# Popular Science Monthly/Volume 57/August 1900/Chapters on the Stars II

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* CHAPTERS ON THE STARS.
By Professor SIMON NEWCOMB, U. S. N.
Constellations and Star Names.
IN ancient times the practice was adopted of imagining the figures of heroes and animals to be so outlined in the heavens as to include in each figure a large group of the brighter stars. In a few cases some vague resemblance may be traced between the configurations of the stars and the features of the object they are supposed to represent; in general, however, the arrangement seems quite arbitrary. One animal or man could be fitted in as well as another. There is no historic record or trace as to the time when the constellations were mapped out, or of the process by which the outlines were traced. The names of heroes, such as Perseus, Cepheus, Hercules, etc., intermingled with the names of goddesses, show that the constellations were probably mapped out during the heroic age. No maps are extant showing exactly how each figure was placed in the constellation; but in the catalogue of stars given by Ptolemy in his 'Almagest,' the positions of particular stars on the supposed body of the hero, goddess or animal are designated with such precision as he had at command, in some fairly precise position of the figure. For example, Aldebaran is said to have formed the eye of the Bull. Two other stars marked the right and left shoulders of Orion, and a small cluster marked the position of his head. A row of three stars in a horizontal line showed his belt, three stars in a vertical line below them his sword. In this way the position of the figure can be reproduced with a fair degree of certainty.
In the well-known constellation Ursa Major, the Great Bear, familiarly known as the Dipper, three stars form the tail of the animal, and four others a part of his body. This formation is not unnatural, yet the figure of a dipper fits the stars much better than that of a bear. In Cassiopeia, which is on the opposite side of the pole from the Dipper, the brighter stars may easily be imagined to form a chair in which a lady may be seated without further difficulty. As a general rule, however, the resemblances of the stars to the figure are so vague that the latter might be interchanged to any extent without detracting from their appropriateness.
In any case, it was impossible so to arrange the figures that they should cover the entire heavens; blank spaces were inevitably left in ​which stars might be found. In order to include every star in some constellation, the figures have been nearly ignored by modern astronomers, and the heavens have been divided up, by somewhat irregular lines, into patches, each of which contains the entire figure as recognized by ancient astronomers. But all are not agreed as to the exact outlines of these extended constellations, and, accordingly, a star is sometimes placed in one constellation by one astronomer and in another constellation by another astronomer.
The confusion thus arising is especially great in the southern hemisphere, where it has been intensified by the subdivision of one of the old constellations. The ancient constellation Argo covered so large a region of the heavens, and included so many conspicuous stars, that it was divided into four, representing various parts of a ship—the sail, the poop, the prow and the hull.
Dr. Gould, while director of the Cordoba Observatory, during the years 1870 to 1880, constructed the 'Uranometria Argentina,' in which all the stars visible to the naked eye more than 10 degrees south of the celestial equator were catalogued and mapped. He made a revision of the boundaries of each constellation in such a way as to introduce greater regularity. The rule generally followed was that the boundaries should, so far as possible, run in either an east and west or a north and south direction on the celestial sphere. They were so drawn that the smallest possible charjge should be made in the notation of the conspicuous stars; that is, the rule was that, if possible, each bright star should be in the same constellation as before. The question whether this new division shall replace the ancient one is one on which no consensus of view has yet been reached by astronomers. Simplicity is undoubtedly introduced by Gould's arrangement; yet, in the course of time, owing to precession, the lines on the sphere which now run north and south or east and west will no longer do so, but will deviate almost to any extent. The only advantage then kept will be that the bounding lines will generally be arcs of great circles.
When the heavens began to be carefully studied, two or three centuries ago, new constellations were introduced by Hevelius and other astronomers to fill the vacant spaces left by the ancient ones of Ptolemy. To some of these, rather fantastic names were given; the Bull of Poniatowski, for example. Some of these new additions have been retained to the present time, but in other cases the space occupied by the proposed new constellation was filled up by extending the boundaries of the older ones.
At the present time the astronomical world, by common consent, recognizes eighty-nine constellations in the entire heavens. In this enumeration Argo is not counted, but its four subdivisions are taken as separate constellations.
names of the stars.
A glance at the heavens will make it evident that the problem of designating a star in such a way as to distinguish it from all its neighbors must be a difficult one. If such be the case with the comparatively small number of stars visible to the naked eye, how must it be with the vast number that can be seen only with the telescope? In the case of the great mass of telescopic stars we have no method of designation except by the position of the star and its magnitude; but with the brighter stars, and, indeed, with all that have been catalogued, other means of identification are available.
It is but natural to give a special name to a conspicuous star. That this was done in very early antiquity we know by the allusion to Arcturus in the Book of Job. At least two such names, Castor and Pollux, have come down to us from classical antiquity, but most of the special names given to the stars in modern times are corruptions of certain Arabic designations. As an example we may mention Aldebaran, a corruption of Al Dabaran—The Follower. There is, however, a tendency to replace these special names by a designation of the stars on a system devised by Bayer early in the seventeenth century.
This system of naming stars is quite analogous to our system of designating persons by a family name and a Christian name. The family name of a star is that of the constellation to which it belongs. The Christian name is a letter of the Greek or Roman alphabet, or a number. As a number of men in different families may have the same Christian name, so the Greek letter or number may be given to a star in any number of constellations without confusion.
The work of Bayer was published under the title of 'Uranometria,' of which the first edition appeared in 1601. This work consists mainly of maps of the stars. In marking the stars with letters on the map, the rule followed seems to have been to give the brighter stars the earlier letters in the alphabet. Were this system followed absolutely, the brighter stars should always be called α; the next in order β, etc. But this is not always the case. Thus in the constellation Gemini, the brighter star is Pollux, which is marked β, while α is the second brightest. What system, if any, Bayer adopted in detail has been a subject of discussion, but does not appear to have been satisfactorily made out. Quite likely Bayer himself did not attempt accurate observations on the brightness of the stars, but followed the indications given by Ptolemy or the Arabian astronomers. As the number of stars to be named in several constellations exceeds the number of letters in the Greek alphabet, Bayer had recourse, after the Greek alphabet was exhausted, to letters of the Roman alphabet. In this case the letter A was used as a capital, in order, doubtless, that it should not be confounded with the Greek α. In other cases smaller italics are ​used. In several catalogues since Bayer, new italic letters have been added by various astronomers. Sometimes these have met with general acceptance, and sometimes not.
Flamsteed was the first Astronomer Royal in England, and observed at Greenwich from 1666 to 1715. Among his principal works is a catalogue of stars in which the positions are given with greater accuracy than had been attained by his predecessors. He slightly altered the Bayer system by introducing numbers instead of Greek letters. This had the advantage that there was no limit to the number of stars which could be designated in each constellation. He assigned numbers to all the brighter stars in the order of their right ascension, irrespective of the letters used by Bayer. These numbers are extensively used to the present day, and will doubtless continue to be the principal designations of the stars to which they refer. It is very common in our modern catalogues to give both the Bayer letter and the Flamsteed number in the case of Bayer stars.
The catalogues by Flamsteed do not include quite all the stars visible to the naked eye, but various uranometries have been published which were intended to include all such stars. In such cases the designations now used frequently correspond to the numbers given in the uranometries of Bode, Argelander and Heis.
In recent times these uranometries have been supplemented by censuses of the stars, which are intended to include all the stars to the ninth or tenth magnitude. I shall speak of these in the next section; at present it will suffice to say that stars are very generally designated by their place in such a census.
There is still here and there some confusion both as to the boundaries of the constellations and as to the names of a few of the stars in them. I have already remarked that, in drawing the imaginary boundaries on a star map, as representing the celestial sphere, different astronomers have placed the lines differently. One of the regions in which this is especially true is in the neighborhood of the north pole, where some astronomers place stars in the constellation Cepheus which others place in Ursa Minor. Hence in the Bayer system the same star may have different names in different catalogues. Again, in extending the names or numbers, some astronomers use names which others de not regard as authorative. The remapping of the southern constellation by Dr. Gould changed the boundaries of most of the southern constellations in a way already mentioned.
I have spoken of the subdivision of the great constellation Argus into four separate ones. Bayer having assigned to the principal stars in this constellation the Greek letters α, β, γ, etc., the general practice among astronomers since the subdivision has been to continue the designation of the stars thus marked as belonging to the constellation ​Argo. Thus, for example, we have Argus, which after the subdivision belonged to the constellation Carina. The variable star η Argus also belongs to the constellation Carina. But in the case of stars not marked by Bayer, the names were assigned according to the subdivided constellations, Vela, Carina, etc. Confusing though this proceeding may appear to be, it is not productive of serious trouble. The main point is that the same star should always have the same name in successive catalogues. Still, however, it has recently become quite common to ignore the constellation Argus altogether and use only the names of its subdivisions. The reader must therefore be on his guard against any mistake arising in this way in the study of astronomical literature.
In star catalogues the position of a star in the heavens is sometimes given in connection with its name. In this case the confusion arising from the same star having different names may be avoided, since a star can always be identified by its right ascension and declination. The fact is that, so far as mere identification is concerned, nothing but the statement of a star's position is really necessary. Unfortunately, the position constantly changes through the precession of the equinoxes, so that this designation of a star is a variable quantity. Hence the special names which we have described are the most convenient to use in the case of well-known stars. In other cases a star is designated by its number in some well-known catalogue. But even here different astronomers choose different catalogues, so that there are still different designations for the same star. The case is one in which action of uniformity of practice is unattainable.
Cataloguing and Numbering the Stars.
A catalogue or list of stars is a work giving for each star listed its magnitude and its position on the celestial sphere, with such other particulars as may be necessary to attain the object of the catalogue. If the latter includes only the more conspicuous stars, it is common to add the name of each star that has one; if none is recognized, the constellation to which the star belongs is frequently given.
The position of a star on the celestial sphere is defined by its right ascension and declination. These correspond to the longitude and latitude of places on the earth, in the following way: Imagine a plane passing through the center of the earth and coinciding with its equator, to extend out so as to intersect the celestial sphere. The line of intersection will be a great circle of the celestial sphere, called the celestial equator. The axis of the earth, being also indefinitely extended in both the north and the south directions, will meet the celestial spheres in two opposite points, known as the north and south celestial poles. The equator will then be a great circle 90° from each ​pole. Then as meridians are drawn from pole to pole on the earth, cutting the equator at different points, so imaginary meridians are conceived as drawn from pole to pole on the celestial sphere. Corresponding to parallels of latitude on the earth we have parallels of declination on the celestial sphere. These are parallel to the equator, and become smaller and smaller as we approach either pole. The correspondence of the terrestrial and celestial circles is this:
To latitude on the earth's surface corresponds declination in the heavens.
To longitude on the earth corresponds right ascension in the heavens.
A little study of these facts will show that the zenith of any point on the earth's surface is always in a declination equal to the latitude of the place. For example, for an observer in Philadelphia, in 40° latitude, the parallel of 40° north declination will always pass through his zenith, and a star of that declination will, in the course of its diurnal motion, also pass through his zenith.
So also to an observer on the equator the celestial sphere always spans the visible celestial hemisphere through the east and west points.
In the case of the right ascension, the relation between the terrestrial and celestial spheres is not constant, because of the diurnal motion, which keeps the terrestrial meridians in constant revolution relative to the celestial meridians. Allowing for this motion, however, the system is the same. As we have on the earth's surface a prime meridian passing from pole to pole through the Greenwich Observatory, so in the heavens a prime meridian passes from one celestial pole to the other through the vernal equinox. Then to define the right ascension of any star we imagine a great circle passing from pole to pole through the star, as we imagine one to pass from pole to pole through a city on the earth of which we wish to designate the longitude. The actual angle which this meridian makes with the prime meridian is the right ascension of the star as it is the longitude of the place on the earth's surface.
There is, however, a difference in the unit of angular measurement commonly used for right ascensions in the heavens and longitude on the earth. In astronomical practice, right ascension is very generally expressed by hours, twenty-four of which make a complete circle, corresponding to the apparent revolution of the celestial sphere in twenty-four hours. The reason of this is that astronomers determine right ascension by the time shown by a clock so regulated as to read 0 hrs., 0 min., 0 sec. when the vernal equinox crosses the meridian. The hour hand of this clock makes a revolution through twenty-four hours during the time that the earth makes one revolution on its axis, and thus returns to 0 hrs., 0 min., 0 sec. when the vernal equinox again crosses the ​meridian. A clock thus regulated is said to show sidereal time. Then the right ascension of any star is equal to the sidereal time at which it crosses the meridian of any point on the earth's surface. Right ascension thus designated in time may be changed to degrees and minutes by multiplying by 15. Thus, one hour is equal to 15°; one minute of time is equal to 15′ of arc, and one second of time to 1″ of arc.
It may be remarked that in astronomical practice terrestrial longitudes are also expressed in time, the longitude of a place being designated by the number of hours it may be east or west of Greenwich. Thus, Washington is said to be 5h. 8m. 15s. west of Greenwich. This, however, is not important for our present purpose.
The first astronomer who attempted to make a catalogue of all the known stars is supposed to be Hipparchus, who flourished about 150 B.C. There is an unverified tradition to the effect that he undertook this work in consequence of the appearance of a new star in the heavens, and a desire to leave on record, for the use of posterity, such information respecting the heavens in his time that any changes which might take place in them could be detected. This catalogue has not come down to us—at least not in its original form.
Ptolemy, the celebrated author of the 'Almagest,' flourished a.d. 150. His great work contains the earliest catalogue of stars which we have. There seems to be a certain probability that this catalogue either may be that of Hipparchus adopted by Ptolemy unchanged, or may be largely derived from Hipparchus. This, however, is little more than a surmise, due to the fact that Ptolemy does not seem to have been a great observer, but based his theories very largely on the observations of his predecessors. The actual number of stars which it contains is 1,030. The positions of these are given in longitude and latitude, and are also described by their places in the figure of the constellation to which each may belong. Not unfrequently the longitude or latitude is a degree or more in error, showing that the instruments with which the position was determined were of rather rough construction.
So far as the writer is aware, no attempt to make a new catalogue of the stars is found until the tenth century. Then arose the Persian astronomer, Abd-Al-Rahman Al-Sufi, commonly known as Al-Sufi, who was born a.d. 903 and lived until 986. Nothing is known of his life except that he was a man celebrated for his learning, especially in astronomy. His only work on the latter subject which has come down to us is a description of the fixed stars, which was translated from the Arabic by Schjellerup and published in 1874 by the St. Petersburg Academy of Science. This work is based mainly on the catalogue of Ptolemy, all the stars of which he claimed to have carefully examined. But he did not add any new stars to Ptolemy's list, nor, it would seem, did he attempt to redetermine their positions. He simply used the ​longitudes and latitudes of Ptolemy, the former being increased by 12° 42' on account of the precession during the interval between his time and that to which Ptolemy's catalogue was reduced. The translator says of his work that it gives a description of the starry heavens at the time of the author and is worthy of the highest confidence. The main body of the work consists of a detailed description of each constellation, mentioning the positions and appearances of the stars which it contains. Here we find the Arabic names of the stars, which were not, however, used as proper names, but seem rather to have been Arabic words representing some real or supposed peculiarity of the separate stars, or arbitrarily applied to them.
Four centuries later arose the celebrated Ulugh Beigh, grandson of Tamerlane, who reigned at Samarcand in the middle of the fifteenth century. Bailey says of him: "Ulugh Beigh was not only a warlike and powerful monarch, but also an eminent promoter of the sciences and of learned men. During his father's lifetime he had attracted to his capital all the most celebrated astronomers from different parts of the world; he erected there an immense college and observatory, in which above a hundred persons were constantly occupied in the pursuits of science, and caused instruments to be constructed of a better form and greater dimensions than any that had hitherto been used for making astronomical observations."
His fate was one which so enlightened a promoter of learning little deserved; he was assassinated by the order of his own son, who desired to succeed him on his throne; and in order to make his position the more secure, also put his only brother to death. A catalogue of the stars bears the name of this monarch; he is supposed to have made many or most of the observations on which it is founded. Posterity will be likely to suppose that a sovereign used the eyes of others more than his own in making the observations. However this may be, his catalogue seems to have been the first in which the positions of the stars given by Ptolemy were carefully revised. He found that there were twenty-seven of Ptolemy's stars too far south to be visible at Samarcand, and that eight others, although diligently looked after, could not be discovered. It is curious that, like Al-Sufi, he does not seem to have added any new stars to Ptolemy's list.
Next in the order of time comes the work of Bayer, whose method of naming the stars has already been described. The main feature of this work consists of maps of all the constellations. Previous to his time, celestial globes, made especially for the use of the navigator, took the place of maps of the stars. The first edition of this book was published in 1603, and is distinguished by the fact that a list of stars in each constellation is printed on the backs of the maps. Bayer did not confine ​himself to the northern hemisphere, but extended his list over the whole celestial sphere, from the north to the south pole.
The catalogue of the celebrated Tycho Brahe, prepared toward the end of the sixteenth century, though of great historic value, is of no special interest to the general reader at the present time. A supplement to it, continuing its list of stars to the south pole, was published by Halley, who made the necessary observations during a journey to St. Helena in 1677.
The catalogue of Hevelius, published in 1690, offers no feature of special interest, except the addition of several new constellations which he placed between those already known. Having the aid of the telescope, he was able to include in his catalogue stars which had been invisible to his predecessors.
Modern catalogues of the stars may be divided into two classes: Those which include only stars of a special class, or stars of which the observer sought to determine the position or magnitude with all attainable precision; and catalogues intended to include all the stars in any given region of the heavens, down to some fixed order of magnitude. It may appear remarkable that no attempt of the latter sort was seriously made until more than two centuries after the telescope had been pointed at the heavens by Galileo. A reason for the absence of such an attempt will be seen in the vast number of stars shown by the telescope, the difficulty of stopping at any given point, and the seeming impossibility of assigning positions to hundreds of thousands of stars. The latter difficulty was overcome by the improved methods of observation devised in modern times.
About the middle of the present century the celebrated Argelander commenced the work of actually cataloguing all the stars of the northern celestial hemisphere to magnitude 9½. This work was termed a Durchmusterung of the northern heavens, a term which has been introduced into astronomy generally to designate a catalogue in which all the stars down to a certain magnitude are supposed to be mustered, as if a census of them were taken. The work fills three quarto volumes and contains more than 310,000 stars, of each of which the magnitude and the right ascension and declination are given. This work was extended by Schönfeld, Argelander's assistant and successor, to 22° of south declination.
In the latitudes in which most of the great observatories of the world are situated, that part of the celestial sphere within 40° or 50° of the south pole always remains below the horizon. Around this invisible region a belt of somewhat indefinite breadth, 10° or more, can be only imperfectly observed, owing to the nearness of the stars to the horizon, and the brevity of the period between their rising and setting. Up to the middle of the nineteenth century, the few observatories ​situated in the southern hemisphere were too ill-endowed to permit of their undertaking a complete census of this invisible region.
The first considerable work emanating from the Cordoba Observatory, under Gould, was a catalogue of all the stars from the south pole to 10° of north declination which could be seen with the naked eye. Another work, which was not issued until after Dr. Gould's death, was devoted to photographs of southern clusters of stars.
The work of the Cordoba Observatory, with which we are more especially concerned in the present connection, consists of a 'Durchmusterung' of the southern heavens, commencing at 22° of south declination, where Schönfeld's work ended, and continued to the south pole. This work is still incomplete, but two volumes have been published by Thome, extending to 41° of south declination. It is expected that the third is approaching completion. This catalogue is, in one point at least, more complete than that of Argelander and Schönfeld, as it contains all the stars down to the tenth magnitude. The two volumes give the positions and magnitudes of no less than 340,000 stars, and therefore more than the catalogue of Argelander gives for the entire northern hemisphere. If the remaining part of the heavens, from 42° to the south pole, is equally rich, it will contain nearly half a million stars, and the entire work will comprise more than 800,000 stars.
The Royal Observatory of the Cape of Good Hope, under the able and energetic direction of Dr. David Gill, has undertaken a work of the same kind, which is remarkable for being based on photography. The history of this work is of great interest. In 1882 Gill secured the aid of photographers at the Cape of Good Hope to take pictures of the brilliant comet of that year, with a large camera. On developing the pictures the remarkable discovery was made that not only all the stars visible to the naked eye, but telescopic stars down to the ninth or tenth magnitude were also found on the negatives. This remarkable result suggested to Gill that here was a new and simple method of cataloguing the stars. It was only necessary to photograph the heavens and then measure the positions of the stars on the glass negatives, which could be done with much greater ease and certainty than measures could be made on the positions of the actual stars, which were in constant apparent motion.
As soon as the necessary arrangements could be made and the apparatus put into successful operation, Gill proceeded to the work of photographing the entire southern heavens from 18° of south declination to the celestial pole. The results of this work are found in the 'Cape Photographic Durchmusterung' a work in three quarto volumes, in which the astronomers of all future time will find a permanent record of the southern heavens towards the end of the nineteenth century. The actual work of taking the photographs extended from 1887 to 1891. ​This, however, was far from being the most difficult part of the enterprise. The most arduous task of measuring the positions of a half-million of stars on the negatives, including the determining of the magnitude of each, was undertaken by Professor J. C. Kapetyn, of the University of Groningen, Holland, and brought to a successful completion in the year 1899.
What the work gives is, in the first place, the magnitude and approximate position of every star photographed. The determining of the magnitude of a star is an important and delicate question. There is no difficulty in determining, from the diameter of the image of the star as seen in the microscope, what its photographic magnitude was at the time of the exposure, as compared with other stars on the same plate. But can we rely upon similar photographic magnitudes on a plate corresponding to similar brightnesses of the stars? In the opinion of Gill and Kapetyn we cannot. The transparency of the air varies from night to night, and on a very clear night the same star will give a stronger image than it will when the air is thick. Besides, slightly different instruments were used in the course of the work. For these reasons a scale of magnitude was determined on each plate by comparing the photographic intensity of the images of a number of stars with the magnitudes as observed with the eye by various observers. Thus on each plate the magnitude was reduced to a visual scale.
It does not follow from this that the magnitudes are visual, and not photographic. It is still true that a blue star will give a much stronger photographic image than a red star of equal visual brightness. In a general way, it may be said that the catalogue includes all the stars to very nearly the tenth magnitude, and on most of the plates stars of 10.5 were included. In fact, now and then is found a star of the eleventh magnitude.
A feature of the work which adds greatly to its value is a careful and exhaustive comparison of its results with previous catalogues of the stars. When a star is found in any other catalogue the latter is indicated. Most interesting is a complete list of catalogued stars which ought to be on the photographic negatives, but were not found there. Every such case was inexhaustibly investigated. Sometimes the star was variable, sometimes it was so red in color that it failed to impress itself on the plate, sometimes there were errors in the catalogue.
The great enterprise of making a photographic map of the heavens now being carried on as an international enterprise, having its headquarters at Paris, is yet wider in its scope than the works we have just described. One point of difference is that it is intended to include all the stars, however faint, that admit of being photographed with the instruments in use. The latter are constructed on a uniform plan, the aperture of each being 34 centimetres, or 13.4 inches, and the focal ​length 343 c.m. Two sets of plates are taken, one to include all the stars that the instrument will photograph near poles, and the other only to take in those to the eleventh magnitude. Of the latter it is intended to prepare a catalogue. Some portions of the German and English catalogues have already been published, and their results will be made use of in the course of the present work.
numbering the stars.
Closely connected with the work of cataloguing the stars is that of enumerating them. In view of what may possibly be associated with any one star—planets with intellectual beings inhabiting them—the question how many stars there are in the heavens is one of perennial interest. But beyond the general statement we have already made, this question does not admit of even an approximate answer. The question which we should be able to answer is this: How many stars are there of each easily visible magnitude? How many of the first magnitude, of the second, of the third, and so on to the smallest that have been measured? Even in this form we cannot answer the question in a way which is at the same time precise and satisfactory. One magnitude merges into another by insensible gradations, so that no two observers will agree as to where the line should be drawn between them. The difficulty is enhanced by the modern system—very necessary, it is true—of regarding magnitude as a continuously varying quantity and estimating it with all possible precision. In adjusting the new system to the old one, it may be assumed that an average star of any given magnitude on the old system would be designated by the corresponding number on the new system. For example, an average star of the fourth magnitude would be called 4.0; one of the fifth, 5.0, etc. Then the brightest stars, which formerly were called of the fourth magnitude, would now be, if the estimate were carried to hundredths, 3.50, while the faintest would be 4.50. What were formerly called stars of the fifth magnitude would range from 4.50 to 5.50, and so on. But we have meet with a difficulty when we come to the sixth magnitude. On the modern system, magnitude 6.0 represents the faintest star visible to the naked eye; but the stars formerly included in this class would, on the average, be somewhat brighter than this, because none could be catalogued except those so visible.
The most complete enumeration of the lucid stars by magnitudes has been made by Pickering ('Annals of the Harvard Observatory,' Vol. XIV). The stars were classified by half magnitudes, calling
| | | | M. | | M. |
| Mag. | 2.0 | all from | 1.75 | to | 2.25 |
| | 2.5 | "" | 2.25 | to | 2.75 |
| | etc., | | | etc. | |
For the northern stars Pickering used the Harvard Photometry; for ​the southern, Gould's 'Uranometria Argentina.' A zone from the equator to 30° south declination is common to both; for this zone I use Gould. The number of each class in the entire sky, north and south of the celestial equator, is as follows:
| | Northern Hemisphere. Pickering. | Southern Hemisphere. Gould. | Total. |
| 1± | 9 | 14 | 23 |
| 2.0 | 17 | 15 | 32 |
| 2.5 | 17 | 24 | 41 |
| 3.0 | 37 | 41 | 78 |
| 3.5 | 61 | 74 | 135 |
| 4.0 | 114 | 126 | 240 |
| 4.5 | 228 | 234 | 462 |
| 5.0 | 450 | 426 | 876 |
| 5.5 | 787 | 681 | 1,468 |
| 6.0 | 789 | 1,189 | 1,978 |
| Sum. | 2,509 | 2,824 | 5,333 |
It would seem from this that the number of lucid stars in the southern celestial hemisphere is 315 greater than in the northern. But this arises wholly from a seemingly greater number of stars of magnitude 6. In the zone 0° to 30° S., Pickering has 214 stars of this class fewer than Gould. Hence it is not likely that there is any really greater richness of the southern sky.
The total number of lucid stars is thus found to be 5,333. But it is not likely that stars of magnitudes 6.1 and 6.2 should be included in this class, though this is done in the above table. From a careful study and comparison of the same data from Pickering and Gould, Schiaparelli enumerated the stars to magnitude 6.0. He found:
| North pole to 30°S | 3,113 | stars. |
| 30° S. to south pole | 1,190 | " |
| Total lucid stars | 4,303 | |
For most purposes a classification by entire magnitudes is more instructive than one by half magnitudes. From the third magnitude downward we may assume that 40 per cent, of the stars of each half magnitude belong to the magnitude next above, and 60 per cent, to that next below. We thus find that of
Total. :
| Mag. 0 and 1 there are | 21 | stars | 21 |
| Mag. 2 there are | 52 | stars | 73 |
| Mag. 3 there are | 157 | stars | 230 |
| Mag. 4 there are | 506 | stars | 736 |
| Mag. 5 there are | 1,740 | stars | 2,476 |
| Mag. 6 there are | 5,171 | stars | 7,647 |
​Here it is to be remarked that under magnitude 6 are included many other than the lucid stars, namely, all down to magnitude 6.4. The last column gives the entire number of stars down to each order of magnitude.
It will be remarked that the number of stars of each order is rather more than three times that of the order next higher. How far does this law extend? Argelander's 'Durchmusterung,' which is supposed to include all stars to magnitude 9.5, gives 315,039 stars for the northern hemisphere, from which it would be inferred that the whole sky contains 630,000 stars to the ninth magnitude. Comparing this with the number 7,647 of stars to the magnitude 6.5, we see that it is forty-fold, so that it would require a ratio of about 3.5 from each magnitude to the next lower. But it is now found that Argelander's list contains, in the greater part of the heavens, all the stars to the tenth magnitude.
On the other hand, Thome's Cordoba 'Durchmusterung' gives 340,-380 stars between the parallels -22° and -42°. This is 0.14725 of the whole sky, so that, on Thome's scale of magnitude, there are about 2,311,000 stars to the tenth magnitude in the sky. This is more than three times the Argelander number to the ninth magnitude.
It would, therefore, seem that the ratio of number for each magnitude must exceed 3, even up to the tenth. If a ratio of only 3 extends four steps farther, the whole number of stars in the sky down to magnitude 14.5 inclusive must approach 200 millions. Until the international photographic chart of the sky is subjected to a detailed examination, it is impossible to make an estimation with any approach to certainty.