean distance, in semidiameters of the equator, is found = 59∙964.

But it is in mean semidiameters of the earth that the moon’s distance is given in the former computation; therefore, to reduce the last measure to the same scale, it must be increased by a 600th part, as the mean radius of the globe is about that much less than the radius of the equator; the distance 59∙964 then becomes 60*063, which agrees with the former number to the small fraction ∙003 of the earth’s radius.

Thus, from the theory of gravity, combined with the time of the moon’s sidereal revolution, her distance from the earth is found to within a very small fraction of the whole.

21. It is therefore a general proposition, derived from the most rigorous induction, that the primary planets gravitate to the sun, and the secondary planets to the primary, with forces which are inversely as the squares of the distances. But since, in all communication of motion, the re-action is equal to the action, when a planet gravitates to the sun, analogy forces us to conclude that the sun gravitates to the planet, in such a manner, that if the momentary approach of the planet to the sun, and of the sun to the planet, were respectively multiplied by the quantity of matter in those bodies, the products, or the quantities of motion, would be equal. Such a mutual tendency, therefore, of the great bodies of our system to the sun, and of the sun to them, doubtless takes place; but whether this be in consequence of an attractive force residing in their centres, as the magnetic force does in certain parts of the loadstone, or if it arise from the mutual attraction of all the particles of the one for all the particles of the other, does not appear from the phenomena hitherto examined. We may, however, observe that the bodies between which this attraction in the inverse ratio of the squares of the distances takes place, are all of a round form, and are either accurately spherical, or nearly approaching to that shape. It will therefore be of use for resolving this question, to inquire whether, if the particles of matter did attract one another with forces inversely as the squares of their distances, the spherical bodies, compounded of such particles, would attract one another according to the same law. If this is found to be the case, it will be reasonable to conclude that the gravitation of large bodies to one another arises from the mutual attraction of their particles to one another.

22. In order to determine the relation between the attraction of a sphere and that of the particles of which it consists, we may consider the sphere as made up of plates [7:4:53]or lamince infinitely great in number, and infinitely small in thickness. The attraction of each of these is to be estimated, and from thence the attraction of the whole may be computed. Let AFBG (fig. 115), therefore, be a circular plate, of which the centre is C; CE a straight line passing through C, and perpendicular to the plane AFBG; E any particle in that line attracted by each particle of the circular plate, as D, with a force inversely as the square of DE, the distance between the particles; it is required to find the whole force with which E is attracted in the direction EC.

If DC be drawn, the force with which D attracts E in the direction ED is inversely as DE 2 or as and that same force, reduced to the direction EC, is as ŋjp

EC EC

× —— = =≈r α . From the centre C, with the radius DC, DE DE 3 let a circle DKH be described, and indefinitely near it the circle dhh; then, since every particle in the ring of matter contained between these circles has its attraction

EC

proportional to gŋ 8 , the attraction of the whole ring will

EC

be as - multiplied into the number of particles, or into the solidity of the ring. But if EC = a, ED = x, and AC = r, CD 2 = x i — a 2 , and the surface of the ring — 2¶txx. If then the thickness of the plate AFBG = m, the solidity of the ring = 2nmχχ, and its attraction in the ,. . τ ,,, . 2¶mXX × a 2<!TamX 1 n „ direction EC is = z = 5—, the fluent 01

x 3 χ ’ which taken so as to vanish when DC = 0, or when x = a, . z , 2nma _ ∕, a ∖ , „ . is 2<vm = 2tm í 1 = the attraction of the x ∖ , χ) circle DKH. Therefore, when x = AE, the whole attraction of the plate, or the whole force which it exerts on the

. 1 τ , . A EC∖

particle E, is 2πm fl— + v )∙

23. Next, let ABD (fig. 116) be a circle of which the centre is C, and E a particle of matter anywhere in the diameter AB produced. Draw ED to any point D in the circumference; draw also DC, and let DF be at right angles to AB. Then, when the whole figure revolves about L’B, the semicircle ADB will generate a sphere, and DF a circle perpendicular to the plane ABD, and having its centre in F. If all the particles of the sphere attract the particle E with forces inversely as the squares of their distances from it, then, by the last proposition, the attraction of the circular plate, of which the centre is F, will be 2m, (1 -≡)∙

Let CE=β, AC=r, ED=ae, EF=t∕, and the attraction above will be 2<rm^1— ŋ; and if x and y be variable, the quantity m in this formula, or the thickness of the circular plate, will be —y, and therefore the attraction of the plate =2ιryi 1— i -i. In order to integrate this quantity, y must be expressed in terms of χ, or χ in terms of y.

Now, because AE=α — r, and AF=v∕— a∙j~r, FB =2r— y-ļ-a — r=a-1-r — y, and AF×FB=(y — α + r) («+r — y')-r 2 — cdA∙2ay — z∕ 2 =DF 2 =x 2 — τ∕ 2 . Hence _ sfi — - 4-

r 2 — a 2 + 2ay = χ 2 , or y = — , and therefore ŷ = ~. By substituting these values of y and y in the expression for the attraction of the circular plate, that . 2^xx( λ a? — r 2 + aΛ

attraction = {1— -— I

a ∖ 2αχ ļ ( 2axx — a 2 x + r 3 x — x i x ∖

a* A

But the attraction of this circular plate may be considered as the fluxion of the attraction of the spherical segment, generated by the revolution of the arch AD, and therefore the fluent of the above fluxionary quantity will give the attraction of that segment. Now, this fluent ( ax i — α⅛ + r⅛ ’ — ⅜ a ,3 ∖ 1 z -, ττ ~

-j- —∣+C. Here C must be so determined that the fluent may be equal to 0, when the arch AD = 0, or when x-=y-a — r. Therefore C

(X ffì ' 4- —

3 5 — ); and the attraction a i ∕

= --f<z+— tt¾+A — 3 ic3 +⅜ β3 — αr2 +⅜ 7 * 3 )∙

This is the attraction of the spherical segment generated by the arch AD, and will become equal to that of the whole sphere when AD=the semicircle ADB, or when x=οa-j-r. This substitution being made, and the terms reduced, the λ 4<rr 3 4iΓr 3 .

attraction is found = -≈-5-. But - h - is the solid content 6a i ó of the sphere; therefore the attraction of the sphere, on any particle E, is as the quantity of matter in the sphere, divided by the square of the distance of its centre from E. Hence also the sphere attracts any particle τvithout it, as if all its matter were united in its centre. The sphere, it is also obvious, would attract another sphere just in the ratio of its quantity of matter, divided by the distance of the centres of the spheres.

24. Thus, supposing that the particles of matter attract one another with forces which are inversely as the squares of the distances, it is certain that the spherical bodies composed of these particles would do so likewise, or would attract one another with forces directly as their quantities of matter, and inversely as the squares of the distances of their centres. Since, therefore, it has been found that round or spherical bodies, such as the sun and the planets, do attract other bodies with forces that are inversely as the squares of the distances, it is reasonable to suppose that these bodies are composed of particles gravitating towards one another, or attracting one another with forces inversely as the squares of the distances. Gravitation, therefore, is not to be considered as a force residing in the centres of the planets, but as a force belonging to all the particles of matter, and as universally diffused throughout the universe.

And as it has been shown that between spherical bodies constituted of such particles, the force of attraction is as the quantity of matter in the attracting body, divided by the square of the distance between its centre and that of the attracted body; if m be the mass or quantity of matter in the former body, and x the distance of the centres, is the value of f, the accelerating force with which it attracts the other body.

25. Hence the masses of any two planets which have bodies revolving round them may be compared with one another. Let a and a! be the mean distances at which satellites revolve about any two planets, m and ml the quantities of matter in those planets, t and i' their periods [7:4:54]A -3

„ , . . , , . , 2ffö- 2<α 2 ’ of revolution; it has been shown that t= - = -, r 7f 2 m 2 3 3

J 1 j , , a 2 d 2 , , α o α' 3 and consequently t : t' :: ——: —ŗ, and m : ml :: — : -ğ-.

m 2 m' 2 The masses, therefore, of any two planets are as the cubes of the mean distances at which their satellites revolve, divided by the squares of the periodic times of those satellites.

26. In this way the masses of the four planets which have satellites may be compared with one another, and with the mass of the sun.

When this calculation is undertaken with the most correct data, it is found that, making

Mass of the Sun 1

that of the Earth =; ξjξ

pτ . i 1

of Jupiter.. — 1070∙5

of Saturn =

πττ 1

of Uranus = 1γg∙1~8'∙

Or if we make the mass of the Earth 1, that of the Sun = 329630, of Jupiter 330∙6, of Saturn 101∙06, and of Uranus 20*3. From this also may be derived the densities of the sun and of the four planets just mentioned. Seen from a distance equal to the mean radius of the earth’s orbit, the diameter of the sun subtends an angle of 1923", that of the earth would subtend 17"∙4, of Jupiter 186"∙8, of Saturn 177"∙7, and of Uranus 74". The real diameters, therefore, are in the proportion of these numbers, and the bulk in the proportion of their cubes. By dividing the quantities of matter by the bulks, we have the densities; and if that of the earth be 4∙713, which is its mean density, that of water being = 1, then

Density of the Sun =1∙1775

of the Earth =4*713

of Jupiter =1∙1678

of Saturn =0∙4055

of Uranus =1∙0348.

The mean density of the earth, in respect of water, is here taken from the experiments made at Schehallien. (Phil. Trans. 1811, p. 376.)

27. It has been already observed, that because action is always accompanied by an equal re-action, when the sun attracts a planet, the planet also attracts the sun, and that the velocities impressed on the bodies by their mutual attraction are in the inverse ratio of their masses.

In consequence of this mutual action the sun and the planet must both move, and must describe orbits about their common centre of gravity, the only point which the mutual action of those bodies has no tendency to put in motion.

In the solar system, therefore, the centre of gravity of the whole is the focus about which all the orbits are described. Thus, if C be that centre (fig. 117), Sthe sun, and P a planet; while P describes the elliptic arch PP ʹ about C, S describes the arch SS ʹ similar to PP ʹ, and having to it the ratio that SC has to CP, or the ratio which the mass of the planet has to the mass of the sun.

The true orbits, therefore, are all described about the same immovable point; but the orbit of any of the planets may be referred to the sun as a centre, by supposing a body placed in that centre equal to the sum of the masses of the sun and of the planet. This is true, because the bodies appear to approach one another, or to recede from one another, with a force that is equal to the sum of the forces with which they tend towards their centre of gravity. Thus, if S denote the mass of the sun, and E that of the earth, the distances from the centre being CP and CS, the orbit which each of the two bodies will appear to describe round the other, is that which would be described about an immovable centre C, with a centripetal force - S + E - sp2 ∙

Thus we have arrived at the knowledge of the principle of universal gravitation, a power which pervades all nature, extending to an unlimited distance, and determining the condition of every body in the universe at any instant, from its state in the former instant, and from the relations in which it stands to all other bodies. Whether this force can be explained upon any principle more general than itself, is yet undecided, though, from the bad success which has hitherto attended all attempts towards that object, it seems probable that such explanation is not within the reach of the human understanding. Thus much, however, we know with certainty, that the law of gravity, as just announced, may be considered as a very accurate expression of all the phenomena of the planetary motions.

Sect. II.— Of the Forces which disturb the Elliptic Motion of the Planets.

1. Of the force by which the Sun disturbs the motion of the Moon round the Earth.

28. The motion of the moon in an elliptic orbit round the earth is disturbed by the action of the sun: the gravity of the moon to the earth is increased at the quadratures, and diminished at the syzygies; and the areas described by the radius vector, except near these four points, are never exactly proportional to the times.

Let ADBC (fig. 118) be the orbit, nearly circular, in which the moon Μ revolves, in the direction CADB, round the earth E. Let S be the sun, and let SE, the radius of the earth’s orbit, be taken to represent the force with which the earth gravitates to the sun.

1 1 SE^

Then:: SE: = the force by which the sun

SE 3 draws the moon in the direction MS. Take MG

SM 2 and let the parallelogram KF be described, having MG for its diagonal, and having its sides parallel to EM and ES. The force MG may be resolved into the two, MF and MK, of which MF, directed towards E, the centre of the earth, increases the gravity of the moon to the earth, and does not hinder the areas described by the radius vector from being proportional to the times.

The other force MK draws the moon in the direction of the line joining the centres of the sun and earth. It is, however, only the excess of this force above the force represented by SE, or that which draws the earth to the sun, which disturbs the relative position of the moon and earth. This is evident, for if KM were just equal to ES, no disturbance of the moon relatively to the sun could arise from it. If, then, ES be taken from MK, the difference HK is the whole force in the direction parallel to SE, by which the sun disturbs the relative position of the moon and earth. Now, if in MK, MN be taken equal to HK, and if NO be drawn perpendicular to the radius vector EM produced, the force MN may be resolved into two, MO and ON, the first lessening the gravity of the moon to the earth; and the second, being parallel to the tangent of the moon’s orbit in Μ, accelerates the moon’s motion from C to A, retards it from A to D, and so alternately in the other two quadrants.

Thus the whole solar force directed to the centre of [7:4:55]the earth is composed of the two parts MF and MO, which are sometimes opposed to one another, but which never affect the uniform description of the areas about E. Near the quadratures the force MO vanishes, and the force MF, which increases the gravity of the moon to the earth, coincides with CE or DE. As the moon approaches the conjunction at A, the force MO prevails over MF, and lessens the gravity of the moon to the sun. In the opposite point of the orbit, when the moon is in opposition at B, the force with which the sun draws the moon is less than that with which the sun draws the earth, so that the effect of the solar force is to separate the moon and earth, or to increase their distance; that is, it is the same as if, conceiving the earth not to be acted on, the sun’s force drew the moon in the direction from E to B. This force is negative, therefore, in respect of the force at A, and the effect in both cases is to draw the moon from the sun, in a direction perpendicular to the line of the quadratures.

29. The analytical values of these forces must be found if a more exact estimate is to be made of their effects. Let SE, considered as constant, = a ; EM, the radius vector of the moon’s orbit, = r ; the angle CEM = φ ; the mass of the sun = m. The force SE, then, which retains the earth in its orbit, is— 5 , and the sun’s force in the di- cd Til rection SM, if ML be drawn perpendicular to ES, is m m nι

SL 2 + LM 2 (a — r sin.p) 2 + r 2 cos∙p~ α 2 —2izr sin.p + r 2 ' The part of this force, which is in the direction ES or TìlCb

MK, is therefore 7-5—-; τr⅝. By raising the

(α 2 —2ćzrsin. p + r 2 ) 2 j denominator to the power —∣-, rejecting the terms which involve the higher powers of r, and multiplying ma by those

7Zi f 3τ ’ ∖

that are left, the force MK comes out =—s 1 + — sin. <p) 7)1> nearly. Taking away from this ES or MHx- 2 , there remains the force MN=-^∙ × 3r sin. φ. a 3 r

Hence the force MO=— ∙ 3r sin. 2 ®; and the force a 3 r • ∙ Ύϊϊ

NO at right angles to the radius vector = — ∙ 3r sin. φ

× cos. φ — ∙ ⅛ sin. 2®; also the force MF =

a 3 2 a 3 rejecting such terms as involve the square and higher powers of r. Therefore MF—MO, or the whole solar force increasing or diminishing at any point, the moon’s TìVT tendency to the earth is — (1 —3 sin. 2 p).

30. At the quadratures where p vanishes, this force is mr . . .

, and is affirmative, increasing the moon s gravity to the earth. At a certain point, between the quadratures and the syzygies, when 3 sin. 2 p = 1, or sin. φ = ~, that is, 3

when φ = 35° 15' 5", the same force becomes equal to 0, and at this point in each quadrant the moon’s gravity to the earth is neither increased nor diminished. From these points to the conjunction and opposition, as sin. φ increases, the quantity 1 — 3 sin. 2 φ is negative, and the moon’s gravity to the earth suffers a diminution. At the opposition and conjunction sin. φ = 1, and therefore the disturbing 2t72>7 , ∙ ∙ ∙ force is —, and by this quantity the moon’s gravita tion is diminished.

The mean quantity of the force which is thus continually directed to or from the centre of the earth may also be easily computed. Since for any point in the moon’s orbit, where the radius vector makes the angle <p with the γ∩γ line of the quadratures, this force = (1—3 sin. 2 p); • ĩïit ∙ ∙ multiplying by p, we have (φ — 3φ sin. 2 5), the fluent of which = — ( — ⅜P + ⅛ sin. φ cos. p), and this, when φ is an entire circumference or four right angles, TìłT ÇT

is — × — —. This is the sum of the forces for an en- a 3 2

tire revolution, and when divided by ¢r, gives the mean DIT force — —which being negative, shows that the solar la 3 force, on the whole, diminishes the gravitation of the moon to the earth.

Thus it appears, that at the quadratures the gravity of the moon to the earth is increased by a quantity equal to the mass of the sun, multiplied into the radius of the moons orbit, and divided by the cube of the sun’s distance from the earth : at the syzygies it is diminished by twice this quantity ; and the effect on the whole is a diminution by one half of the same quantity. TsbT

If — be reduced to its numerical value, supposing the moon’s gravitation to the earth to be 1, it is found = — nearly. Hence the mean disturbing force of the sun is nearly = of the moon’s gravity to the earth.

31. From the disturbing force of the sun arise two kinds of inequalities which affect the lunar motions; the one kind affects the form and position of the orbit of that planet, the other immediately affects the motion of the planet in the orbit. When any of these inequalities is expressed numerically, the measure of it so obtained is, in the language of astronomy, called an Equation.

32. The line in which the plane of the moon’s orbit cuts the ecliptic is called the line of the nodes; and this line is subject to change its position continually, in such a manner as to go back annually 19° 19' 42"∙3. The way in which this effect is produced may be thus conceived. That part of the solar force which is parallel to the line joining the centres of the sun and earth, is not in the plane of the moon’s orbit, except when the sun itself is in that plane, or when the line of the nodes, being produced, passes through the sun. In all other cases it is oblique to the plane of the orbit, and may be resolved into two forces, one of which is at right angles to that plane, and is directed towards the ecliptic. This force of course draws the moon continually towards the ecliptic, or produces a continual deflection of the moon from the plane of her own orbit towards that of the earth. Hence the moon meets the plane of the ecliptic sooner than it would have done if that force had not acted. At every half-revolution, therefore, the point in which the earth meets the ecliptic advances in a direction contrary to that of the moon’s motion, or contrary to the order of the signs. This retrograde motion is such that, in its mean quantity, it amounts to 19° 19' 42"∙3 in a year. The manner of de [7:4:56]ducing it from the theory of gravity is explained by Newton, Princip. lib. iii. prop. 31. This motion is subject to many inequalities, depending on the changes in the quantity and direction of the solar force.

If the earth and the sun were at rest, the effect of the deflecting force just described would be to produce a retrograde motion of the line of the nodes till that line was brought to pass through the sun, and of consequence the plane of the moon’s orbit to do the same, after which they would both remain in their position, there being no longer any force tending to produce a change in either. The motion of the earth carries the lines of the nodes out of this position, and produces, by that means, its continual retrogradation.

33. The same force produces a small variation in the inclination of the moon’s orbit, giving it an alternate increase and decrease within very narrow limits.

34. The line of the moon’s apsides, that is, the longer axis of her orbit, has also a slow angular motion round the centre of the earth, which is progressive, or in the same direction with the motions of the moon itself. To conceive the cause of this phenomenon, we may begin with supposing the moon at the lower apsis, or perigee; and it is plain, if that planet were urged by no other force than its gravitation to the earth, that after the radius vector had moved over 180°, the moon would be at the higher apsis, where its motion would be at right angles to the said radius. But as the mean disturbing force in the direction of the radius vector tends, on the whole, to diminish the gravitation of the moon to the earth, the portion of her path, described in any instant, will be less bent or deflected from the tangent, than if this disturbing force did not exist. The actual path of the moon, therefore, will be less incurvated than the elliptic orbit that would be described under the influence of gravity alone, and will not be brought to intersect the radius vector at right angles, till this last have moved over an arch of more than 180°.

Hence the solar force, by lessening the moon’s gravity to the earth, produces a progressive motion in the apsides of the lunar orbit. If the disturbing force had increased the moon’s gravity to the earth, the motion of the apsides would have been in antecedentia.

The precise quantity of the motion of the apsides is not however easily determined. Newton left this part of the lunar theory almost untouched; and the only investigation he has entered into having any reference to it, assigned a measure only the half of that which is known from observation to belong to it. Several years afterwards, when Clairaut attempted a more accurate investigation of the lunar inequalities than was to be obtained by the method which Newton had followed, he at first encountered the same difficulty, and found that his calculus gave the motion of the apogee only half of the real quantity. He began, therefore, to suspect that gravity does not follow so simple a law as the inverse of the squares of the distances, but one which is more complex, and such as cannot be expressed but by a formula of two terms. The second of these terms he supposed to be inversely as the fourth power of the distance, and proceeded to inquire what must be the co-efficient of that term, in order to make this new supposition represent the true motion of the apsides. In order to this, he found it necessary to carry his approximation farther than he had yet done, and to include terms which he had before neglected. When these terms were included, he found that the coefficient he was seeking for came out equal to 0; the necessary inference from which was, that there was no such term; and that the Newtonian law of gravity, when the approximation was carried far enough, was quite sufficient to explain the motion of the apsides. This doubt concerning the law of gravity terminated, therefore, in the confirmation of it.

35. When the doubts excited by Clairaut’s first attempt were made known, and before his final solution of the difficulty was fully understood, there were several mathematicians who, still following the method of Newton, endeavoured to deduce the true motion of the moon’s apsides from the theory of gravity. Among those who were most successful in this attempt were Dorn. Walmesley, and afterwards Dr Matthew Stewart, professor of mathematics in the university of Edinburgh. In his Mathematical and Physical Tracts he has demonstrated this remarkable theorem:

Let r be the radius of the moon’s orbit, supposing it to be a circle, and the moon to be acted on only by F, her gravity to the earth. If the mean disturbing force by which the sun diminishes the moon’s gravity be f then will the greatest distance to which the moon will recede

F

from the earth be r × = i —÷ i ,; and the cube of this dis-f—5/

tance will be to the cube of τ, in the duplicate ratio of the angle described by the moon from one apsis to the same apsis again, to four right angles.

Hence the angle described by the radius vector from

(F 3f∖ 3.

F— bf) ’

This proposition, which is demonstrated by Dr Stewart in the fourth of his Tracts, in a manner somewhat prolix, on account of his rigorous adherence to the methods of the ancient geometry, but in a way perfectly clear and elementary, is employed by him to deduce the mean disturbing force from the motion of the apsides as ascertained by observation. But when the mean disturbing force is known from other phenomena, the same proposition may be employed to deduce the motion of the apsides from that force. Accordingly, if the disturbing force be taken = the motion of the apsides will come out = 3° P 20" for a sidereal revolution of the moon, very near the quantity actually observed.

36. Having determined the sun’s mean disturbing force from the motion of the apsides, Dr Stewart proceeded to determine from the former of these the sun’s distance from the earth. The resultof a very nice investigation gave the sun’s parallax 6"∙9, a quantity that is no doubt too small, and makes of course the sun’s distance too great. It is indeed but an inconsiderable part of the sun’s disturbing force into which the parallax enters as an element, and therefore any deduction founded on it must be liable to this inaccuracy, that a small error in the data will produce a great one in the result.

37. After the inequalities which are conceived as belonging to the moon’s orbit, come those which directly affect the place of the moon in that orbit. The most considerable of these, after what is called the equation of the centre, arising from the elliptic figure of the lunar orbit, and independent of all disturbance, is the equation or inequality called the evection, which was discovered by the Greek astronomers. This depends on the position of the transverse axis of the moon’s orbit in respect of the line of the syzygies. When that axis is in the line just mentioned, because the quantity by which the solar force diminishes the gravitation of the moon in the syzygies is, caeterisparibus, proportional to her distance from the earth, it is greatest when the moon is in the apogee, and least when in the perigee. In this situation of the orbit, therefore, the greatest diminution is made from the quantity of the moon’s gravitation which is already the least, and [7:4:57] the least from that which is already the greatest; the gravitation at the perigee, and therefore the difference, is augmented, and the orbit appears to have its eccentricity increased. When the line of the apsides is in the quadratures, the contrary happens; the gravitation at the apogee is most augmented, and at the perigee least; the difference is therefore diminished, and the eccentricity of the lunar orbit seems also to be diminished. This is conformable to observation; and when the evection is accurately deduced from the theory of gravitation, it appears = (1° 20' 29"∙9) sin. (2 ( ¢ —©)— aj where ♯ is the mean longitude of the moon, Θ that of the sun, and a the mean anomaly of the moon counted from the perigee.

38. The moon’s variation is an inequality which was discovered by Tycho, and found to depend on the angular distance of that planet from the sun. It is derived from that part of the sun’s disturbing force which is at right angles to the radius vector, and which accelerates the motion of the moon from the quadratures to the syzygies, and retards it from the syzygies to the quadratures. The effect of this force is found, from the theory of gravity, to be represented by three terms, which, if δ be the angular distance of the moon from the sun, are, + (35' 41"∙9) sin. 2 δ + ( 0' 2" ) sin. 3 δ + ( 0' 14" ) sin. 4 ∆.

39. The lunar inequality, called the annual equation, arises from the variation of the sun’s disturbing force according to the place which the earth occupies in its orbit. It is shown above that the sun’s disturbing force is, coeteris paribus, as the cube of his distance from the earth; so that when the earth is in its perihelion this force is the greatest, and at the aphelion the least, its effect varying at the same rate with the equation of the sun’s centre, or having everywhere the same ratio to that equation. Hence this equation is nearly (IP 12") × sin. sun’s mean anomaly, with a contrary sign to the equation of the sun’s centre.

40. These inequalities are all phenomena which were observed before the explanation of them was known. To them may be added a fourth inequality, known by the name of the moon’s acceleration. It appeared to astronomers as a continual increase in the velocity of the moon, or in the rate of her mean motion, amounting to about 10" in a century, and its effect, like that of all other constant accelerations, accumulating as the squares of the times. It did not seem to"be periodical, like the other lunar inequalities, but to be a constant increase of the velocity, and a corresponding diminution of the periodical time of the moon, which must in the end change entirely the relation of that body to the earth.

It is but within these few years that Laplace discovered it to be a periodic inequality, though requiring, in order to accomplish the series of its changes, a length of time which science has not yet ventured to calculate. For many centuries to come it may be expressed by this formula, taking n to denote the number of centuries reckoned from the year 1700, viz.

10"∙7232 × n 2 + 0"∙019361 × n 3 .

The first term includes all that was known from observation previously to the discovery of Laplace. This, however, must be considered not as the true form of the equation, which must include the sines or cosines of certain angles, but merely a provisional formula, to serve till the true one can be rigorously assigned.

This inequality has in its cause a great affinity to the annual equation.

Whatever changes the form of the earth’s orbit, has an effect on the disturbing force of that body on the moon, which is in the inverse ratio of the cube of the distance between the sun and earth. But it is found that though the mean distance remains invariable, the eccentricity of the earth’s orbit changes, on account of the action of the other planets, and in fact has been diminishing, from a more remote antiquity than that to which the history of astronomy extends. From this cause Laplace has deduced the supposed acceleration of the moon’s mean motion.

41. All these inequalities have been pointed out by observations, and have been explained in the most satisfactory manner by the principle of universal gravitation. But when all these were reduced into equations and arranged in tables, yet the places of the moon calculated from them were never quite exact; and there seemed a cause of error or a mass of small inequalities unknown in their magnitude and form, to which this inaccuracy was to be ascribed, and which operated, as it may be said, like a mist which concealed the true place of the moon from the calculator, and prevented his results from agreeing completely with those of the observer. The most likely way to discover these inequalities, if they arose from gravity, was to push the approximation to the moon’s place still farther, and to try if the terms hitherto neglected in the approximation would not, when taken into account, afford a complete analysis of the circle of confusion which might be said to surround the moon on all occasions.

The problem on which mathematicians now entered, and which Clairaut, already mentioned, Euler, and D’Alembert, all three resolved nearly about the same time, has been called the Problem of Three JBodies. The thing proposed is, three bodies which attract one another with forces directly as their quantities of matter, and inversely as the squares of their distances, being given, and any motions whatever being impressed on them, to find the orbits which they will describe round their common centre of gravity. It is, however, only in certain cases that this general problem admits of solution, and one of these is, when one of the bodies is at a vast distance from the other two. This is exactly the case with the moon and earth in respect of the sun, the orbit of the earth being nearly the same as if there only existed the sun and earth, and the orbit of the moon relative to the earth being nearly the same if there were only the moon and earth. This solution of the problem, however, in this direct way, leads to far more exact conclusions than can be obtained from the more simple but more indirect method which Newton followed. The general view which leads to the most exact estimate of the merit of the two solutions is, that the motions of the moon, when analytically and fully expressed, necessarily form a number of different series, each of which converges with more or less rapidity. The prosecution of the direct method allows the terms of these series to be computed to an indefinite extent, or till the quantities omitted are too small to affect observation. The method of Newton can go no farther than to compute the first, or at most a few of the leading terms of each of the series. Its accuracy is therefore limited; that of the other knows no limits. Though this be a true estimate of the value of the methods, yet that of the original inventor possesses infinite merit, as having first led the way to this arduous investigation, and as still serving to carry the imagination better along with it than the other, and to keep the mechanical principles more directly in view.

The complete solution of the problem of the three bodies has accordingly discovered a great number of new equations, each individually small, which would sometimes nearly destroy one another, and, at other times, having many of them the same sign, would accumulate to a considerable amount. This was the triumph of the theory, and the strongest evidence of its truth. The effect [7:4:58]of these irregularities varied so much, and depended on; so many elements, that it may be doubted whether the most accurate and most constant observation would ever have enabled astronomers to discover their precise quantities, and to separate them from one another.

The tables of the moon, in the state to which they are now brought, contain twenty-eight equations for the longitude, twelve for the latitude, and thirteen for the horizontal parallax of the moon. Of the first of these, twenty-three have been deduced from theory alone; of the second, nine; and of the third, eleven. This applies to the tables of Burg; those since published by Burckhardt contain more equations, and are still more accurate.

2. Of the Disturbance in the Motion of the Primary Planets, produced by their action on one another.

42. It is evidently necessary, in this inquiry, to know the quantities of matter in the different planets, or, which comes to the same, the intensity of the attraction of each at a given distance from its centre. With respect to those planets which have satellites, the Earth, Jupiter, Saturn, and Uranus, their masses or quantities of matter have been already determined. The masses of Venus and Mars have been estimated by Laplace, from the effects which they appear to produce on the earth’s motion. The mass of Mercury has been estimated on the supposition that the densities of that planet and of the earth are inversely as their mean distances from the sun. This law holds with respect to the Earth, Jupiter, and Saturn, and analogy renders it probable that the same law includes the other planets. Thus, the mass of the Sun being 1, that of Mer- cur - v i ≡ 2Ō258ĪÕ’ ° f VenUS 40⅛71 , and ° f MarS tóàäŌ i the masses of the other planets being as already stated.

43. The effects of the action of the planets on one another is more difficult to be investigated than the effects of the sun’s action on the moon, because the disturbing forces are not only more numerous, but because the distance of the disturbing from the disturbed body is not so great that the quantities divided by higher powers of that distance can be so safely rejected. The general principle, however, according to which the solar action on the moon was resolved into forces either in the direction of the radius vector or at right angles to it, is applicable to both questions.

Thus, supposing P and P' (fig. 119) to be two planets revolving in orbits, nearly circular, about the sun at S; in order to find how the motion of P' is affected by the action of P, let PP', PS, and P'S be drawn, and let the line A denote the force with which P attracts a particle of matter at the distance PS, then the force with which it attracts

PS 2 a particle at the distance PP', will be A × p/pġ · Let PS 2 .

P'R = A × = s = i0 5 and if P'R be resolved into two forces, rr j

P'M and P'N, the one in the direction of the radius vector PS, and the other parallel to PS, take NO = A; then the remaining forces OP'and P'M are those which disturb the motion of P', as was proved in the case of the moon. The former of these, OP', may be resolved into OQ and P'Q, of which P'Q diminishes the gravity of the planet to the sun, and OQ accelerates its motion in a direction perpendicular to the radius vector. Therefore, as the force P'M always increases the planet’s gravity to the sun, P'M—P'Q is the whole force increasing or diminishing the gravity of P' to S; and the force directly employed in increasing or diminishing the angular motion of P about S, is OQ or P'T. The analytical values of these quantities may be found, as in the theory of the moon, though not with equal simplicity, because SP cannot always be supposed great in respect of SP'.

44. In consequence of these actions, the orbit of every planet may be considered as an ellipsis, which is under- ^ going slowly certain changes in its form, magnitude, and ’ position, or in what are called its elements. By the elements of the orbit of any heavenly body, are meant the quantities that determine the position and magnitude of that orbit, viz. the position of the line of the nodes, the inclination of the plane of the orbit to the plane of the ecliptic, the position of the line of the apsides, the eccentricity, and the mean distance. These are all quantities independent of one another, and from them may be deduced all other circumstances with respect to the elliptic orbit. Of these five elements, which would be invariable if the planet only gravitated to the sun, all except the mean distance are subject to slow but perpetual changes.

45. The line of the nodes, in every one of the planets, has a retrograde angular motion, which goes on continually, and of which the amount, when calculated as due to each planet, agrees very well with observation. The plane of the orbit also varies its inclination, by certain small periodical changes, which alternately increase and diminish it, as in the case of the moon. The line of the apsides, from the same cause as in the planet just mentioned, has a continued motion forward, or according to the order of the signs. Thus, in Mercury the node goes back about 7"∙82 annually. The aphelion goes forward about 5"∙84 in a year, and the inclination of the orbit in the course of a century increases about 18"∙18, which, in the course of succeeding ages, will be compensated by an equal diminution, so as to preserve it always nearly of the same quantity. In the same planet the equation of the centre, which depends on the eccentricity, increases about 1"∙6 in a century, indicating a small increase of eccentricity. These variations in the orbit of Mercury arise from the action of Venus, the Earth, Mars, Jupiter, and Saturn; the effects of the first of these planets, on account of its vicinity, being by much the most considerable. The mean distance, however, of Mercury from the Sun, does not, any more than that of the other planets, undergo any change whatever.

46. Similar conclusions apply also to the orbit of Venus. The orbit of the earth also is subject to similar changes, the line of the apsides moving forward annually at the rate of 11"∙8, in respect to the fixed stars. The earth’s eccentricity is also diminishing, and the secular variation of the greatest equation of the centre is — 17"∙66.

The motion of the earth is subject to another inequality on account of the action of the moon; for, to speak strictly, it is not the centre of the earth, but the centre of gravity of the moon and earth, which describes equal areas in equal times about the centre of the sun. It is evident that, on this account, the earth will be sometimes advanced before, and sometimes will fall behind, the point which describes the circumference of the ellipsis, in conformity with the general law of the planetary motions. From the same cause also, as the moon does not move in the plane of the ecliptic, the earth will be forced out of that plane, in order to preserve a position diametrically opposite to the moon. These irregularities, however, are inconsiderable. By observers on the earth’s surface, they are transferred to the sun, but in an opposite direction. The sun, therefore, has a motion in longitude, by which he alternately advances before the point which describes the elliptic orbit in the heavens, and falls behind it; and also a motion in latitude, by which he alternately ascends above and descends below the plane of the ecliptic. As the mass of the moon, however, is not more than part of that of the earth, the distance of the centre of gravity of the moon and earth from the centre of the latter must be [7:4:59]less than a semídiameter, and therefore the inequality thus •produced in the sun’s longitude must be less than his ' horizontal parallax. The alteration in latitude can hardly amount to a second. This inequality in the sun’s motion is called the menstrual parallax, and was first mentioned by Smeaton, Phil. Trans. 1768.

47. In the orbit of Mars the node moves backward 23"∙3 annually, and the line of the apsides moves forward 15"∙8, both in respect of the fixed stars. The eccentricity of the orbit is increasing, and the secular variation of the greatest equation of the centre is + 37".

In the case of this planet, however, the elliptic orbit is not only changed by these quantities, but the place of the planet in that orbit is sensibly affected by the action of Venus, Jupiter, and the Earth. The effect of the action of Venus is expressed by this formula, 5"∙7 sin. (long. ¢ — 3 long. J ); of the earth, 7"∙2 sin. (long. 0 — long. Ş ). Several inequalities are produced in Jupiter.

48. The inequalities of the small planets Vesta, Juno, Ceres, and Pallas, have not yet been computed. The disturbances which they must suffer from Mars and Jupiter are no doubt considerable, and, on account of their vicinity, though their masses are small, they may somewhat disturb the motions of one another. Their action on the other bodies in the system is probably insensible.

As two of these planets have nearly the same periodic time, they must preserve nearly the same distance and the same aspect with regard to one another. This offers a new case in the computation of disturbing forces, and may produce equations of longer periods than are yet known in our system.

The motion of the apsides and the change of eccentricity in the orbits of Jupiter and Saturn are chiefly produced by their action on one another, but a part also depends on the action of the other planets. The node of Jupiter moves backward annually 15"∙8, and his aphelion forward 6"∙96. The secular change in the inclination of the orbit is 22"∙6, and in the first and last of these inequalities the action of Venus has the principal share. The equation of the centre increases 56"∙25 in a century, of which nearly the whole arises from the action of Saturn. In Saturn again the node goes back at the rate of 19"∙4 annually, and the aphelion forward at the rate of 19"∙4; the secular change of the inclination is — 15"∙5, and the secular diminution of the equation of the centre 2'∙1.

There is, besides these variations in the orbits, an inequality in the motion of each of these planets, which it has been found very difficult to explain, and has only lately been fully accounted for, according to the theory of gravity, by the profound investigations of Laplace. These inequalities are both of a long period, viz. 918 ’ 76 years, which is the time that they take to run through all their changes. If n express a number of years reckoned from the beginning of 1750, S the mean longitude of Saturn, and I that of Jupiter, reckoned from the same time, then the equation which must be applied to the mean longitude of Jupiter, or the amount of this inequality, is + (20'49"∙5-n × 0"∙042733) ×

sin. (5 S — 2 I + 5° 34' 8" — n × 58"∙88); and that which must be applied to S is

— (48'44"-n × 0"∙1) ×

sin. (5 S — 2 I + 5° 34 8" — n × 58"∙88).

These two equations are to one another nearly in the ratio of 3 to 7. The reason of the long period above mentioned is, that the argument 5S — 21 — η × 58''∙88 requires all that time to increase from 0 to 360°.

Uranus, on account of his great distance, suffers hardly any disturbance in his motion, but from Saturn and Jupiter. The node moves backward at the rate of 36" annually, and the aphelion forward at that of 2"∙55. The eccentricity is diminishing, and the secular variation of the greatest equation of the centre is 11"∙03.

There is also an inequality in the longitude of this’ planet, depending on the action of Saturn. If S be the longitude of this last planet, U the longitude of Uranus, and A the longitude of the aphelion of Saturn, the inequality in question amounts to 2' 30" × sin. (S — 2 U + A).

49. Of all these inequalities, and of many other smaller ones which theory has discovered, it must be observed that they are periodical, each returning after a certain time to run through the same series of changes which it had formerly exhibited. Another general remarh is, that one element in every planetary orbit, viz. the mean distance, is exempted from all change ; and since on the mean distance depends the time of revolution, that time remains also unchanged. From the invariability of the mean distance, and the periodical revolution of all the inequalities, it follows that the actual condition of the planetary system can never deviate far from the mean, about which, we may, therefore, conceive it to be continually making small oscillations, which in the course of ages compensate one another, and therefore produce nothing like disorder or permanent change. It is in this manner that the stability of the planetary system is provided for by the τνisdom of its Author. 50. Comets, in describing their elliptic orbits round the sun, have been found to be disturbed by the action of the larger planets, Jupiter and Saturn; but the great eccentricity of their orbits makes it impossible, in the present state of mathematical science, to assign the quantity of that disturbance for an indefinite number of revolutions, though it may be done for a limited portion of time, by considering the orbit as an ellipsis, the elements of which are continually changing. This is the method of Lagrange, and is followed in the Mécanique Celeste, Part ii. chap, ix. Dr Halley, when he predicted the return of the comet of 1682, took into consideration the action of Jupiter, and concluded that it would increase the periodic time of the comet a little more than a year; he therefore fixed the time of the re-appearance to the end of the year 1758, or the beginning of 1759. He professed, however, to have made this calculation hastily, or, as he expresses it, levi calamo. (Synopsis of the Astronomy of Comets.)

The effects both of Jupiter and Saturn on the return of the same comet were afterwards calculated more accurately by Clairaut, who found that it would be retarded 511 days by the action of the former planet, and 100 by the action of the latter; in consequence of which, the return of the comet to its perihelion would be on the 15th of April 1759. He admitted at the same time that he might be out a month in his calculation. The comet actually reached its perihelion on the 13th of March, just 33 days earlier than was predicted; affording, in this way, a very striking verification of the theory of gravity, and the calculation of disturbing forces. The same comet may be expected again about the year 1835.

In some instances, the effect which the planets produce on the motion of comets are far more considerable than in this example. A comet which was observed in 1770 had a motion which could not be reconciled to a parabolic orbit, but which could be represented by an elliptic orbit of no great eccentricity, in which it revolved in the space of five years and eight months. This comet, however, which had never been seen in any former revolution, has never been seen in any subsequent one. On tracing the path of this comet, Mr Burckhardt found that between the year 1767 and 1770 it had come very near to Jupiter, and had done so again in 1779. He therefore conjectured, that the action of Jupiter may [7:4:60] have so altered the original orbit as to render the comet •for a time visible from the earth; and that the same ' cause may have so changed it, after one revolution, as to restore the comet to the same region in which it had formerly moved. This is the greatest instance of disturbance which has yet been discovered among the bodies of our system, and furnishes a very happy, as well as an unexpected, confirmation of the theory of gravity.

Though the comets arc so much disturbed by the action of the planets, yet it does not appear that their reaction produces any sensible effect. The comet of 1770 came so near to the earth as to have its periodic time increased by 2∙246 days, according to Laplace’s computation; and if it had been equal in mass to the earth it would have augmented the length of the year by not less than two hours and forty-eight minutes. It is certain that no such augmentation took place, and therefore that the disturbing force by which the comet diminished the gravity of the earth is insensible, and the mass of the comet, therefore, less than 3 -⅛θth of the mass of the earth. The same comet also passed through the system of the satellites of Jupiter without causing any derangement of their motions. Hence it is reasonable to conclude, that no material or even sensible alteration has ever been produced in our system by the action of a comet.

3. Of the disturbances which the satellites of Jupiter suffer from their action on one another.

51. The same resolution of the forces by which one satellite acts upon another, into two, one directed to the centre of the primary, and the other at right angles to it, serves to explain the irregularities which had been observed in their motions, and to reduce under known laws several other inequalities, of which the existence only is indicated by observation.

An instance of this we have in the very remarkable relation which takes place between the mean motions of the first three satellites; the mean motion of the first satellite, together with twice the mean motion of the third, being equal to three times the mean motion of the second. Laplace has shown that, if the primitive mean motions of these satellites were nearly in this proportion, the mutual action of these bodies on one another must in time have brought about an accurate conformity to it.

The first satellite moves nearly in the plane of Jupiter’s equator, and has no eccentricity, except what is communicated from the third and fourth, the irregularities of one of these small bodies producing similar irregularities in those that are contiguous to it. The first satellite has, beside, an inequality, chiefly produced by the action of the second, and circumscribed by a period of 437∙659 days.

52. The orbit of the second satellite moves on a fixed plane, to which it is inclined at an angle of 27' 13", and on which its nodes have a retrograde motion, so that they complete a revolution in 29∙914 years. The motion of the nodes of this satellite is one of the principal data used for determining the masses of the satellites themselves, which are so necessary to be known for computing their disturbances. This satellite has no eccentricity but that which it derives from the action of the third and fourth. 1he third satellite moves on a fixed plane, to which it is inclined at an angle of 12' 20", and its nodes make a tropical revolution backwards in 141∙739 years. The equator of Jupiter is inclined to the plane of his orbit at an angle of 3° 5' 30". The fixed planes on which the planes of the orbits move are determined by theory, and could not have been discovered by observation alone.

The orbit of the third satellite is eccentric, but appears to have two distinct equations of the centre; one which really arises from its own eccentricity, and another which theory shows to be an emanation from the equation of the centre of the fourth satellite. The first equation is referable to an apsis which has an annual motion of 2° 36' 39" forward in respect of the fixed stars; the second equation is referable to the apsides of the fourth satellite.

These two equations may be considered as forming one equation of the centre, referable to an apsis that has an irregular motion. The two equations coincided in 1682, and the sum of their maxima was 13' 16". In 1777 the equations were opposed, and their difference was 5' 6".

The two last inequalities were perceived by Mr Wargentin, by observation alone; but their exact amount, and the law which they observe in their changes, he could not discover. The orbit of the fourth satellite moves on a fixed plane, to which it is inclined at an angle of 14' 58"; and its nodes complete a sidereal revolution backward in 531 years. The fixed plane on which the orbit moves is inclined at an angle of 24' 33" to the equator of Jupiter.; the orbit is sensibly elliptical, and its greater axis has an annual motion of 42' 58"∙7. The motion of this axis is one of the principal data from which the quantities of matter of the different satellites have been determined.

If the mass of Jupiter be supposed unity, the mass of the 1st satellite = ∙0000173281

of the 2d = ∙0000232355

of the 3d = ∙0000884972

of the 4th = ∙0000426591

If the mass of the earth be supposed unity, that of the third satellite will be found = ∙027337; and as the mass of the moon is ~ğ.Q ( ý — ‘012517, the quantity of matter in the third satellite is about twice as great as that in the moon. The fourth satellite is therefore nearly equal to the moon, the second about one half, and the first somewhat more than one third.

53. The general result of this investigation concerning the inequalities in the motion of the planets, both primary and secondary, is, that in every one of these orbits two things remain secure against all disturbance, the mean distance and the mean motion, or, which is the same, the transverse axis of the orbit, and the time of the planet’s revolution. Another result is, that all the inequalities in the planetary motions are periodical, and observe such laws that each of them, after a certain time, runs through the same series of changes. This last conclusion follows from the fact, that every inequality is expressed by terms of the form A sin. nt or A cos. nt, where A is a constant co-efficient, and n a certain multiplier of t the time, so that nt is an arch of a circle, which increases proportionally, to the time. Now, in this expression, though nt is capable of indefinite increase, yet, since nt never can exceed the radius, or 1, the maximum of the inequality is A. Accordingly, the value of the term A sin. nt first increases from 0 to A, and then decreases from A to 0; after which it becomes negative, extends to — A; and passes from thence to 0 again. If, when the inequality was affirmative, it was an addition to the mean motion, when negative it will become a diminution of it; and the sum of all these increments and decrements, after nt has passed over an entire circumference, or 360°, is equal to 0; so that at the endof that period the planet is in the same position as if it had moved on regularly all the while at the rate of the mean motion. As this happens to every one of the inequalities, the deviation of the system from its mean state can never go beyond certain limits, each inequality in a certain course of time destroying its own effect.

It would be far otherwise if into the value of any inequalities a term entered of the form A × nt, A tan. nt, [7:4:61] A

—. The inequalities so expressed would continually sin. nt increase with the time, so as to go beyond any assignable limit, and of consequence to destroy entirely the order of any system to which they belonged.

Lagrange and Laplace, who discovered and demonstrated that no such terms as these last can enter into the expression of the disturbances which the planets produce by their action on one another, made known one of the most important truths in physical science. They proved that the planetary system is stable, and that it does not involve any principle of destruction in itself, but is calculated to endure for ever, or till the action of an external power shall put a period to its existence. After the knowledge of the principle of gravitation, this may be fairly considered as the greatest discovery to which men have been led by the study of the heavens.

The accurate compensation, just remarked, depends on three conditions, belonging to the primitive or original constitution of our system, but not necessarily determined, as far as we know, by any physical principle. The first of these conditions is, that the eccentricities of the orbits are all inconsiderable, or contained within very narrow limits, not exceeding in any instance one tenth or one eighth part of the mean distance. The second condition is, that the planets all move an the same direction, or from west to east. This is true both of the primary and secondary planets, with the exception only of the satellites of Uranus, which may be accounted retrograde; but their planes being nearly at right angles to the orbit of their primary, the direction of their motion, whether retrograde or otherwise, can have little effect. Lastly, the planes of the orbits of the planets are not much inclined to one another. This is true of all the larger planets, though it does not hold of some of the new and smaller ones; of which, however, the action on the whole system must be altogether insensible.

Unless these three conditions were united in the constitution of the solar system, terms of the kind just mentioned, admitting of indefinite increase, might enter into the expression of the inequalities, which would indicate a gradual and unlimited departure from the original order and constitution of the universe. Now, the three conditions just enumerated do not necessarily arise out of the nature of motion, or of gravitation, or from the action of any physical cause with which we are acquainted. Neither can they be considered as arising from chance; for the probability is almost infinite to one, that, without a cause particularly directed to that object, such a conformity could not have arisen in the motions of thirty-one different bodies, scattered over the whole extent of the solar system. The only explanation, therefore, which remains is, that all this is the work of intelligence and design, directing the original constitution of our system, and impressing such motions on the parts as are calculated to give stability to the whole. (n.)

For some further discussions connected with Physical Astronomy, see the articles Comets ; Earth, Figure of; Precession of the Equinoxes ; and Tides.

PART IV.

PRACTICAL ASTRONOMY.

introduction. Practical Astronomy may be considered as comprehending the observations which must originally have been made to determine the facts which have now been embodied in a system; also those which are continually making for its farther extension and improvement;—the observations which are required to make the science useful in the affairs of life; and the rules and calculations which must be applied to the observations, to obtain from them the required result.

The division of labour in this, as in other subjects, has been attended with advantage. One class of astronomers now survey the heavens by telescopes of more or less power. These, within the last fifty years, have enriched the science by the discovery of five new planets in addition to the six known from the most remote ages. The late Sir William Herschel took the lead in this noble labour, and he has been followed by many others. By their exertions various comets have been observed; and the periodic times and orbits of three have now been determined with such certainty as to enable us to predict their return. A second class observe the heavenly bodies with instruments which combine increased powers of seeing, with the means of measuring minute angular distances or considerable angles with great accuracy. The established national observatories, and some private observatories, conducted with not less zeal and intelligence, are engaged in this important labour. These employ transit instruments, clocks, mural or meridian circles, quadrants, equa-toral or azimuth and altitude instruments, of the best workmanship. Their object is less to add new facts than to improve and extend those which are already known.

By the observations made in observatories, the science is continually advancing towards perfection; but, like all knowledge founded on observation, there will ever be room for further improvement.

Although it is only a few individuals, comparatively speaking, that can possess telescopes of such power as to give reasonable hopes of extending astronomical knowledge by the discovery of new phenomena, yet, by the liberality of governments, the united efforts of learned bodies, and the fortunate possession of leisure, pecuniary means, and zeal for the improvement of the science evinced by some private gentlemen, telescopes of extraordinary dimensions and excellence have been constructed, and considerable additions have of late years been made to our knowledge of the heavens in regions almost beyond the flight of human imagination.

The great instruments, such as are placed in national observatories, are also in general beyond the reach of private astronomers; yet a few wealthy amateurs of the science have rivalled these establishments, both in the acquisition of instruments and in their application. With instruments of an inferior construction little can be done towards improving the primary elements of astronomy, because the results they give are less correct and less to be depended on than those obtained by superior instruments. When the measure of an angle is determined to the accuracy of a second, it is of no use to have another measure which may err as much as ten seconds. However, it cannot be doubted that the best instruments now employed will in time be superseded by others more perfect. After a certain degree of accuracy has been obtained, it is only by slow degrees and much labour that we can go beyond it; and then, like differentials of different orders, every [7:4:62]addition to the accuracy of a result is small in comparison to that which went before it.

Another branch of practical astronomy, of no small importance, applies the knowledge which the labour of many years, or even ages, has accumulated, to the improvement of geography, and to navigation, and the wants of society. For this, instruments are also required. These are the portable transit instrument and chronometer; portable azimuth and altitude circles; reflecting and repeating circles; sextants, quadrants, and other minor instruments, down to the convenient garden sun-dial. These instruments are within the reach of ordinary cultivators of astronomy, and they are the most useful, because the advantage they yield is immediate and obvious: for by their aid the exact position on the earth of every point visited by a traveller may be determined; ships may be conducted in safety and with certainty to remote regions; and true time may be ascertained in all places. A third department of practical astronomy may comprehend the calculations required for reducing the observations made in observatories; that is, in disengaging them from the effects of refraction, parallax, nutation, aberration, instrumental errors, &c. and fitting them for their place in the annals of the science. The time spent in making an observation is in fact often small in comparison to that required for its reduction. The observation is a process purely mechanical, requiring only steady attention, and an expertness in observing a moving body, and in estimating small portions of time by the beat of the clock, which may be acquired by practice. But the reduction requires the application of much theoretical knowledge of the science, and an acquaintance with some of the refinements of modern analysis; at least these must be possessed by the person who frames the rules and directs the operations, although a less degree of knowledge may suffice for the mere numerical calculation.

The observations of the traveller and navigator require also the application of mathematical science. Plane and spherical trigonometry are indispensable for the intelligent navigator; and the latter of these is required for the simplest observations, such, for example, as are made by a quadrant or sextant, to find true time. It frequently happens that the navigator is deficient in theory, contenting himself with the practice of the calculation of his lunars, as he calls observations made to ascertain the distance of the moon from the stars. This is an evil which, we fear, sometimes leads to the loss of many lives and much property.

Another kind of calculations is required in the construction of Almanacs and Ephemerides. These are made under the direction of men well versed in the theory, although the mere labour of numerical calculation can be executed by correct arithmeticians according to prescribed rules; and, to insure accuracy, each is commonly performed by two persons, and their results compared by a third.

The attentive reader of the preceding treatise will readily understand that much practical astronomy is involved in it; necessarily indeed, for the theory and the practice cannot be entirely disengaged from each other. Thus, it is shown (chap. II. sect. 1.) how the obliquity of the ecliptic may be derived from observations of the sun’s declination on three successive days about the time of the summer solstice; and again, in section 2, a formula is given for finding the eccentricity of the earth’s orbit from the greatest equation of the centre, as determined from observations; also another for determining the true anomaly from the mean: these and various others, in chap. III. and elsewhere, may be regarded as applications of practical astronomy. We shall therefore here only farther give some examples of the more important problems in what may be called spherical astronomy. ' ’

The important and very general problem of determining what will be the position of any proposed body of the sys-∙ l tern at any given time, past or to come, belongs to the division of the subject which we are now treating. Its correct solution, however, requires an apparatus of tables far beyond what can be comprised in the space to which this article must be limited. Even the single case of the moon would require a very considerable number of tables and precepts. Such a body of tables is, however, not necessary even in works professedly on astronomy. The British Nautical Almanac, the Connaissance des Terns, and the Berlin Ephemeris (Berliner Astronomisches Jahrbuclι), are the sources from which the practical astronomer usually derives a knowledge of the phenomena he expects to happen, and for the observation of which he must prepare beforehand. The same observation applies to various other subsidiary tables, of which there are professed collections. One of the most copious is the first of two large volumes, forming a work on practical astronomy, by the Rev. Dr Pearson; this contains tables expressly intended for the reduction of astronomical observations, and many others of great importance to the professed astronomer. There is a smaller collection, entitled Tables to be used with the Nautical Almanac, by the Rev. W. Lax, professor of astronomy and geometry in the University of Cambridge. This is most essential to the navigator, for whose use also a much larger work was published, viz. A complete Collection of Tables for Navigation and Nautical Astronomy, by Joseph de Mendoza Rios. There is another and later work, published in 1829 in France, entitled Nouvelles Tables Astronomiques et Hydrographiques, par V. Bagay, professeur d’hydrographie. This must be valuable to the French navigator; but it is not less so to practical astronomers of all countries, on account of its portable size, considering that it contains a table of logarithmic sines and tangents to every second of the quadrant. We must also notice with much approbation a neat and comprehensive collection of astronomical tables and formulae by Mr Baily, late president of the Astronomical Society of London. This work is chiefly adapted to the practical astronomer on land. There is one part of practical astronomy which at all times has, in a particular manner, excited the attention of mankind; we mean the determination of the exact time of new and full moon, and the prediction of eclipses. To determine these with as much accuracy as is sufficient to gratify ordinary students of practical astronomy, tables have been purposely constructed, and are given in the sequel; with the addition of plain precepts, applicable without any considerable degree of mathematical knowledge. By means of them the young astronomer may find the time of an eclipse of the sun and moon, and exhibit its appearance by a geometrical construction. The places of upwards of 500 fixed stars, and some tables useful in practical astronomy, are also given; and the principal instruments of astronomy are described and exhibited by engravings in the superior style which distinguishes the present from all former editions of this work.

CHAP. I.

PROBLEMS IN PRACTICAL ASTRONOMY.

Problem I.— Given the right ascension and declination of a star or planet, together with the obliquity of the. ecliptic, to find the star’s longitude and latitude.

Let P (fig. 120, Plate XCIII.) be the pole of the equator O<yQ, E the pole of the ecliptic I<γΕ, and S the place of the star. Let PR, the circle of latitude passing through [7:4:63]S, meet the equator in R; and EM, the circle of declina- ∙tion, meet the ecliptic in Μ; also let ɤE, ɤP, be arcs of great circles passing through the equinoctial point <p and the poles of the ecliptic and equator respectively. Let us now make

L = ɤM the longitude of S, l = SM the latitude, AR = ɤ R the right ascension,

D = SR the declination^

ω = LɤQ the obliquity of the ecliptic.

Since op Μ = ɤL — ML = 90° — ML, and ML is the measure of the angle SEP, we have ɤM=L = 90° — SEP, and, consequently, sin. L = cos. SEP. Again, since ɤ R — υ >Q RQ = 90° — RQ, and RQ is the measure of the angle RPQ, we have w>R = AR = 90°-RPQ; whence sin. AR = cos. RPQ. = — cos. EPS. In like manner we have l = SM = 90° — ES, whence sin. I = cos. ES; and also D = SR — 90° — PS, whence sin. D = cos. PS. Thus the problem depends on the solution of the oblique angled spherical triangle PES.

From the known properties of spherical triangles, we have cot. PS sin. EP = cot. PES sin. EPS + cos. EP. cos. EPS; therefore, by substituting the values just found, and observing that EP = ω, we have tan. D sin. ω = tan. L cos. AR — cos. ω sin. AR ; whence τ tan. D sin. ω + sin. AR cos. ω tan. L = 75 ∙

cos. Alt

Assume an angle <p such that tan. φ — sin. AR cot. D, then sin. D tan. ø = sin. AR cos. D, τx sin. AR sin. AR cos. φ or tan. D = —-; tan. ļ> sin. φ whence, by substituting,

(sin. ω ∖

———h cos. w , tan. φ ∕ and by reducing, τ sin. (ω + ø) yr , zi x tan. L = τ — tan. AR (1.)

sin. φ a formula from which the longitude L is easily computed by means of the logarithmic tables.

To find the latitude l, we have cos. ES = cos. EP cos. PS + sin. EP sin. PS cos. EPS, which gives sin. l = cos. ω sin. D — sin. ω cos. D sin. AR.

Assume as before, tan. <p = cot. D sin. AR, then, by substituting, sin. l — sin. D (cos. ω — sin. ω tan. p), that is,

, ∙ ∏ /cos. ω cos. ø — sin. ω sin. ø\

sin. l = sin. D (l~ ļ

∖ cos. φ J whence

, sin. D cos. (ω + ¢) zλ . sin. l — ï—!_JA f2. )

cos. <ţ> v 7 Problem II.— Given the longitude and latitude of a star or planet, together with the obliquity of the ecliptic, to determine its right ascension and declination.

The spherical triangle EPS gives the relation cot. ES sin. EP = cot. EPS sin. PES + cos. EP cos. PES, that is, tan. l sin. ω — — tan. AR cos. L + cos. ω sin. L, whence

, r , cos. ω sin. L — tan. l sin. ω tan. AR = -= . , cos. L

Assume tan. ∙ψ = sin. L cot. l ; then

, sin. L sin. L cos. ∙ψ

tan. l = - = ;; ; tan. ∙ψ sin. ∙ψ

and by substituting this value of tan. l in the above equation, it becomes

√∏ i τ ∕ ∙ c ° s ∙ ’ Ψ∖

tan. AR = tan. L I cos. ω — sin. ω ~ ļ,

∖ sin. ∙ψ∕

that is, tan. AE = tan. L (cos · " sin ∙ ⅜~ sin ∙ a c ° s ∙ ≠), sin. ∙ψ

whence zr , tan. L sin. (∙ψ — ω) z x tan. AR — ; —+- 7 fl.)

sin. ∙ψ * v 7

To find the declination, we have cos. PS = cos. EP cos. ES + sin. EP sin. ES cos. PES; that is, sin. D. — cos. ω sin. l + sin. ω cos. l sin. L,

— (cos. ω + sin. ω cot. l sin. L) sin. l,

. . , . . tan. + η and by substituting — : —-÷- for cot. Z, ; b sin. L

sin. D — (cos. ω + sin. ω tan. +) sin l, (cos, ω cos, ∙ψ + sin. ω sin. ∙ψ) sin. l cos. ∙ψ ’

therefore . ta cos. (∙ψ — ω) . , sin. D — v —— 1 sin. l (2.)

cos. + v 7 Corollary. — In the case of the sun, the latitude l becomes zero, and the formulae are considerably simplified. Thus, let T be the sun’s place in the ecliptic; the relations subsisting between the longitude, right ascension, declination, and obliquity, are given by means of the right-angled spherical triangle ɤ TR. The following are the formulae:

τ sin. D -∣

sin. L ——-. , sin. ω ļ Zļ X

τ tan. AR s × '> tan. L= , cos. ω J

sin. AR = cot. ω tan. D, ì ,∩ κ

tan. AR = cos. ω tan. L, J × 'l sin. D=sin. ω sin. L I .

tan. D=tan. ω sin. AR, J ×

Problem III.— Given the latitude of the observatory, the polar distance of a star, and its hour-angle at the pole, to find its zenith distance, azimuth, and angle of variation.

Let P (fig. 121) be the pole of the equator, A the zenith of the place, and V the position of the star: we have then given the two sides, AP and PV, of an oblique angled spherical triangle, together with the included angle APV, to determine the third side and the remaining angles. Make λ=AP the complement of the latitude,

Δ=PV the polar distance of the star, which is here supposed to be north,

P=APV the hour-angle at the pole,

Z=AV the zenith distance,

A=VAP the azimuth,

V=AVP the angle of variation.

1. To find Z we have cos. Z=cos. λ cos. ∆ + sin. λ sin. Δ cos. P.

Assume tan. <p=tan. Δ cos. P; then by substituting, cos. Z=cos. Δ (cos. λ+sin. λ tan. p); whence, by reducing, [7:4:64]cos. (λ—ρ) , λ cos. Z=cos. Δ - (1.)

cos. φ

2. To find A, we have from spherical trigonometry the formula

. cot. Δ sin. λ τ , . cot. A =; ∏ cot. P COS. λ.

sin. P

Assume cot. ∙ψ = -, whence cot. ∆=cot∙ ∙ψ cos. P, cos. P

and, by substituting, cot. A = cot. P (cot. ∙ψ sin. λ — cos. λ)

(cos. ∙ψ sin. λ — sin. -ψ cos. λ∖ X ), sin. + ∕

whence

. cot. P sin. (λ — ∙ψ) /o x sin. -ψ

3. To find V we have

. , r cot. λ sin. Δ t τ , cot. V =; √τ COS. Δ cot. P.

sin. P

. , cot. λ κ τι . 1

Assume cot. y = —, or cot. λ = cot. y cos. P, then, zw cos. P

by substituting and reducing as above, we have cot. v -∞t-P⅛∙(⅛-¾) w sin. χ '

Problem IV.— Given the declination and zenith distance of a star, and the latitude of the observatory, to determine the hour-angle.

This is the case in the solution of spherical triangles, in which the three sides are given to find one of the angles. The formula, therefore, from which P is computed (see Spherical Trigonometry) is tan 2 π>- sin ∙ ⅜(Z÷λ-∆) sin. ⅜(Z-λ+∆)

* 2 sin. ∣(Z + λ+∆) sin. ∣(λ + Δ—Z)‘

Problem V.— Given the latitude of a place, and the sun’s declination, to find, 1st, the time of sunrise ; 2d, the suns amplitude at rising ; 3d, the time when the sun is due east ⅛th, the sun’s altitude when in that position.

In fig. 122 let AZP be the meridian, in which Z is the zenith, and P the pole; let ZO be the prime vertical, and AOB the horizon, A being the south, 0 the east, and B the north points; let SS' be the part of the parallel described by the sun between his rising and passing the prime vertical, and PS PS' hour circles.

Let L = the latitude of the place = PB,

D = the sun’s declination = 90° — PS, v = the time from midnight to sunrise = hour angle SPB, x = the sun’s amplitude at rising = OS, y = the time from midnight to the sun’s coming on the prime vertical = hour-angle S'PB, z = the sun’s altitude when in the prime vertical

= OS'.

In the spherical triangle PBS, right angled at B, c , „ tan. PB τ - c , cos. PS cos. SPB =- 1 -, cos. BS = = vb ; tan. PS cos. PB

and in the spherical triangle PZS', right angled at Z, onc , _ tan. PZ . cos. PS'

cos. ZPS = - , cos. ZS' = 57 =.

tan. PS' cos. PZ

Now PB=L, PZ = 90° — L, PS=PS' = 90° — D, BS = 90° — x, ZS' = 90° — z∙, hence we have these four formulae:

cos. v — tan. L, tan. D (1.)∙

• _ sin ∙ d ∕ 9 ∖

cos. L

cos. y — cot. L, tan. D (3.)

. sin. D ,, .

sin. z = — (4.)

sin. L '

The hour-angles v and y must be converted into time by allowing 1õ° to one fiour.

Problem VI.— To find how much the rising of the sun or a star is advanced by refraction.

Let ZPB be the meridian (fig. 123), Z the zenith, P the pole, and OB the horizon. Let SS', the parallel described by the sun, meet the horizon in S. Were it not for the effect of refraction, the sun or star would appear to rise at S, and at rising the hour-angle from midnight would be SPB; but it is elevated by refraction (which takes place in a vertical circle), and appears to rise at D, while in fact it is below the horizon at S' somewhere in the vertical circle ZD produced downward, and the time of rising is accelerated by the small angle SPS'∙

Let L denote the latitude of the place = 90 — PZ; D the declination of the star = 90— PS; H the hour-angle BPS, reckoned from midnight; r the arc DS of the vertical ZS, the effect of refraction; and x the angle SPS, the acceleration of time of rising.

In the spherical triangle ZPS', cos. ZS' = cos. ZP cos. PS' + sin. ZP sin. PS' cos. ZPS', Now, cos. ZS' = cos. (90 + r y ) = — sin. r, and cos. ZPS' = — cos. BPS' = — cos. (H — x) ; therefore

— sin. r = sin. L sin. D — cos. L. cos. D cos. (H — χ).

In like manner, in the triangle ZPS, in which cos. ZS = cos. 90° =0,

0 = sin. L sin. D — cos. L cos. D cos. H; hence, by subtraction, sin. r = cos. L cos. D [cos. (H— x) — cos. H],

1 ∕ττ ∖ ττ _ sin ∙ r and cos. (H — χ) — cos. H — = —; v z cos. L cos. D

but cos. (H — ae) — cos. H = 2 sin. ⅜ x sin. — 0 (Algebra, § 239, D); i , r o ∙ i ∙ ∕ττ a A — sln ∙ τ therefore, 2 sin. ⅛ x sin. H — -) = v τ∖∙

ύ ∖ 2J cos. L cos. D

Now, x and r being small angles, we may consider (φ∖

H — —} = sin. H.

-r Τ

We have then x = ——— √ —: sin. H cos. L cos. D

and here H is determined by the formula ‘ cos. H = tan. L tan. D.

The value of x just found is only a near approximation; let it be denoted by a/, and let H' = H + a?', and we shall have more nearly

_ r ‘ sin. H' cos. L cos. D’

q∙ ττ . -r , sin. L sm. D

öince cos. H = tan. L tan. D = —; cos. L cos. D

. 1 c ∙ ο tt cos. 2 L cos. 2 D — sin. 2 L sin. 2 D therefore, sin. 2 H = — .

cos. 2 L cos. 2 D

Again, cos. 2 L cos. 2 D — sin. 2 L sin. 2 D

= (cos. L cos. D—sin. L sin. D)(cos. L cos. D + sin. L sin. D) = cbs. (L + D) cos. (L — D).
[7:4:65]

. cos. (L + D) cos. (L — D) . Therefore, sin. 2 H _ JJJLcos? D ’

. r and x = √lcos. (L + D) cos. (L —D)ĵ’

The value of r is variable; but in general it is about 33 minutes of a degree, or 132 seconds of time. Therefore the acceleration by refraction is nearly, in time, 132* √ , [cos. (L + D) cos. (L — D)]

Problem VII.— To find the length of the twilight.

Let ZPB (fig. 123) be the meridian, Z the zenith, P the pole, and SDB the horizon. Let S'S be the arc of the parallel described by the sun between the beginning of the twilight, when he is at S', and its end at sunrise, when he reaches the horizon at S.

The twilight begins when DS', the sun’s depression below the horizon, is about 18°.

Let a denote the arc, DS the depression, L the latitude of the place = 90° — ZP, D the sun’s declination = 90° — PS, H the hour angle SPB (from midnight), x the angle SPS'.

Then, cos. ZS' = cos. ZP cos. PS' + sin. ZP sin. PS' cos. ZFS'; that is, because cos. ZS' = — sin. a, and cos. ZPS =

-— cos. SPB,

— sin. a= — sin. L sin. D — cos. L cos. D cos. (H — x) ; hence we obtain zττ X -r tλ ι sin. a cos. (H — x) = tan. L tan. D + ~->∙∙ τ —.

x j cos. L cos. D

Now, by Problem V. cos. H = tan. L tan. D (1)

Let p be such an angle, that sin. a cos. <p = t y.; r cos. L cos. D

Then cos. (H — x) = cos. H + cos. φ

= 2 cos. ⅜ (H + p) cos. ⅜ (H — φ) (2)

Now H is determined by formula (1); therefore x, the angle described about the pole while the twilight lasts, is determined by formula (2).

Problem VIII.— Given the right ascensions and declinations of two celestial bodies, to compute their angular distance.

Let S and Μ (fig. 124) be the two stars (for example the sun and moon), and let Z be the zenith, and P the pole of the equator. In the triangle SPM, the side SM represents the angular distance of the two stars; PS and PM are respectively their polar distances, or the complements of their declinations, and therefore given; and the angle SPM is also given, being the difference of ZPS and ZPM, the right ascensions of the given stars. Let PS = Δ, PM = ∆', the angle SPM = P, and SM, the distance sought, =.d. We have then cos. cZ∙∙= cos. Δ cos. Δ'+ sin. ∆ sin. ∆'.cos. P.

Assume tan. φ — tan. Δ cos. P; then, by substituting and reducing, as in Problem IL, we obtain 7 cos. Δ cos. ( Δ' — ®) , cos. d = · X- i-2-. cos. <p Problem IX.— To determine the latitude of a place.

1∙ Of the various methods which are employed for determining the latitude, that which depends on the observation of the double transits, or upper and lower culmination of a circumpolar star, is perhaps the best, being independent of the star s dcdinatioιτ and of the effects of aberration and nutation. The accuracy of the result depends, indeed, on the allowance made for refraction; but unless the observed star at the lower culmination passes within 15° or 20° of the horizon, the errors of the tables will be very inconsiderable.

Let Z be the observed zenith distance, and R the refraction of the star at its lower culmination; Z' and R' the same quantities at its upper culmination, and λ the correct zenith distance of the pole, or the co-latitude; then λ = ⅛(Z + Z') + ⅜(R + R').

The quantities R and R' must be taken from the tables, regard being had to the state of the barometer and thermometer.

2. Another method of determining the latitude, which, by reason of the facility of observation and computation, is extremely commodious, and therefore much employed, especially by voyagers, depends on observations of the meridional zenith distances of the sun or a star. Let P (fig. 125) be the pole, Z the zenith, C the intersection of the meridian and equator, and S the sun or star; then, l being the required latitude,

Z = ZC =ZS→≡SC =Z→zD, according as the star is situated S or S', that is, above or below the equator.

If the star is to the north of the zenith, and above the pole, as at a, then

Z = ZC -Ca — Zo = DZ, and if below the pole, at a’, then Cα' = 180°--D, and l = ZC = C«'— Zd’ = 180° — (Z + D).

Problem X.— To determine the difference of longitude between two points on the eartĩìs surface.

The different methods which have been proposed for the solution of this problem, one of the most difficult in practical astronomy, are the following: Is/, The eclipses of Jupiter’s satellites; 2d , the eclipses of the moon; 3d , the eclipses of the sun; 4∕A, the occultations of fixed stars by the moon; and, bth, the comparison of the moon’s transits over the meridian with those of certain fixed stars selected for the purpose. Of these five methods, the two first give results affected by many causes of uncertainty; and the third can seldom be practised, because solar eclipses occur very rarely for any given point on the earth’s surface. Of the remaining two, the last (which is recommended by Mr Baily) seems entitled to the preference, on account of its being independent of great accuracy in the rate of the clock or the position of the transit instrument.

The difference of the longitudes of the two stations is supposed to be nearly known from the chronometer, or by other means; ħnd the object is to correct or determine the error of the first approximation.

Let A and B be the two stations, of which A. is supposed to be the most westerly, and put r = the difference (in sidereal time) of the transit of the moon’s limb and the star previously agreed on at A; r’ = the same difference at B; t — the apparent Greenwich time of the culmination of the moon at A; t’ — the apparent Greenwich time of the culmination of the moon at B; d = D’s true declinationi computed for the time τ — D’s true radius J t ; d, ι j — the same quantities computed for the time t’ ; s = the length of the true solar day in seconds; m = î ’s motion in AR in half that interval, expressed in seconds of space; [7:4:66]χ = the assumed difference of longitude, in time; (% + e ) = t h e correct difference of longitude.

The angle comprised between the two horary circles which pass respectively through the centre and limb of the moon, at the station A, is — 7 -which being reduced cos. d to time, becomes — 7 , and expresses the sidereal time

15 cos. d in which the moon’s semi-diameter passes the meridian.

Hence r z+= r —, is the observed difference of the AE

15 cos. d of the star and moon’s centre at A, the upper sign being taken when the first or western limb of the moon is observed, and the under when it is the eastern limb, the star being supposed to precede the moon. In the same φ ∙ ∙ ∙ ∙

manner — 7 is the sidereal time in which the semι-

15 cos. d diameter of the moon passes the meridian at B, and rpł ∙

τ —ł— the observed difference of the AE of the

15 cos. d' star and moon’s centre. We have, therefore, by subtracting these two expressions, ι r ι ’

(τ τ ) 15 cos. d ~ i ~ 15 cos. d! for the observed difference of the AR of the moon’s centre, during the time elapsed between the two observations. Put this difference equal to δ, then γ j + ∆ is the difference between t and t’ in sidereal time, which becomes z . 8640 ,,. 1 .

(% + ∆) —-— when expressed in mean solar time.

Hence we have z x 8640 ∕ = ∕'+ (χ+ ∆)- consequently, when the apparent Greenwich time at one of the observatories is known, it is also known at the other observatory. Now, let a and a! be the moon’s AR in space, computed for the times t and t' respectively (taken from the ephemeris); then the formula for the correction of the assumed difference of longitudes will evidently be e = ĭ 15 δ — (α — α') ļ- ——;

I 7 J 2 m and this added to γ j gives χ + e, the corrected difference of longitudes.

Problem XI.— To find the meridional zenith distance of the sun {or a star), from observations made near the meridian.

Let P (fig. 124) be the pole, Z the zenith, and S the sun. Make I = 90° — PZ = the latitude of the place;

D = 90° — PS = the declination;

P = ZPS the hour angle;

Z = l — D = the true zenith distance;

Z' = -ZS the observed zenith distance; x = Z' — Z the correction.

The triangle, PZS gives cos. ZS = cos. ZP cos. PS + sin. ZP sin. PS cos. ZPS, or cos. Z' = sin. I sin. D + cos. I cos. D cos. P; but cos. {l — D) = cos. I cos. D + sin. l sin. D, and cos. P = 1 — 2 sin. 2 ⅜ P, therefore, ' '

cos. Z' = cos. {l — D) — 2 cos. I cos. D sin. 2 P, or cos. Z' = cos. Z — 2 cos. I cos. D sin. 2 ⅜ P.

Now, by the trigonometrical formula (Algebra, p. 240), cos. Z — cos. Z' ~ 2 sin. ⅜ (Z' — Z) sin. ⅜ (Z' + Z), therefore

2 sin. ; (Z' — Z) sin. 1∙(Z' + Z) = 2 l cos.D sin. 2 ’ P. j But Z' — Z = x, and Z' + Z = 2 Z + χ, therefore,

2 sin. I χ sin. (Z + ⅜ x) = 2 cos. I cos. D sin. 2 ⅛ P, that is,

2 sin. I χ cos. ļ x sin. Z + 2 sin. 2 į x cos Z

= 2 cos. l cos. D sin. 2 - 2 - P, or 2 sin. ļ x cos. 2 - x + 2 sin. 2 ⅛ χ co t. Z

2 cos. l cos. D . o 1 τ , = τ— ξz sm. 2 J P.

sm. Z i

In order to resolve this equation, make cot. Z = a, Eθ-'- /_5¶ĵ· .^P sin. 2 ⅜ P = b, and divide both sides by sin. Z 2 j

2 cos. 2 į x; it then becomes tan. -V x + a tan. 2 ļ x = - — = b { 1 + tan. 2 A χ ∖

2 — 2 cos. 2 ⅛ x v ‘ 27

Whence 9 1 1 , b tan/ A∙ x + τ tan. ⅛ x = r ; 2 , a — b 2 a — b and, consequently, 1 _ 1 1

tan. 2 X _ 2( . a _ 0 ) — 2 (α _ y 1 + 4 b ( a - b γ Developing this expression, and rejecting all the powers of b higher than the cube, we have tan. ^x = b — alA + (1 + 2α 2 ) 5 3 —, &c.

Now the series which expresses the arc in terms of its tangent (see art. Algebra, p. 270), gives ⅛x — tan. fx — ⅛ tan. 3 -ļx + ļ-tan. 5 ⅜x — , &c. Therefore, by substituting, and rejecting the terms containing higher powers of b than the cube, įx = b — alA + (f + 2α 2 )5 3 —, &c. that is, x = 25 — 2α5 2 + (⅜ + 4α s )5 3 -—, &c. and therefore, on restoring the values of a and b, and dividing by sin. 1", 2

„/cos. I cos. D∖ sin. 2 ⅜P _ r ,∕cos. I cos. D∖ x = 2 — 77 — ∙ ——⅞- — 2 cot. Z — )

∖, sin. Z J sin. P r ∖, sin. Z ∕

sinΛ⅜P „ / c o s.žcos.D y sin.%LP _

. sin. 1" } V sin. Z ) sin. P ’

The last term of this series is scarcely sensible in any case; it is therefore only necessary to compute the first two.

If, instead of eliminating Z' by means of the equation Z' — Z = x, we had eliminated Z, the resulting expression would have been

„/cos. I cos. D∖ sin. 2 ⅜P *=- 2 (-χ⅛zH -⅛¾- + 2oot Z-(g⅛ ιc °+y ⅞ΔF-, &c .

∖ sin. Z ʹ J sin. 1"

Problem XII.— Given the times of two observed equal altitudes of the sun, to find the true time of his meridional passage.

Let P (fig. 126) be the pole, Z the zenith, ZPM the meridian, and A and C the places of the sun, before and after his meridional passage, when his zenith distances ZA and ZC are observed to be equal. In consequence of the variation of the sun’s declination while he passes from A to C, the hour angles APM and MPC are unequal. Make BPM = APM = P, and let BPC be denoted by <ţ>. Now, APC = 2P + p; half of which is P + l 2 φ∙, therefore the true time of the meridional passage will be found by subtracting ⅜p from the mean of the times of observed equal altitudes.

To find the value of <ţ>, we have the equation cos. ZA =≡ cos. ZP cos. PA + sin. ZP sin. PA cos. APM; [7:4:67]that is (employing the denominations of last problem), cos. Z = sin. I sin. D + cos. I cos. D cos. P (1)

Now, letΔ = B6 be the small change of declination corresponding to the variation of the hour-angle from P to P + φ, and let D + Δ be substituted for D, and P + p for P in this equation. It then becomes cos. Z =sin. I sin. (D + ∆) + cos. I cos. (D + ∆) cos. (P+P)∙ But as Δ and φ are very small arcs, their cosines may be made equal to the radius, and their sines equal to the arcs themselves; therefore, on expanding the above equation, it becomes cos. Z =sin. I sin. D+cos. I cos. D cos. P.

+ (sin. I cos. D — cos. I sin. D cos. P) Δ

— cos. I cos. D sin. P. φ. + cos. I sin. D sin. P. Δ. φ ; whence, in consequence of equation (1), and rejecting the term multiplied by Δ. p,

(sin. I cos. D — cos. I sin. D cos. P) ∆=cos. I cos. D. sin. P.p; therefore, a ∕ tan. I _ ∖

ø = Δ ——— — tan. D cot. P . r ∖sin. P ∕

Let t be the sidereal time of the first observation, t’ that of the second, and t’ — t = 2θ. Also let the diurnal variation of D be denoted by v, expressed like the arcs φ and Δ in seconds; we have then 24+: 2á:: v : Δ, whence Δ = ⅛0v, and consequently

, . /tan. I ,a .A

ø = τ<A ν ——— — tan. D cot. P .

r 1 ∖sin. P j

Now, to convert φ into seconds of time, it is only necessary to divide the number to which it is equal by 15; therefore, expressed in time, φ becomes 1 . /tan. I i υλ . τ A ø = - τ ⅛Jι√ -—ŋ — tan. D cot. P ). r 100 ∖sm. P J

For the angle P we may substitute ö (half the time elapsed between the two observations) converted into degrees, and suppose D to be the value of the declination at the instant of the meridional passage; the value of φ being so small as not to be affected by these substitutions. Therefore, ultimately,

, , . /tan. I _ A

⅜P = s⅜(A∖-; — tan ∙ D cot * 6 )> which, subtracted from û, gives the true time of the meridional passage.

Problem XIII.— To compute the angle of the vertical, or the difference between the apparent and geocentric latitude arising from the spheroidal figure of the earth (fig. 127).

Let AMP be a quadrant of the elliptic meridian, C being the centre, and P the pole. The straight line VME, perpendicular to the ecliptic in Μ, determines the apparent zenith V of the place Μ, while ZMC drawn through C determines the true zenith Z. Hence the angle ADM is the apparent latitude of Μ, and ACM its geocentric latitude; and the angle ZMV between the true and apparent zenith is the angle of the vertical.

Let x and y be the rectangular co-ordinates of the point Μ, the origin being at C, and make AC = m, CP = n, ADM = I, and ZMV = v.

The equation of the ellipse gives

τ∕ 2 = ⅝ 2 x 2 , nd whence 1 J d d = ~rn 2 χdχi therefore, y rd dx x~ nd ’ dy'

But = tan. ACM, and — ⅛ = tan. ADM = tan. I; χ dy therefore tan. ACM = —5 tan. I. Now ZMV = DMC m 2

= ADM — ACM, therefore tan. ZMV = tan. (ADM λ ∩ 1 ∙ 1 -x tan. ADM — tan. ACM

— ACM) = 1 + tan adm tan acm ; that is,

∖

1 mdj an ' (nd — rd) tan. I tan ∙ ν rd 9 and + η 2 tan. 2 1

1 -ļ 5 tan? I m~ (and — rd) sin. I cos. I (nd — id) sin. I cos. I m 2 cos. 2 I + rd sin. 2 I nd — ∙(ηd — rd) sin∙ 2 Z"

Now, let m — n = 1, or η = m — 1; then, by substituting, (2 m — 1) sin. I cos I tan v = - - 5 -τ~- 1λ ∙ 0-7> and — (2m — 1) sin/ t or, neglecting the terms multiplied by and recollect ing that sin. I cos. I = į sin. 2 I, sin. 2 I ( 1 ∖ 1

tan. v — — 2 ) x ,

2 m ∖ mJ, 2 . ,

1 sm. 2 I in whence sin. 2 I ( 1 ∖ ∕ , 2 . ∖

tan. v — — Į 2 (Id sin. 2 1 ) ;

2 m ∖ mJ ∖ m J that is, sin. 2 I ∕ 1 , 2 . „ A

tan. v = I 1 — χ 1 sm.- 1 1

m ∖ 2 m m J sin. 2At 1 ,1 1 2 7 i 1 ∙ 2 7 ∖

= ( 1 — I cos. 2 1∙ ∖ sin. 2 1 ). an∖ 2 m m m m ∕ but cos. 2 I — sin. 2 I = cos. 2 1; therefore, sin. 2 I ∕ π , 1 — 2 cos. 2 Z∖

tan. v = ( 1 d x I; m ∖ 2 m J and 2 sin. 2 I cos. 2 1 = 4 sin. 4 I, therefore, sin. 2 Z 1 sin. 2Z—sin. 4 Z

tan. v = 1 x—., .

m 2 rn-

The last term of this expression cannot in any case amount to 2", so that the angle of the vertical is very nearly proportional to the sine of twice the latitude.

Problem XIV.— To compute the parallax of the moon, or a planet, in altitude.

Let P = the horizontal parallax, p = the parallax of altitude, Z = the apparent zenith distance.

It was shown in Chap. I. Sect. 2 of Theoretical Astronomy that sin. p — sin. P sin. Z; now let Z' = the true zenith distance, then Z = Z' + p, and consequently sin.p = sin. P sin. (Z ’ +∕>)

= sin. P (sin. Z' cos. p + cos. Z ʹ sin. p), therefore tan. p = sin. P (sin. Z' + cos. Z' tan. p) tan. p (1 — sin. P cos. Z') = sin. P sin Z ʹ, whence sin. P sin. Z' tan. » = x .—— x-.

1 — sin. P cos. Z'

But by a well-known series, p = tan. p — ⅛∙ tan. 3 p + ⅛ tan. 5 p — , &c. Substituting therefore the above expression for tan. p, and reducing the powers to series, we find sin. P sin. Z' , sin. 2 P sin. 2Z' , sin. 3 P sin. 3Z'

1 sin. 1" sin. 2" sin. 3"
[7:4:68]

The first two terms of this series are in every case sufficient for the computation ofjo.

Problem XV.— To compute the parallax in right ascen-sioìl.

In fig. 128 let P be the'pole, Z the zenith, A the true place of the moon or a planet, depressed on the vertical circle through the effects of parallax to B. Having joined PA and PB by arcs of great circles, the corresponding variation in right ascension will be represented by the angle APB. Make

P = the horizontal parallax; p = AB, the parallax of altitude;

∏ = APB, the parallax in right ascension; l = 90° — ZP, latitude of place;

Δ = PA, the polar distance;

N = ZA, the true zenith distance;

N++ = ZB, the apparent zenith distance;

H = ZPA the hour angle.

In the parallactic triangle APB we have sin. PA: sin. AB:: sin. ABP: sin. APB; that is, sin. A: sin. p :: sin. ABP: sin. ∏, but sin. p — sin. P sin. (N + p). See Theoretical Astronomy, Chap. I. Sect. 2.

Therefore,

. sin. P sin. (N + ø) sin. ABP sin. ∏ = ~ —

sin. ∆

Now, in the triangle ZBP, we have sin. ZB: sin. ZP:: sin. ZPB: sin. ZBP (ABP); that is, sin. (N + p) : cos. l :: sin. (H + ∏): sin. ABP. Therefore, sin ABP - -° s ∙ ζ sin, (∏ + π ) sin. ABP - sin . (N + , whence

. ττ sin. P cos. l . zττ , . sin. Π =; sin. (H + ∏).

sin. A '

ι , t , sin. P cos. l

Make : — A; then sin. A

sin. Π = A sin. (H + ∏) = A (sin. H cos. ∏ + cos. H sin. ∏); whence tan. ∏ = A (sin. H + cos. H. tan. ∏), and tan. ∏ (1 — A cos. H) — A sin. H; that is,

„ A sin. H tan. ∏ = - - σ ,

1 — A cos. H

whence we have the following series,

„ . sin. H ,„sin. 2II ,

∏ = A -÷— + A g -. + , &c.

sin. 1" sin. 2"

Problem XVI.— To compute the parallax in declination.

Make ¶ = PB — PA (fig. 129) = the parallax in declination.

From the triangle ZPA we get cos PZA — ċos ' a P — cos ^ cos. AZ sin. PZ sin. AZ ’

and from ZPB

1 ,< zι , cos. BP — cos. PZ cos. BZ

COS. PZB =; ∙

sin. PZ sin. BZ.

therefore cos. AP — cos. PZ cos . AZ __ cos. BP — cos. PZ cos.BZ sin. AZ sin. BZ ’ that is (retaining the notation of last problem), cos. A — sin. I cos. N _ cos. (∆ + w) — sin. l cos. (N +p) sin. N ~ sin. (N + p) whence

. , x cos. A sin. (N + ŵ)

COS. (A + V) = ; èf —

× ʹ sm. N

sin, l [cos. N sin. (N +p) — sin. N cos. (N +p)] sin. N

cos. A sin. (N + 7>) sin, l sin, p sin. N sin. N ’

cos. A sin. (N + p) sin, l sin. P sin. (N’÷ p)

~~ sin. N sin. N.

therefore z , . sin. (N+jp) r ∙ ł ∙ m ∕i∖ cos. (∆+<r) = — sįn [cos. A·—sin. Zsin. P]....(1)

Now, from the property of spherical triangles, Sin. ZA: sin. PA:: sin. ZPA: sin. AZP, therefore

. rr - sin. PA sin. ZPA sin. AZP = τ—≈rr .

sin. ZA

For the same reason τ , r , τ -- sin. PB sin. ZPB sin.BZP = r— vp ; sin. ZB

therefore sin. PA sin. ZPA __ sin. PB sin. ZPB sin. ZA sin. ZB

that is, sin. A sin. H _ sin, (∆ + ¾r) sin. (H+∏∖

sin. N ~ sin. (N+^) whence sin. (N+∕j) _ sin, (∆ + w) sin. (H+ ∏) sin. N ~ sin. A sin. H

By substituting this in equation (1) there results cos. (∆+<Γ) = ⅛+±⅜ i °∙ (H + π > x sin. A sin. H

[cos. A — sin. l sin. P] whence z , . sin.(H + ∏) f sin. Z sin. Pì

cot. (∆+σ) =— ? -I 7 i cot. a ,—y.

κ z sin. II I sin. A J

, sin. l sin. P 1

Make : = cot. α, then sin. A

. sin. (H + ∏) z .

cot. (A + σ) = r— ýy — ' (cot. A — cot. a), v z sin. H

that is, z 1 . sin. (H + ∏) sin. (α—∆) cot.(∆÷w) = ∖——+ >- a - 7 , x sin. H sin. α sin. A

from which, as A is known, cot. (∆ + w) maybe computed, and thence w. A more convenient formula, however, may be obtained by proceeding as follows. (See Delambre, Abrégé d Astronomie, p. 154).

From the equation z , . sin. (H + ∏) 7 sin. l sin. P∖

cot. (∆+%) = ÷~FT— ( cot. A - — )

v z sin. H ∖ sin. A ∕

we have sin. H cot. (∆+π) , sin. l sin. P

cot. A = : — r -A 4:; sin. (H + ∏) sin. A whence

. , x sin. H cot. (∆+%) z t . cot. A — cot. (∆ + cr)= -.— 7φτ ~—r— , -—cot∙(∆+σ z sin. (H + ∏) x l sin. l sin. P , z . .

4;; but cot. A — cot. (∆ + <5-) =

sin. A ∖ ∙ ʹ [7:4:69]sin. (∆+ ff) cos. Δ — cos· (∆ + ff) sin» Δ sin« ff

• sin. (∆ + ff) sin. Δ ~ sin. (∆ + ff) sin. Δ ’

therefore sin. ff sin. I sin. P

sin. (∆+ ff) sin. Δ sin. Δ.

— cot. (∆+ff)[sin. (H + ∏) — sin. H] sin. (H + ∏)

Now, by the trigonometrical formulae, the difference of the sines of two arcs is equal to twice the sine of half their difference multiplied by the cosine of half their sum, therefore sin. (H + ∏) — sin. H = 2 sin. ∙∣∙ ∏ cos. (H+⅜Π), whence sin. ff . sin. l sin. P

sin. (∆+ff) sin. Δ~^ sin. Δ

2 sin. ⅜∏cos. (H + ⅜ ∏) cot. (∆ + ff) sin. (H + ∏)

and sin ¶ — sin. I sin. P sin. (∆ +ff)

2sin. ∙Lnsin.∆cos, (H + ⅛∏) cos, (∆ + ff) sin. (H + ∏) ’

λ . 1 sin. ∏

or, since 2 sin. ⅛∏ = ;—, ʹ cos. ⅜ ∏

sin. ff = sin. ĩ sin. P sin. (∆ + ff)

sin. ∏ sin. Δ cos. (H+∣ ∏) cos. (∆ + π)

cos. ļ- ∏ sin. (H + ∏)

But it was shown in the last problem that sin. P cos. I . zττ , . sin. ∏ =; sin. ( H + ∏), sin Δ. 7 whence sin.∏ sin. Δ . ŋ ,

“TVļ—: r = sin. P. cos. l; sin. (H + ∏) ’ ‘ ’

consequently, sin. ff = sin. I sin. P sin. (∆ + ff)

_ sin. P cos. I cos. (H + ∣ ∏) cos, (∆ + ff) cos. ⅜ ∏

at i * cos∙(H + ⅜∏)cot∙Z 1 , , .

Make tan. a: = ¶-—; then, by substituting, cos. ∣ ∏ ’ j °’

sin. ff = sin.Zsin. P [sin∙(∆+ff)—tan.a?cos. (∆ + w)], or sin. ff =

sin. Z sin. P . , z ι x .

— [sin. (Δ + ff) cos. x — cos. (∆ + ff) sin. aŋ, sin. Z sin. P . z , .

sin. ff = sin. (∆ + <r — x), cos. x x 7 sin. ff =

sin.Zsin.P, . , x 1 , . .

—; [sin. (Δ — x) cos. ff + cos. (Δ — x) sin. ff];

COS. łZz whence, tan. ff =

sin.Zsin.P, . , . , ,

[sin. (Δ — x) + cos. (Δ— x) tan. ff];

COS. X therefore (making ⅛ n, s * n, . β) cos. X 7

B sin. (∆ — x) tan ∙* = r≡B-^√∑τ=τy con ≡ e <ι u →

ff — B sin ∙ + β 2 sin ∙ 2 ( δ ~ a 0 sin. 1" sin. 2"

+ B 3 - s -- - 3 ∕ δ 7^ - +,&c. sin. 3"

Problem XVII.— To compute the altitude and longitude of the nonagesimal. t The .nonagesimal is the point of the ecliptic where that circle intersects the vertical plane passing through its pole. It is consequently the highest point of the ecliptic above the horizon, or 90° from the horizon measured on the ecliptic.

Let HH' (fig. 129) be the horizon, EO the ecliptic, EQ the equator, Z the zenith, P the pole of the equator, and P' the pole of the ecliptic. The great circle passing through P' and P intersects EO and EQ at right angles in C and D, whence EC = ED — 90° = MQ, and therefore EM = DQ. In like manner PD = 90° — P'C, consequently P'P = CD the obliquity of the ecliptic.

The great circle PZNI which passes through the pole of the ecliptic and the zenith is a circle of latitude and also a vertical circle; hence the angles at N and I are right angles, and 0 is the pole of PZNI; consequently 01 = ON = 90°. Now since ON = 90°, the point N is the nonagesimal, and its altitude IN = 90° — ZN = ZP' = the complement of the altitude of the pole of the ecliptic, or the co-latitude of the zenith. In like manner its longitude EN = 90° — NC = 90° — ZP'P. Now in order to compute ZP' and the angle ZP'P, we have given, in the triangle ZPP', the side ZP = the co-latitude, PP' = the obliquity, and also the angle ZPP', for ZPP' = 180° — ZPD = 180° — MD = 180° — (ED — EM) = 180° — 90° + EXI = 90° + EM, and EM (which is the right ascension of the zenith, or as it is technically called, the right ascension of the mid-heaven) is given, being equal to the sidereal time of observation converted into degrees.

Let R = EM the right ascension of the zenith, λ = PZ theco-latitudeofthe place (reduced by problem XIII.), ω= PP' the obliquity of the ecliptic, K = P'Z the co-latitude of the zenith or altitude of the nonagesimal,N = 90° — ZP'P the longitude of the nonagesimal.

By the trigonometrical formulae, cos. K = cos. λ cos. ω + sin. λ sin. ω cos. (90° + R) = cos. λ cos. ω + sin. λ sin. ω sin. R.

Assume tan. φ = tan. asin. R, then, by substituting and reducing, we obtain

τ z ∙ cθS∙ X z i ∖

cos. K = cos. ( ω + ø).

cos. <ρ v 7

To find the longitude L, we have , τ cot. λ sin. ω , r , r , , τ , x tan. N = -— 77 - — — cot. (90 + R) cos. ω

sin. (90° + R) ∖ τ ∕ cot. λ sin. ω v ,

= — + tan. R cos. ω, cos. R

τ> x x , tan ∙ ĩ> 1 * % sin. E

But tan. λ = ——+ whence cot. λ = , sin. R tan. p whence, by substituting^

(sin. ω 1 ∖

+ cos. ω , tan. p J and on reducing, tan. R . z . x tan. N = - 1 sin. (« + 0), sin. p x Problem XVIII.— To compute the parallax in longitude and latitude.

Let A (fig. 129) be the place of the star situated on the vertical ZAR; Eα is its longitude, and P'A the complement of its latitude, or its distance from the pole of the ecliptic. Suppose that, through the effects of parallax, it is depressed from A to B; its longitude then becomes EZ», and its distance from the pole of the ecliptic P'B, so that ab is the variation in longitude, and P ’ B — PA the variation in latitude, which it is required to compute. Now it is evident that these quantities will be given in terms of the different parts of the triangle AP'B, exactly in the same manner as the parallax in right ascension and declination has been found from the triangle APB [7:4:70] (fig. 128). The angle ZPA, which was before denoted by H, now becomes ZP'A = Nα = Eα — EN = longitude of the star — longitude of the nonagesimal, and ZPB becomes ZP'B = NZ» = Nα + ab. Retaining, therefore, the notation employed in the three last problems, and making L = Eα the longitude of the star, ∆' = PA its distance from the pole of the ecliptic, k — ZP ’ = 90° — K, ∏' = ab its parallax in longitude, »' = P'B — P'A its parallax in latitude, we shall have ∏' and w' from the same formulae as ∏ and r κ in Problems XV. and XVI., by changing l into A, H into L — N, and Δ into Δ'. Hence (Problem XV.)

sin. P cos. h . z , , τ ι . x sin. ∏ = : -—, sm. (L — N + ∏'), sin. Δ'

,, . z , sin. P cos. h 1 , and by putting C = -.—, we deduce ; r o sin. Δ'

C sin. (L — N) 1 tan. ∏' = ~ 1 -+ 7f —~ t - and

1 —Ccos.(L — N)

. sin. (L — N) sin. 2 (L — N)

∏ = sin. C ⅛—=—- + C 2 t δ -q7 +, &c.

sm. 1" sin. 2"

In like manner the formula for the parallax in declination given in Problem XVI., viz.

, . sin. (H + ∏)l’ i a sin. l sin. P“ļ

v 1 7 sin. H L sin ∙ δ I becomes ’, , l ,. sin.(L—N+∏') l’ sin.Asin.P~∣

COt.(∆'+√)= ÷— 7 ⅞r- cot. Δ' 7 —-; v 7 sin.(L — N) ļ_ sin. ∆' J

from which, by proceeding in the same manner as in Problem XVI., and making

, cos. (L — N + ⅛ ∏') cot. h tan. a 7 = - r -—~

cos. į ∏' we deduce tan. w sin. h sin. P Γ^ . , , ,. z , ,. ”1

= —ļ sin. (∆ — A ) + cos. (∆' — a 7 ) tan. +

COS· ∙v I

. . . sin. h sin. P

whence, on making -= ;— = D, o cos. so

, D sin. (∆ — a/) tan. π' = —— Ą_,

1—Dcos.(∆' — a/) '

consequently < = d sin ∙O'-O + d ≡!∏l+¾ξA +; &c . sm. 1" sin. 2"

Problem XIX.— Given the apparent altitudes of the moon and the sun, or a star, and the apparent distance between them, to find the true distance.

This is a problem of great importance in Practical Astronomy, because the observed distance between the moon and the sun or a star is the surest means the navigator has to determine his longitude.

To obtain the apparent distance and altitudes, it is convenient to have three observers: one, the most expert, takes the apparent distance of the limb of the moon from that of the sun, or from the star; another observes the moon’s altitude at the moment of the observation of the distance; and a third takes the star’s altitude. For greater accuracy, these simultaneous operations ought to be repeated several times, and a fourth assistant may, by a good watch, note the intervals of time between them. A mean of the whole will then be obtained, and the corresponding time by the watch, by which the true time at that place will oe nearly known. If there be only one observer, he must take the altitudes immediately before and after the distance, and endeavour to allow for the change of altitude during the time between the observations.

The observed distance between the limbs of the moon and sun must be increased by their semidiameters to obtain the distance of their centres. If the distance from ä star is taken, because its diameter is insensible, the dis- j tance is to be increased by the moon’s semidiameter only.

The observed altitudes thus found are affected by parallax and refraction. The moon’s horizontal parallax is given in the Nautical Almanac : from this the parallax at the time of observation may be found by Prob. XIV., but the navigator avails himself of aid from The Requisite Tables, which shorten the process of calculation. The sun’s parallax is almost insensible, but it may be taken into account; that of a star is accounted nothing. The altitudes are diminished by the parallaxes, but they are increased by refraction. The sun’s altitude is more increased by the latter than diminished by the former; but the reverse happens with the moon. At sea the altitudes must be also corrected for the dip of the horizon (that is, for the height of the observer above the surface of the sea), and the refraction for the height of the thermometer.

Supposing all this done, let S and Μ be the true places of the sun or star and moon (fig. 130), Z the zenith; because parallax and refraction take place only in vertical circles, the apparent place of the sun, s, will be in the vertical ZS above S, and that of the moon, m, in ZM below Μ. The apparent distance will be sm.

Let A — 90° — ZM be the moon’s true altitude; a — 90° — ZS the star’s true altitude; H = 90° — Zm the moon’s apparent altitude; h = 90° — r Ls the star’s apparent altitude; D = SM the true, and d — sm the apparent distance.

By spherical trigonometry, in the triangles SZM, s r Lm, r cos. D — sin. A sin. a

⅛ r ∕vτĴ cos ∙ A cos ∙ α os. SZxl-< _ cos . g — s ; n . jļ s ; n . j i v ' - cos. H cos. h

From these equal values of cqs. SZM we find tλ z > ∙ ττ ∙ 7 ∖COS. A cos. a . . . . cos D = (cos.α—sm. H sm. h) ŋ 7 + sm. A sm. a x cos. H cos. A

l’ -7! Z Ţτ ł 7 ∖ -rτr 7 -ι COS. A. COS. a = l’cos. d + cos. (H + h) — cos. H cos. h Ί w , x 7 j cos. H cos. h

+ sin. A sin. a

= 2 cos. ⅜ (H + h + d) cos.⅜ (H + h — ^ cos ∙A cos ∙g z cos ∙ H cos ∙ h

— (cos. A cos. a — sin. A sin. a).

But the last term = cos. (A + a); subtract now both sides from 1; then, observing that by the calculus of sines (see Algebra)

1 — cos. D = 2 sin. 2 ļ D,

1 + cos. (A + a) = 2 cos. 2 ⅜ (A+α), we have, after dividing by 2, and putting p _ cos. A cos, a cos.H cos. h ’ sin. 2 ⅜ D = cos. 2 ⅜ (A + a)

— cos. I (H + ä + d) cos. ⅜ (H+ä — d) × F =cos∙⅛(A+< 1 )l’ 1- ⅜(H+∕+)co 1 ⅜ (H+⅞→ Τ⅜ Ί

L cos. 2 ⅛ (A + α) J

Let ô be such an angle that s ; n s'ğ— c0s ∙⅜ (H+A+<Z) cos.⅜ (H+7⅜ — c?)cos. A cos, a cos. 2 ⅜ (A + űt) cos. H cos. h and in this expression the value of F is put instead of it, then we have sin. 2 ⅜ D = cos. 2 ∣∙ (A + α) cos. 2 Û ; and sin. ⅜ D = cos. ⅜ (A + ß) cos. û.

The value of cos. D in the first formula may serve to determine D, but not conveniently; by the various steps of the analytic process it is transformed into another, viz.
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the last, whicn gives sin. ∣ D, and consequently D readily by logarithmic calculation.

The final result is the rule found by Borda: as a practical rule it is very convenient, because no attention is necessary to the signs of the trigonometrical quantities. On this account it is well adapted to seamen.

Problem XX.— To find the longitude of a ship at sea by an observation of the distance of the moon from the sun or a star.

From the observed distance, the true distance as it would appear if it could be seen from the earth’s centre, may be found by the last problem: Now by the Nautical Almanacthis is given for every third hour, Greenwich time, therefore, by an easy calculation, the time at which the observation was made, as it would be given by a watch showing Greenwich time, may be obtained. But the time of observation as reckoned at the ship may be found from the ship’s latitude, the moon’s or star’s zenith distances (found by the observation), and their polar distances as given by the Almanac. Therefore the difference between the time of the observation as estimated at the ship and the corresponding Greenwich time becomes known: This difference is the ships longitude. Example. At sea, June 5,.1793, about an hour and a half after noon, in 10° 46' 50'' south latitude, and 149° longitude, by account, by means of a set of lunar observations made at a height of about 20 feet above the surface of the sea, t was found that

Distance of nearest limbs of © a ∏d D ∙∙∙ ft θ c, 26' 46"

Altitude of lowest limb of ø 48 16 10

Altitude of upper limb of D 27 53 30.

Hence the longitude of the ship is required.

Eeduction of the apparent to the true altitude.

Dist. of nearest limbs of ø and '↑> 83° 26' 46"

Semidiam. of 0 ļ f n Almanac í θ 15 46 of J J Į 0 14 54

Augmentation of the latter, prop, to altitude... 0 0 7

Apparent distance (rf) of centres 83° 57' 33"

Altitude of sun’s lower limb 48° 16' 10"

Subtract for dip of horizon 0 4 24

48° 11' 46"

Semidiameteŕ 0 15 46

Apparent altitude of sun’s centre (ä) 48° 27' 32"

Refr. —par. — correct, for therm — 0 0 43

True altitude of sun’s centre (a) 48° 26' 49"

Altitude of moon’s upper limb 27° 53' 30'

Correct for dip — 0 4 24

. . 27 c 49' 6"

Semidiameter 0 15 1

Apparent altitude of moon’s centre (H) 27° 34/ 5"

Par. — refr. + corr. for therm 0 46 43

True alt. of moon’s centre (A) 28° 20' 48"

deduction of the apparent to the true distance by Prob. XIX.

d 83° 57'33"

h 48 27 32 ar. com. cos. ∙1783835

H 27 34 5 ar. com. cos. ∙0523390

sum 159° 59' 10"

⅜sum 79 59 35 cos. 9∙2399686

d— ⅜ sum 3 57 58 cos. 9∙9989587

a 18 26 49 cos. 9∙8217187

A 28 20 48. cos. 9∙9445275

A + a = 76° 47' 37". 39∙2358960

⅜(A + α) = 38 23 48 2 log. cos. 19∙7883324

2)19∙4475636 7 ů = 31 57 33 cos. û 9∙723 l ∕818

sin. ^9⅛5875 ⅜(A + n)= 38 23 48 cos.9∙8941662

1 D = 41 40 27; sin. 9∙8227537

D = 83 20 55 nearly.

Computation of time at Greenwich.

By Nautical Almanac for 1793 tλ . p _ fat 15 hours 83° 6' 1"

Dist. D from Θ ∙∙1 at lb 84 , 28 2 θ

Increase of dist. in 3 hours = 1° 22' 25"

D = 83° 20 ’ 55"

Dist. ļ) from 0 at 15 hours 83 6 1

0° 14' 54"

Hence it appears that in 3 hours the distance of the moon from the sun was increased by 1° 22' 25" = 4595", and that between the time of making the observation, and the 15th hour, Greenwich time, the increase of distance was 0° 14' 51" = 891." Now for small intervals the distances will increase nearly as the times, therefore we have 4945": 894":: 3h.: 0h. ∙ 5425 = 32m. 33s.

Thus it appears that the observation must have been made 32 minutes 33 seconds after the 15th hour, Greenwich time.

We have yet to find the correct time of the observation as estimated at the ship. To determine this, we know the sun’s altitude 48° 26' 49", and the sun’s declination, which is given in the Almanac for Greenwich, noon; and from this the declination at the time of the observation is found by making allowance for the difference of longitude and the hour from noon at the ship, both known nearly. The declination thus found is 23° 22' 48" north. The ship’s latitude, 10° 16' 40", is also known. Hence to find the hour angle by Problem IV. we have λ the colat. = 79° 43' 20",

Δ the pol. dist. = 113 22 48, z the zenith dist. = 41 33 11.

To find P the hour angle, the computation (see formula of Prob. IV.) may stand thus:—< sin. ļ- (z + λ — ∆) 3° 56' 51" 8∙8378559

sin. L (z — λ + Δ) 37 36 19 9∙7854851

' sin. ⅛ (λ + Δ + z) 117 19 39 ar. comp. 0 ’ 0513929

sin. ļ- (λ + ∆ — z) 75 46 28 ar. comp. 0∙0135256

2)18∙6882595 tan. (⅜ P = 12° 27' 17fc") 9∙3441297

P = 24° 54' 35" = 1 h 39® 38≡∙3

time at Greenwich = 15 32 33 ∙0

Long, from Greenwich westward 13 h 52∞ 54*∙7

Therefore, longitude east of Greenwich is 10 h 7 m 4 e *3 We have put down the result to seconds, as it comes out by the calculation; but such accuracy is not attainable in nautical practice. Delambre says, “even with the best instruments, the most skilful navigators find anomalies for which they cannot account. The lunar tables are in a state of continual improvement, yet we cannot be sure that there is not an error of 20" in a distance, which will produce an error of 40" of time. To this possible error of theory must be joined that of the observation, which may be still greater. The error of the time may therefore amount to 80", which is equivalent to 20', or ⅜d of a degree. This should be the maximum of the error; and, in general, this degree of accuracy is sufficient. Indeed a greater accuracy than ⅜d or ith of a degree may sometimes be obtained, but still it is uncertain.
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CHAP. IL

CALCULATION OF NEW AND FULL MOON, AND ECLIPSES.

Sect. I.— Tables for Computing New and Full Moon, and the Elements for Eclipses.
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Sect. H.— Application of the Tables, and Projection of Eclipses. Problem I.— To calculate the time of true new or full moon for any period within the limits of the nineteenth century. Precept 1. Write out the time of mean new moon in January for the proposed year from Table I., together with the mean anomalies of the moon and sun, and the moon’s mean distance from her ascending node, applying to each of these quantities the secular equation found by its side by addition or subtraction, according as it has the sign + or —. If you want the time of full moon in January, add the half-lunation at the foot of Table IV., with its anomalies, &c. to the former numbers if the new moon falls before the 15th of January; but if it falls after, subtract the half-lunation, with the anomalies, &c. belonging to it, from the former numbers, and write down the respective sums or remainders.

2. In these additions or subtractions, observe that 60 seconds make a minute, 60 minutes make a degree, 30 degrees make a sign, and 12 signs make a circle. When you exceed 12 signs in addition, reject 12, and set down the remainder. When the number of signs to be subtracted is greater than the number you subtract from, add 12 signs to the lesser number, and then you will have a remainder to set down. In the tables, signs are marked S, degrees D, minutes Μ, and seconds S.

3. When the required new or full moon is in any given month after January, write out from Table IV. such one of the mean lunations, with the anomalies, &c. as, added to the time of mean new or full moon in January, will make the mean new or full moon to fall within the given month, setting them below the number taken out for January.

4. Add all these together, and in leap-years (which in Table I. have the letter B annexed to them) subtract one day from the time of mean new or full moon when it happens after 28th February. You will then have the time of the required mean new or full moon, with the mean anomalies, and the moon’s mean distance from the ascending node, which are the arguments for finding the proper equations.

5. With the signs and degrees of the moon’s mean anomaly enter Table V., and therewith take out the first equation for reducing the mean syzigy to the true; taking care to make proportions in the table for the odd minutes and seconds of anomaly, as the table gives the equation only to whole degrees.

Observe in this and every other case of finding equations, that, if the signs are at the head of the table, their degrees are at the left hand, and are reckoned downwards; the equation being in the body of the table under or over Jhe signs in a collateral line with the degrees. The signs + and — at the head or foot of the tables where the signs are found, show whether the equation is to be added to the time of mean new or full moon, or to be subtracted from it.

6. With the signs and degrees of the sun’s mean anomaly enter Table VI. and take out the second equation for reducing the time of mean to that of new or full moon, with a proportional part of its secular variation in the column adjoining, corresponding to the number of years elapsed since 1801, the whole variation being adapted for a period of 100 years.

7. Add together the mean anomalies of the sun and moon, and with the sum enter Table VII. and take out the third equation. For this and the following equations it will be sufficient to compute the arguments to minutes, neglecting the seconds.

8. Subtract the sun’s mean anomaly from the moon’s mean anomaly, and with the remainder enter Table VIII. and take out the fourth equation.

9. Subtract the moon’s mean anomaly from twice the moon’s distance from the ascending node, and with the remainder enter Table IX. and take out the fifth equation.

10. The moon’s mean distance from the ascending node is the argument of Table X., with which take out the sixth equation.

11. To twice the moon’s mean anomaly add the sun’s mean anomaly, and with the sum enter Table XI. and take out the seventh equation.

12. From twice the moon’s mean anomaly subtract the sun’s mean anomaly, and with the remainder enter Table XII. and take out the eighth equation.

13. These are all the equations for reducing the time of mean new moon to the time of true new moon; but for full moon other two equations are required, the argument for equation ninth being the moon’s mean anomaly, which equation is exhibited in Table XIII.; and the argument of the tenth equation being the sun’s mean anomaly, the equation being exhibited in Table XIV.

14. Add together the equations which have the sign of addition, and also those which have the sign of subtraction, and subtract the lesser sum from the greater, giving to the remainder the sign of the greater; and add or subtract the remainder, according as its sign denotes, to or from the time of mean new or full moon, and you have the time of true new or full moon required.

These tables are adapted to the meridian of Greenwich observatory; and for any other place, its longitude in time is to be added to or subtracted from the time given by the tables, according as it is to the east or west of Greenwich, and the time as reckoned at the given place is obtained. The tables begin the day at noon, and reckon forward from thence to the noon following. Thus January the 31st, at 22 hours 30 minutes 25 seconds of tabular time, is February 1st (in common reckoning), at 30 minutes 25 seconds after 10 o’clock in the morning. It is to be further observed, that the time obtained from the tables is mean time, or that shown by a well-regulated clock or watch. But to make it agree with solar or apparent time, or that given by a sun-dial, which is necessary in the computation of solar eclipses, you must apply the equation of time contained in Table XXVIII. as afterwards directed.

The method of calculating the time of any new or full moon, without the limits of the nineteenth century, will be shown farther on; and a few examples compared with the precepts will make the whole work plain.
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Problem IL— To calculate the time of new and full moon in a given year and month of any particular century between the Christian era and the nineteenth century. Note. — Prior to the sixteenth century the times are supposed to be reckoned according to the Julian calendar, or old style. Between it and the nineteenth century they may be reckoned according to either the Julian or Gregorian calendar, or old or new style.

Precept 1. Find a year of the same number in the nineteenth century, with that of the year in the century proposed, and take out the time of mean new moon in January for that year, with the mean anomalies, and the moon’s mean distance from the node at that time, as already taught, neglecting the secular equations contained in Table I.

Precept 2. Take from Table II. as many complete centuries of years having the sign — prefixed, and titled either old style or new style, according to the given date, as, when subtracted from the above said year in the nineteeth century, will answer to the given year, and take out the time of mean new moon and its anomalies, &c. belonging to the said centuries, and add them to those for the year in the nineteenth century, and the sums, after applying to them the secular equations taken from Table III., making proportions for the odd years, will be the times and anomalies. &c. of mean new moon in January or February, according as the time is less or more than 31 days, in the given yeaι of the century proposed. Then work in all respects foi the time of true new or full moon, as shown in the above precepts and examples.
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Problem III.— To calculate the true time of new or full moon in any given year and month before the Christian era. Precept 1. Find a year in the nineteenth century which, being added to the given number of years before Christ diminished by one, shall make a number of complete centuries.

2. Find this number of centuries in Table II. and add the time and anomalies belonging to it to those of the above-íound year of the nineteenth century, applying the secular equations for the given year in Table HL, and the sums will denote the time and anomalies, &c. of mean new moon in January or February of the given year before Christ. Then for the true time of new or full moon proceed as above taught, for any year between the Christian era and the nineteenth century, observing that the given year before Christ is or is not leap-year, according as the above-mentioned year in the nineteenth century is leap-year or not.
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Problem IV.— To calculate the time of true new or full moon, according to the Gregorian calendar or new style, in any given year or month of the 20∕7z or 21si century. Precept 1. Find a year of the same number in the nineteenth century with that of the year proposed, and take out the mean time and anomalies, &c. of new moon for that year in Table I., omitting the secular equations.

2. Take so many years from Table II. having the sign + prefixed, as, when added to the above-mentioned year in the nineteenth century, will answer to the given year in which the new or full moon is required; and take out the time of new moon, with its anomalies, for these complete centuries.

3. Add these together, and to the sum apply the secular equations for the given year found in Table HL, then work in all respects as above shown.
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Problem V.— To find the true anomalies of the sun and moon, and the moans true distance from the ascending node at the true time of new or full moon.

With the sum of equations already found for reducing the time of mean new or full moon to that of the true, enter Table XV., and take therefrom the sun’s mean motion from the moon’s perigee, the change of the sun’s mean anomaly, and the sun’s mean motion from the ascending node, and apply these quantities to the mean anomalies and mean distance from the node at the time of mean new or full moon by addition or subtraction, according as the sum of the equations has the sign + or —. Then with the sun’s corrected mean anomaly as argument, take from Table XVI. the equation of the sun’s centre, correcting it for the secular variation as directed at the bottom of the table, and add or subtract the same to or from the corrected mean anomalies and mean distance from the node, and there will be obtained the true anomalies of the sun and moon, and the moon’s true distance from the ascending node at the time of true new moon or full moon.
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Elements far the Projection of Solar Eclipses.

When at the time of true new moon the moon’s true distance from the ascending node is between 11 s ll 0 50' and 0 s 18° 10′, or 5 s ll 0 50' and 6 s 18° 10', there may at that time be an eclipse of the sun to some place on the earth’s surface; but if it is beyond those limits there can be no eclipse. At the new moon in May 1836, the moon’s distance from the node being within the limits, there may be an eclipse of the sun at that time.

It being ascertained that there may be an eclipse, the elements for predicting it are to be obtained as explained in the following example of the solar eclipse in May 1836, which is to be predicted, as it will happen at Edinburgh, in latitude 55° 57' N.

I. The moons latitude at the true time of new moon. The moon’s true distance from the ascending node at the time of true new moon is the argument of Table XVII.; the moon’s true anomaly minus her true distance from the ascending node is the argument of Table XVIII.; this argument plus the moon’s true anomaly is the argument of Table XIX.; the moon’s true anomaly plus her true distance from the ascending node is the argument of Table XX.; the sun’s true anomaly plus the moon’s true distance from the ascending node is the argument of Table XXL; and the sun’s true anomaly minus the moon’s true distance from the ascending node is the argument of Table XXII. With these arguments enter the respective tables, and take out the proper quantities, and add together such as have the sign +, and also such as have the sign —, and the difference of the two sums is the moon’s latitude required, north if the greater sum has the sign +, south if it has —.
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H. The inclination of the moon’s relative orbit to the ecliptic is found in Table XVII., with the moon’s true distance from the ascending node for argument. In the present instance the inclination is 5° 43', left, signifying that the axis of the moon’s orbit is to the left hand of the northern axis of the ecliptic.

III. The semidiameter of the earths dish is equal to the difference of the moon’s horizontal parallax corrected for the latitude of the given place, and the sun’s horizontal parallax, which may always be assumed equal to nine seconds. The moon’s equatorial horizontal parallax is found in Table XXIII., with her true anomaly for argument; and the correction to be subtracted therefrom is obtained in Table XXIV., with the latitude of the place and equatorial parallax as joint arguments. Thus, in the present instance, the moon’s equatorial horizontal parallax is 54' 26"

The correction to be subtracted 8"

Leaving for reduced horizontal parallax 54' 18"

which being again diminished by the sun’s horizontal parallax, there remains 54/ 9" for the semidiameter of the earth’s disk.

IV. The sun’s semidiameter is also obtained in Table XXII., with his mean anomaly as argument. We have, therefore, sun’s semi diameter 15' 51".

V. The moon’s semidiameter is likewise obtained in Table XXII., with her true anomaly as argument. Hence, moon’s semidiameter = 14' 50".

VI. The moons horary motion from the sun is equal to the difference of the sun’s and moon’s horary motions; both of which are found in Table XXII., with the sun’s mean anomaly and the moon’s true anomaly as arguments. Hence

Sun’s horary motion 2' 25*

Moon’s horary motion 30′ 9"

Moon’s horary motion from the sun 27' 44"

VII. The sun’s true longitude at the true time of new moon is equal to the longitude of the sun’s perigee added to his true anomaly. The longitude of the perigee is obtained by taking from Table XXV. the epoch of the first year of the century to which the given year belongs, and adding thereto the motion in Table XXVI. answering to the remaining number of years, months, and days, but subtracting therefrom a proportional part of the secular variation of the precession of the equinoxes found in Table XXV. opposite the first year of the given century, corresponding to the remaining number of years.

Example.

Longitude of Perigee.

S. D. M. S.

1801 9 9 30 5

30 years 30 57

5 do 5 9

May 20

15 days 2

Longitude of perigee... 9 10 6 33

Sun’s true anomaly 4 14 36 0

Sun’s true longitude 1 24 42 33

VIII. The sun’s declination is found in Table XXVII., with the sun’s true longitude for argument, and is to be corrected by the secular variation given in the same table, as directed at the bottom. The declination is north or south according as it bears the sign + or —, and, in the table, is given to tenths of a minute.

Example.

S. D. Μ. S. D. Μ.

Sun’s true longitude 1 24 42 33 Declination +18 58Ό

o 1 ∙ . — 0∙6×35 no

Secular variation ^θθ = — 0∙^

Sun’s declination, north +18 57 ’ 8

IX. The apparent time of new moon. The mean time the equation of time found in Table XXVIII., with the of new moon, already found and reduced to the place for sun’s true longitude for argument, and corrected for the which the prediction is to be made, is to be converted into secular variation given in the same table, as in the follow-apparent time, by applying thereto, according to the sign, ing example:

Example. H> m s>

Mean time of true new moon at Edinburgh on May 15, 1836 1 51 11

s. D. Μ. μ. s.

Sun’s true longitude 1 24 43 Equation of time, +3 591

c 1 . . — 10 s ×35 J- + 3 56

Secular variation θθ = —∙>l

Apparent time of true new moon 1 55 7

X. The reduced latitude of the given place is obtained by Edinburgh being in north latitude 55° 57'

subtracting from the true latitude the correction in the The correction is ....... 10′ ∙8

last column of Table XXIV. answering to the latitude. —

In this particular case, Reduced latitude 55° 46'∙2 [7:4:92]To Project an Eclipse of the Sun geometrically.

Take from a scale of any convenient length as many equal parts as the semidiameter of the eartlfs disk contains minutes of a degree, which for Edinburgh at the time of the eclipse in May 1836 is 54' 10", or 54⅜'. Then with this quantity as a radius describe the semicircle AHB upon the centre C (Plate XCIV. fig. 136); which semicircle shall represent the northern half of the earth’s enlightened disk as seen from the sun. If the given place were in south latitude, the southern half of the earth’s disk must be represented.

Upon the centre C raise the straight line CH perpendicular to the diameter ACB; so ACB shall be a part of the ecliptic, and CH its axis.

Being provided with a sector,^[5. Although a sector be a convenient instrument in these projections, yet it is not absolutely necessary. The intelligent student jvul be able to lay off an arc of any number .of degrees, also to make an angle of a given number of degrees, in various ways. ] open it to the radius CA in the line of chords; and taking from thence the chord of the sun’s greatest declination or the obliquity of the ecliptic for the given time (in the present instance 23° 28') in your compasses, set it off both ways from H to g, and to h in the periphery of the semi-disk; and draw the straight line g V h, in which the north pole of the disk will be always found.

When the sun’s longitude is between O s . and VI s . the north pole of the earth is enlightened by the sun; but whilst the sun is in the other six signs, the south pole is enlightened and the north pole is in the dark.

And when the sun’s longitude is between IX s . and III s . the northern half of the earth’s axis C XII. P lies to the right hand of the axis of the ecliptic as seen from the sun; and to the left hand while the sun is in the other six signs. It is evident that a contrary rule prevails with regard to the southern half of the earth’s axis.

Open the sector till the radius of the sines be equal to the length of V h , and take the sine of the difference of the sun’s longitude from III. s or IX. s , whichever it is nearest (in the present instance 35° 17'), in your compasses from the line of the sines, and set off that distance from V to P in the line gVh, because the northern half of the earth’s axis lies to the right hand of the axis of the ecliptic in this case; and draw the straight line C XII. P for the earth’s axis, of which P is the north pole. If the earth’s axis had lain to the left hand of the axis of the ecliptic, the distance VP would have been set off from V towards g.

To draw the parallel of latitude of the given place, Edinburgh, or the path of that place, on the earth’s enlightened disk, as seen from the sun from sunrise till sunset, take the following method :—

Find the sum and difference of the reduced latitude 55° 46' and the sun’s declination 18° 58', which are 74° 44/ and 36° 48'. Take these arcs from the line of sines on the sector, CA being radius, and set them off from C to the two points, each marked XII. in the line of the earth’s axis.

Bisect XII. — XII., and through the point K draw the line VI. K VI. perpendicular to the axis. Then, making C A or CB the radius of a line of sines on the sector, take the co-latitude of Edinburgh, 34° 14', from the sines in your compasses, and set it both ways from K to VI. and VI. These hours will be just in the edge of the disk at the equinoxes; but at no other time in the whole year.

With the extent K VI. taken into your compasses, set, one foot in K as a centre, and with the other foot describe the semicircle VI., 7, 8, 9, 10, &c., and divide it into 12 equal parts. Then from these points of division draw lines parallel to the earth’s axis C XII. P.

With the extent K XII. as a radius, describe the quadrantal arc XII. f and divide it into six equal parts, as XII. α, ab, be, cd, de, and er, and through the division points, a, b, c, d, e, draw the lines VII. e V., VIII. d 1V., IX. c HL, X. b IL, and XI. a I., all parallel to VI. K VI., and meeting the former lines in the points VII., VIII., IX., X., XL, V., IV., HL, IL, and I.; which points shall mark the several situations of Edinburgh on the earth’s disk, at these hours respectively as seen from the sun; and the elliptic curve VI., VII., VIII., &c. being drawn through these points, shall represent the parallel of latitude, or path of Edinburgh, as seen from the sun from six in the morning to six in the afternoon. On continuing the lines VII. p, VIII. o, &c. IV. u, V. x, &c. on the other side of VI. K VI., and setting off p V. equal to VII. p, ο IV. equal to VIII. o, &c., and continuing the elliptic curve through the points V. IV. &c., VII. V III. &c., the path of Edinburgh, as seen from the sun before six in the morning and after six in the afternoon, will be had; but it is needless to draw the curve farther than the points where it meets the periphery of the earth’s disk, which represent the times of sunrise and sunsetting.

N. B. If the sun’s declination had been south, the diurnal path of Edinburgh would have been on the upper side of the line VI. K VI. If the latitude of the given place were south, in which case the southern half of the earth’s disk would be represented, the diurnal path between six in the morning and six in the afternoon would be between the line VI. K VI. and the centre of the disk, when the sun’s declination was south, and the contrary when north. It is requisite to divide the horary spaces into quarters as in the figure, and if possible into minutes also.

In the present case the northern half of the axis of the moon’s relative orbit lies to the left hand of the axis of the ecliptic. Make CB the radius of a line of chords on the sector, and taking therefrom the chord of 5° 43', the inclination of the moon’s relative orbit to the ecliptic, set it off from H to Μ on the left hand of CH, the axis of the ecliptic; then draw CM for the axis of the moon’s orbit, and take the moon’s latitude 26' 2" from the scale CA in your compasses, and set it from C to y in the line CH, and through y draw the straight line NyS at right angles to the axis of the moon’s orbit CM for the path of the penumbra’s centre over the earth’s disk.

Take the moon’s horary motion from the sun, 27' 44", in your compasses, from the scale CA (every division of which is a minute of a degree), and with that extent make marks along the path of the penumbra’s centre, and divide each space from mark to mark into 60 equal parts or horary minutes by dots; and set the hours to every 60th minute in such a manner that the dot signifying the instant of new moon by the tables may fall into the point Z, half-way between the axis of the moon’s orbit and the axis of the ecliptic; and then the rest of the dots will show the points of the earth’s disk, where the penumbra’s centre is at the instants denoted by them in its transit over the earth.

Apply one side of a square to the line of the penumbra’s path, and move the square backwards and forwards, until the other side of it cuts the same hour and minute (as at 3 hours and 21∙ minutes) both in the path of Edinburgh and in the path of the penumbra’s centre; and the particular minute or instant so pointed out is the instant of the greatest obscuration of the sun, at the place for which the construction is made, namely, Edinburgh in the present example.
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Take the sun’s semidiameter, 15' 51", in your compasses, from the scale CA, and setting one foot on the path of Edinburgh, at the point answering to the instant of the greatest obscuration, namely, at 2∣∙ minutes past three, with the other foot describe the circle UY, which represents the sun’s disk as seen from Edinburgh at the greatest obscuration. Then take the moon’s semidiameter, 14' 50", in your compasses, from the same scale, and setting one foot on the path of the penumbra’s centre at the point 2⅜ minutes past three, describe the circle TX for the moon’s disk as seen from Edinburgh at the time when the eclipse is at the greatest, and the portion of the sun’s disk which is hid or cut off by the moon’s will show the quantity of the eclipse at that time; which quantity may be measured on a line equal to the sun’s diameter, and divided into 12 equal parts for digits. As the moon’s disk is entirely contained within the sun’s, the eclipse as seen from Edinburgh will be annular.

Lastly, take the sum of the semidiameters of the sun and moon, 30' 41", from the scale CA, in your compasses; and setting one foot in the line of the penumbra’s centre path, on the left hand from the axis of the ecliptic, direct the other foot toward the path of Edinburgh, and carry that extent backwards and forwards till both the points of the compasses fall into the same instants in both the paths, and these instants will denote the time when the eclipse begins at Edinburgh. Then do the like on the right hand of the axis of the ecliptic; and where the points of the compasses fall into the same instants in both the paths, they will show at what time the eclipse ends at Edinburgh.

These trials give 35∣ minutes after one in the afternoon for the beginning of the eclipse at Edinburgh at the points N and 0, 2⅛ minutes after three for the time of greatest obscuration, and 23∣ minutes after four at R and S for the time when the eclipse ends, according to apparent time. To have the mean time, or that shown by well-regulated clocks and watches, apply the equation of time in the contrary manner to that used for converting the mean time of new rfioon into apparent time. Therefore, in the present instance, subtract 3 minutes 56 seconds, or 4 minutes approximately, from the apparent times, and we have

H. Μ.

Beginning of eclipse at Edinburgh, 1 31⅜ p. Μ.

Greatest obscuration 2 58∣ —

End of eclipse 4 19∣ —

all according to mean time.

The Projection of Lunar Eclipses.

When the moon’s mean distance from either of her nodes at the time of mean full moon is less than 13° 21' there may be an eclipse of the moon; but if greater, there cannot be an eclipse.

We find by Example II. that at the time of mean full moon in September 1830, the moon’s mean distance from the descending node is only l 0 20' 33", which being so much less than the limit, there will then be an eclipse.

By Problem V. find the true anomalies of the sun and moon, and the moon’s true distance’from the ascending node, at the true time of full moon.
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The elements for constructing an eclipse of the moon are eight in number, as follows:

1. The true time of full moon, and at that time; 2. the moon’s horizontal parallax; 3. the sun’s semidiameter; 4. the moon’s; 5. the semidiameter of the earth’s shadow at the moon; 6. the moon’s latitude; 7. the angle of the moon’s visible path with the ecliptic; 8. the moon’s true horary motion from the sun. Therefore,

1. To find the true time of full moon. Work as already taught in the precepts. Thus we have the true time of full moon in September 1830 (see Example II.) to be the 2d day at 10 h . 36 m . 47 s . mean time at Greenwich.

2. To find the moon’s horizontal parallax. Enter Table XXIII. with the moon’s true anomaly 10 s . 6° 29' 25", and thereby take out her horizontal parallax; which, by making the requisite proportions, will be found to be 59' 52".

3. 4. To find the semidiameters of the sun and moon. Enter Table XXIII. with the sun’s mean anomaly and moon’s true anomaly (8 s . 1° 33' 0" and 10 s . 6° 29' 25"), and thereby take out their respective semidiameters, the sun’s 15' 54", and the moon’s 16' 19".

5. To find the semidiameter of the earth’s shadow at the moon. Add the sun’s horizontal parallax, which is always 9", to the moon’s, which in the present case is 59' 52"; the sum is 60' 1", from which subtract the sun’s semidiameter, 15' 54", and the remainder, 44' 7", being increased by 50" for the effect of the earth’s atmosphere, we have 44' 57" for the semidiameter of the earth’s shadow, which tlie moon then passes through.
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6. To find the moons latitude. Proceed as already directed under the prediction of solar eclipses. Thus, g φ 0 f l, g φ - g β

Argument, Table XVII. 5 29 50 14 52

XVIII. 4 6 39 10

XIX. 2 13 9 25

XX. 4 6 20 14

XXI. 1 29 43 r ....17

XXII. 2 0 2 28

+ 2 16 —- 10

—10

Moon’s latitude north + 2 6

7. To find the angle of the moons visible path with the ecliptic. Enter Table XVII. with the moon’s true distance from the ascending node for argument, and the angle is found to be 5° 44', left, signifying that the axis of the moon’s orbit is to the left hand of the northern axis of the ecliptic.

8. To find the moons true horary motion from the sun. With the true anomaly of the moon and the mean anomaly of the sun, take out their horary motions from Table XXIII., and the sun’s horary motion subtracted from the moon’s leaves remaining the moon’s true horary motion from the sun; in the present case 33' 58".

These elements being found for the construction of the moon’s eclipse in September 1830, proceed as follows:

Draw the line ACB for part of the ecliptic, and CD perpendicular thereto for the northern part of its axis, the moon having north latitude. (Plate XCIV. fig. 137.)

Add the semidiameters of the moon’s and earth’s shadow together, which in the eclipse make 61' 16"; and take this in your compasses, from a scale of equal parts, and setting one foot on the point C as a centre, with the other foot describe the arch ADB, in one point of which the moon’s centre will be at the beginning of the eclipse, and in another at the end thereof.

Take the semidiameter of the earth’s shadow, 44' 57", in your compasses, from the scale, and setting one foot in the centre C, with the other foot describe the semicircle KLM for the northern half of the earth’s shadow; because the moon’s latitude is north in this eclipse.

Subtract the semidiameter of the moon from the semidiameter of the earth’s shadow, and the remainder is 28' 38", which take in your compasses from the scale, and setting one foot on the point C as a centre, with the other foot describe the arch OPQ; in one point of which the moon’s centre will be at the beginning of total darkness, and in another at the end thereof.

Draw the line CE on the left hand of the northern axis of the ecliptic, and making an angle of 5° 44' therewith, which line represents the northern part of the axis of the moon’s orbit, the moon’s latitude being north.

Take the moon’s latitude, 2' 6", from the scale with your compasses, and set it from C to G in the axis of the ecliptic, and through the point G draw the straight line RSGTU, at right angles to the axis of the moon’s orbit, for the path of the moon’s centre. Then F, in the line CE, is the point in the earth’s shadow where the moon’s centre is at the middle of the eclipse; G, the point where her centre is at the instant of her ecliptical conjunction; and the middle between them, the point where her centre is at the time of true full moon by the tables.

Take the moon’s horary motion from the sun, 33' 58", in your compasses, from the scale, and with that extent make marks along the line of the moon’s path; then divide each space from mark to mark into 60 equal parts or horary minutes, and set the hours to the proper dots, in such a manner, that the dot signifying the instant of full moon (36 minutes and 47 seconds after ten) may be midway between the points F and G.

The point U, where the moon’s path intersects on the right hand the arch described with the sum of the semidiameters of the moon and earth’s shadow, denotes the instant when the eclipse begins, namely, at 52 minutes after eight; the point T, where the moon’s path intersects on the right hand the arch described, with the difference of the semidiameters, denotes the instant when the moon’s total darkness begins, namely, at 48 minutes after nine; the point F denotes the middle of the eclipse at 39 minutes after 10; the point S, where the moon’s path intersects on the left hand the arch described, with the difference of the semidiameters, denotes the end of the moon’s total darkness at 30 minutes after eleven; and the point R, where the moon’s path intersects the arch described, with the sum of the semidiameters, denotes the end of the eclipse at 26 minutes after 12, all mean time according to the meridian of Greenwich. If the times reckoned by any other meridian are required, apply the longitude from Greenwich to the Greenwich times, by addition or subtraction, according as the place is east or west of Greenwich.

On F as a centre, with a radius equal to the moon’s semidiameter, describe a circle which represents the moon’s disk at the middle of the eclipse.

The line VX denotes the quantity eclipsed at the middle of the eclipse, which may be measured on a line equal to the moon’s diameter, and divided into equal parts for digits. In the present case, the eclipse being total, the quantity eclipsed is said to be greater than the moon’s diameter, and is found to be 21∣ digits.
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CHAP. HI.

CATALOGUE OF FIXED STARS, AND TABLES OF REDUCTION.
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The preceding catalogue has been selected from one containing the approximate right ascensions and north polar distances of 720 stars, from observations made with two microscopes. This, however, is considered by the astronomer royal, Mr Pond, only as a first approximation to a more perfect catalogue, which is now forming at Greenwich, from observations made with two mural circles and six microscopes. A German astronomer, Bode, has published a catalogue of 17,240 stars, with their right ascensions and declinations, as a sequel to his Uranogra- phia, which exhibits engraved figures of the constellations. The Memoirs of the Astronomical Society of London contain a catalogue of nearly 3000 stars, with accompanying tables, by Mr Baily, who has stated every particular relating to it in a most elaborate preface. This is fully adequate to the wants of practical astronomers: but the principal astronomers in Europe have undertaken a survey of the heavens, in which each has chosen a section; and the completion of the labour will farther advance this part of the science.
[table]

The numbers in this table express the acceleration of the fixed stars. Its use is to convert any interval of sidereal into mean solar time.

Rule. — Take from the table the acceleration corresponding to the hours, the minutes, and the seconds ; subtract their sum from the given sidereal interval ; the remainder will express its value in mean solar time. Example.— Convert 7h. 14 m ∙ 51 s ∙ sidereal time into solar time.

H. M. S. M. 3.

The acceleration on...7 0 0∙00 is 1 8∙807

14 0∙00 2∙294 5l∙00 ∙139

From sidereal time....7 14 51∙00 1 11*240

Subtract acceleration. 1 11∙24

Mean solar time 7 12 39∙76, the answer.

The reduction of sidereal to solar time is constantly wanted in the solution of the following important problem in Practical Astronomy, viz.

To find the mean solar hour when any star in the preceding catalogue will be on the meridian at a given place, on a given day, in a given year. Solution —The right ascension of the star at the given time is the sidereal hour of its transit. The sun’s R. A. in the Nautical Almanac, for the given time, is the sidereal hour of its transit at Greenwich on that day; from this, and the longitude of the given place, the sun’s R. A., that is, the sidereal hour of its transit at that place, may be found. Then the sidereal interval between the sun’s transit and that of the star will be the difference between the R. A. of the star and that of the sun. This converted into solar time will give the mean solar interval between the sun’s transit and that of the star: to this the equation of time must be applied, and the result will be the hour of the star’s transit reckoned from mean moon.

Example. —At what hour will α Aquilae pass the meridian of Edinburgh on 1st October 1831?

The proposed time is about 1^ year after the epoch of the catalogue. The correction for change of R. A. by precession will therefore be + 2 s ∙93 × 1∙75 = + 5 s ∙12, and the star’s R. A. will be 19 h ∙ 42 m ∙ 34 s ∙5. Again, the sun’s R. A. at Greenwich for the given day is 12 h ∙ 27 m ∙ 41 s ∙3, and its daily increase ∣js 3 m ∙ 37 s ∙5. Now the longitude of Edinburgh is, in time, 12 m ∙ 44 s ∙ west from Greenwich, and corresponding to this, the change in the sun’s R. A. will be 1 s ∙8; therefore the sun’s R. A. when on the meridian of Edinburgh will be 12 h ∙ 27 m ∙ 43 s ∙1.

H. μ. s.

From R. A. star 19 42 34∙5

Subtract R. A. sun 12 27 43T

Difference in sidereal time 7 14 51∙4

Correct for acceleration 1 1P9

From apparent time 7 13 39∙5

Subtract equation of time 10 9∙3

Time of star’s transit 7 ∙3 30∙2

Hence it appears that the star will pass the meridian of Edinburgh at 7 h ∙ 3 m ∙ 30 s ∙ after mean noon. Here no corrections are made on the R. A. of the star for aberration and nutation; but these are too small to be sensible, except by good astronomical instruments.

By this problem the astronomer finds the error and rate of his clock, taking in, however, aberration and nutation, which need not be noticed in ordinary estimations of time.
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Explanation of the Table of Refractions.

This table is computed upon principles explained by the late Dr Young in the Philosophical Transactions for 1819; and it appears to agree more perfectly with the latest observations than any other table before published.

The apparent altitude being found in the first column, the second shows the refraction when the barometer stands at 30 inches, which is its mean height on the level of the sea, and the thermometer at 50° of Fahrenheit. The third column contains the difference to be subtracted or added for every minute of altitude, reckoned from the nearest number in the first column. The fourth shows the number of seconds to be added for every inch that the height of the barometer exceeds 30, or to be subtracted for each inch that it wants of 30: and the last contains the number of seconds to be subtracted for each degree that the thermometer stands above 50°, or to be added for each degree that its height wants of 50°.

If great accuracy be required, we must also deduct from the observed height of the barometer ∙003 inch for each degree that the thermometer near it is above 50°, and add an equal quantity for an equal depression. In fact, however, the table, as it now stands, is found to require the temperature to be estimated from the height of the thermometer within; and if we employed the height of the thermometer without, which would be more consistent with the theory, it would probably be necessary to suppose the standard temperature of the table 48° only (or rather 47°), instead of 50".

Examples.

1. At 7° 18' 13", barometer 29∙87, thermometer 6ß", the refraction is ß' 52"∙2β, from twenty-two observations of Bradley.

2. At 19° 18' 19", barometer 30Ό45, thermometer 34°, the refraction is 2'5 from three observations of Bradley.

. 3. At 13° 43', barometer 29 ∙85, thermometer 45°, the refraction is 3'55 from 15β observations of Air Pond.

1. Alt. 7’20'11. 7' 8" Diff. Alt. "∙9 B. 14"∙3 Th."·

+ 1∙62 1' 47" = l z ∙8 — ∙13 -

7 9’62 + 1 ∙G2 1 ∙86 1

16’74

G 52∙88

6 52’26

Error.. 0’62

2. Alt. 19° R. 2' 47"∙7 Diff. Alt. "∙16 B. 5"∙61 Th — - 2 ∙93 18' 19" = 18 ∙3 + ∙045

2 44 ∙77 _ 2-93 ∙252

•25

. 5 ‘44

Error l"∙0 2 50 ∙4G

3. Alt. 13° 40' R. 3' 55"∙5 Diff. Alt. "∙29 B. 7"∙89 Th.

+ __ ^36 3 ∙15 _

3 55 ∙8G 87 1 ∙18 +

3 55 ∙85 ∙87

Error ∙01 2 ∙05 +