# Popular Astronomy: A Series of Lectures Delivered at Ipswich/Lecture 5

LECTURE V.
IN last evening's lecture I treated more especially of the measures of the distance of the moon and the sun from the earth; and first of the distance of the moon from the earth. I endeavoured to point out to you that the method of parallax, by which the distance of the moon from the earth is measured, might be illustrated by the combined operation of the two eyes in the head, by which the distance of objects near us may be approximately estimated; but that in measuring the distance of the moon from the earth, instead of making use of the two eyes in the head, we make use of two observatories on the earth, placed at a considerable distance from each other, from which places we observe the same celestial object. I then pointed out to you two places that are peculiarly adapted for this measure; one of these is the Observatory at the Cape of Good Hope, and the other is Greenwich or Cambridge, or any other European Observatory. At the European Observatory, by means of the mural circle, we observe the apparent angular distance of the moon from the North Pole, at the time when it passes the meridian; and we also observe on the same day, at the Observatory at the Cape of Good Hope, what is the apparent ​angular distance of the moon from the South Pole of the heavens. The reason why we must make the observations in this way is, because at the European Observatory we can see the North Pole, while at the Cape of Good Hope, they can only see the South Pole, The angle between the two celestial poles is necessarily 180 degrees exactly, because the directions of the lines drawn from the two places to the celestial poles are parallel to the same line, namely, the axis of the earth. I then remarked that the moon, as viewed from the Cape of Good Hope, does not pass the meridian precisely at the same time as when viewed at the Observatory at Greenwich or Cambridge. But the difference of time is well known; and we have only to take into account the change in the moon's place during the interval which, elapses between the observations at the two places. That is done with very great accuracy, and we can reduce the angular distance of the moon from the South Pole, to what it would have been if it had been observed at the same time as at Greenwich. Thus we have, for the same instant of time, got the apparent angular distance of the moon from the North Pole, as seen at Greenwich or Cambridge, and the apparent angular distance of the moon from the South Pole, as seen from the Cape of Good Hope. The sum of these two angular distances, if a star were observed, would be 180 degrees; but when the moon is observed, the sum of these two angular distances is found to be more than 180 degrees, and the excess above 180 degrees is the effect of parallax. It is the same as the angle which is made at the moon by two lines, one drawn from the European Observatory and the other from the Cape of Good Hope. Here we have got everything in much the same state as when ​measuring any distant object by means of a base line. For, from our knowledge of the form and dimensions of the earth (Figure 40), we know the length and position of the line GC; and the observations made at Greenwich and at the Cape of Good Hope give us the angles MGC and MCG; and thus we have the elements for computing the lines GM and CM, and then the distance EM can be found with little trouble.
I then added, that there is one cause of uncertainty, which is refraction, and which produces its effect in this way: we refer the observation of the moon at Greenwich to the North Pole of the heavens; and we refer the observation made at the Cape of Good Hope to the South Pole of the heavens. In deducing the real places of the moon from the apparent places, it is necessary to take into account the quantity of refraction which enters in these two cases. There is one calculation of refraction amounting to a great many seconds, or, perhaps a minute or two, to be taken into account in the observations made at Greenwich; and another calculation of refraction, perhaps amounting to a like quantity, to be taken into account at the Cape of Good Hope. As I said before, refraction is the plague of astronomers, and owing to it, there is always a little uncertainty in the measurement of large angles on the celestial meridian. On that account it is desirable, if possible, to diminish that refraction. If we suppose that there is a star S, Figure 40, at a distance so great that its position is sensibly the same when seen from any part of the earth, and if two observers select the star by previous concert; and if the person at G observes with his instrument the place of the moon as well as that of the star, he finds how many degrees, minutes, and seconds, ​the moon is below the star. This is the angle SGM, and the comparison by which it is determined is almost independent of refraction. By similar observations at C, the angle SCM is found with equal accuracy. And the difference between these angles gives the angle CMG with great accuracy. And it is this angle upon which the distance of the moon mainly depends.
I then explained that the calculation, in point of fact, is not made by treating MGC as one triangle in a survey, but by dividing the angle GMC into two parts by the line EM, and then assuming for trial a value of the distance EM, and computing the angle EMG, and with the same assumption computing the angle EMC, and adding them together, and finding whether this sum agrees with the observed angle GMC; if it does not agree, the assumption of distance must be varied till it does agree. There is no difficulty in each of these computations; because, from the dimensions of the earth, it it easy to find the inclination of the line GE to the vertical H'GE'; and therefore from the observed angle H'GM the angle HGM is found; also the length GE is known; and the length EM is assumed for trial; and then the calculation of EMG is easy.
It is now proper to mention that astronomers very seldom refer to the actual length EM in yards or miles. I explained that the angle EMG is called the parallax of the moon at G, and the angle EMC is the parallax of the moon at C. Now, (referring for the present to the place G only,) if the line GM were perpendicular to the line EG, that is, if the moon were in the horizon as viewed from G, the parallax would be greater than in any other position of the moon, (supposing the distance EM not to be ​altered.) This is called the horizontal parallax. And as the earth is not spherical, and therefore different places are at different distances from the earth's centre, the horizontal parallaxes will not be the same at different places; it is therefore convenient to fix on some one place as a standard. The place fixed on by the consent of astronomers is the equator; and the horizontal parallax of the moon at the equator is called the horizontal equatoreal parallax. This is the quantity used by astronomers in relation to the moon's distance; it is convenient for their calculations, and it amounts to the same thing as using the distance; for if the distance is known, the horizontal equatoreal parallax is known; or if the horizontal equatoreal parallax is known, the distance is known. The moon's horizontal equatoreal parallax varies (according to the moon's distance), from 54 minutes of a degree to 611/4 minutes; these correspond respectively to the distances 252,390 miles and 222,430 miles.
I then stated that when every observation of the moon made at any one place is corrected for parallax, so as to inform us what would be the position of the moon as viewed from the centre of the earth, it is found that her orbit is sensibly a plane; and this conclusion may then be properly used as a basis for determining with greater accuracy the moon's horizontal parallax at that place. For the moon's angular distance from the Pole when she is nearest to it will fall short of 90 degrees, just as much as it will exceed 90 degrees when she is farthest from the Pole, provided that the proper correction for parallax is made so as to reduce the observed place to what it would have been as seen from the earth's centre. If the condition which I have mentioned is not satisfied, it ​is a proof that the assumed value of horizontal parallax is wrong, and a new trial must be made. I mentioned to you that this method is interesting, not only as being in use at the present time, but also because it is the method first used by Greek Astronomers.
Both in the last and in the present century, the distance of the moon has been found with great accuracy by these methods. Its mean value may be roughly stated to be 240,000 miles, or about thirty times the breadth of the earth. It seems a long way, and it is a long distance to measure, considering that it is ascertained by the use of a yard measure. You will observe that it is really and truly measured so. A yard measure was used to measure the base in the trigonometrical survey; by means of this, and a series of triangles, a long line was measured on the earth; by knowing its length, and by making observations with the Zenith Sector, the form and dimensions of the earth were found; and by knowing the size of the earth, and by observing the angles which relate to parallax, the moon's distance is found. You see we have thus proceeded step by step from the yard measure to the moon's distance. We may thus picture to ourselves the distance of the moon. If a railway carriage travelled at the rate of 1000 miles a day, it would at that rate be eight months reaching the moon.
I then proceeded to point out to you by what means the measure of the sun's distance had been attempted; and first I pointed out to you some methods which have failed. In the first place I remarked that the distance of the sun from the earth might apparently be measured, just in the same way as the distance of the moon from the earth, by observations at two distant observatories. This method ​practically is inapplicable, because the uncertainty of refraction is great, for the air is in a heated state, and it is therefore in a state unfavourable for the observation; and because no star can be observed near the sun. In this case, a small uncertainty produces a far greater effect than in the case of the moon. Suppose, for instance, that there is an uncertainty in the observation which will produce an uncertainty of one second in the horizontal parallax. The smaller the parallax is, the greater is the distance in the same proportion. If, then, by this error, the moon's horizontal parallax is altered from 57 minutes to 57 minutes 1 second, the measure of distance is altered by only 1/3491 part of the whole, or by 70 miles. But if the sun's horizontal parallax is altered from 9 seconds to 10 seconds; the measure of distance is altered by 1/10 part of the whole, or by more than 9 millions of miles. On this account, the method of simple parallax entirely fails in ascertaining the distance of the sun.
I then mentioned another very ingenious method, one founded on the observation of the place of the moon when it is "dichotomized." This is a Greek expression used to denote that state of the moon when it is half illuminated. If we can fix upon that time exactly, we shall know that the angle at the moon made by lines drawn to the sun and the earth is a right angle, and if we can then measure the angle at the earth between the sun and the moon, and subtract that angle from 90 degrees, we shall have the angle at the sun made by lines drawn to the earth and moon; and from this we shall be able to compute the proportion between the sun's distance and the moon's distance. The method fails because the surface of the moon is so exceedingly rough.
​I then pointed out a third method, which has been tolerably successful, (although it is not the one regarded as being the most accurate), namely, by observing the parallax of Mars, which moves round the sun in an orbit between those of the earth and Jupiter. It is founded upon these considerations. First, that from seeing how much the planets go right or left of the sun, or how much their motion deviates from motion in a circle round the earth, the proportion of the distance of Mars from the sun to the distance of the earth from the sun is known quite independently of any knowledge of the absolute distances; and that in fact this proportion was known with considerable accuracy many centuries ago. In the year 1700 it was nearly as well known as it is now. Secondly, the proportion of the distances from the sun being known, it followed that the proportion of the distance of Mars from the earth to the distance of the earth from the sun was known. Suppose then, at a certain time, (I am obliged to say at a certain time because the orbit of Mars is very eccentric,) when the sun, the earth, and Mars, are in a straight line, suppose at that time we know the distance of Mars from the sun is four-thirds of the distance of the earth from the sun, it follows from that, that the distance of Mars from the earth at that time, is one-third of the distance of the sun from the earth. If we can by any method find the distance of Mars from the earth at that time, and if we multiply it by three, we shall get the distance of the sun from the earth. Thirdly, the distance of Mars from the earth can be obtained by the simple method of parallax, as in the first method for the moon; and with considerable accuracy. At two observatories, as in Europe and at the Cape of Good Hope, the position of Mars when on the ​meridian, may be compared with some fixed star, the same star being observed at the European Observatory and at the Observatory at the Cape of Good Hope. I am afraid that it will seem that I am dealing in generalities in this matter; but all that I can endeavour to do is to make the principles of the computation intelligible. It is nearly impossible to go into details. The method which I have just described is not, however, the best method, although it has been used with tolerable success.
The fourth method is, by observing the transit of Venus over the sun's disc. The transit of Venus occurs rarely. In explaining this, it was necessary to point out that the orbits of the different planets are inclined to each other, as in Figure 42, where EE' represents the orbit of the earth and VV' that of Venus. You will observe that at V, Venus is considerably elevated above the plane in which the earth moves. If the conjunction takes place at V, that is, if the sun, Venus, and the earth, are nearly in the line at S, V, E, still, however, they are not and cannot be exactly in the same direction; and if a spectator upon the earth looks at Venus, he will see her considerably above the sun. If the conjunction takes place at V', she will be seen below the sun. But if the conjunction takes place at V", Venus, the earth, and the sun, are exactly in a line when the conjunction takes place; at this time a spectator on the earth will see Venus on the sun's face as a black spot. I believe it is visible to the naked eye. With a telescope it is seen extremely well I have seen Mercury, which is a much smaller body, on the sun; and a very beautiful black spot it is.
In the observation of which I am going to speak, it is necessary to know beforehand the time when ​the conjunction will take place, that is to say, the time when Venus will be seen on the sun's face. In eight years Venus goes round the sun thirteen times with very considerable accuracy, but still not with perfect accuracy. Suppose, then that a conjunction of Venus and the earth takes place, at a particular position of the two planets; eight years after that time there will be another conjunction, nearly but not precisely at the same place. In eight years after that there will be another conjunction at a point still more distant than the first, and thus the points of conjunction will recede gradually from V", and it will be a long time before a conjunction occurs again, either at V" or on the opposite side. Venus was seen on the sun's face in 1761 and 1769, at the position opposite to V": she will next be seen on the sun's face in 1874 and 1882, at the position V".
I then proceeded to point out the principle of the method in which these conjunctions are used for determining the linear distance of the sun from the earth. Figures 43 and 44 represent the state of things at a transit of Venus on two assumptions: Figure 43 on the assumption that the distance of the earth from the sun is one hundred millions of miles, and Figure 44 on the assumption that the distance of the earth from the sun is fifty millions of miles. Venus moves in her orbit faster than the earth, and, in consequence, as the earth moves towards a or a’, and Venus moves faster towards v or v’, she will be seen when between the sun and the earth to move across the sun in the direction CD or C'D'. I remarked that, before the observation is made, one linear measure only is known, namely, the size of the earth; but that we do not know the distance of Venus from the sun, or the distance of the earth from ​the sun, but we know that proportion to be as 72 to 100 nearly; and, similarly that we do not know the absolute breadth of the sun, but we know that, whatever the distance of the sun may be, the breadth of the sun bears a certain proportion to that distance, namely, that it is nearly the hundredth part of the distance.
Then I pointed out that if the transit of Venus be observed at two points of the earth, A and B, or A' and B', which are 7000 miles apart, Venus will appear to describe on the sun's face the line CD or C'D' as seen from A or A', and EF or E'F as seen from B or B', and that the distance between the lines EF and CD, or between E'F and C'D', will be the same quantity in linear measure, namely, 18,000 miles, whatever be the supposition as to the distance of the sun; and therefore, as the general position of the lines on the sun's face must be the same on any supposition, and therefore they will cut the edge of the sun's disc at nearly the same angle, the difference of the lengths of the two paths CD and EF, or C'D' and E'F must be the same number of miles; but, as the linear breadth of the sun is not the same on the two suppositions, and therefore the linear length of the lines CD or EF is about double the length of C'D' or E'F, it follows that the difference of their lengths bears a smaller proportion to their whole length in Figure 43 than in Figure 44; and therefore, the difference of the times occupied by the apparent passage of Venus, as seen from A and from B, bears a smaller proportion to the whole time on the assumption of Figure 43 than on that of Figure 44. It is plain that here we have something which will guide us immediately to a decision on the distance of the earth from the sun, if we can but make observations at two stations, as A and B. For, observing with telescopes and clocks the ​entrance and departure of Venus on the sun's face at both places, and therefore ascertaining the whole duration of the passage at both places, and consequently the difference of durations; if that difference bears the same proportion to the whole time as what we have computed on the assumption of Figure 43, then the assumption of distance in Figure 43 is a true one; if the proportion of the difference to the whole time is the same as that computed in Figure 44, then the assumption of distance in Figure 44 is true; if neither of these agree, another assumption may be made which will come near the truth.
I then pointed out that the difference of durations (on any supposition of distance) may be much increased by choosing two stations, one as at A (Figure 45), such that it has nearly turned to the shady side of the earth when the transit is commencing, (or in other words, such that the transit begins shortly before sunset,) and has just turned to the illuminated side when the transit is ending, (or in other words, such that the transit ends shortly after sunrise,) as the former circumstance makes the beginning of the transit earlier, and the latter makes the end of it later, and therefore, the time occupied by the apparent passage along CD, which is the longer, is still more increased; and by choosing another station B, such that the transit begins in the morning and ends in the afternoon, in which case the time occupied by the passage along EF, and which is already the shorter, is still more diminished. And thus, upon any supposition of the sun's distance, the difference of the durations of the transit is increased, and therefore, in comparing the observed difference of durations with the computed difference, a small error of observed durations will be a smaller proportional part of the ​whole, and therefore, the result for the sun's distance will be more accurate. I also mentioned that, in the transit of 1769, Wardhoe and other places in Lapland answered very well to the former of these conditions, and Otaheite and other places in the Pacific answered well to the latter; and that these in fact were the places of observation upon which the measure of the sun's distance principally depends.
I may now mention that, although the principles of the method are stated with most perfect correctness in the explanation given above, and although the process must be thus contemplated by an astronomer, in order to enable him to select stations in the most advantageous positions, yet an astronomer's calculation is not made in that form. His calculation is made entirely by the method of parallax. The process, strictly speaking, is algebraical; but it may be correctly described in the following manner. He assumes a certain value in seconds for the sun's horizontal equatoreal parallax; then from the known proportion of the distances of Venus and the sun, he computes the horizontal equatoreal parallax of Venus (the parallax being greater as the distance is less). Thus, if the sun's horizontal equatoreal parallax be assumed at ten seconds, and if it be known that at that time the distances of Venus and the sun from the earth are in the proportion of 28 to 100, then he must take the horizontal equatoreal parallax of Venus at thirty-five seconds and five-sevenths. Then, from knowing the earth's form, he computes the horizontal parallax of each at the place of observation; and then from knowing the apparent elevation of the sun and Venus, he calculates the actual parallax of each at the time of observation, that is, how much each of them is apparently depressed by parallax. Venus being ​nearer than the sun is apparently more depressed than the sun, and is therefore moved by parallax off the sun's limb if she is lower, or upon the sun's limb if she is higher. From this he calculates, by a troublesome mathematical process, how much the time of Venus' entering or leaving the limb has been either accelerated or retarded by the effect of parallax. And this is done for both observations at every station. And thus he calculates what will be the difference of durations of the transit at different stations. All this is upon the supposition that the sun's horizontal equatoreal parallax has a certain value, say ten seconds. Then he compares the computed difference of durations with the observed difference of durations. If they agree, the sun's horizontal equatoreal parallax is ten seconds. If not, the the value will be found by this proportion. As the computed difference of durations is to the observed difference of durations, so is the assumed horizontal equatoreal parallax (ten seconds) to the true horizontal equatoreal parallax. In this manner it was found that the sun's horizontal equatoreal parallax, when the earth is at its mean distance is eight seconds and six-tenths very nearly; and this corresponds to the distance which I have stated before, of about 95,300,000 miles.
It was supposed by Dr. Halley, when he gave the outline of this method of observation, that the difference of durations could be determined accurately within 1/500 part of the whole difference, and that, consequently, the sun's horizontal equatoreal parallax could be found within 1/500 part of the whole, and therefore that the sun's distance could be ascertained to within 1/500 part. This is a most remarkable degree of certainty, considering how great the distance is, and that it is measured (so to speak) by one step ​only from the dimensions of the earth, which, if the measure were conducted in any ordinary way, would be too small a base for so great a measure.
I said, that the distance of the moon from the earth is 240,000 miles, and that if a railway carriage were to travel at the rate of 1000 miles a day, it would be eight months in reaching the moon. But that is nothing compared with the length of time it would occupy a locomotive to reach the sun from the earth; if travelling at the rate of 1000 miles a day, it would require 260 years to reach it.
I will now proceed with explanations of some of the higher branches of Astronomy, which, though difficult, will be found most valuable: first, as instances of very important applications of the principle of gravitation; secondly, as showing the nature of some of the corrections to observations, which it is necessary to understand, in order to see clearly the different steps that must be made before we can arrive at a measure of the distance of the fixed stars.
of separate particles: attracting every particle, independently of the others. It is essential to bear this in mind. The next thing is, that it attracts every particle with a force which depends upon the distance of the attracted particle from that body which is the cause of attraction; and the nearer that body of the more strongly the particle is pulled by the attracting body. You will see then, that the sun attracts those, parts of the earth next to it with a greater force than those parts near the centre of the earth. At A, Figure 46, the sun's attraction is stronger than at C, and therefore the sun is always acting upon the part A nearest to it, as if it were pulling it away from the earth's centre. This is not merely because the whole force which the sun exerts upon A is directed towards S, because if the sun pulled the centre and the surface of the earth equally, it would not tend to separate them; but it is because it pulls the part at A more than it pulls the central part C, and thus it tends to pull it away from the centre of the earth ​towards S. In like manner, if the sun were to pull the centre C of the earth and the part B with equal force, it would not tend either to push B towards the centre or to draw it away from the centre; but, as it pulls the centre more powerfully than it pulls B, it does tend to separate them, not by pulling the opposite side B from the centre, but by pulling the centre from the opposite side B. The general effect of the sun's attraction, therefore, as tending to affect the different parts of the earth, is this: that it tends to pull the nearest parts towards the sun, and to push the most distant parts from the sun.
If the earth were a perfect sphere, this would be a matter of no consequence—it would produce tides of the sea, but it would not affect the motion of the solid parts. But the earth is not a sphere; it is flattened like a turnip, or has the form of which I have spoken to you under the description of a spheroid. Moreover, the axis of the earth is not perpendicular to the ecliptic; the earth's equator is inclined to the line joining the earth's centre with the sun at all times, excepting at the equinoxes.
Let us now consider the position of the earth at the winter solstice, represented in Figure 46. The North Pole is distant from the sun, the South Pole is turned towards the sun. This spheroidal earth, at this time, has its protuberance, not turned exactly towards the sun, but elevated above it. As I said before, the attraction of the sun is pulling the part D of the earth more strongly than it pulls the centre. What is the tendency of that action? The immediate tendency of that action is to bring the part D towards a, supposing a to be in the horizontal circle passing through S. In like manner, in consequence of the sun attracting the centre of the earth more than it ​attracts the protuberance E, which, amounts to the same thing as pushing the protuberance E away from the sun, there is a tendency to bring E towards b. It is very important that you should see this clearly: that if I were standing in the place S where the sun is, and if I had a line fastened by a hook to the place D, and if I pulled it, I should tend to bring that part towards a. The immediate tendency of this pull, therefore is, so to change the position of the earth that its axis will become more nearly perpendicular to the plane of the ecliptic. You might suppose then, that the effect of that pulling will be to change the inclination of the earth's axis to the line which connects the earth and sun. No such thing; the effect is entirely modified by the rotation of the earth. Undoubtedly, if the earth were not revolving, and if the earth were of a spheroidal shape, the attraction of the sun would tend to pull it into such a position that the axis of the earth would become perpendicular to the line SC; or (if in the position of the winter solstice) it would become perpendicular to the plane of the ecliptic; but, in consequence of the rotation of the earth, the attraction produces a perfectly different effect. Let us consider the motion of a mountain in the earth's protuberance, which, passing through the point c on the distant side of the earth, would, in the semi-revolution of half-a-day, describe the arc c D e, if the sun did not act on it, (c and e being the points at which this circle c D e intersects the plane of the ecliptic, or the plane of the circle a b that passes through the sun). While the protuberant mountain is describing the path c D e it is constantly nearer to the sun than the earth's centre is; the difference of the sun's actions therefore tends to pull that mountain towards S, and, therefore, (as it cannot be separated ​from the earth,) to pull it downwards, giving to the earth such a tilting movement as I have already spoken of; it will therefore, through the mountain's whole course, from c, make it describe a lower curve than it would otherwise have described, and will make it describe the curve c f g instead of the curve c D e. What will the result of that be? As it mounts from c to f, the sun's downward pull draws it towards the ecliptic., and consequently, renders its path less steep than it would otherwise be. At f, it will be a very little lower than it would otherwise have been at D; but as the sun's downward pull still acts upon it till it comes to g, the steepness of its path between f and g is increased more than belongs naturally to its elevation at f, and becomes in fact the same as it was at c, or very nearly so; so that the inclination of the path to the plane of the circle a b is the same as at first. But instead of crossing the circle a b at e, it will cross at g: in other words, it will, in consequence of the sun's action, come to the crossing place earlier than it would have come had the sun not acted. Now, consider what will be the motion of this protuberant mountain in the remaining half of its rotation, from g towards c again. In this part of its rotation, it is further from the sun than the earth's centre is; therefore the sun's action does in fact tend to push it away (as I have already explained); and as it cannot be separated from the earth, this force tends to push the mountain upwards towards the circle b, tilting the earth in the same direction as before; the mountain therefore, in this part, will move in a path higher than it would have moved in if not suffering the sun's action, and therefore it will come to its intersection with the circle a b sooner than it would if not subject to the sun's action. The inclination of ​its path, (just as in the former half of its rotation) will not be altered. Thus the effect produced by this action of the sun, in both halves of the rotation of this mountain, is, that it comes to the place of intersection with the plane of the circle a b, or with the plane of the ecliptic, sooner than it otherwise would. And whatever number of points or mountains in the protuberant part of the earth we consider, we shall find the same effect for every one; and therefore, the effect of the sun's action upon the entire protuberance will be the same; that is, its inclination to the circle a b or the plane of the ecliptic will not be altered, but the places in which it crosses that plane will be perpetually altering, in such a direction as to meet the direction of rotation of the earth.
I have spoken of this as if the protuberance were the only part to be considered. But in reality, this protuberance is attached to the remaining spherical part of the earth; and the action of the sun on the different masses of that spherical part balances exactly; so that, as regards it, we need not consider the sun's action at all The effect therefore of that spherical part will be, to impede the motion which the protuberant part would otherwise have; not to destroy it, but to diminish it.
On the whole, therefore, the effect of the sun's action on the spheroidal earth will be, that the points at which the earth's equator intersects the plane of the ecliptic move very slowly in the direction opposite to that in which the earth revolves; but the inclination is not altered.
All that I have said here applies to the position of the earth at the winter solstice. But if we consider the earth at the summer solstice, as in Figure 47, we shall find that the effect of the sun's action is exactly ​the same. The sun's greater attraction upon any part of the protuberance when nearest to the sun, at which time that part is, in the case of Figure 47, below the ecliptic, tends to raise it towards the plane of the ecliptic, and therefore it cuts the ecliptic sooner than otherwise it would. And the sun's smaller attraction upon it when furthest from the sun, producing the effect of a pushing force upon it when above the ecliptic, tends to made it descend to cut the plane of the ecliptic sooner than it otherwise would, and therefore, in both halves of the diurnal rotation (as at the winter solstice) the place of intersection of the earth's equator with the ecliptic will move in the direction opposite to the earth's rotation.
At the equinoxes, the plane of the earth's equator passes through the sun, and then the sun's action does not tend to tilt the earth at all, and consequently does not tend to alter the position of its equator at all; but at all other times the sun's action produces a motion, greater or less, of the intersection of the earth's equator with the plane of the ecliptic, in a direction opposite to the direction of the earth's rotation. And this is the motion called the Precession of the Equinoxes.
It is to be observed, that the principal part of precession is not produced by the sun, but by the moon. The moon's mass is not a twenty-millionth part of the sun's; but she is four hundred times as near as the sun. Still she does not pull the earth, as a mass, with more than a hundred-and-twentieth part of the sun's force. But, because the difference of the distances of the different parts of the earth from the moon bears a greater proportion to the whole distance than for the sun, the differential effects of the moon in pulling the near parts of the earth from the earth's ​centre, and in pushing the distant parts of the earth from the earth's centre, is about treble the effect of the sun. And as the precession depends entirely on this differential effect, the precession produced by the moon is about treble that produced by the sun.
It will be well for you here to consider the consequences of this precession on the position of the earth's axis. In Figure 48, after a certain number
Fig. 48.
of days or years, the position of the earth's equator has changed from e f h to g k l its inclination to the circle a b in the plane of the ecliptic remaining the same as before. The earth's axis of revolution must be always perpendicular to the plane of the earth's equator. From this it will be seen that the earth's axis has changed its position from such a direction as CP to such a direction as Cp. If we draw a line CQ ​perpendicular to the plane of the ecliptic, the inclination of the earth's axis to CQ will be the same as the inclination of the earth's equator to the ecliptic; and that, as we have already seen, undergoes no alteration. Consequently the inclination of CP or Cp to CQ is always the same. Therefore, we may represent the motion of the earth's axis by saying that it turns slowly round an axis perpendicular to the ecliptic, but keeping the same general inclination to it, in the direction in which the hands of a watch turn, (as viewed from the outside of a celestial globe,) or in what astronomers call a retrograde direction. Now, the Pole of the heavens is the point in the heavens to which the earth's axis is directed, and therefore that Pole is not absolutely invariable, but turns slowly in a circle in a retrograde direction (or in the same direction as the hands of a watch, as viewed from the outside of a celestial globe) round another point to which the line CQ is directed. The latter point is called the Pole of the ecliptic.
This motion of the earth's axis admits of a most remarkable illustration in the motion of a spinning top; and the more remarkable because the forces which act in that case are of an opposite character to the forces which act on the earth, and the effect which they produce is of an opposite character to the effect produced on the earth. The earth's axis being inclined to the line CQ, in Figures 46, 47, and 48, we have seen that the immediate tendency of the sun's force upon the earth is in all cases to bring the earth's axis CP nearer to CQ, and that if the earth had no motion of rotation, this force would bring the earth's axis CP nearer to CQ; but that in consequence of the earth having a motion of rotation, the effect really produced is, that the earth's axis CP revolves slowly round CQ ​in the direction opposite to the direction of rotation.
Now, in Figure 49, let CP be the axis of a spinning
top, CQ the vertical line; the immediate tendency of the force of gravity is to bring the axis CP further from CQ (or to make the top fall) and if the top were not spinning, it would make CP recede further from CQ; but it will be found that, in consequence of its spinning, the inclination of CP to CQ does not sensibly alter (till the spinning motion is retarded by friction), but CP revolves slowly round CQ in the same direction as the direction of rotation. So that if Figure 50 represent a view of the top from above, if the top be spun in the direction marked by the arrow, its axis will reel in the direction PRS. This, as I have said, is strictly analagous to the precessional motion of the earth, although, from the immediate tendency of the forces being of an opposite kind, the ultimate effect is of an opposite kind.
The following astronomical effects of this precession deserve attention.
First, the celestial equator, (which, as I said in a former lecture, is the great circle in the heavens, which at every point is 90 degrees from the Pole,) is in fact ​in the same plane as the equator of the earth, whose axis (perpendicular to its equator, or making an angle of 90 degrees with every part of the equator) is directed to the Pole. Therefore, as the earth's equator changes its intersections with the plane of the ecliptic, the celestial equator also changes its plane and changes its intersections with the ecliptic. These intersections, as I mentioned, are called the first point of Aries and the first point of Libra: and one of the co-ordinates by which the place of a star or other body is defined, technically called the Right Ascension, is the interval of time between the passage of the first point of Aries over the meridian, and the passage of the star or other object over the meridian. But as this first point of Aries travels to the right, it passes the meridian every successive year earlier (with respect to the stars) than it would have done if it had been stationary; and therefore, the right ascensions of stars (for the most part) increase a little every successive year.
Secondly, as the place of the Celestial Pole changes from year to year, the North Polar distances of the stars change from year to year; some of them increase and some diminish. The annual amount of precession, although a formidable quantity in delicate astronomical observations, is a very small quantity for ordinary observers.
The annual motion of the first point of Aries is about fifty seconds in a year: it will require about 26,000 years to perform the entire revolution. The change in the distance of a star from the North Pole does not in any case amount to twenty-one seconds in a year. But these are quantities so large that we must be perfectly acquainted with their laws and magnitudes when we treat of small changes in the places of stars not exceeding one or two seconds.
​The next thing which I have to mention, as one of the calculations which must be applied to observations in order to obtain an accurate result from them, is the nutation of the earth's axis. The nutation of the earth's axis may be described as arising in this way. I have explained that the action of the sun will produce a motion in the earth's equator and in the position of the earth's axis, principally at and near to the times of the winter solstice and the summer solstice, when either the North Pole or the South Pole of the earth is turned towards the sun. It matters not whether the North or South Pole is turned towards the sun, for the tendency is to produce equal motion of the axis, and in the same direction in both cases. But at the time of the equinoxes, when neither Pole is turned towards the sun, the sun's attraction has no tendency to produce precession. From this you may easily see that the precession of the equinoxes goes on more rapidly in summer and in winter than in the intermediate months; there is therefore a certain irregularity in the precession, and this irregularity is one part of the solar nutation. The other part arises from the circumstance that the inclination of the earth's axis to the ecliptic though unaltered on the whole, yet suffers slight changes from one instant to another; as clearly appears from the foregoing explanation.
I stated also that the moon produces a considerable part of the precession. There is a very small irregularity of precession produced by the moon in different parts of her monthly revolution, similar to that produced by the sun in different parts of his apparent yearly revolution. But the principal irregularity arises in this manner. The moon does not move in the plane of the ecliptic, but in an orbit inclined to the plane of the ecliptic; and this orbit (from an effect ​of the sun's attraction, exactly similar to the motion of the earth's equator produced by the sun's attraction) moves so as to change the place of its intersection with the ecliptic, performing a complete revolution in 19 years. Therefore, during nearly half of this time, the moon's orbit is inclined to the ecliptic, in the same way as the earth's equator, (but not so much,) as at m n, Figure 51; and therefore the moon's path is
Fig. 51.
little inclined to the earth's equator. And during nearly the other half of the time, the moon's orbit is inclined to the ecliptic in the way opposite to the earth's equator, as at p q, Figure 51, and the moon's path is much inclined to the earth's equator. In the former state, the moon's force to tilt the earth is small, and the precession goes on slowly; in the latter, the moon's force to tilt the earth is great, and the precession goes on rapidly. The consequence of this is, that there is a very sensible irregularity in the motion of the earth's axis every 19 years, and the place of the Pole is irregularly shifted in different ways through more than 18 seconds. This is lunar nutation. As the right ascensions and North Polar distances of all stars and other objects are affected by this irregular disturbance of the Pole, it is necessary to take it into account in comparing observations made at one time with observations made at another time.
These corrections, then, (precession and nutation,) ​are two of the irregularities of which it is necessary to take account in obtaining accurate results from observations; but there is a third, which is greater in its magnitude than nutation, and totally different in its nature. It is the aberration of light. It was long ago made out that vision is produced by something coming from the object to the eye, and that this something comes from the object to the eye with a definite velocity. Now, in consequence of this light coming from the object to the eye with a definite velocity; and in consequence of the earth's moving with a definite velocity; by the combination of these two things, there is produced a disturbance in the visible place of every object not connected with the earth which we look at. Perhaps one of the simplest ways of giving an idea of the effect of this combination, in relation to the aberration of light, will be to refer you to the chance experiment which suggested the theory of aberration to one of my predecessors (Dr. Bradley), by whom in fact, the aberration of light was discovered and reduced to law. He says, he was being rowed on the Thames, in a boat which had a small mast with a vane at the top. At one time the boat was stationary, and he observed, by the position of the vane, the direction in which the wind was blowing. The men commenced pulling with their oars, and he observed that, at the very time they commenced pulling, the vane changed its position. He asked the watermen what made the vane change its position? The men said they had often observed the same thing before, but did not pretend to explain the cause. Dr. Bradley reflected upon it, and was led by it to the theory of aberration of light. I may here offer a slight illustion of it, which every person may observe if he walks out on a rainy day. If you can choose a day when ​the drops are large, then if you stand still for a moment, and observe the direction in which the drops are falling, when there is little or no wind, you will see that the drops fall vertically downwards; but if you walk forward, you will see the drops fall as if they were meeting you; and if you walk backward, you will immediately observe the drops of rain falling as if they were coming from behind you. This is an accurate illustration of the principle of the aberration of light, I will now offer another.
Fig. 52.
In Figure 52, let A be a gun in a battery, from which a shot is fired at a ship DE that is passing. Let ABC be the course of the shot. The shot enters the ship's side at B, and passes out at the other side at C. But in the meantime the ship has moved from the position DE to the position d e, and the part B where the shot entered has been carried to b. Now, if the ship were a sentient and reflecting being, when it perceived that the path which the shot made through it, entering at b and going out at C, was in the inclined direction b C, it would say, "The shot came from somewhere ahead." You will see in this, the effect of the combination of the movements; that the shot appears to have come from a part further ahead than it would have seemed to come from if the ship had been at rest. And you will also see that the inclination of the apparent direction of the shot b C to the true direction BC depends on the proportion of b B to BC, that is, on the proportion of the velocity of the ship to the velocity of the shot. The greater ​is the velocity of the shot, the smaller will be the space b B described by the ship while the shot is passing across her, and therefore the smaller will be the angle b CB between the apparent direction of the shot and its real direction.
The same thing happens with regard to the effect of the motion of the earth on the apparent path of light, and it will produce an apparent change in the places of the stars. And if we find that there is such an apparent change, it will be a certain proof that the earth is in motion; but if we find the change to be small, it will prove that the velocity of light is much greater than that of the earth.
Now I will point out to you the visible effect of the aberration of light upon the place of a star. The immediate interpretation of the consideration which I have mentioned is this. In whatever direction the earth is moving, the apparent position of any star which we are looking at, is displaced in the direction towards which the earth is moving. In Figure 53, let C be the sun, E',E",E"',E"", the earth in four successive positions of its orbit (viewed in perspective), its motion at each place being in the direction of the arrow drawn there; S the true place of a star. Then, in consequence of the aberration, when the earth is at E', as its motion is in the direction of the arrow drawn from E', the light coming from the star will enter the eye of a spectator or the tube of a telescope, not as if it came from S, but as if it came from s′, the line Ss’ being parallel to the arrow at E'; and therefore the observer, when the earth is at E', does not see the star at S but at s’. In like manner, when the earth is at E" he sees the star at s"; when at E"' he sees the star at s′″; and when at E"" he sees the star at s″″. Thus you will see that, in ​every position of the earth, the star's place is affected by the aberration of light; and from this cause every
Fig. 53.
star apparently describes a small circle every year parallel to the earth's orbit. It is a minute circle; its angular diameter, as seen without any foreshortening, is found to be about forty seconds. This quantity is, however, so serious that it cannot be omitted in the computation of any observation whatever.
From the measure of the apparent semi-diameter of the small circle described by the star, corresponding to the angle BCb in Figure 52, we are able to compute the proportion of the earth's velocity to the velocity of light; and we find that the velocity of light is about 10,000 times as great as the earth's velocity in its orbit, or about 200,000 miles in a second, In other words, light travels a distance equal to eight times the circumference of the earth between two beats of a ​clock. This is a prodigious velocity, but the measure of it is very certain.
These three quantities, (precession, nutation, and aberration,) are the corrections to a star's apparent place, which it is necessary for us to take into account in every observation of a star, at whatever part of the earth it is observed; and besides these, it is necessary at every place to apply the proper correction for refraction, which may be different at every different place of observation. Having obtained these elements of calculation, we can proceed at once with the measure of the distance of the fixed stars.
In Figure 54, let E',E",E'",E"", be four positions
Fig. 54.
of the earth in its orbit (seen in perspective), P a place of observation, S a star, (the earth being in such a ​part of its rotation that the meridian of P passes through the star); also let S' be another star in the plane of the earth's orbit, and in the direction corresponding nearly to the earth's solstitial position. And suppose (in conformity with the assertion that we have made all along, but which we shall now subject to the severest proof) that the earth's axis remains strictly parallel to itself in its motion round the sun, with no other motion than those which we have described as produced by precession and nutation. And suppose that with a mural circle at P we observe the zenith-distances of the stars S and S' when they pass the meridian of P, and apply the proper corrections for refraction; and then, by applying corrections for the effects of aberration, we find the place in which the star would have been seen if unaffected by the earth's velocity: and by applying corrections for precession and nutation, we find the zenith-distances which the stars would have had if the position of the earth's axis had not been affected by precession and nutation. Now, if our assumption (that the earth's axis has no motion but those depending on precession and nutation) be correct, the result of the observation of the star S', whatever be its distance, will be, that its corrected zenith-distance when observed on the meridian will be the same whether the earth be at E', E", E'", or E"". This is found to be strictly in agreement with the results deduced from actual observation, so that it is certain that the earth's axis has no motion but those depending on precession and nutation. Moreover, for the vast majority of stars in all parts of the heavens, when the same corrections are applied, the corrected meridional zenith-distances are found to be the same whatever be the position of the earth in its orbit; ​and this proves, both, that the earth's axis has no motion except those of precession and nutation, and that the stars are at an inconceivable distance.
But there may be other stars as S, whose distance we have some reason for conjecturing to be not so enormously great. Now, the only way in which we can measure its distance is one strictly analogous to that used for measuring the distance of the moon; with this difference, that we cannot observe from two places at once. On account of the immense distance of the stars, it would be necessary to observe the place of the star from two positions, as far distant as the breadth of the earth's orbit; but we cannot do that. We can, however, observe the position of the star from the earth when the earth is in two positions, as E' and E"', on opposite sides of the earth's orbit; that is, at times half a year apart.
I have used, as an elucidation of parallax, the effect of the two eyes in the head. If you have your head in any fixed position, and you shut one eye, you cannot determine accurately the distance of an object; but if you open both eyes the distance is seen immediately. But with one eye a person can judge of distance very well if he moves his head. In like manner, one observer on the earth can observe the distance of a star, provided he takes advantage of the change of places at different times; that is, provided he allows his eye to be moved round for him by the revolution of the earth round the sun; it is, however, necessary for us to be fully possessed of every element for correction of the star's place, so as to clear it of every source of change, except the difference of apparent place depending on the star's distance and the earth's place in its orbit. This is the reason why I have deferred the mention of this measure until I had ​mentioned the subjects of precession, nutation, and aberration. The star's places most sensibly change from circumstances unconnected with the parallax. We must know these accurately beforehand; and knowing these accurately beforehand, we proceed as follows.
We observe the star with the mural circle while the earth is in the position E'. We apply the corrections for precession, nutation, aberration, and refraction; and we shall know what the corrected position of that star should be half a year hence, as observed when the earth is at E'". Now, suppose we go to this second state of things, and when the earth is at E"', we observe the meridional zenith distance of the star, we correct it for refraction, precession, nutation, and aberration. Now, do these two corrected zenith-distances agree? Are the stars (after all these corrections are applied) seen exactly in the same direction when the earth is at E' and when it is at E'"? All calculations for these accidental causes of disturbance being affected, the result is this:—for the vast majority of stars we do not discover any sensible difference; the difference is, at any rate, exceedingly small; the stars are so far off that, for the vast majority of them, we can see no difference in the directions of the line E'S and the line E'"S. There are some stars, however, that are not at so great a distance, so that the inclination of these lines to each other can be ascertained; but the angle is exceedingly small, and is measured with much difficulty. In the southern hemisphere, there is the bright star of the Centaur, (Alpha Centauri,) for which it would seem that the inclination of the two lines from the opposite sides of the earth's orbit to the star, is an angle of two seconds and no more. ​An angle of two seconds is that in which a circle 6/10 of an inch in diameter would be seen at the distance of a mile. This is the star which shows the greatest parallax of all. The parallax of the bright star of Lyra is not more than a quarter of a second. Struve, at the Observatory of St. Petersburgh, has deduced, as he thinks, from observations, that for stars of the second magnitude the general average of parallax is 1/10 of a second. This is so small an angle that it is almost impossible to answer for it. Supposing, however that it is 1/10 of a second, then the distance of the star from the sun is two million times as great as the distance of the earth from the sun. It seems almost inconceivable that we should be able to measure, even in a rough way, a distance so great.
I will only mention one more thing. There is one correction upon which I said there was a little doubt, and that is that troublesome thing, refraction. It is one of those things which throw a doubt upon every observation of a delicate kind. Refraction enters here, because we must necessarily observe the zenith-distance of the star; and in comparing observations of zenith-distance at opposite times of the year, there is this unfortunate circumstance: the same star which is observed on the meridian in the day-time at winter, will be seen on the meridian at night in the summer; or the star which is observed in the night in the winter, will be observed in the day-time in the summer, when the state of the air is very different; so that the amount of refraction at the two observations will be very different, and we cannot determine the correction to the zenith-distance accurately, so as to answer for 1/10 of a second between the observations. Under these circumstances, this determination of a difference between the observations at E and E'", ​amounting to only 2/10 of a, second, is more than I can undertake to answer for. In consequence of that uncertainty, another method has been introduced, admitting of far greater accuracy; it is "by comparing two stars whose declinations are nearly the same. And here we fall upon another method, very similar to that which is used for measuring the distance of the moon. Suppose we have two stars S and S", and suppose that the star S" is at such an immeasurable distance that we cannot see in it any change of position. But suppose that I think it possible that the star S has a sensible parallax, these two stars being seen nearly in the same direction. I have already mentioned that we have obtained the parallax of the moon with the greatest accuracy, by comparing it with a fixed star, which is seen nearly in the same direction. We get rid of the uncertainty of refraction in this case, as the moon and the star are seen near to each other, and are therefore affected with almost exactly the same refraction. In like manner, if we compare a near star with a distant star, seen nearly in the same direction, we get rid of the uncertainty of refraction; and we also get rid of precession, nutation, and aberration; because they produce sensibly the same effect on both stars. Now, if I suppose S to be near, and S" to be at such an enormous distance that it will have no sensible parallax, then when the earth revolves round the sun, I have only at E' to observe the angle S"E'S between the two stars, and then in another position E’” to observe the angle S"E"'S between the two stars; and because E'S" is sensibly parallel to E'"S", the difference between these two measured angles, is the angle E'SE"'.
This is the method which the celebrated Bessel, of Königsberg, used for determining the distance of the ​small star, known by the name of No. 61, in the constellation of Cygnus. He found that the change in the place of that star, as viewed from E' or from E'", produced by parallax, is about 6/10 of a second; and this corresponds to a distance of 660,000 times the radius of the earth's orbit, or 63,000,000,000,000 miles. Enormous as this distance is, I state it as my deliberate opinion, founded upon a careful examination of the whole of the process of observation and calculation, that it is ascertained with what may be called in such a problem, considerable accuracy.
The distance of the stars of the second magnitude, founded upon Struve's conclusions to which I have already alluded, is not far from two millions of times as great as the distance of the sun from the earth. In this determination I have much less confidence. The distance of Alpha Centauri, if reliance may be placed on the observation, is only two hundred thousand times as great as the distance of the sun from the earth.