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Recapitulation of Lecture II.—Apparent motion of the planets.—Greek Theory of Planetary Motions.—Epicycles, Deferents, &c.—Copernican Theory.—Kepler's Elliptic Theory.—Theory of Central Forces.—Laws of Motion.—Composition and Resolution of Forces.—Motion of a Planet in its orbit deduced from these laws.—Measure of distance by Parallax.

The subject of the lecture of yesterday evening was the dimensions, the figure, and the rotation of the earth. I then thought it necessary to put before you some details of evidence relative to the rotation of the earth; and in again entering upon that subject, but not exactly in the same order in which I took it last night, I shall feel it necessary to remark that it is highly important, in beginning a subject like this, to divest yourselves as far as you possibly can, of notions acquired in the ordinary routine of education. Every person here has, without doubt, been brought up in the belief that the earth is in motion; and because they have had this belief instilled into their minds from their earliest infancy, they may have concluded that it is necessarily and obviously true. This is a thing most dangerous, and instances are not wanting to prove that in every branch of science, absurdities have arisen from it. I may mention one which just at this moment occurs to my mind, and which influenced the mind of one of the earliest philosophers of the Royal Society of London. At the beginning of that Society, as at the beginning of most Societies, although some care might have been taken that no absurdities should creep in, it was difficult to avoid them entirely. And in a paper which was intended to prove the truth of the Copernican theory, for which purpose the writer (I do not remember why) thought it necessary to prove that the stars are at tolerably equal distances, he begins by establishing the latter proposition by means of the following assumption: "Now we all know that hell is the centre of the earth." It seems perfectly absurd at the present time that anybody should start with a proposition like that to work out a physical theory. Yet it is equally absurd to assume at once that the earth is in motion, and for that reason I have, been anxious to convey to you the evidence by which it is proved generally that the earth is in motion. And I shall now proceed to recapitulate, in as few words as I can, the main points of what was said yesterday in regard to the earth.

I endeavoured to point out to you the method of measuring the earth, and I told you that we wanted the means of measuring hundreds or thousands of miles. In some instances it is obvious that to measure a long meridional arc, in the most direct line that the earth's curvature permits, is an impossibility. The way is, to measure a short line which I call the base line, being a few miles in length; and great trouble is necessary to give even to this measurement the requisite accuracy. When the base is measured, we plant theodolites at its two ends, and by means of these we observe a signal upon a hill, or any other distant place. It is then usual to carry a third theodolite to the signal station: this is not absolutely necessary, but it is done as a matter of prudence, to verify the observations in case of suspicion of error. Now, having the base of the triangle, and the two angles next that base, there is then no difficulty in laying them down on paper, or in calculating the other sides of the triangle. Then we may use one of these computed sides as a measured base, and if from its two ends we can see some other signal, we can observe it with our theodolites, and compute its distance in the same manner: and in this way the triangulation goes on. Thus, in Figure 17, a series of triangles was formed, extending from Shanklin Down, in the Isle of Wight, to Clifton, in the South of Yorkshire.

I called your attention particularly to the remark, that this is the first instance in which we use the yard measure, which is done by actual application in measuring the length of the base, and by computation from this in measuring the length of every one of the sides of the triangles; and thus we do really get the different distances in the triangulation, by the use of a yard measure.

I then mentioned to you that, supposing we had extended the survey over a very long distance, the next thing was to make use of the Zenith Sector, Figure 19, which consists in its important feature, of a telescope with a graduated arc CE attached to it, turning on two pivots AB, and with a plumb-line suspended from, or passing over, one pivot B, and crossing the graduated arc. (The Mural Circle may be used for the same purpose, but the Zenith Sector is rather more convenient.) We have then to consider that, whatever the form of the earth may be, using the expression as applying to the fluid part of the earth, we must suppose also from the nature of a fluid, that the direction of a plumb-line is perpendicular to the surface, and therefore, if we suppose the earth to be fluid, the plumb-line will be always perpendicular to its surface. If, then, we plant the Zenith Sector at A, Figure 18 or 20, the plumb-line will hang in a direction perpendicular to the surface at A. But if at B, the plumb-line must hang in the direction perpendicular to the surface at B: therefore if at A we observe a star nearly overhead, then the plumb-line will fall over the point G of the arc; but if we carry the Zenith Sector to B, and turn the telescope to the same star, the plumb-line will fall on the point g of the arc. Inasmuch, therefore, as the telescope, from being directed to the same star, which is excessively distant, takes the same direction in different places; and, inasmuch as the plumb-line takes different directions in different places; by means of these we get the variable positions of the plumb-line referred to the invariable position of the telescope. I then called your attention to Figures 20 and 21, and said, if we suppose the vertical lines at A and B to be carried down till they meet at H, the angle made by these two verticals, or by the two plumb-lines, would be the difference of the Zenith-distances of the star as observed at A and B; that is to say, the difference of the two angles made by the telescope with the plumb-line, first at A, and secondly at B. Having got the angle of these two lines, AH and BH, and the length of the line AB which connects their ends, we are enabled to calculate the length AH or BH, or the number of miles of distance of their intersection H. This is, in point of fact, the semi-diameter which must be taken in order to sweep the curvature of the arc AB; or, if you please, we may put the result in this shape: we may say that, having travelled 830 miles, we find the inclination of the verticals to be 12 degrees, and therefore we should have to travel 69 miles to make the inclination of the plumb-lines one degree, and that is commonly expressed by saying a degree on the earth's surface is equal to 69 miles.

I then pointed out to you the principal lines which have been accurately measured. All these lead to the conclusion, that towards the Poles of the earth you have to travel 69½ miles in order to pass over the space where the direction of the vertical changes by one degree, but that near the equator you have to go only 68¾ miles, in order to pass over the space where the direction of the vertical changes one degree.

Now, I call your attention to the interpretation of this circumstance: it shows that the earth's dimension is greater in the direction of the equatoreal axis, as shown in Figure 21. It is necessary to consider that the direction of the vertical is not to the centre of the earth—it is perpendicular to the surface; and the intersection of the two verticals at H or h does not give the distance from the centre, and does not depend on the distance from the centre, but on the curvature at each place. And inasmuch as, when near the Pole, you have to travel the greater distance in order to go through the same change of the direction of the earth's surface, it proves this: that the earth is less curved at the Pole than near the equator, and that you come to a shape something like Figure 21. About AB the surface is comparatively flat; about ab the curvature is sharpened; and at the Cape of Good Hope, or about A′B′, it is flattened again. So that we come to the conclusion, so far as our measures go, that the form of the earth is somewhat turnip-shaped, or is what we call an oblate spheroid.

But there is another kind of evidence derived from the measure of arcs of longitude, obtained by ascertaining the difference of time at which the transits of stars are seen at different places. Thus, in Figure 22, the star S is not on the meridian at the two places K and L at the same moment of time. Now, we want to measure the difference of time at which a star passes over the meridian at the two places: and this is done by using some means of comparing the times of the two clocks at the two stations—thus ascertaining how much one clock is before or behind the other. And inasmuch as we have the transit instrument, we can determine the absolute times at which the same star passes at both stations; so that by observing transits of the star with the clock at one place K, and by observing transits of the same star with the clock at the other place L, and comparing the clocks, we have the means of ascertaining the absolute difference of time of transit; and when we have done that, we can tell how great a fraction of the revolution of the earth has been performed. Clocks may be compared by observation of instantaneous signals, such as the flashes of gunpowder fired on elevated stations. There is one long arc, commencing in the neighbourhood of Padua, in Italy, crossing the Alps, and terminating at Marennes, near Bordeaux, in France. Intermediate places, A,B,C,D, &c., were chosen for clock stations, in such positions that one set of signals on an intermediate mountain could be seen both at A and at B; another set of signals on another mountain could be seen both at B and at C, and so on to the end of the arc; and thus clock A was compared with clock B, clock B with clock C, &c., to the end of the arc. Another method of performing the same operation is that which has been used on the arc from Greenwich to Valentia, in the South-west of Ireland. Thirty chronometers were carried backwards and forwards twenty-two times, and in this manner the clocks were compared with great accuracy. Surveys have also been carried on by triangulation, connecting the extreme east and west stations (as Padua and Marennes, or Greenwich and Valentia) on the same principles as the surveys in the north and south direction. Thus, then, from the comparison of the clocks, and the observations of transits, we have the means of knowing the fraction of a revolution which the earth has performed, from the time when a star passes over the meridian of one place, to the time when the same star passes over the meridian at another place. Thus the difference of these times at Greenwich and Yalentia was found to exceed 41 minutes 23 seconds. Now, the problem becomes this: if in 41 minutes 23 seconds so many miles pass under the meridian of the star, how many miles will pass under the meridian in 24 hours? This is a mere question in the rule of three. The whole girth of that particular part of the earth may thus be obtained.

We have thus got the measures of the meridian in various parts, giving us the length which it is necessary to go for a degree; we have got two grand measures of parallel (as they are called), and also some smaller ones, giving us the girth of the earth in different parts. The question then is, what sort of figure do they belong to? Do they belong to a spheroid? Upon trying this we find they do belong to a spheroid, so that by giving certain dimensions to the spheroid, the measures of all the different arcs will be very well represented. The diameter passing through the Poles must be about 7899 miles; that passing through the equator about 7926 miles. Therefore we may consider it as established, that the form of the earth is spheroidal.

The next thing we have to consider is, what inference we are to draw from that, in reference to the movement of the earth. By a rough experiment it was shown, that if we take any circular substance that is susceptible of a change of shape, and whirl it round an axis, it will change from a circle into an oval; we think, therefore, that even supposing we had nothing else to guide us, there is good reason to infer, from the oval shape of the earth, that it does turn upon its axis. But, in addition to this, we see the sun, moon, and stars, every day turning from east to west. We know (by the duration of the lunar eclipses) that the distance of the moon is considerable; and (by the fact, that solar eclipses and occultations of stars do not extend over all the earth) that the distances of the sun and stars are very much greater; we also see that the system of stars appears to move all in a piece; we judge it unlikely that these distant bodies should thus revolve round the earth every day. And we conclude that the apparent movement is caused by the earth's turning from west to east. It is worth mentioning, that the planet Jupiter, the largest planet of the system, turns visibly round its axis in a shorter time than the earth. You may suppose, then, that Jupiter is much more flattened by the velocity of his rapid rotation than the earth; and indeed you can see it at once with a telescope without the aid of a micrometer, the equatoreal being to the Polar diameter as 16 to 15, nearly—a proportion which makes the former measure 5000 miles greater than the latter.

I then spoke of the apparent movement of the sun amongst the stars; and in speaking of their movements, I endeavoured to impress upon you how much you can observe for yourselves. You can learn more by your own observations than by the Lectures I can deliver, or by all the books you can read. In speaking of the apparent motion of the sun amongst the stars, I told you that in summer the sun is longer above the horizon and goes higher than in winter; the sun also rises more to the north in summer than in winter. Thus it describes a daily circle nearer to the North Pole of the heavens in summer than in winter; or, in other words, its place among the stars is nearer to the North Pole in summer than in winter. But there is also another set of facts, though not quite so familiarly known as these, which you can observe for yourselves. If you watch the appearance of the stars at a certain hour every night, you will find that these stars are to be found in a position on the night of one month, different from the position they are in on the corresponding night of the following month. You will observe from one month to another (if you always look at the same hour of the night) that they will travel away to the west. These motions are referred to the sun, by our habit of using solar time; that is to say, at the same solar hour, or when the sun is at the same distance from the meridian, the stars are travelling away to the right; or, in other words, the sun travels away to the left amongst the stars. And at the same time, as I have mentioned, it changes its distance from the North Pole.

Now, I will endeavour to point out to you how this is more accurately observed. I remarked that the Mural Circle is used to determine how far the Polar Star appears above the north horizon, at its highest and lowest positions, and thus to determine the height of the Pole above the north horizon. It is also used to determine how far the sun is above the south horizon; and from these two measures the sun's angular distance from the Pole is obtained. When we were speaking of the transit instrument, I described its use in this manner: suppose we observe the time when some well-known star passes the meridian, (for instance, the bright star of Aquila, adopted for that purpose by one of my predecessors, Dr. Maskelyne,) and also the time when the sun passes the meridian. The sun passes the meridian so many hours, minutes, and seconds after the star. We bring the place of the star on the celestial globe under the meridian, and then we turn the globe through the corresponding angle; then we know that the sun's place will be somewhere under the meridian. At the time that it passes, it has a certain elevation above the horizon, and therefore a certain distance from the Pole, expressed in degrees and minutes, which is found by the Mural Circle: we take this number of degrees and minutes along the meridian from the Pole; and thus we find the place where the sun was at the time of observation. We make a mark on the globe at that place. We repeat these observations every day of the year; we get measures of the same kind; and we find a series of places such as those shown in Figure 24 When we come to examine all these as laid down together, which may be done roughly on a globe, or more accurately by calculation, we shall find that they are all lying in a great circle. You must understand what is meant by a great circle: it is a circle dividing the sphere into equal parts. Its plane therefore passes through the centre of the sphere. Now, our use of the celestial globe or sphere is founded on the assumption that it is a representation of the heavens, on the supposition that the eye of the observer is at the centre of the sphere. This circle, then, in which the sun appears to move, being one whose plane passes through the centre of the sphere, or through the eye of the observer; it comes to this, that the sun appears to move around the earth in a plane, or that the earth moves around the sun in a plane.

If the sun moves round the earth, we have only to suppose that the earth stands with its centre stationary, but that it is whirling round its axis, and that the sun travels round and round. If you suppose that the earth travels round the sun, it is necessary to suppose that the earth's axis retains its parallelism to itself without any respect to the sun. Now, is it likely that the axis of the earth would remain parallel to itself without respect to the sun? In my former lecture, I called your attention to the motion of a quoit and a top, in order to show you the strong tendency which rotatory motion has to maintain the position of its axis unaltered. It is the quoit whose motion has the most striking analogy to the motion of the earth. The quoit is not impeded by contact with the floor as the top is. This has been made the subject of mathematical investigation, as well as of experiment, and the result of both is, that the earth, if revolving round the sun, would carry its axis of rotation always parallel to one line, as we see it does.

To this I may add one remark. Geologists have observed that important changes have taken place in the climates of different parts of the earth. Some have supposed that the axis of the earth must have changed its position; but there is no greater impossibility than that the axis of. the earth should change its position. So strong is the tendency of rotation to preserve the position of the axis unaltered, both in parallelism to itself, and with respect to its position in the rotating body, that we may assert boldly that the earth's axis has always been in the same general position within the earth ever since the earth first received motion. And its position, as regards the place among the stars to which it points, is affected only by the very slow motion called Precession of the Equinoxes (of which more will be said hereafter); and even this does not affect its inclination to the plane of the ecliptic.

Having got through this part of my subject, I will now proceed to speak of the apparent motion of the planets. The movements of the planets are extremely complex. At the present time,[1] Venus is what is called the morning star; she is to be seen before sunrise. If you watch her movements through the stars, you will see that her motion is in the same direction as the motion of the sun. You will remember what I said in regard to the motion of the sun: that it moves in regard to the stars, in a direction opposite to the movement of the hands of a watch. You cannot see the stars surrounding the sun, though astronomers can see them with their telescopes; but by watching the stars from month to month, you find that the stars appear to move away from the sun towards the right, or that the sun moves among the stars towards the left. That is called direct motion. If you look at Venus at the present time, you will see that she is moving in a direct motion faster than the sun. She will go behind the sun, and after that she will be seen as an evening star; she will then be going away from the sun; she will go away for a certain distance, but more and more slowly, and the sun will be approaching towards her; she will become stationary; then she will turn backwards and seem to meet the sun.

Can we make a reasonable theory to account for this? We can do it more easily if we refer the apparent motion of Yenus to the sun and not to the stars. Venus sometimes passes the sun in going from left to right (relatively to the sun), and sometimes going from right to left (relatively to the sun); and her extreme angular distance from the sun towards the right is almost exactly the same as her extreme angular distance towards the left. The Greek astronomers began with a good assumption; they laid down at once the notion which they conceived must be the most natural and most proper, which was this: that every planet revolved in a circle. They then supposed that the earth is fixed, and that the sun moves. They supposed that a bar, or something equivalent, is connected at one end with the earth, and that on some part it carries the sun; and as they saw that the planet Venus was apparently sometimes on one side of the sun, and sometimes on the other side, they said that the planet Venus moves in a circle, whose centre is on the same bar. Whether they have expressed themselves distinctly concerning this bar I cannot say, but all their notions of the position of the centre of the orbit of Venus come to the same thing. Then, suppose that Venus is revolving round the centre at the same time that the bar is moving, we then get a perfect representation of the apparent motion of Venus and the sun, as seen from the earth. These suppositions will be represented on Figure 26 by supposing E to be the fixed earth; E v S m n a bar turning in a circle, having one end fixed as at E ; S

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Fig. 26.

the sun carried by it; v , the centre of the orbit in which Venus revolves; V being the planet Venus, connected with v by a bar (real or imaginary), and thus describing a circle round v, while v itself is carried on the bar round the earth.

They supposed that Mercury (see Me in Figure 26) revolves in another circle, and that its centre is on the same bar, but perhaps beyond the sun, as at m. They did not, however, pretend to judge exactly where these centres are; all that they were certain of was this: that the centre of the motion of each planet is on the same bar that supports the sun. Now, you may easily see that, on these suppositions, the planets being viewed from the earth, Venus is at one time to the right, and at another time to the left of the sun, and the sun is carried round the earth in one year. The same is the case with regard to Mercury.

With regard to Mars, they found out that its motion can be represented extremely well, by supposing that this same bar carries another centre as n, around which Mars revolves as at Ma, carried by an arm so long that it projects beyond the earth, so that its orbit completely surrounds the earth, as well as the sun, in describing its whole motion.

It is, however, rather difficult for us to conceive how the centre of this motion can be carried. There is no bar that we can see. It was, however, necessary for them to suppose that there is a bar which is attached at one end to the earth, and which carries the sun, and carries also the centres of the motion of the other planets. It does appear strange that any reasonable man could entertain such a theory as this. It is, however certain that they did entertain such a notion; and there is one thing which seems to me to give something of a clue to it: in speaking to-day and yesterday of the faults of education, I said that we take things for granted without evidence; mankind in general adopt things instilled into them in early youth as truths, without sufficient examination; and I now add that philosophers are much influenced by the common belief of the common people. There is one passage in Herodotus, where he is endeavouring to account for certain phenomena in Egypt, which I have often read, and which, so far as I can see, can only mean this: "That certain periodical winds do carry the sun from north to south, and that thus the change of seasons is produced." I think it likely that Herodotus (who was a learned man for that time) believed that the sun was something in the atmosphere little better than a cloud, perhaps not so important as an aurora borealis, and that it might be carried along by the winds. We know, also, that at a time not very distant from that, a Greek philosopher, named Anaxagoras, dissented from this notion, saying—"That the sun was solid, and as big as the country of Greece," and that he was persecuted for saying so. Having these things before us, I am not much surprised that the Greek astronomers considered the sun as completely subordinate to the earth, and therefore supposed it to revolve round the earth; and when they had once adopted this idea, they were compelled to take the complicated and unnatural explanation which I have given of the motion of the planets.

The motions of the planet Mars, however, still presented some discordance, and there were some smaller discordances with regard to all the other planets. Then were invented those things known by the name of epicycles, deferents, &c. of which the nature may be thus explained. By the contrivance of which I have previously spoken, (and which is represented in figure 26), they found that the movement of the point Ma at the end of the rod n Ma would nearly, but not exactly, represent the motion of Mars. To make it represent the motion more exactly, they supposed that another small rod MN was carried by the longer rod, jointed at M, and turning round in a different time. To make it still more exact, they supposed another shorter rod carried at N, and that its extremity carried the planet Mars; and so for the other planets. Of all the complications of systems that ever man devised, there never was one like this Ptolemaic system. The celebrated King of Castile, Alphonse, the greatest patron of Astronomy in his age, alluding to this theory of epicycles, said "If he had been consulted at the Creation, he could have done the thing better." It was merely expressing his absolute inability to receive, as a possible explanation of nature, such a complexity of things.

But there was one consideration so simple, that it seems astonishing that it did not occur to people before. When we suppose the earth fixed as at E, eFigure 27, and take Venus (for instance) revolving

Popular astronomy airy(1881) page 96 figure 27.png

Fig. 27.

round a centre, we may alter the place of that centre and its distance from the earth, as much as we please, and we shall then get the same appearances, provided we alter the dimensions of the orbit of Venus in the same proportion. As, for instance, in Figure 27, suppose E to be the earth, and suppose the small circle in which V is to be the orbit of Venus, the sun being at S; then, in revolving in her orbit, Venus appears to go to a certain extent to the right and to the left of the sun. But we might take any other point on the bar, even the point S itself, for the centre of the orbit of Venus, provided we give Venus a larger circle to revolve in. In the large orbit in which V is seen, Venus will appear (as viewed from the earth) to move to the right or left of the sun; and if we do but make the orbit large enough, it will, as viewed from E, appear to move just as much to the right or left of the sun as if it moved in the small orbit. We may then fix the centre of the orbit of Venus where we please. When we have got thus far, we may easily make another step. Suppose w assume the centre of the orbit of Venus to be the same point as the centre of the sun: we shall not have so much complexity. Suppose now we assume also that the centres of the orbits of the other planets are in the centre of the sun: we have seen that we can thus account for the motion of Venus, by giving proper dimensions to her orbit; and we can do the same thing for Mercury, and for Mars, and for everyone of the planets. Just observe the state of things we have got to, as in Figure 28; instead of having

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Fig. 28.

different centres of motion for different planets, we have got them all in the centre of the sun; and the sun turns round the earth, carrying the orbits of the planets with it. That is a considerable simplification. In this state, I believe, the theory was received by the great Danish Astronomer, Tycho Brahé.

But now, instead of supposing the sun to be travelling, being itself the centre of the other orbits, and by some imaginary power causing the planets to revolve round itself as their travelling centre, suppose we say that the earth revolves round the sun, and that the sun is a fixed, or nearly fixed body, and that all the planets including the earth, go round the sun; that is, in Figure 28, instead of supposing S with the whole system of orbits to travel round E, suppose Me, V, E, Ma, and others, to travel in separate orbits round S. The appearances of the planets, as viewed from the earth, will be represented exactly as well as before. How it could then happen that a theory like that of the Greek astronomers could still be received as true, after the publication of the simple explanation which I have now given, is beyond my conception. It did, however, very much change the relative importance of the sun and the earth; it made the sun the most important body of all, and the earth one of the least important; and perhaps it, was this which really occasioned the difficulty of receiving it. This great step in the explanation of the planetary motions was made by Copernicus, an ecclesiastic of the Romish Church, a Canon of Thorn, a city of Prussia. The work in which he published it is dedicated to the Pope. At that time it would appear that there was no disinclination in the Romish Church to receive new astronomical theories. But in no long time after, when Galileo, a philosopher of Florence, taught the same theory, he was brought to trial by the Romish Church, then in full power, and he was compelled to renounce the theory. How these two different courses of the Romish Church are to be reconciled I do not know, but the fact is so.

Soon after the time of Copernicus, the telescope was invented by Galileo. One of the most important discoveries made with it was, that the planets do not always appear to be round, and that they obviously are not fully illuminated at all times. The planet Venus puts on all the phases of the moon. When the planet Yenus is at that part of her orbit at which, in conformity with our theory, she is beyond the sun, as at V', Figure 28, we then see her as round as the full moon; and when the planet Venus is at those parts at which, in conformity with our theory, she is almost between us and the sun, as at V", it is found, by observation with the telescope, that she then puts on the phases of a young moon. These are precisely the appearances that would be seen if the theory is true. This is a most important confirmation, which was wanting in the time of Copernicus, and which with us is so convincing, that any one who has seen Venus will not doubt the truth of the theory.

The great step made by Copernicus was the assumption that the sun is the centre of the motion of all the planets (including the earth). But he could not get rid of the epicycles. As in Figure 26, where Mars is carried at N, at the extremity of a small arm, jointed on to a longer arm and revolving round the joint; so it was still necessary to suppose that each of the planets, as well as the earth, was carried by a similar apparatus; and even this did not represent the movements with perfect accuracy. This was reserved for Kepler—to explain, who not so much from his own observations as by examining accurately the observations which Tycho Brahé had made of the planets, and especially the planet Mars, and comparing them with his own—ascertained that the whole would be represented to the utmost accuracy by supposing, that Mars moves in an ellipse. It is impossible now to explain in a few words how Kepler came to that conclusion; generally speaking, it was by the method of trial and error. The number of suppositions he made to account for the motions of the planets is beyond belief: that the planets turned round centres at a little distance from the sun; that their epicycles, deferents, &c., turned on points at a little distance from the ends of the bars to which they were jointed, &c. It is by this kind of investigation, by trial and error, that truth is established. The way in which he has published his adoption of this theory, is very striking. After trying every device with epicycles, eccentrics, and deferents, that he could think of, and computing the apparent places of Mars from these different assumptions, and comparing them with the places really observed by Tycho, he found that he could not bring them nearer to Tycho's observations than by eight minutes of a degree. He then said boldly that it was impossible that so good an observer as Tycho could be wrong by eight minutes, and added, "out of these eight minutes we will construct a new theory that will explain the motions of all the planets." He then proceeded to explain the theory of motion in ellipses.

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Fig. 29.

I shall now speak of elliptic motion. I must first state what an ellipse is. There are different ways of describing an ellipse. I dare say there are many mechanical persons near me who are acquainted with a carpenter's trammel. An ellipse, or an oval may be described in this way, in Figure 29. Suppose AB, DE, to represent the longer and shorter diameters of the ellipse, at right angles to each other; then, if we have a bar GFH, with two pins FG fixed in it, so arranged that the pin F shall always move along the long diameter AB, and the pin G shall always move along the short diameter DE; then any point H of the bar describes an accurate ellipse. This is the principle used in carpenters' trammels and oval chucks. It describes an accurate ellipse exactly similar to that described by another method, of which I am going to speak, but it has no relation to the various parts of the ellipse upon which I am going to remark.

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Fig. 30.

In Figure 30, if we stick a pin in a board at S, and call that point a focus; and if we stick another pin in the board at H, and call that a focus; if we then fasten a string by its two ends to these two pins, keeping it always stretched by the point of a pencil as at P, and carry the pencil round, it will describe an ellipse. S and H are the two focusses of the ellipse; but in all the treatment of astronomical theory, we have only to do with one of them.

If the ellipse in Figure 30 be the orbit of a planet, S will be the place of the sun. The sun is at one focus of the ellipse described by every planet Every planet describes a different ellipse. The degree of flatness of every ellipse is different for every planet; the direction of the long diameter of the ellipse is different for every planet; there is every possible variety among them.

Now, one of the important things that Kepler made out was this: that the orbits of planets are ellipses. Another important thing made out was this: that the planets describing these ellipses move with very different velocities at different times. Each planet, when in that part of its orbit which is nearest to the sun, travels quickly, and when in that part which is furthest from the sun, travels slowly. The way in which he expressed the law of motion is this: if in one part of the orbit I draw two lines SK, SL, from the sun, inclosing a certain space, (when I say "inclosing a certain space," I mean inclosing a superficial area, containing a certain number of acres, or of squares miles,) and if in another part of the orbit I draw two lines Sk, Sl, and if the two lines Sk, Sl, inclose the same number of acres as are inclosed by SK, SL, then the planet will be just as long moving over the long arc KL, as in moving over the short arc kl. From that law of equal areas in equal times, you will see that the planet is moving much more rapidly between K and L, than between k and l.

Having spoken so much of the motion in an ellipse, I will now proceed to speak of the cause of that motion; the force of the sun's attraction, which acts upon every planet. We are now coming to a thing totally different from what we have had before. We have spoken of the form of the orbits of planets, and the proportion of their speed in the different parts of their orbits, as determined from the observation of the planets; and I will now proceed to the consideration of the causes of these motions, in reference to the mechanical theory first propounded by Sir Isaac Newton, and received by every person who possesses a competent acquaintance with the subject. The theory is this: that if we suppose the planets to be once set in motion, (by some cause which we do not pretend to know,) then the attraction of the sun accounts for the curved form of their orbits, and for all their motions in those orbits. Now, in speaking of this, I must observe that there is a term frequently used by persons not acquainted with its real meaning; persons speak of "projectile force" as if such a thing were constantly in action. The planets are in motion; they have been put in motion somehow; but there is no force to maintain their motion afterwards, that we know of, and there is no necessity for us to suppose the existence of a force which keeps up that motion. But, having been once started with a certain velocity, it is necessary to suppose that there is a force constantly pulling them towards the sun. The planets will sometimes go away from the sun, the sun will pull them back, and afterwards they will go away again, and be again pulled back, and so on.

In order to explain this, I must proceed with a very rude experiment on what is called the second law of motion. The first law of motion is simply this: if a body be once set in motion, and if it have a certain velocity given to it, it will continue to move (if not acted on by any force) in a straight line with unabated velocity. We cannot make experiments proving this law in the simple form in which I have mentioned it, but we can make experiments on it in combination with other laws; and we are compelled to believe that the law is true, that if a body were started in motion, and if nothing were acting upon it, it would continue in the same motion. The second law of motion is that which may be illustrated by a very rough experiment. Suppose a body to be projected horizontally, like a cannon ball, or like a stone thrown horizontally, you will observe that it begins to curve in its path, under the attraction of gravity, and it falls to the ground. The second law of motion is, that the force of gravity draws it just as far from the place which it would have reached, if no gravity were acting, as the force of gravity would draw it in the same time from the position of rest. Suppose, for instance, that a body is thrown in the direction of AB, (Figure 31,)

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Fig. 31.

with a speed which would have carried it from A to B in one second of time; and suppose I know from experiment that it would have dropped from A to C in one second of time; then the second law of motion is this: that at the end of one second of time the body will really be found at D, having, by the action of gravity, been pulled away from the place B, which it would have reached with the original direction and the original velocity, just as much as if pulled away from the state of rest.

The law may be illustrated by experiments in this manner: AB, Figure 32, is a board; CD, an arm moving upon it, turning on a hinge at C, and driven by a spring E; at the end D of the arm is a hollow, with its opening in the side of the arm large enough to contain a small ball, so that when the arm is driven by the spring E, the ball will be thrown horizontally from the hollow at D ; at F is another chamber opening downwards, its lower opening being

Popular astronomy airy(1881) page 105 figure 32.png

Fig. 32.

stopped by a board G, which will be knocked away by a blow of the arm CD; then it is plain that if one ball be put in D and another in F, the very same movement which throws one ball forward causes the other ball to drop at the same instant; and if the second law of motion be true, one of them will fall down vertically to the floor at H at the same instant at which the other, which is projected forward reaches the floor at K. And this does really happen so; the two balls do reach the floor at the same instant. What it proves is this: that if a ball is thrown horizontally it falls from that horizontal line down to the ground just in the same period of time as a ball which dropped from a state of rest.

I have described this experiment as applicable only to a horizontal throw; but it is equally applicable to an inclined throw, if the floor upon which the balls fall be inclined exactly in the same degree, as shown in Figure 33, the ball which drops down to H, and the ball which is thrown in the inclined

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Fig. 33.

direction and reaches the floor at K, will arrive at the floor at the same time.

Now, it is important to observe what are the circumstances on which the curvature of the path of a projected ball depends. In the first place, if anything were to increase the force of gravity, the track of the ball would be more curved. Thus, in Figure

Popular astronomy airy(1881) page 106 figure 34.png

Fig. 34.

34, if the velocity with which the ball is projected would carry it in a second of time to B, and if gravity were so strong that in one second the body would fall from rest to C, then the ball would describe the curve AD; but if gravity were so much increased that a ball would fall from rest to C' in one second of time, then the ball would describe the curve AD', which is more curved than AD. In the next place, if two balls are projected with different velocities, without any alteration in the force of gravity, the path of that ball which is projected with the smaller velocity will be more curved. Thus, in Figure 35, if the force of gravity were such as

Popular astronomy airy(1881) page 107 figure 35.png

Fig. 35.

would make a ball fall from rest to C in one second, and if two balls are projected, one with a smaller velocity which would carry it to B in one second, and the other with a greater velocity which would carry it to B' in one second, then the former ball (or that projected with the smaller velocity) will reach D in one second, describing a very curved path AD; while the latter (or that projected with the greater velocity) will reach D', describing the path AD', which is much more nearly straight than AD. Everybody knows the motion of a stone thrown from the hand; its path is much curved, and it reaches the ground before it has gone far. But if you watch the motion of a cannon ball, which you may do if you stand behind a cannon when it is fired, as you can then see the ball from the time that it leaves the cannon's mouth to a distance of a half a mile or more, you will perceive that its path is curved, but very much less curved than the path of a stone ; in fact it is nearly straight, but still not quite straight. The ball dropped downwards through the same space as the stone in one second of time ; but the ball travelled further in the horizontal direction than the stone in a second of time. Now, from this consideration we shall be able to explain something of the most puzzling matters in the motion of planets.

First, however, we must proceed to another consideration, which is called the resolution of forces. This may be illustrated by a model, represented in Figure 36. Suppose A and B to be two pullies fixed

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Fig. 36

upon an upright frame, and suppose two cords to pass over them, carrying the two weights C and D at their ends ; and where they meet at E, let a third cord be attached, carrying the weight F ; then you will see that the tension or pull produced by this one weight F, acting at the place E, does really support two tensions in different directions acting at the same point, namely, the tension produced by the weight C acting in the direction EA, and the tension produced by the weight D acting in the direction EB. Thus we may correctly say, that one pull in the direction EF does exert two pulls in different directions, AE and BE, for it really does keep the two cords strained to such a degree as to support the two other weights. We may say, on the other hand, that these two outside weights C and D support the middle one. The three pulls of the cords keep the point E in equilibrium; but they will support it only in one determinate position, according to the amount of weight which is hung to each cord. If I put another weight upon C, the position of the point E and the direction of the cords will immediately change; showing that for one proportion of the weights or tensions, there is only one set of angles between the different directions in which the tensions will keep the point E at rest; and, conversely, one set of angles of inclination requires the tensions to be in one certain proportion, in order that E may be kept at rest. Regarding the action of the two tensions in the directions EA, EB, as supporting the one tension in the direction EF, this may be considered as an instance of the combination of forces; and regarding the one tension in the direction EF as supporting two in the directions AE, EB, this may be considered as an instance of the resolution of forces, the one force in the direction EF being resolved into two forces in the directions AE, BE, and producing in all respects the same effects as two forces in the directions AE, BE. It may seem strange that a force acting in one direction can produce two forces acting in two different directions; yet we have plenty of familiar examples of the same thing. For instance, in driving a wedge by means of one force, we produce two force's in different directions. I mention that, as a case which must be notorious to everybody, and one which may well furnish food for thinking. By pushing at the back of the wedge with a small force you do exert two great forces at the sides of the wedge; and in like manner, by pulling at E (Figure 36) in a downward direction, you may exert a force even greater than your downward pull at the two inclined directions; and both these are accurate instances of what is called the resolution of forces. It must be understood, therefore, that having got a force in any one direction, we may say that instead of one force we have two forces acting in any two directions suited to the nature of the case, whose magnitudes are determined by certain laws depending upon the angles of inclination; and we may use those two forces instead of the one force which we had originally.

From this consideration, in combination with the considerations which I stated, relative to the dependence of curvature of path upon velocity and deflecting force, I shall endeavour to give you a little idea of the motion of a planet in its orbit. The thing that I wish specially to explain to you is, how it happens that when a planet has once begun to approach to the sun, it does not go quite to the sun, but after a time recedes again from it. If you understand this, you will understand the rest. I will suppose, if you please, that in Figure 30, a planet is moving from l, through M, towards L. The attraction of the sun pulls it in the direction of the line MS. Upon the principle of the resolution of forces, of which I have just spoken, we may consider the force in the direction of MS to be resolved into two, one of which is in the direction of NM, perpendicular to the orbit, and the other is in the direction of OM, parallel to that part of the orbit. Now, observe this carefully. That part of the force which is in the direction NM, perpendicular to the orbit, produces an effect similar to that which gravity produces in the motion of a cannon ball : it makes the orbit curved. But that part which acts in the direction of OM, parallel to the orbit, produces a different effect; it accelerates the planet's motion in its orbit. Thus, in going from l towards L, the planet is made to go quicker and quicker. If you suppose the diagram (Figure 30) turned in such a manner that MS is vertical, S being downwards, you will see that the planet is under the same circumstances as a ball rolling down a hill. If a ball is going down a hill, as at M, Figure 37, the force of gravity, which is in the direction MS, may be

Popular astronomy airy(1881) page 111 figure 37.png
Fig. 37.

resolved into two parts, one of which is a force in the direction NM, perpendicular to the hill side, and merely presses the ball towards , the hill the other is a force in the direction OM, making it to go the faster down the hill. In this manner, as long as the planet goes from k through M towards K, it is going quicker and quicker. This accounts for the difference of speed in different parts of the orbit, which I mentioned before. Now, remember how it has been explained that the curvature of a planet's orbit, or the curvature of the path of a cannon ball, depends upon two circumstances; one is the velocity with which it is going, and the other is the force which acts in such a direction as to bend its path. The greater its speed, the less its path is curved; consequently, as the planet is going so exceedingly quick in the neighbourhood of K, its orbit may be very little curved there, even though the sun is there pulling it with a very great force. The effect of the planet's path being so little curved there, is, that the planet passes the sun and begins to recede from it. But it does not recede perpetually. Suppose, for instance, that it has reached the point M', and we examine the nature of the forces which act on it there. The force of the sun in the direction M'S may be resolved into two, in the directions N'M', O'M', of which the former only curves the orbit, while the latter retards the planet in its movement in the orbit. Therefore, as the planet recedes from the sun, it goes more and more slowly, till at last its velocity may be diminished so much, that the power of the sun, reduced as it is there, is enabled to bring it back again. That is the way the planet goes, revolving in its orbit, alternately approaching to, and receding from, the sun. Of course, in a series of lectures like these, I cannot go into every detail; but enough has been said to show that a planet when approaching the sun will not necessarily fall to the sun, and when receding from the sun will not recede beyond the hope of return. This is a stumbling-block to many a young astronomer who has not considered the subject well, but the remarks I have here made, will, on consideration, be found to be perfectly clear, and there is no doubt of their application.

From what I have shown, you will see that there is a tendency in the planet to go off again when it has come nearest to the sun; but whether it will absolutely go off again depends upon another circumstance: it depends upon the amount of force when the planet is nearest to the sun; as, though the speed be great, it may happen that the force is very great also, and it may happen that the force is so great, that after all we cannot, merely upon general considerations, answer for its coming back again. I wish to point out the general explanation, but it is quite impossible here to enter fully into these particular details, and to show to you whether, when the planet is coming near to the sun, the force will not be too great to allow it to recede again; or whether when it is going away from the sun, the force will be great enough to bridle it in or not. That is a thing for which you must trust me for a moment. If, as we assume in the law of gravitation, when the distance of the planet from the sun is doubled or trebled, the force of the sun is reduced te one-fourth or to one-ninth, and so on; if that be the law of force, then the velocity of the planet, on coming near to the sun, is so increased that the tendency to recede increases in a greater proportion than the force, and it is certain that the body will begin to recede. But this would not be the case with all laws of force; if we supposed that when the distance was doubled the force was one-sixteenth instead of one-fourth, this law of alternate recess and approach would not be true. Mathematical investigations are made to ascertain whether certain conditions are fulfilled. A planet is invariably moving quickest at that part of its orbit where it is nearest to the sun, and in consequence of that increased velocity of motion, it is able to overcome a degree of attractive force which it would not overcome if its velocity were not so increased. Under certain circumstances it would go out and come back again, and so on; that is the case with regard to the law of gravitation.

I will now take a few minutes only for the next section. We will depart from the consideration of mechanical forces, and consider the measure of distances. The thing which I wish to explain to you is, how can we measure the distance of the moon from the earth? The distance of the moon is measured by the method of Parallax. This is a technical word of which we are obliged to make perpetual use in Astronomy. I will explain in as few words, and in as familiar a manner as I can, what parallax is. There is an experiment pleasing and profitable, and which I have made in my youth, and which I have no doubt most of you have made in your time. It is this: if you place your head in a corner of a room, or on a high-backed chair, and if you close one eye and allow another person to put a lighted candle upon a table, and if you then try to snuff the candle with one eye shut, you will find that you cannot do it; in all probability you will fail nine times out of ten. You will hold the snuffers too near or too distant; you cannot form any estimation of the distance. But if you open the other eye the charm is broken; or if, without opening your other eye, you move your head sensibly, you are enabled to judge of the distance. I will not speak of the effect of motion of the head at present, but will call your attention to the circumstance, that when your head is perfectly still you will be unable, with a single eye, to judge with accuracy of the distance of the candle. In Figure 38, let A and B be the two eyes, C an object which is viewed first with the eye A only. This eye alone has no means of estimating the distance of C. All that it can tell is, that it is in the direction of the

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Fig. 38.

line AC; but there is no phenomenon of vision by which it can judge accurately of its distance in that line AC. Suppose, now, the other eye B is turned to C, then there is a circumstance introduced which is affected by the distance, namely, the difference of direction of the two eyes. While the object is at C, the two eyes are turned very little inwards to see it; but if the object is brought very close, as for instance to D, then the two eyes have to be turned considerably inwards to see it; and from that effort of turning the eye, we acquire some notion of the distance. We cannot lay down any accurate rule for the estimation of the distance; but we see clearly enough in this explanation, and we feel distinctly enough when we make the experiment, that the estimation of distance does depend upon this difference of direction of the eyes. When the object is brought very near, the feeling becomes very annoying. This is the principle upon which is founded this experiment of which I have spoken. Now, this difference of direction of the two eyes, is a veritable parallax; and this is what we mean by parallax, that it is the difference of direction of an object as seen in two different places. The two different places in the experiment which I have illustrated, are the two eyes in the head. This is the way in which, as will he seen, the distance of the moon from the earth is to be found. The two eyes in the head will be two observatories; and will be supposed to be placed at a considerable distance from each other on the earth. Without any exception at all, the principle is precisely the same. You will thus see how, by the observation of the difference of directions, the distance of the moon from the earth may be obtained. I shall give a more detailed explanation of this in the course of my next lecture.

  1. 15th March, 1848.