Popular Astronomy: A Series of Lectures Delivered at Ipswich/Lecture 6
IN the lecture of yesterday evening, the first subject to which I alluded was the Precession of the Equinoxes, in reference to its mechanical causes. This is a thing so important, partly in itself and partly in connection with the causes which produce it, that I have no hesitation in speaking of it again. The thing which I particularly intended to convey to you, was this: that if we consider the attraction of the sun upon the earth, and if we consider that the earth is not a sphere, but has a flattened turnip-like shape which we call an oblate spheroid; if we also remark the laws which apply to gravitation, namely, that the force which the sun exerts is greater the nearer the body is to it, and that the law of gravitation is to be understood as applying to every particle, not to the body as a mass; that it attracts the earth not as a whole, but as a number of parts separately; if we mark these things, we find that the sun attracts that swelling part of the earth which is nearest to it more powerfully than it attracts the central part of the earth; and that it attracts the centre of the earth more powerfully than the parts further off. The sun, therefore, is endeavouring to pull, as it were, the nearest part of the earth from the centre towards the sun; and it is endeavouring to pull the centre from the distant parts towards the sun, which is the same as saying, that it is pushing this distant part from the sun. Now, if the earth were a complete sphere, the pulling at any part above the ecliptic would not disturb its motion; because there would be a corresponding pull on the corresponding part below the ecliptic; but, inasmuch as the earth is not a complete sphere, but has this flattened, turnip-like shape, protuberant at the middle; at the time of the solstice this protuberant part in the direction of the sun being above the ecliptic, then the extraordinary pull which the sun makes on that place is not balanced by a corresponding pull at the part below the ecliptic, because there is no protuberant part there on which the sun can act; and the action of the sun on that part which is above the plane of the ecliptic tends to pull it down towards that plane.
I remarked that the first effect of this would seem to be, to change the inclination of the axis on which the earth revolves, and to bring that axis more nearly perpendicular to the ecliptic. And that would be the effect, if the earth were not revolving on its axis; but in consequence of the revolution of the earth round its axis, a totally different effect is produced. I illustrated that by calling your attention to the motion of a single point, as for instance, a mountain at the earth's equator. While that mountain is near the sun and above the ecliptic, the force of which I have spoken tends to pull it downwards. And as the rotation goes on, the forces still tend to pull it downwards and downwards, until at last it comes to meet the plane of the ecliptic at a point sooner than it otherwise would. This takes place with regard to the attraction of the sun on every part of this protuberance. The tendency of this force is to bring every part which happens to be above the ecliptic lower and lower towards the ecliptic; and to make its path intersect the circle a e b, Figure 48, sooner than it otherwise would. It amounts to this, that at the earth's equator the motion of each point is affected by such forces, that it tends constantly to come to its intersection with the plane of the ecliptic sooner than it otherwise would; or, to speak in other words, that intersection travels backwards to meet the rotatory motion of the earth. The same thing (as I fully explained) will happen if we consider the action of the sun on the distant parts of the earth, which I represented as being equivalent to a pushing force.
I then mentioned to you that the moon produces a larger part of precession than the sun does, although the moon is so very much smaller than the sun, (only 1 part of the sun). She is, however, 400 times nearer than the sun; and this makes her whole attraction, in proportion to her mass, 160,000 times as great as the sun's; still her whole attraction is only 1 of that of the sun. But the important thing to be remarked in the explanation above given is, that precession is not produced by the whole attraction of the sun or moon upon the earth, but by the difference between the attractions which they exert upon the earth's centre and upon the earth's nearest surface. For the moon, the proportion of the distance of these parts in nearly as 60 to 59, and then the Difference of the attractions is about 1 of the whole attraction of the moon. But for the sun, the proportion of distances is nearly as 24,000 to 23,999, and then the difference of the attractions is about 1 of the whole attraction of the sun. The consequence is, that that difference of attractions of the moon is actually three times as great as that of the sun, and the precession produced by the moon is three times as great as that produced by the sun.
This is the mechanical explanation of the precession of the equinoxes. It was discovered as a fact by Hipparchus, a Greek Astronomer, one hundred and fifty years before the Christian era; it has been recognized ever since by all astronomers, and is now known with very great accuracy; and, in all probability when Sir Isaac Newton first applied the theory of gravitation to the explanation of the movements of the solar system, the explanation of this discovery of Hipparchus' was one that struck his mind, and that of his contemporaries, more than any other.
The next subject which I pointed out as an important one in connection with observations, was nutation, and I described it in this way: that nutation is a want of uniformity in precession. For the explanation of precession, I had taken the position of the earth at one of the solstices. At this time the earth's equator is much inclined to the line connecting the sun with the earth's centre, and the precession of the equinoxes is going on very rapidly. But if the earth were in the equinoxial position, then the sun would shine equally on the North and South Poles, and the protuberance of the earth would be directed exactly to the sun; and the action of the sun upon that protuberance would not tend to change the position of the globe. From these causes it will be seen that precession is not uniform. But in our calculations for the application of a correction to the places of the stars, as dependent on precession, it is convenient to begin in the first place by using a precession increasing uniformly with the time. And therefore, inasmuch as precession does not increase uniformly with the time, we are obliged to apply a correction to the precession computed as uniformly increasing, in order to take into account the inequality (both in the place of intersection of the equator with the ecliptic, and in the inclination of the equator to the ecliptic) with which precession goes on at different times; and that correction is the quantity called solar nutation.
There is, however, a much more important want of uniformity called lunar nutation, which I described in this way. The precession produced by the moon depends on the inclination of the moon's orbit to the earth's equator; and this inclination is not uniform. For the moon revolves in an orbit inclined to the ecliptic; and the sun attracts the earth and the moon, and disturbs the motion of the moon with regard to the earth when he acts unequally on the two, nearly in the same way as the sun disturbs the motion of the supposed mountain at the earth's equator; and the effect produced is similar, namely, that the intersection of the moon's orbit with the ecliptic travels backwards; and thus, at periods nearly ten years apart, it is alternately more inclined and less inclined to the earth's equator. And thus for nearly ten years the precession is going on too fast, and for an equal period it is going on too slowly; and thus a considerable inequality is produced in the motion of the intersection of the equator with the ecliptic. Moreover, for nearly ten years the moon's orbit is so inclined that the moon's action tends to diminish the inclination of the earth's equator to the ecliptic, and for an equal time it tends to increase the inclination of the earth's equator to the ecliptic; and thus a considerable inequality is produced in the inclination of the earth's equator to the ecliptic.
The general effects of precession will be more easily conceived if, instead of considering the intersection of the equator with the ecliptic, we consider the motion of the earth's axis; as the change in the intersection of the equator with the ecliptic must produce a change in the position of the earth's axis as directed towards the stars. In consequence of this, the real Celestial Pole does change among the stars. The bright star of the Little Bear, for instance, now our Polar Star, is not at the same distance from the Pole now at which it was one hundred years ago. In the time of Dr. Bradley, the Polar Star was more than two degrees from the Pole; now it is one and a half. In the course of a great many centuries the Celestial Pole describes a circle among the stars, and different stars successively take the position of the Polar star. About 4500 years ago, the Polar Star was the bright star in the constellation Draco.
The third subject which I mentioned was the aberration of light; an effect produced by the combination of the earth's motion with the motion of light. I endeavoured to illustrate this in several ways. One was this: if in a summer shower you stand still and watch the rain, you will see it falling in its proper direction; but if you walk forward, you will see the drops falling in an inclined direction, as if they were meeting you; and if you step backwards you will see immediately that the drops of rain appear to be falling as if they were coming from behind you. As another illustration, I supposed that a ship is sailing past a battery, and that a shot is fired at the ship, and I remarked that the direction which the shot takes through the ship is not in a direction exactly corresponding to that in which it is fired, but has an inclined direction, which inclination depends upon this, that after the shot has entered the first side of the ship, and before it comes out at the second side, the ship has advanced sensibly. The magnitude of the inclination depends therefore on the proportion of the velocity of the ship to the velocity of the shot. From this it is plain that if we know the extent to which the apparent direction of that line of the motion of the shot was changed when passing through the ship, we shall have the means of computing the proportion of the velocity of the ship to the velocity of, the shot. Now this is a case strictly analogous to the motion of light. The earth is travelling along, and whilst it is so travelling along, light comes upon it from different objects, for instance from the stars. And the effect is the same as in the case of the ship; that in consequence of this motion of the earth, the light appears to come, not from the real place of the star, but from an ideal place of the star, which is in advance, as estimated by the direction of, the earth's motion. If we know in what direction the earth is moving, the light of the star appears to come from a point more in that direction than it should.
I then endeavoured to point out to you the influence which this would have on the apparent places of the stars. We have an earth revolving in an orbit round the sun. The place of the star then will not appear always the same, but will always be found in a circle, whose centre is the true place of the star, the line from the true place to the apparent place being always in the direction in which the earth is moving. If we can observe the star in different seasons of the year, we can infer from our observations how much the place of the star is perverted by this effect of aberration; we shall see how much the apparent path of the light is inclined to the true path of the light, as in the analogous instance of the breach made through the ship. Thus we have the means of comparing the velocity of the ship with the velocity of the shot, or the velocity of the earth with the velocity of light. And the result of the observation is this: that the place of the star is disturbed one way or the other in different directions at different seasons of the year, twenty seconds and one-third. The inference from this is, that the velocity of light is ten thousand times as great as the velocity of the earth in its orbit. The velocity of light is perhaps the most inconceivable of all things; the velocity is so enormous, 200,000 miles in a second.
But these are things which we must often look at with suspicion. What I have stated, seems at first an indirect way of getting at these results. Even by a person properly conversant with these matters, such results are hardly received without additional confirmation. There are phenomena which give confirmation, which I will now explain. Jupiter has four satellites. Their orbits are, in proportion to his diameter, comparatively small. Our moon is at such a distance from the earth that she is not eclipsed very often; her distance being about thirty times the breadth of the earth. Jupiter's satellites are comparatively close to him; so that three out of the four are eclipsed every time they go round. On watching the appearances of Jupiter, one of the most remarkable things observed is, the eclipses of the satellites, (first seen by Galileo.) When the earth is in one position with respect to Jupiter, we see the satellites go into the shadow; that is, we see them disappear without any apparent cause. In another position we see them come out of the shadow; that is, we see them begin to appear in the dark space at a short distance from Jupiter. Figure 55 is adapted
to the supposition that the satellites are coming out of the shadow. Suppose that C is the sun; E' ,E", the earth in two positions; J', J″, Jupiter in two corresponding positions. The time which is most favourable for the observation of Jupiter's satellites is that when the earth is nearly between the sun and Jupiter, as at E', because then Jupiter is seen nearly the whole night. In a short time after the invention of telescopes, Galileo and other astronomers observed the satellites, and found that their eclipses could be observed with great accuracy, and registered them with great care. They were able in no long time to form tables and calculations of the eclipses of Jupiter's satellites. These occurred principally when the earth and Jupiter were in such a position as E'J'. The earth went travelling on in its orbit, and came to such a position as E". Jupiter, who is very slow in his motions, travelled perhaps as far as J" in his orbit. And now came the remarkable thing: it was found that, when the earth came to such a position as E", the tables and preliminary calculations upon which had been founded the predictions of the eclipses of the satellites would not apply. The eclipses of the satellites invariably occurred later than they ought to have done. This occurred year after year, and it was a long time before people could guess at the cause. Every time the earth came to that part of its orbit in which it is nearest to Jupiter, the eclipses of the satellites, happened as predicted: every time the earth approached the part of its orbit furthest from Jupiter, the eclipses of the satellites occured later than predicted. At last a very celebrated man, a Dane, of the name of Römer, gave the explanation, that in these latter observations the earth was further off from Jupiter than at the time when those observations were made on which the tables and calculations were founded: and therefore the light from Jupiter had to travel over a path longer by very nearly the breadth of the earth's orbit. Upon this, calculations were made, and the result was this: that the time occupied by the passage of light across the semi-diameter of the earth's orbit is 8m. 18s.; and therefore the time occupied by the passage of light across the whole breadth of the earth's orbit is 16m. 36s. Upon applying corrections, proportionably to the distance, to the observations made in other positions, it was found that they all harmonized perfectly well, and no doubt was left of the truth of the result, that the time the light occupies in travelling from the sun to the earth is 8m. 18s. The question for us now is this: does this determination of the velocity of light agree with the deduction made from the aberrations of the stars? We found that the light travels ten thousand times faster than the earth moves in its orbit: if the light occupy 8m. 18s. in coming from the sun to the earth, does that imply a speed ten thousand times as great as the speed of the earth? The fact is, that the two calculations, though perfectly independent, support each other with the greatest nicety; and there is no doubt of the correctness of the measure of the velocity of light.
The subject on which I then proceeded at the Lecture yesterday was the measure of the distances of some of the fixed stars; and I observed in the first place, that it was necessary for me to premise these various things, namely, the explanation of precession, nutation, and aberration, and for this reason: that the apparent places of the stars are disturbed by them to a very sensible degree, both in right-ascension and North Polar distance; and that the very utmost accuracy is necessary in everything relating to the observations upon which the measure of the distances of the stars are to be founded. I assume that we now know the meaning of the term "parallax." Between the apparent places of the moon, as seen at one point of the earth and as seen at another point of the earth, there may be a difference of a degree and half, or more. Now when we have a degree and half of difference, an error of a second is of no particular consequence. The parallax of the sun, as found in the way described in a former lecture, is a very much smaller quantity, between 8 and 9 seconds; that is, there is a difference of 8 or 9 seconds in the sun's places, as seen at the centre of the earth and on the surface. A second or two either way becomes of great importance. But when we come to treat of the distances of the stars, we find that the parallax which can be exhibited, even in the difference of the position of the stars as seen when the earth is at different parts of its orbit, (which is frequently called the annual parallax,) is not certainly in any case two seconds; and in every case but one is certainly less than a single second. An error of a fraction of a second is here of very great importance; it deranges the whole of the results. It is therefore of the utmost importance to take into account the quantity of precession, which amounts to 50 seconds in the year; nutation, which amounts to 9 seconds one way or another; and aberration, which amounts to 20 seconds in one direction or another. All this we must know perfectly well before we enter into the question as to the parallax of the stars.
Now, with regard to the observations of the distances of the stars, I remarked that those observations were, in character, not exactly similar to the observations which are made for ascertaining the distances of the sun and the moon and Mars, which we have spoken of before. In all other cases we were able to plant Observatories at different parts of the earth, and from these different parts of the earth to make observations at the same instant on the subject in question, whether it was the distance of the sun, or of the moon, or of Mars, that was to be measured. But we cannot do so with respect to the stars, and for this reason: the mere observation of the stars from two points of the earth does not present any sensible difference whatever. We have no reason to believe that the apparent places of a star, as seen from one part or another of the earth, are different by the ten-thousandth part of a second. We can, however, do this: we can make observations on the place of a star, when the earth is in two positions widely different. If S is a star, in Figure 54, whose distance we wish to estimate, we can do it if we can observe it when the earth is in the position E', and also in the position E'". There is half a year between these observations; but still, if we can take into account all the changes in the star's apparent place that happen in the course of half a year, we shall be able to get some notion of the real change in the position of that star which arises solely from the different position of the earth in its orbit. Now let us see what we have to do. The position of the star at each of these times is affected by the three causes I have mentioned: precession, nutation, and aberration. The way in which we must reduce the observation of the star is this: we must make such a correction to the place in which we do really see the star, as will reduce it to the place in which we should have seen the star at a certain time, provided that the variable parts of the corrections were done away with. For instance, I observe the star on the 1st of February; precession has been going on for many centuries; I do not, however, reduce the place of the star very far back, I only apply the correction which is due to the change between January 1 and February 1, and thus refer the star's place to the beginning of the year. I then remark that nutation has sensibly disturbed the star's place. I do not, however, apply the correction (as regards nutation) to show where the star would have been on the 1st of January; but I apply a correction to show where the star would have been seen if there were no such thing as nutation at all. I take the same steps as regards aberration; that is to say, I correct the place of the star so as to shew where it would have been seen if there were no such thing as aberration. I have then got my observation of the star corrected for these various causes in such a manner, that its place is totally freed from the disturbances of a periodical nature, and from the change of precession which it has undergone from the 1st January. About the 1st August I repeat the observation. I then apply to this observation, corrections of the same kind; that is to say, I correct it for the change which, by precession, the place of the star has undergone since the 1st January. I apply the correction for nutation on August 1, so as to show what its place would have been if no nutation existed; and I then apply the correction for aberration on August 1, so as to show what its place would have been if no aberration existed; and then from the observation made on August 1, I have the position of the star as it would have been seen at this part of the earth's orbit; all the periodical causes of disturbance being removed, and the precession existing in just the same state as it did on the 1st January. There then remains only one disturbing cause, and it is that which depends on the position of the earth in the different parts of its orbit. In order to find whether the place of the star is really affected by that cause, the step most convenient is to observe the North Polar distance of the star. If the magnitude of the earth's orbit be not anything sensible, as viewed from the star, then the North Polar distance of the star, corrected as I have mentioned, will be the same in these two positions of the earth. But if the North Polar distance of the star, duly corrected, be not the same at these times, then it is to be inferred that there is a sensible difference in the direction of the lines E'S and E"'S.
Now you will observe, that there is a good deal assumed in this. In the first place, when we observe the North Polar distance of the star, the thing to which we refer it, is the position of the Pole in the heavens. The Pole of the heavens is in reality that point of the heavens defined by continuing the earth's axis in a straight line to the region of the stars, so that when we observe the North Polar distance of the star, we do really determine the position of that star in relation to the position of the earth's axis. We assume, therefore, that we can account fully and accurately for the change of position which the earth's axis has undergone, and which arises from precession and nutation. Now there is always a very minute uncertainty about these, which it is desirable to get rid of. Another cause of uncertainty arises from aberration; on this account there is always an uncertainty of a fraction of a second, which enters into the observation. There is also in most cases another cause of uncertainty; it is that of which I have spoken so frequently, refraction, which is such a trouble to astronomers. Nearly every observation which we make upon the positions of the stars is affected by refraction, and after making all proper allowance, we cannot always answer for the results. These considerations serve to cast some doubts on the observations made for determining the distances of the stars. Still there is one star of which the parallax seems to be determined with considerable accuracy, and that is the bright star of the Centaur, Alpha Centauri. It appears certain that this star has an annual parallax of one second (using the term "annual parallax" to denote the extreme difference of apparent positions of a star, as seen from the sun on the one hand, and from the earth on the other hand), which amounts to this: the distance of the star is 200,000 times greater than the distance of the sun from the earth. That is a thing, however, which requires many observations for its verification.
I then mentioned another way in which the distances of the stars may be ascertained; a method which is free from all those defects of which I have spoken. This method is by the observation of two stars, of which one is believed to be very much nearer to the sun than the other. For then we may assume that the distant star will have no sensible change of place from parallax, depending on the position of the earth in its orbit. And then, in observing the stars from the various parts of the earth's orbit, we can compare the apparent place of that star which we believe to be the nearer with the place of the other. Practically this is of importance. The refraction, precession, nutation, and aberration, are sensibly the same; and there is no uncertainty whatever from the computation of the various quantities which cast so much uncertainty on the results derived from other observations. This is the method pursued by Bessel in determining the distance of the star 61 Cygni. He measured the angular distance of this star from two small stars near it, by means of an instrument called the Heliometer, well known on the Continent, but of which there was at that time no specimen in England. With this he determined the parallax of the star 61 Cygni to be one-third of a second; that amounts to the same as saying that the distance is 600,000 times greater than the distance of the earth from the sun. It is deserving of attention that 61 Cygni is a double star; but we know from long observation that the two stars partake of the same motions, and probably are a connected system like the earth, and moon, and therefore we speak of them and of their distance as if they were only one star.
I have here spoken repeatedly of our supposition that some stars are nearer to us than others. The grounds of this supposition are generally the amount of what is called the proper motion of the stars. Upon comparing the places of the stars, as we observe them in different years, and applying the corrections for precession, nutation, and aberration, so as to reduce every observation of every star to what it ought to exhibit on the first day in the year, agreeably to the common practice of astronomers, we find that a great number of the stars have what is called proper motion. We are obliged to give up the idea of fixity entirely. The term "fixed stars" is a good term for young astronomers to use; but the vast majority of the stars which have been well observed, seem to have a motion of their own, and that is known by the term proper motion. In all good catalogues of stars there is a reserved column, distinguishing the proper motions of the stars, showing the direction in which the stars appear to be moving through other stars, and the amount of their motion in a year. This has only been discovered after many years' observation; it is in every case a small quantity; but still in most instances the quantity has been correctly ascertained. Those of Sirius and Arcturus are pretty large; but the largest known are those of two small stars, 61 Cygni (whose motion is nearly three seconds in a year), and a star known by the name of Groombridge, 1830, (whose motion is nearly four seconds in a year.) The attention of astronomers has therefore been directed to both those stars, and it appears certain that the former has sensible parallax, and probable that the latter has parallax of a somewhat smaller amount. In closing this account of the method of measuring the distance of the stars, I will only remind you that I have redeemed my pledge of showing how the distance of the stars is measured by means of a yard measure, and I will very briefly recapitulate the principal steps. By means of a yard measure a base-line in a survey was measured; from this, by the triangulations and computations of a survey, an arc of meridian on the earth was measured; from this, with proper observations with the Zenith Sector, the surveys being also repeated on different parts of the earth, the earth's form and dimensions were ascertained; from these, and a previous independent knowledge of the proportions of the distances of the earth and other planets from the sun, with observations of the transit of Venus, the sun's distance is determined; and from this, with observations leading to the parallax of the stars, the distance of the stars is determined. And every step of the process can be distinctly referred to its basis, that is, the yard measure.
Before dispatching the subject of observation of stars, I will make one remark. The proper motions of many are very irregular in direction and magnitude; but with regard to some others there is a rude regularity which may be conceived in this way. I speak of it in connection with what is supposed to be the motion of the solar system in space. Suppose that I am walking through a crowd of people, or through a forest, if I keep my attention on those objects that are exactly in front they do not appear to change their places; but if I look at the objects to the right or to the left, they appear to be spreading away to the right or to the left. Even if I did not know that I was moving myself, yet by seeing these objects spreading away, I should infer with tolerable certainty that I was moving in a certain direction. Now if it should appear that, taking the stars generally, we can fix on any direction and see that the stars in that direction do not appear to be moving, but that the stars right and left appear to be moving away from that point, then there is good reason to infer that we are travelling towards that point. This speculation was first started by Sir William Herschel. He found a point in the heavens, in the constellation Hercules, possessing this property, that a great majority of the stars about this constellation had not any sensible proper motion, but that the stars right and left of it had apparently motion to the right and left respectively. He inferred from this that the solar system was travelling in a body to that point, and this notion has been generally received amongst astronomers. I believe that every astronomer, who has examined it carefully, has come to a conclusion very nearly the same as that come to by Sir William Herschel, that the whole solar system is moving bodily towards that point in the constellation Hercules. But it is a thing on which the computation is not very accurate, and it will probably remain inaccurate for many years to come. This is the last subject which I have to mention in regard to the fixed stars.
I shall now proceed to the last division of my lectures: a general view of the evidence that applies to the theory of gravitation, with which is inseparably connected the determination of the masses of the different bodies of the solar system. And here I must observe that, on entering minutely into this subject, it is impossible to take one thing alone. When I take the theory of gravitation, I must begin by taking the evidence relating to the laws of motion, for these were described and defined long before the theory of gravitation was expounded. Now the laws of motion, in the shape in which they have been commonly expressed, are these: in the first place, if a body be started in motion, and if no force act upon it, that body will continue in motion in the same direction and with the same velocity. Of all things in the world, this is the most difficult to prove immediately. It is obvious that we cannot put a body in motion so that it shall go on in one unvaried direction, and that it shall go on for ever, for we cannot put it in motion in a place where no force will act on it; and we cannot observe it through infinite space and infinite time. This is one of those instances in which we can examine a law only in connection with other laws. We must investigate by profound mathematical process what will be the effect of combining this law with others, so that we may observe whether the results, which are produced practically, agree with the results which we have found from the mathematical process.
For instance, one of the cases which we can observe is, that where rotatory motion is continued for a very long time, and where the velocity very slowly diminishes, as the motion of a wheel or spinning top. Contrivances have been made for the purpose of spinning tops and wheels in the exhausted receivers of air pumps. They go on spinning and spinning, and the motion seems as if it would never come to an end. Now, how does this apply to the first law of motion, that a body moves in the same straight line and with the same velocity? All I can tell you is this: if it is true that each part of a body would, if unconstrained, move steadily in a straight line, and if (by the connection of the parts) each part is constrained to move in a circle, then it appears by mathematical investigation that the body will revolve with a uniform, velocity; but if it were not true, then the body would not move in a circle with a uniform velocity. There is another instance which is perhaps still more remarkable. I allude to the motion of a pendulum. The motion of a pendulum backwards and forwards is a result of the first law of motion, taken in combination with the disturbing force of gravity; this motion of the pendulum being the most permanent of all that are the subjects of ordinary experiments. If a pendulum be properly constructed, mounted with a steel edge (like that of the best balances) moving on a flat plate of hard agate, and if it be set in the exhausted receiver of an air pump, it will go on for 24 or 30 hours, without the action of anything to keep up its motion. But still it seems a very strange thing to infer, from this backward and forward motion, the law which asserts that the pendulum would go on continually in the same direction in a straight line, if there were nothing to disturb or counteract the operation of that law. Upon making the proper mathematical investigation, it is found that the only way of explaining the motion of the pendulum is, by saying that it would go on in a straight line continually, if it were not acted on by certain causes which we are able to take into account.
It is right you should understand how the matter stands. When we speak of the evidence of these things, we cannot give the evidence simply as it applies to any one law, but we can give it in combination with other laws.
The second law of motion we have endeavoured to illustrate (page 104) by apparatus, which showed that when a ball, was allowed to fall freely it was carried to the ground in the same time as a ball projected horizontally. The third law of motion relates to the effect of pressure, with which I have no occasion however to trouble you at present.
Having said so much on these subjects, we now come to gravitation. It is necessary to make this mention of the laws of motion first, because the movements connected with gravitation are but an instance of the application of the laws of motion to the movements produced by a certain force. The planets and satellites are in motion, and, according to the first law of motion, they would move on in straight lines, if they were not bent-aside by some force. This force, according to the theory of gravitation, is the attraction of another body. Let us now examine whether such a force, following the law of decreasing as the square of the distance increases, will account for some of these motions; and we will begin, as Newton did, with the moon.
The moon's motion with respect to the earth is influenced (according to this theory) almost entirely by the attraction of the earth; because though the sun attracts both the earth and the moon, yet it attracts them nearly in the same degree, and therefore produces little disturbance in their relative motions. And though the moon attracts the earth, still the moon is so much smaller than the earth, that we may omit the consideration of that at present. We shall however hereafter allude to the effects of both these circumstances. Now the principle of calculation in this and all similar cases will be the following: in Figure 56, let MN be the arc which the moon describes in her course round the
earth E, in one hour., or one minute,
Fig. 56. or one second, or any other short time that we may choose to fix on (we shall, for the present, take one second). If no force had acted on the moon, she would have moved in a straight line Mm. Therefore the force with which the earth has attracted the moon, has drawn her from m to N, or through the space mN. We must therefore compute the length mN we shall then know how far the earth's attraction draws the moon in one second; we also know how far the earth's attraction makes any-thing at the surface of the earth fall in one second; and the proportion of these will give the proportion of the earth's attraction in these two different places. Now, considering the moon as moving in a circle whose semi-diameter is her mean distance, EM is 238,800 miles. Also the whole circumference of the moon's orbit is 1,500,450 miles; and her periodic time is 27 days, 7 hours, 43 minutes; hence the length of the line Mm, which would have been described in one second if no force had acted, is 0.5356 of a mile. With these two lengths of the sides of the right-angled triangle EMm, by the usual rule of squaring the two sides, adding the squares together, and extracting the square root of the sum, the hypotenuse Em is found to be 238800.0000008459 miles. Therefore the line mN is 0.0000008459 of a mile, or 0.0536 of an inch; and this is the space through, which the earth's attraction draws the moon in one second.
Now at a place on the earth's surface, which is 3,959 miles from the earth's centre, it is found by experiment, that a stone falls 193 inches in one second. And it is found (by a difficult mathematical investigation), that if our theory is true in this respect, "that the attraction of the stone to the earth is produced by the attraction to every particle of the earth," (see page 175,) then the attraction of the whole earth (considered to be a sphere) will be the same as if the whole matter of the earth were collected at its centre. Thus the question upon which the explanation of the moon's motion by gravitation must depend is this: the earth's attraction at the distance of 3,959 miles draws a body 193 inches in one second, and the earth's attraction at the distance of 238,800 miles draws the moon 0.0536 inch in one second. Are these effects of the earth's attraction inversely as the squares of the distances? They are, almost exactly. To make them exactly so, the space through which the earth draws the moon should be 0.05305 inch. Now it is found (by a process which I cannot hope fully to explain to you) that the two circumstances which I mentioned, namely the moon's action on the earth and the sun's disturbing force, do exactly explain this small difference; so that it is certain that the attraction of the earth which causes a stone to fall, and the attraction of the earth which bends the moon's path from a straight line to a circle, are really the same attraction, only diminished for the moon in the inverse proportion of the square of her distance.
I have used, for the time through which I compare the attracting power of the earth in the two cases (upon the moon and upon the stone), one second, because the experiment of the distance through which a stone falls in one second, and its result are the most familiar to our minds. If I had used one minute, I should have found for the space through which the earth draws the moon 0.0030452 of a mile; or if I had used one hour, I should have found 10.963 miles; and I should have had corresponding numbers for the space through which a stone would fall in the same time. These numbers it is to be observed, increase in the proportion of the squares of the times; and so do those for the fall of a stone (for a stone falls in two seconds four times as far as in one second; in three seconds it falls nine times as far as in one second; and so on).
I will now compare the spaces through which the sun's attraction draws the planets in one hour, and as an instance I will take the earth and Jupiter. In Figure 57, let EF be the path described by the earth in one hour, Ee the path in a straight line which the earth would have described in one hour if nothing had disturbed it. JK the path described by Jupiter in one hour. Jj the path which Jupiter would have described in one hour if nothing had disturbed it. Then eF is the space through which the sun's attraction has drawn the earth in one hour, and jK is the space through which the sun's attraction has drawn Jupiter in one hour; and we shall proceed to find the proportion of eF to jK. Now taking CE as 95,000,000 miles, the circumference of the earth's orbit is 596,900,000 miles, which the earth describes in 365.26 days; and therefore the line Ee which is the earth's motion in one hour, is 68,091 miles. Adding the square of CE to the square of Ee, and extracting the square root of the sum, we find that CE is 95000024.402 miles; and therefore eE, the space through which the sun draws the earth in an hour, is 24.402 miles. For Jupiter, CJ is 494,000,000 miles; the circumference of its orbit is therefore 3,104,000,000 miles; which is described in 4332,62 days; therefore Jj, the motion in one hour, is 29,850 miles; and the length of Cj, found in the same manner, is 494000000.9019 miles; and jK, the space through which the sun draws Jupiter in one hour, is 0.9019 miles. Hence, the attractive force of the sun on the earth is to the attractive force of the sun on Jupiter, in the proportion of 24.402 to 0.9019. But if we compute from the rule of the inverse square of the distances, what would be the proportion of the force of the sun on the earth to the force of the sun on Jupiter, we find that it is the proportion of 24.402 to 0.9024. These proportions may be regarded as exactly the same, the trifling difference between them arising mainly from the circumstance, that I have only used round numbers for the distance of the two planets from the sun. And thus for these two planets it is true that the strength of the sun's attraction is inversely proportional to the square of the distance of the attracted body from the sun.
If I had compared any two planets, I should have arrived at exactly the same agreement. And generally, I may state (though I cannot at present demonstrate it to you), that whenever this rule (see page 126) is found to hold "the squares of the periodic times of several bodies moving round a central body are proportional to the cubes of the distances of the several bodies from that central body," then it will be found, by a process exactly similar to that which we have gone through, that the effects of the central body's attraction at the different distances are inversely as the squares of the distances. Now, this law (that the squares of the times are proportional to the cubes of the distances) was discovered by Kepler, long before the theory of gravitation was invented, to hold in regard to the times and distances of the planets in their revolutions round the sun. Moreover, in regard to the four satellites of Jupiter, the same law holds. For we are able without difficulty to observe their periodic times; we are able also (by observing the transits and the difference of Polar distances of Jupiter and each satellite, or by other methods) to ascertain their apparent angular distance from Jupiter; and from this, knowing the distance of Jupiter from the earth in miles, we can compute the distance of each satellite from Jupiter in miles; and we find that the squares of their times are proportional to the cubes of their distances; and therefore the attraction of Jupiter upon his several satellites is inversely proportional to the squares of their distances from him. In like manner, it is found that the attraction of Saturn upon his seven satellites is inversely proportional to the squares of their distance from him; and, as far as we can examine, the same law holds with regard to the attraction of Uranus on his satellites. Thus for every body which we know, around which other bodies revolve, the force of attraction of the central body on the different bodies that revolve round it is inversely proportional to the squares of their distances.
But, though it is thus established that the attraction of the central body on the different bodies follows that law, is it true that its attraction on the same body alters in the inverse proportion of the square of the distances, when the distance of that body is altered? It is quite certain. But so difficult are the mathematical operations by which this is proved, that I can do little more than refer you (as I have done once before) to the results. The following, however, are the principal steps.
The planets do not revolve in circles but in ellipses, (see page 126,) and therefore, the distance of each planet from the sun undergoes considerable alteration. Kepler's second law of planetary motion was this: that if we draw a line from the sun to a planet, that line passes over equal areas in every successive hour; that area not being the same for different planets, but being constantly the same for the same planet; or, which is the same thing, it describes areas proportional to the times. Now, if we assume the first and second laws of motion to be true, we find that this equal description of areas compels us to admit that the planet is attracted towards the sun; but it does not give us any information as to the law of the attractive force. But the circumstance that the planets move in ellipses, with the sun in one focus of each ellipse, settles this question. It has already been proved that the attractive force must be directed to the sun, that is, to the focus of each ellipse; and then it is proved by a mathematical investigation that if a planet moves in an ellipse, and if the force is directed to the focus of the ellipse, that force in different parts of the orbit must be inversely as the square of the distance from the sun. Thus it is proved that the attraction of the sun on each planet, at its different distances, is inversely proportional to the square of the distance. The same thing also is proved with regard to those planets which have satellites; for several of the known orbits of satellites are elliptical (the others being circular).
There is, however, another very remarkable set of bodies, each of which in its motions sometimes goes nearer to the sun than any other known body, and sometimes passes further from the sun than any other known body; I mean the comets, the explanation of whose motions is one of the most remarkable of Newton's discoveries. A very few comets (not more than five or six,) it is now known, move in very long ellipses, and return periodically to our sight; and to these the same remarks apply which have been above applied to the motion of planets. But at the time when Newton investigated the motions of comets, the idea of periodical comets was totally unknown, and Newton's investigations in regard to comets proceeded entirely on the supposition that the comet did not return. It is difficult for me to attempt to explain here how the orbit of a comet is investigated; the best way perhaps will be to give you something like a history of the thing.
When Newton had investigated the forces which apply to the motion in an ellipse, it was very natural that he should endeavour to see whether the same law of force (namely, that the force is inversely as the square of the distance) which accounts for motion in an ellipse, would account for motion in any other curve. You will see easily that there are two things upon which the motions of a planet depend. One is the force of the attraction of the sun; the other the velocity with which the planet is set going. It is quite conceivable that if a planet were started with very great velocity, it might go away and never come back. The idea which Newton suggested was, that the motion of a comet was of that kind. And, upon pursuing the investigation, he found that a body subject to the attraction of a central body (as the sun) might, if the force varied inversely as the square of the distance, describe the curve called the parabola, (see page 131); but no other law of force would account for the description of such a curve. The form of the parabola is represented in Figure 58, C being the sun;
Fig. 58. and this curve it is evident, possesses two of the peculiarities which most markedly distinguish the motions of comets; it comes very near to the sun at one part, and it goes off to an indefinitely great distance at other parts.
Now, when Newton had found out that the same laws of gravitation which were established from the consideration of elliptic motion would account for motion in a parabola, he began to try whether the parabola would not represent the motion of a cornet. It was found, that by taking a parabola of certain dimensions, and in a certain position, the motions of the comet which had then been observed most accurately could be represented with the utmost precision. Since that time, the same investigation has been repeated for hundreds of comets, and it has been found in every instance that the comet's movements could be exactly represented by supposing it to move in a parabola of proper dimensions and in the proper position, the sun being always situated at a certain point called the focus of the parabola. This investigation tends most powerfully to confirm the law of gravitation; showing that the same moving object, which at one time is very near to the sun and at another time is inconceivably distant from it, is subject to an attraction of the sun varying inversely as the square of the distance.
But if it is true that every particle of matter attracts every other particle of matter, with a force varying inversely as the square of the distance, the effects of these attractions will be shown in other ways besides influencing the periodic revolutions of planets round the sun, or of satellites round their primaries. For instance, the sun attracts both the earth and the moon, and as they are always either at different distances from the sun or lie in different directions from the sun, they will be differently attracted by the sun; and hence their relative motions will be disturbed. Thus arise the perturbations of the moon's apparent motion. These perturbations naturally divide themselves into several classes; and they had been discovered from observation and divided into these classes, long before the theory of gravitation was invented. One of the first triumphs of the theory was the complete explanation of these classes of perturbation of the moon; and the suggestion of many others, which have been verified by the observations made since that time with more accurate instruments.
Of these applications of the theory of gravitation to explain the different perturbations of the moon a great deal might be said. It is a subject involved in mathematical perplexity beyond anything else that I know. But there is one perturbation of the moon which is of so singular a character that probably I may be able to give you some notion of it. It is that which is called the Moon's Variation.
Fig. 59. In Figure 59, suppose E to be the earth, M'M"M'"M"" the moon's orbit, and C the sun. The sun, by the law of gravitation, attracts bodies which are near with greater force than those which are far distant from it. Therefore, when the moon is at M' the sun attracts the moon more than the earth, and tends to pull the moon away from the earth. When the moon is at M'" the sun attracts the earth more than the moon, and therefore tends to pull the earth from the moon, producing the same effect as at M' or tending to separate them. When the moon is at M" the force of the sun on the moon is nearly the same as the force of the sun upon the earth, but it is in a different direction. If the sun pulls the earth through the space Ee, and if it also pulls the moon through the space M"m, these attractions tend to bring the earth and the moon nearer together, because the two bodies are moved as it were along the sides of a wedge which grows narrower and narrower. Thus, at M' and M'" the action of the sun tends to separate the earth and the moon, and at M" and M"" the action tends to bring the earth and the moon together.
You might perhaps infer from this that the moon's orbit is elongated in the direction M' M'″. No such thing: the effect is exactly the opposite. The fact really is, that the moon's orbit is elongated in the direction M" M"". And if you consider what has been said before about the curvature of the orbit, and examine the subject for yourselves, you will see that it must be so. The moon I will suppose is travelling from M"" to M'. All this time the sun is attracting her more than the earth, and therefore increasing her velocity till she reaches M'. When she is passing from M' to M" the sun is pulling her back, and her velocity is diminished till she reaches M". From this point her velocity increases again till she reaches M'", and then diminishes again till she reaches M"". Therefore, when the moon is nearest to the sun, and furthest from the sun, she is moving with the greatest velocity; when she is at those parts of her orbit at which her distance from the sun is equal to the earth's distance from the sun, she is moving with the least velocity. I mentioned in a former lecture (see page 106) that the curvature of the orbit depends on two considerations: one is the velocity; and the greater the velocity is, the less the orbit will be curved: the other is the force; and the less the force is, the less the orbit will be curved. The consequence is this: that as the velocity is greatest at M' and M'", and the force directed to the earth is least (because the sun's disturbing force there diminishes the earth's attraction,) the orbit must be the least curved there. At M" and M"" the velocity has been considerably diminished; the force which draws the moon towards the earth is greatest there (because the sun's disturbing force there increases the earth's attraction), and therefore the orbit must be most curved there. The only way of reconciling these conclusions is by saying that the orbit is lengthened in the direction M" M""; a conclusion opposite to what we should have supposed if we had not investigated closely this remarkable phenomenon. It will easily be understood that the amount of this effect is modified in some degree by the change which the earth's attraction undergoes in consequence of the change of the moon's distance, (the earth's attractive force varying inversely as the square of the moon's distance) but still the reasoning applies with perfect accuracy to the kind of alteration which is produced in the moon's orbit.
This particular inequality was discovered by Tycho Brahe before gravitation was known; and it underwent an examination and was explained by Newton as a result of gravitation. There are other perturbations even more important than this, (the Progression of the Apse, the Evection, the Annual Equation,) of which I only mention the names; they were discovered before gravitation was known, and they were most fully and accurately explained by gravitation.
The next point which I shall mention is this: that the planets disturb one another generally. For it is to be remarked, that the attraction of planets is not confined to the sun, although, in consequence of the sun's very great magnitude and very great attraction, accurate and long-continued observations may be necessary for discovering the comparatively small effect of the planets. But the law of gravitation asserts that every particle of matter attracts every other particle, and therefore every planet attracts every other planet; and therefore the motions of the planets are not exactly the same as if only the sun attracted them. The differences of the real movements from the movements computed on the supposition that only the sun attracts the planets are the perturbations or disturbances of the planets. These disturbances are exceedingly complicated. In fact there is nothing in science which presents the degree of complication that these perturbations of the planets and their satellites present. There is one kind of disturbances, however, of which possibly some notion may be given; they are the most remarkable in Jupiter and Saturn. There are many books, written as late as the beginning of the present century, in which the motions of Jupiter and Saturn are spoken of as irreconcilable with the theory of gravitation. It was one of the grand discoveries of La Place, that the great disturbances of those two planets are caused by what is called the "inequality of long period," requiring some hundreds of years to go through all its changes.
Fig. 60. Let Figure 60 represent the orbits of Jupiter and Saturn. You must observe that they are both ellipses, and the positions of their axes do not correspond. Now, the thing which La Place pointed out as regards these planets, affecting their perturbations, is one which applies more or less to several other planets; it is this: that the periodic times of Jupiter and Saturn are very nearly in the proportion of two small numbers, namely 2 to 5. Upon the proximity of that proportion depend entirely some of the peculiarities of their disturbances. And the effect of this will be seen if we consider in what part of their orbits their successive conjunctions will happen.
Inasmuch as these periodic times are in proportion of 2 to 5, it follows that when Saturn is describing two-thirds of a revolution in its orbit, Jupiter is describing almost exactly five-thirds of a revolution in its orbit. And therefore, if the two planets have been in conjunction, then about twenty years afterwards Saturn has described two-thirds of a revolution, and Jupiter a whole revolution and two-thirds, and the planets will be in conjunction again, but not in the same parts of their orbits as before, but in parts more advanced by two-thirds of a revolution. Thus in Figure 60, suppose 1, 1, to be the place of the first conjunction of which we are speaking. Saturn describes two-thirds of his orbit as far as the figure 2. Jupiter goes on describing a whole revolution and two-thirds of a revolution, and arrives at the same time at the figure 2 in his orbit, and the planets are in conjunction at. 2, 2. Saturn goes on describing two-thirds of the orbit again, and comes to figure 3. Jupiter goes on describing a whole revolution and two-thirds of another, and he comes to figure 3, and they are in conjunction there. The next time they are in conjunction at figure 4, the next at figure 5, and the next at figure 6, and so on. These conjunctions occur in this manner from the circumstance that the periodic times are nearly in the proportion of 2 to 5; there are three points of the orbit at nearly equal distances at which the conjunctions occur.
But we will suppose that they occurred exactly at three equidistant points, and that time after time they happened exactly at the same points. It is plain that in that case there would be a remarkable effect of the disturbances, particularly at those parts of the orbit 1,1, 2,2, 3,3, &c., where Jupiter and Saturn are nearer to each other than at other times. They are very large planets, each of them bigger than all the rest of the solar system, except the sun; they exercise very great attractive force each upon the other: and therefore they would disturb each other, in a very great degree and in a very curious way, if their conjunctions occurred exactly at the same place.
Now these conjunctions do not occur exactly at the same place. The periodic times are nearly in the proportion of 2 to 5, but not exactly in that proportion. In consequence of the periodic times being not exactly in the proportion of 2 to 5, their places of conjunction travel on, until after a certain time the points of conjunction of the series 1,4,7, &c., would have travelled on until they met the series 3,6,9, &c. Not fewer than 900 years are required for this change.
Now so long as three conjunctions take place at any definite set of points, the effect on the orbits is of one kind. As they travel on, the effect is of another description (because, from the eccentricity of their orbits, the distance between the planets at conjunction is not the same), and so they go on changing slowly until the points of the series 1,4,7, &c., are extended so far as to join the series 3,6,9, &c., and then the conjunctions of the two planets occur at the same points of their orbits as at first, and the effect of each planet in disturbing the other is the same as at first; and thus we have the same thing recurring over and over again for ages. During one-half of each period of 900 years, the effect that one planet has upon the other is that its orbit has been slowly changing, and then during the other half, it comes back to the same thing again. Suppose that, during half the 900 years, one planet has been causing the other to move a little quicker, and that during the other half of that 900 years it has been causing it to move a little slower; although that change may be extremely small as regards the velocity of the planets, yet as that velocity has 450 years to produce its effect in one way, and an equal time to produce its effect in the opposite way, it does produce a considerable irregularity. If the place of Saturn be calculated on the supposition that its periodic time is always the same, then at one time its real place will be behind its computed place by about one degree, and 450 years later its real place will be before its computed place by about one degree, so that in 450 years it will seem to have gained 2 degrees. The corresponding disturbances of Jupiter are not quite so large.
These are the most remarkable of all the planetary disturbances, their magnitude being greater than any other, on account of the magnitude of the planets, and the eccentricity of their orbits. There are, however, others of the same kind. One of these was discovered by myself; it depends upon the circumstance, that eight times the periodic time of the earth is very nearly equal to thirteen times the periodic time of Venus. I am afraid I have not conveyed to you any very definite notions of these things; but the foregoing is, I think, the best that can be done. In cases of this kind it is only possible to give a glimmering of what I desire to convey. I wish to impress upon your minds the fundamental circumstances on which these remarkable perturbations depend, and to what they tend, so that you may be able to think and in some measure to investigate for yourselves. I would observe that I have attempted to do all which I believe can be done in the way of popular explanation, in a book which I published some years ago, entitled Gravitation, which was re-published as an article in the Penny Cyclopœdia.
I must however remind you, that I have attempted to explain only one limited class of perturbations. There are some which may be described as a slow increase and decrease of the eccentricities of the orbits, and a slow change in the direction of the longer axes of the orbits; but there are others of which no intelligible account can be given to you.
In order, however, to bring these theories into actual calculation, it is necessary to know, not only the general tendency of the disturbances, but also their actual magnitude. In the perturbations produced by the earth, by Jupiter, and by Saturn, there is no difficulty in doing this. I have already shown you how we can calculate the number of miles through which the earth's attraction draws the moon in one hour. We are certain from most careful experiments made by Newton and (in the present century) by Bessel, that the earth's attraction draws every body at the earth's surface through the same space in the same time; or in other words, that a ball of lead, a cricket ball, and a feather, will fall to the ground with equal speed, if the resistance of the air is removed. We say, therefore, that the earth's attraction would draw a planet through the same space as the moon, provided the planet were at the moon's distance; and for the greater distance of the planet, we must, on the law of gravitation, diminish that space in the inverse proportion of the square of the distance. Now I have already shown you how to compute the space through which the sun draws a planet in one hour; and therefore the problem now is, to compute the motion of a planet, knowing exactly how far and in what direction the sun will draw it in one hour, and also knowing exactly how far and in what direction the earth will draw it in one hour. Without pretending to explain to you how this computation is made, it will be evident to you that we have thus the bases of accurate computation.
In like manner we can from observations of Jupiter's satellites, compute how far Jupiter draws one of his satellites in one hour, and therefore how far Jupiter would draw a planet at the same distance in one hour; and then by the law of gravitation we can compute (by the proportion of inverse squares of the distances) how far Jupiter will draw a planet at any distance in one hour; and this is to be combined, in computation, with the space through which the sun will draw the planet in one hour. In like manner, by similar observation of Saturn's satellites, and similar reasoning, we can find how far Saturn will draw any planet in one hour, and we can combine this with the space through which the sun would draw it in one hour. Thus we are enabled to compute completely the perturbations which these three planets produce in any other planets; and then comes the critical question. Do the planets' motions, as computed with these disturbances, agree with what we see in actual observation? They do agree most perfectly. Perhaps the best proof which I can give of the care with which astronomers have looked to this matter, is the following: the measures of distances of Jupiter's satellites in use till within the last 16 years had not been made with due accuracy, and. in consequence the perturbations produced by Jupiter had all been computed too small by about 1 part. So great a discordance manifested itself between the computed and the observed motions of some of the planets (especially the small asteroids whose orbits are between those of Mars and Jupiter, more particularly Juno), and also in the motions of one of the periodical comets, called Encke's comet, that many of the German astronomers expressed themselves doubtful of the truth of the law of gravitation. I made, and continued at proper intervals, for four years, a new set of observations of Jupiter's satellites, and I had the good fortune to find that the satellites were further from Jupiter than was supposed, that the space through which Jupiter drew them in an hour was greater than was supposed, and that the perturbations ought to be increased by about 1 part. These measures of mine were verified by continental observers. On using the corrected perturbations, the computed and the observed places of the planets agreed perfectly.
For the perturbations produced by Mercury, Venus, and Mars, which have no satellites, we have no similar foundation for our computations; and here we can only go on a method of trial and error. For instance, assuming for calculation that one of these planets has as great a disturbing power as the earth, we can compute how much it will disturb another planet's motion in every position; and if the disturbing power be altered in any proportion, we know that the disturbance of the other planets motion will be altered in that proportion. We therefore find by trial what must be the proportion to make the calculated place of the disturbed planet agree most exactly with its observed place; and then, having settled to our satisfaction the measure of the disturbing power of Venus, or Mars, &c., we can try in all subsequent observations whether it makes the computed places agree equally well. It is found that they do agree perfectly well.
I shall only add to this that the motions of our moon are sensibly disturbed by the planet Venus. An irregularity which has been discovered by observation, and has puzzled all astronomers for fifty years, was explained about two years ago by Professor Hansen, of Gotha, on the theory of gravitation, as a very curious effect of the attraction of Venus.
We have thus a mass of irresistible evidence to prove that the attraction of the sun upon the planets and upon our moon, of the planets upon their satellites, and of the planets one upon another, do follow the law of gravitation. But now comes another question: how do we know that these attractions are produced by every particle of matter in each of these different bodies, as is asserted by the law of gravitation? To prove this I must refer you to a totally different set of computations and observations. I must make a comparison. of the results of theory with the facts of observation, in some of the cases in which it is necessary to consider one body as attracting separately every particle of another, or to consider every particle of one body as separately attracting another body, or to consider every particle as separately attracting every particle.
The first subject to which I shall allude, is the precession of the equinoxes and nutation, which are produced entirely by the attraction which one body (the sun or the moon) exerts separately upon every separate particle on the earth. Upon these I have already spoken, (see page 175); and there will be no need for me to detain you further at present, because you will have been sufficiently aware that there is general conformity between the results which I obtained upon Newton's theory and the results obtained by actual observation. With regard to the numerical agreement, I shall make some remarks presently, when I speak of the mass of the moon.
I will now speak of the ellipticity of the earth; and this, it will be found, is a case in which it is necessary to consider every particle as attracting every particle. First of all you will remember that when the hoop, Figure 23, is put in motion round the vertical spindle, it changes its form. Now in order to explain this, there is a term commonly used which I believe I have not in these lectures hitherto uttered; the reason is, that I do not like it; I allude to the term "centrifugal force." In order to explain why this hoop expands horizontally when it is whirled round the vertical axis, I must recall to your minds the first law of motion. The first law of motion as applied to the hoop is this: if the part a of the hoop is put in motion horizontally, it would go on in a horizontal straight line if it could. No matter what may be the nature of the force which puts a in motion, it has no tendency to move in a circle; and if it were set free, as a stone from a sling, it would immediately fly off in a straight line. And by motion in a straight line, it would go further and further still from the central bar. In order to keep it at the same distance from the central bar, a restraining force is necessary. The term "centrifugal force" has been used to express the tendencies of the various parts of that hoop to acquire greater distances from the central bar. It is a bad term, because in reality there is no force. Perhaps it would be better to say "centrifugal tendency," tendency to recede from the centre, which will in all cases require a force to control it. Now this centrifugal tendency tends to change the figure of the earth; but the consideration of the centrifugal tendency alone is not sufficient to give us the means of calculating what the form of the earth will be.
Newton was the first person who made a calculation of the figure of the earth, on the theory of gravitation. He took the following supposition as the only one on which his theory could be applied: he assumed the earth to be a fluid, or at least to be so far fluid in all parts below the surface, that its form would be the same as if it were entirely fluid. This fluid matter he assumed to be equally dense in every part, so that it was composed of no heavier matter at the centre than at the circumference. For trial of his theory, he supposed the fluid earth to be a spheroid; he then computed the attraction of the whole spheroid upon every one of its component particles of fluid; with this he combined the centrifugal tendency; and then he examined whether, by giving a proper degree of ellipticity to the assumed spheroid, the forces computed on this supposition would be such as would keep the fluid in the spheroidal form which he had supposed to be the earth's form. Now upon the theory of gravitation it is evident that the attraction of a sphere is not the same thing as the attraction of a spheroid. It is necessary to compute what the attraction of this spheroid is, before we can enter into the effect of its combination with the centrifugal tendency. This is the result: suppose that the spheroid AB, Figure 61, is not revolving at all ; still even in that case the attraction of the spheroid upon a body at the part A of the earth is greater than the attraction upon a body at the part B of the earth. But besides this, when we suppose the earth to revolve round the axis Aa, there is the centrifugal tendency of which I have spoken, which does not affect the body at the part A of the earth in the axis of rotation, but which affects the body at B at a great distance from the axis of rotation. We have to consider then that at the Poles of the earth there is an attraction which may
Fig. 61.Fig. 62.
be computed when we assume that the earth is in the form of a spheroid; and at the equator there is an attraction which may also be computed, and is found to be smaller than that at the Pole, and which is still further diminished by the centrifugal tendency.
Thus the whole effective attraction at the Pole is sensibly greater than the whole effective attraction at the equator. This is not unfrequently expressed by saying that "a body weighs more at the Pole than at the equator." And this statement is correct, if it be received with the proper caution. If we carried a pair of scales with proper weights from the Pole to the equator, the same weights which balanced a stone at the Pole would balance it at the equator, because the effect of gravity on both is altered in the same degree. But if we carried a spring-balance from the Pole to the equator, the spring would be more bent by the weight of the same stone at the Pole than at the equator. There is also another effect, to which I shall shortly allude, that a stone would fall further in one second at the Pole than at the equator.
Having computed the effective attractions at the Pole and at the equator, we must now examine what is the consideration to be applied in order to discover whether, with a certain supposition of ellipticity of the earth, this homogeneous fluid will be in equilibrium. The way in which Sir Isaac Newton proceeded is the same as that adopted by every other person who treats of the theory of fluids. You may conceive a cylindrical tube AE, open at both ends, to be put down from the Pole to the centre. Suppose you put down a similar pipe BE from the equator to the centre; and suppose that they communicate at the centre E—these imaginary pipes will not at all disturb the state of rest of the fluid, if it be at rest—by means of each of these pipes we shall ascertain the state of pressure of the fluid at the centre E. By the "state of pressure" I mean , the measure of that compression of the fluid at E, which would enable it to burst any shell that enclosed it at E, if there were no opposing pressure on the outside of the shell; and this measure is to be understood as expressed by so many pounds per square inch; just as we measure the pressure of water in the cylinder of a Bramah's press, by so many pounds per square inch, meaning by that, the pressure on every square inch of its case, tending to burst it.
Now in order to find the pressure produced by the fluid in the column AE, it is not sufficient to know the length of that column, but we must also know the attraction which acts on every part of it. In ordinary cases we speak of the pressure of "a head of water," and we measure it by the depth of the water, and that measure is accurate, because gravity acts equally on all the water in such depths as we have to treat of in ordinary cases; but if there were any part of the water on which gravity did not act at all, that part would add nothing to the pressure; or if there were any part on which gravity acted with only half its usual force, that part would contribute only half its proportion to the pressure. We must therefore ascertain, not only the lengths of different parts of the columns of fluid AE and BE, but also the proportions of the attractions acting on those different parts.
Now we have just seen that the attraction, as diminished by the centrifugal tendency, is less at B than at A; and I may now state as a result of mathematical investigation, that the attraction diminished by the centrifugal tendency is less at the middle of EB than at the middle of EA, and so at very corresponding part of their lengths. Therefore when we estimate the pressure produced by the fluid in the column AE, we have to consider that there is a short column of fluid, of which every part is pulled downwards by a large attraction; and for the column BE, we have to consider that there is a longer column of fluid, of which every part is pulled downwards by a smaller attraction. And therefore (on the principles which I have just explained in reference to a head of water) by choosing a proper ellipticity of the spheroid, or in other words, choosing a proper proportion of length of the columns AE and BE, the pressures per square inch which these columns produce at the centre E may be made exactly equal.
And now comes into account the fundamental property of fluids, namely, the equality of pressure in all directions. When the fluid is in a state of rest, the pressure per square inch at E must be the same, whether it is estimated by the pressure which it exerts in sustaining the column EA, or by the pressure which it exerts in sustaining the column EB. This is the fundamental property of fluids, upon which (as a matter of science) I shall not speak at greater length; I shall merely remind those who have to do with steam boilers, or Bramah's presses, or other engines in which fluids are in a state of violent compression, that there is equal tendency to burst upwards, downwards, or sideways. Therefore when we have gone through the investigation, taking into account what is the attraction of the spheroid on every part of the fluid in each column, what is the amount of the centrifugal tendency in each part of the column EB, and what is the length of each of the pipes, on an assumed ellipticity of the earth; and when we have thus found, by considering each pipe separately, the pressure at E where the two pipes join, we must have the two pressures equal. And if the ellipticity which we have assumed for the earth will not make these two pressures equal, we shall know that is a wrong ellipticity, and we must try another, till we find one which will make the pressures equal. If we use a proper algebraical process, we can diminish the trouble of this so far as to reduce it all to a single trial; but still the principle of the process is exactly the same. And when we have found one ellipticity which does make the pressures equal, we are sure that we have got the right ellipticity for the earth; still limited, however, by our original supposition, that the earth is a fluid of equal density in every part, and is not more dense at the centre than near the surface; and if that original supposition be wrong, our conclusion will be wrong.
It was in this manner, assuming the earth to be a fluid no more dense at the centre than near the surface, and proceeding in every way as I have described, that Sir Isaac Newton inferred that the form of the earth would be a spheroid, in which the length of the shorter diameter or axis on which it turns, is to the length of the longer or equatoreal diameter, in the proportion of 229 to 230. And this, perhaps, may be considered as one of the most wonderful investigations in modern science.
With this proportion, as you will have perceived, is intimately connected the proportion of gravity at A and B, that is to say, of the attraction at A to the attraction diminished by centrifugal tendency at B. You must accept it as a result of mathematical investigation, of which I can give you no further explanation at present, that if you compare a point f in the line EA, with a point g in the line EB, such that Ef : EA : : Eg : EB, you then find the whole effective force at f, to bear the same proportion to the whole effective force at g, which that at A bears to that at B. You must also remember what I have said about the pressure at E depending not only on the length of the pressing column of fluid, but also on the effective force upon each part of it. Then you will easily see that the effective force or gravity at B, must be to the gravity at A, in the proportion of 229 to 230. All this depends on the supposition that the fluid is of equal density thoughout.
The next point is, how can we verify this by observation? How can we find whether a body appears heavier upon one part of the earth than upon another? I have already said that it will not do to take a pair of scales and weigh the body with weights. The next suggestion is to weigh it with a spring-balance. It is not beyond possibility that a spring-balance might be made which would be sufficiently delicate for this purpose. The principal difficulty perhaps would be experienced in overcoming the effects of change of temperature in altering the elasticity of the spring: but if this could be done, and if the spring balance otherwise could be constructed with very great delicacy, the gravity at different parts of the earth could be compared. But the method which actually is used for this purpose, is one depending on the effect of gravity in producing motion. Theoretically, this effect of gravity may be measured by observing how far a stone falls in one second; but practically it is more accurately ascertained by the use of a pendulum. This is susceptible of very great accuracy indeed.
The pendulum is made of metal; it turns with a hard steel prism, having a very fine edge, upon hard plates of agate, or some very hard stone. It swings like the pendulum of a clock. But you must observe that a clock pendulum will not do for this purpose, because there are other forces besides gravity acting on the pendulum, that is to say, the clock weights acting through the train of the clock wheels. It is necessary to have a detached pendulum. Now I wish to know how many vibrations that pendulum would make in a day. It is troublesome to find out. Possibly the pendulum will not swing for a whole day.
In the experiments made in an expedition directed by the Spanish Government, a man was stationed on each side of the pendulum, to count 60 vibrations at a time; and they continued to count the vibrations as long as the pendulum continued sensibly in motion. When they had got through a great number of 60's, they observed the time which a clock showed. It was a very tedious method indeed.
In every other instance in modern times, the vibrations of the detached pendulum have been compared with the vibrations of a clock pendulum. The mode adopted in the English and French expeditions was this: a detached pendulum is placed in front of a clock; a person is watching with a telescope; he watches when the two pendulums are going the same way; he remarks whether the vibrations of the detached pendulum recur quicker or slower than those of the clock pendulum; he sees that the vibrations separate more and more, till the two pendulums actually move in opposite ways; after this, they begin to move more nearly in the same way, and at length move exactly in the same way; perhaps the number of vibrations elapsed between these two agreements of motion may be 500. If you can determine the time when the two pendulums swing the same way, you find how long it is before one pendulum gains two vibrations upon the other. Then the calculation is this: suppose that the detached pendulum is going slower than the clock pendulum; and suppose that 71 minutes elapse between two agreements of motion of the pendulum; then this shows that while the clock has gone 71 minutes, or while its pendulum has made 450 vibrations, the detached pendulum has made only 448 vibrations. Now, the clock is going day and night, and by means of observation with the transit instrument, you can find how many hours, minutes, and seconds, the clock hands pass over in one day, or how many vibrations the clock pendulum makes in one day. Then, as the detached pendulum makes 448 vibrations for every 450 made by the clock pendulum, you find at once how many vibrations the detached pendulum makes in 24 hours.
Some corrections for the effect of temperature in altering the length of the pendulum, and for other circumstances, are necessary; but I cannot enter upon the details of these at present. The method which I have described is exceedingly delicate. There is no difficulty in ascertaining by it the number of vibrations which the detached pendulum will make in a day, with no greater error than one-tenth of a vibration, or with an error not exceeding one eight-hundred-thousandth part of the whole.
This same pendulum is then carried to different parts of the earth, and is observed at every place in the same manner and with the same accuracy. The most important of our modern expeditions were those conducted by Colonel Sabine and Captain Foster. Each of these officers was entrusted by the Government with a ship, for the purpose of going to different parts of the earth, in order to observe the same pendulums at different places. Colonel Sabine went as far as a point in Spitzbergen, near the Pole; and both officers went to many places near the equator; to the West India Islands, to South America, and to South Africa. In this manner it was found that the number of vibrations which a pendulum makes per diem is not the same in different parts of the earth. When near the Pole, the pendulum makes about 240 vibrations in a day more than when near the equator. It is easily seen that this is a consequence of the force of gravity being greater there. If gravity be very small indeed, the motion of the pendulum will be exceedingly sluggish. The greater is the force of gravity which acts upon a pendulum when out of its central position, the more briskly it pulls the pendulum down towards its central position, and the shorter time is occupied by every vibration, and the greater is the number of vibrations made in one day. Thus we have the means of measuring the gravity at different parts of the earth.
Now, the practical inference from the experiments is this: the proportion of the force of gravity at the Pole to the force of gravity (that is, attraction diminished by the centrifugal tendency) at the equator, is not as 230 to 229, as Newton stated, but is very nearly the proportion of 180 to 179. Now, here we have a remarkable departure from Newton's results. He proved that, if the earth were of equal density throughout, the proportion of the two axes would be as 229 to 230; and the proportion of gravity at the Pole and the equator would be as 230 to 229. We find from trigonometrical surveys, and observations with the Zenith Sector, as you may remember, that the proportion of the earth's axes is as 299 to 300; and we have now found, from experiments with the pendulum, that the proportion of gravity at the Pole to gravity at the equator, is as 180 to 179. This shows that Newton was wrong.
The question then is, in what was he wrong? Now, it must be remarked, that Newton's calculation was founded entirely on the supposition that the earth is of equal density thoughout. When we consider the matter, it is very unlikely that, if the interior of the earth is fluid, its density is equal in every part. Accordingly in the last century, investigations were made, supposing the earth to be of different densities in different parts, and specially that the density increased as we approach nearer to the centre. The principal investigation (to which in fact, nothing important has been added in later times) was made by an eminent French mathematician, named Clairaut. The supposition on which he went (and which is really the only kind of supposition to be made at all in this investigation) was, that the earth consists of strata of different densities, but that each stratum is in some degree elliptical; the ellipticity of one stratum being different from that of another; and the investigations leave these ellipticities to be determined by considerations connected with the equilibrium of fluids.
For instance, in Figure 62, setting aside the water and floating matter on the top; suppose that F is the region of lava, if you please, that G is the region of melted iron, and that H is the region of melted platinum. Suppose you conceive one tube to be drawn from the Pole, and another from the equator, meeting at E; you make this condition, that when you have investigated properly the gravity, (arising from the attraction of every particle of the spheroid upon any one particle, and modified by the centrifugal tendency from the axis Aa ,) acting on the different substances at the different parts of these tubes, and when you have found the pressure of the fluids in the various tubes, by taking into account these several circumstances—the gravitation, with the centrifugal tendencies, the lengths of the different portions of the pipes, and the density of the fluid in each of those portions of the pipes on which the forces are acting; when you have taken these into consideration, you find the pressure of the fluid at the place E where the two pipes meet; then, by the principle of the equality of the pressures of fluids in all directions, you must have the two pressures at E equal, or the fluid will not be in a state of rest. Suppose then, that we have assumed such a degree of ellipticity for the external surface of the earth, and such ellipticities for the different strata, that this condition of equality of pressure at E is satisfied; still we have not done all that is necessary. It is necessary that, if we suppose two or more tubes of any shape whatever, drawn from any points of the surface to any point of the fluid, as for instance, the point K, or the point L, in Figure 62, the pressures at K produced by the fluids in the different tubes abutting at K shall be equal; and similarly, that the pressures at L produced by the fluids in the different tubes abutting at L shall be equal.
These considerations make the problem rather complicated. However, it can be completely solved, whatever be the succession of densities of the different strata; and the result is this. According to the law of density of the successive strata, the law of the ellipticities of the successive strata would be different, and the amount of the ellipticity of the earth's surface would be different. Except you know what is the structure of the interior, you cannot say what the ellipticity of the earth will be; but whatever that law of internal structure may be, you can say that there is a certain relation between the ellipticity of the earth and the degree of alteration of gravity from the poles to the equator. Suppose you take a vulgar fraction to express the proportion of the whole diameter by which the earth is flattened at the Poles, and suppose you take another vulgar fraction to express the proportion of the whole gravity by which the gravity is diminished as you go from the Poles to the equator; if you add the two fractions together, whatever be the succession of densities of the different fluid strata of the earth, the sum of those two fractions will be 1. This particular value 1 depends upon the velocity of the earth's rotation; if the earth revolved in a longer or shorter time than 24 hours of sidereal time, the sum must be a different quantity.
We are enabled thus by the pendulum experiments, which give us the law of change of gravity, to infer what is the ellipticity of the earth, provided the law of gravitation be true, (for that has been the basis of the whole investigation.) When, by means of the pendulum, we have got the variation of the law of gravity, we have only to express it by a fraction, and to subtract that fraction from 1, and we get another fraction which expresses the compression of the earth, or the difference between the two axes divided by one of them. And if the compression or ellipticity of the earth which we find by this process (depending entirely on the law of gravitation) agrees with that which we find from trigonometrical surveys and the use of the Zenith Sector, (in which the law of gravitation is not concerned at all,) this will be a strong proof of the correctness of the law of gravitation. Now, the proportion of gravity at the Poles and the equator is found to be about 180 : 179, so that the diminution of gravity in going from the Poles to the equator is about 1 part. And if we subtract the fraction 1 from the fraction 1, the remainder scarcely differs from 1, showing that according to this theory the ellipticity of the earth ought to be 1, or the proportion of the earth's diameters ought to be as 300 : 299. And this is exactly the same proportion which has been found from triangulation surveys and Zenith Sector, as described to you in a former lecture. This, therefore is a very remarkable proof of the correctness of the theory of gravitation, when applied with proper attention to all the circumstances.
There is another very curious method of determining the ellipticity of the earth, which also depends upon the theory of gravitation. I have said that the attraction of a spheroid upon any external body is not the same as the attraction of a sphere; and therefore the attraction of the earth upon the moon is not the same as if the earth were a sphere. There is therefore a small irregularity in the motions of the moon depending on the earth's ellipticity; and it is very remarkable that, whatever be the succession of densities of the strata of the earth, this irregularity is found upon the theory of gravitation to depend upon nothing but the ellipticity of the earth's surface. And therefore, if we observe the moon's motions so carefully as to discover the amount of this irregularity, and if we make the proper calculation from it, we can find the ellipticity of the earth. The ellipticity thus determined agrees well with that found from the surveys: and thus another proof is given of the correctness of the theory of gravitation. It may not be amiss to state here that the motions of Jupiter's Satellites are much disturbed by the ellipticity of Jupiter's body.
There is another inference from these theoretical investigations of the figure of the earth, which it is proper to mention. Though we do not know what the law of the earth's internal structure is, yet we can assume some law of densities of successive strata gradually changing from the surface to the centre which shall give a value for the earth's ellipticity agreeing with the results which I have mentioned: and from this law we can find what the mean density of the earth is. The inference was thus made by Clairaut and his successors, that the mean or average density of the earth is about twice as great as at the surface, and that in some parts at and near to the centre, the density must be considerably greater than that mean density. Remarking that the mean density of the earths and rocks at the earth's surface, taking one with another, is about twice and 6 that of water, it was inferred that the mean density of the earth is more than five times the density of water. After this, another experiment was made, applying to the determination of the earth's mean density I have mentioned the liberality of George the Third in supplying funds for the observation of the transit of Venus. The same monarch (as I believe) supplied the funds for another experiment of great importance; it was the Schehallien experiment. Probably some of my auditors who have travelled in the highlands of Scotland, have seen the Schehallien Mountain; it may be observed from the banks of Loch Tay. If you go from Killin to Taymouth, it is on the left hand. Now this mountain was selected for observations of a very remarkable kind. It was argued that if the theory of gravitation were true, (that is to say, if attraction were produced not by a tendency to the centre of the earth, or to any special point, but to every particle of the earth's structure,) then by the fundamental law of gravitation, the attraction of a mountain would be a sensible thing; for a mountain is a part of the earth, with this difference only, that though the mountain is small in comparison with the earth, yet you get so close to the mountain, that its effect may be very sensible as compared with the effect produced by the rest of the earth. Some parts of the earth are 8,000 miles from us, and their attraction will be comparatively small. It was therefore thought worth while to ascertain whether the attraction of a mountain would be sensibly felt; and the Schehallien observations were a noble experiment towards the attainment of that result.
The Schehallien Mountain ranges east and west; it was possible to make astronomical observations on the north and south sides; and it was also possible to connect the two places of observation by triangulation. Supposing Figure 63 to represent a section
of the mountain north and south, N the northern, S the southern observing station. Observations were made at N and S upon stars with the Zenith Sector; the same instrument of which we have spoken so frequently, in reference to the determination of the elements of the earth's figure. By the use of the Zenith Sector, the difference of the directions of gravity at these two stations was found, exactly in the same manner as the difference of the directions of gravity in two stations of a meridional survey, Figure 18.
The direction of gravity at each station, you will observe, is the result of the gravity of the whole earth (as considered for a moment independently of the mountain), combined with the attraction of the mountain. And this is the consequence: supposing that at N, if there were no mountain, the direction of the gravity would be NE; then introducing the supposition of the mountain, the attraction of the mountain would pull the plumb-line sideways towards the centre of the mountain, and the direction of the gravity would be Ne. And in like manner, supposing that if there were no mountain, the direction of the gravity at S would be SF; then, introducing the mountain, the effect of its attraction is to pull the plumb-line towards the centre of the mountain, and the direction of gravity would be Sf.
Observe, then, the effect of the mountain; at N the direction of gravity is Ne instead of NE, and at S, the direction of gravity is Sf instead of SF; that is to say, the two directions which are taken by the plumb-line of the Zenith Sector, make a greater angle than they would if the mountain were not there.
Now, then, we come to the thing which we have to try. We know the general dimensions of the earth; we know what the inclination of the plumb-line at N and S would be if there were no mountain in the case. We know that this is a general rule: if we step 100 feet (nearly) forward, the direction of the plumb-line changes one second. If, then, we can find the distance from our observing station at N to that at S, then we can tell from that distance how much the directions of the plumb-line at N and S would be inclined if there were no mountain; and we can compare that inclination with the inclination observed by means of the Zenith Sector.
Accordingly the observations were made in exactly the same manner as the observations made for determining the figure of the earth. The Zenith Sector was carried to N, and certain stars were observed; the Zenith Sector was then carried to S, and the same stars were observed at that place. By means of these observations of the stars, the actual inclinations of the plumb-line at the two places were found. The next thing done was to carry a survey by triangulation across the mountain. This was done in the most careful way in which the best surveyors of the time could accomplish the task. The result was, that the distance between the stations was found such that, supposing that there was no mountain in the case, the inclination of the two plumb-lines ought to be 41 seconds. It was found practically from the observations by the Zenith Sector, that the inclination of the two plumb-lines actually was 53 seconds.
The difference between the two was the effect of the mountain. The mountain had pulled the plumb-line at one station in one direction, and at the other station in the opposite direction, to such a degree, that instead of the two plumb-lines making an angle of 41 seconds, they made an angle of 53 seconds; or, in other words, that the sum of the effects of the two attractions of the mountain, on opposite sides, was 12 seconds.
The next thing was, to draw from this observation a determination of the mean density of the earth. The general form of the process was this: the mountain was surveyed, mapped, levelled, and measured, in every way, so completely, that a model of it might have been made; it was then (for the sake of calculation) conceived to be divided into prisms of various forms: the attraction of every one of these was computed, on the supposition that the mountain had the same density as the mean density of the earth; and by means of this, the attraction of the whole mountain was found on the same supposition.
Thus it was found, that if the density of the mountain had been the same as the mean density of the earth, the sum of the effects of the attractions of the mountain at N and S would have been about 1 part of gravity. But the observed sum of effects was 12 seconds, which corresponds to 1 part of gravity. Hence the density of the mountain is only about 5 of the earth's mean density; or the earth's mean density is nearly double of the mountain's density. The nature of the rocks composing the mountain was carefully examined, and their density as compared with that of water was ascertained; and thus the mean density of the earth was found to be something less than five times the density of water: a result agreeing nearly with that found from the assumption of the law of density of the earth's strata, connected with the observed variation of gravity, and observed ellipticity.
This was the nature of the celebrated Schehallien experiment, which was so extremely creditable to the parties by whom it was promoted and undertaken, and so important in its results.
After this, another set of experiments was made; first by Mr. Henry Cavendish, a rich man, much attached to science, and who made many important contributions to chemistry, and other branches of natural philosophy (from whom the experiment of which I am speaking received the name of the Cavendish Experiment); afterwards by a Dr. Reich; and finally, in a very much more complete way, by Mr. Francis Baily, as the active member of a committee of the Astronomical Society of London, to whom funds were supplied by the British Government. It is an experiment of a different kind—a sort of domestic experiment—one of those experiments which can be made in your own observing rooms at home, and which are, in many respects, preferable to those made on the hill sides of Scotland.
The shape in which the apparatus is represented in Figure 64, is that in which it was used by Mr. Baily. There are two small balls A,B, (generally about two inches in diameter,) carried on a rod ACB, suspended by a single wire DE, or by two wires at a small distance from each other. By means of a telescope, the positions of these balls were observed from a distance. It was of the utmost consequence that the observer should not go near, not only to prevent his shaking the apparatus, but also because the warmth of the body would create currents of air that would disturb everything very much, even though the balls were enclosed in double boxes, lined with gilt paper, to prevent as much as possible the influence of such currents. When the position of the small balls had been observed, large balls of lead, F,G, about twelve inches in diameter, which moved upon a turning frame, were brought near to them; but still they were separated from each other by half-a-dozen thicknesses of deal boxes, so that no effect could be produced except by the attraction of the large balls. Observations were then made to see how much these smaller balls were attracted out of their places by the large ones. By another movement of the turning frame, the larger balls could be brought to the position HK. In every case, the motion of the small balls produced by the attraction of the larger ones, was undeniably apparent. The small balls were always put into a state of vibration by this attraction; then by observing the extreme distances to which they swing both ways, and taking the middle place between those extreme distances, we find the place at which the attraction of the large balls would hold them steady.
Suppose, now, the attraction of the large balls was found to pull the small balls an inch away from their former place of rest: then comes the question what amount of dead pull does that show? The steps by which this is computed are curious.
First I must tell you that it has long been known (from experiment), that when a rod carrying balls is suspended in this manner by a wire, the space through which the balls will be pulled sideways is exactly in proportion to the force which pulls them sideways. In this respect the law of forces acting on the suspended rod, is exactly similar to the law of forces acting sideways on a pendulum vibrating in a moderately small arc, for the motion of a pendulum is thus produced. If the pressure caused by the weight of the pendulum-bob, which acts vertically, is resolved into two parts, of which one part is in the direction of the pendulum rod, and the other acts sideways upon the pendulum, the former does not affect the movement of the pendulum at all, and the latter, which produces the movement, is proportional to the distance of the pendulum from its place of rest, and therefore is similar in its law to the law of the force of twist of the suspending wire by which a rod with balls is supported (which force of twist is the same thing as the force which pulls the balls aside, because it exactly resists that force). Moreover, the force which acts sideways on the pendulum-bob, is in the same proportion to the whole weight of the bob, as the displacement sideways is to the length of the pendulum. Now the length of a pendulum which vibrates in a second, is 39·139 inches; and for such a pendulum, if it is pulled one inch sideways, the dead pull sideways (as I have just explained) will be 1 part of its weight: and thus we know that, for any balls or other things which vibrate in one second, the dead pull sideways corresponding to an inch of displacement is 1 part of their weight.
Then it is known as a general theorem regarding vibrations, that to make the vibrations twice as slow, we must have forces (for the same distances of displacement) four times as small; and so in proportion to the inverse square of the times of vibration. Thus if balls or anything else vibrate once in ten seconds, the dead pull sideways corresponding to an inch of displacement is 1 of their weight. So that in fact, all that we now want for our calculation, is the time of vibration of the suspended balls. This is very easily observed; and then on the principles already explained, there is no difficulty in computing the dead pull sideways corresponding to a sideways displacement of one inch; and then (by altering this in the proportion of the observed displacement, whatever it may be) the sideways dead pull or attraction corresponding to any observed displacement is readily found. The delicacy of this method of observing and computing the attraction of the large balls may be judged from this circumstance: that the whole attraction amounted to only about 1 part of the weight of the small balls, and that the uncertainty in the measure of this very small quantity did not amount probably to 1 or 1 of the whole.
Then the next step was this: knowing the size of the large balls and their distances from the small balls in the experiment, and knowing also the size of the earth, and the distance of the small balls from the centre of the earth, we can calculate what would be the proportion of the attraction of the large balls on the small balls to the attraction of the earth on the small balls (that is the weight of the small balls), if the leaden balls had the same density as the mean density of the earth. It was found that this would produce a smaller attraction than that computed from the observations. Consequently the mean density of the earth is less than the density of lead in the same proportion; and thus the mean density of the earth is found to be 5·67 times the density of water.
The near agreement of this result with that found from the Schehallien experiment, and that found from the theory of the figure of the earth, (taking the observed ellipticity of the earth in combination with such a law of density as would produce that ellipticity,) shows, beyond doubt, that the same law of gravitation which regulates the attraction of the sun upon the planets, and the attraction of the earth upon the moon, does also apply to the attraction of a leaden ball upon another ball within a foot of it. In regard to the slight difference of results, it is probable that the result of the Cavendish experiment is the more accurate of the two, simply because there is always some uncertainty upon the constitution of the rocks and mineral veins forming the interior of such a mountain as Schehallien.
I have gone into this subject, "the evidence for the theory of gravitation," at some length. First, because the law of gravitation is the most extensive in its application of all known laws; for we are certain that it applies to every body, and to every portion of a body, in the Solar System, and we have strong reason to believe that it applies to the mutual action of those stars which are observed to revolve in binary systems, the two stars revolving each round the other. Secondly, because the explanation of the methods of calculating its effects, in several instances of varied character, leads us to the consideration of several very interesting principles and applications of them. And thirdly, because the assumption of this law is necessary for the estimation of the weight of the bodies of the Solar System, the part of my subject to which I now proceed.
I must first allude to the weight of the earth, because the weights of all other bodies of the Solar System are necessarily referred to it as a standard. Taking the dimensions of the earth as I have stated them before, the number of cubic miles in the earth is about 259,800,000.000; each cubic mile contains 147,200,000,000 cubic feet; and each cubic foot, upon the average, weighs 5·67 times as much as a cubic foot of water, or 354 lbs. 6 oz. avoirdupois. I will leave the combination of these numbers to you, and will only remark at present, that I have shown you how the first step is made in referring the weights of the bodies of the Solar System, to the pound weight avoirdupois.
Next, I shall proceeed with the estimation of the weight of the sun as compared with the weight of the earth. And this I shall do by comparing the attraction produced by the sun with the attraction produced by the earth at the same distance. And here is involved an important principle, namely that the weight of a body is proportional to the attraction which it exerts. In order to explain this, it is necessary to remark, that every calculation of perturbation in the Solar System requires us to suppose, that the attraction of one body A upon another body B is not a mysterious influence by which the presence of A causes a movement in B, without any reciprocal influence upon A, but is a real mechanical action which exerts equal strains upon both, just as if they were connected by a contracting spring. Thus, every strain which a large body A produces upon a small body B; is accompanied by an equal strain, produced by the small body B upon the large body A; and both A and B will be disturbed; but A will not be disturbed so much as B, because its mass is greater. In the computations of perturbations, for instance the perturbations of Saturn by Jupiter, it is necessary to consider that Jupiter attracts the sun according to the same law, (as regards the motion produced in it,) by which it attracts Saturn; else the computed disturbance of Saturn would not at all answer to the observed disturbance. If, then, the sun attracts Jupiter and a comet (when at equal distances from the sun) so as to produce the same motion in them, this shows that the mechanical pull upon Jupiter is greater than the mechanical pull upon the comet in the same proportion in which the mass of Jupiter is greater than the mass of the comet; and therefore, (considering the reciprocal mechanical actions upon the sun as equal to the mechanical actions of the sun upon them,) Jupiter's pull upon the sun is greater than the comet's pull upon the sun in the same proportion as their masses; and the movements which they produce are in the same ratio. I may add that the same principle is involved in every investigation relating to the figure of the earth. Assuming this principle then, I shall proceed to compare the attractions which the sun and the earth would exert upon a body at equal distances from them.
In former computations in this lecture, I found that in Figure 56, the earth draws the moon through 10·963 miles in one hour, the moon being at the distance of 238,800 miles from the earth; and in Figure 57, that the sun draws the earth through 24·402 miles in one hour, the earth being at the distance of 95,000,000 miles from the sun. In order to make these attractions comparable, we must reduce them both to the same distance; and we shall therefore first say, if the earth draw the moon through 10·963 miles in an hour when at the distance of 238,800 miles, how far would it draw the moon in an hour if it were at the distance of 95,000,000 miles? Diminishing 10·963 in the proportion of the inverse squares of the distances, we find that the earth would draw the moon through 0·00006927 mile or 4·389 inches in an hour, if it were at the distance of 95,000,000 miles. Comparing this with 24·402 miles through which the sun draws the earth or moon when at the same distance, we find that the sun's attraction is 352,280 times as great as the earth's, and therefore, that the sun's mass is 352,280 times as great as the earth's. You can, if you please, combine this with the numbers which I gave before, to express the weight of the sun in pounds.
The angular diameter of the sun, as viewed from the earth, is 32 minutes of a degree. Computing from this the sun's diameter, we find that the sun's bulk is 1,400,070 times as great as the earth's bulk. Therefore the sun's mean density is only about 1 of the earth's mean density, or about 1·4 times the density of water.
The principle which has been used above for comparing the mass of the earth with that of the sun, is used without the smallest alteration for comparing the mass of Jupiter, Saturn, Uranus, or Neptune, with that of the sun; and in all cases where the satellites can be easily observed, it can be applied with very great accuracy. For those planets which have no satellites there is considerable uncertainty. The only way in which they are determined is by the perturbation of other planets. For instance, we see that in certain positions, the earth is disturbed by Mars a few seconds, say six or eight. We compute what would be the amount of perturbations if the planet Mars were as big, or half as big, as the earth, and we alter the supposition till we find a mass which will produce perturbations equal to those which we observe. Thus we go through a process which is one of trial and error. In this manner the masses of Mars and Venus are determined. That of Mars is not very certain; that of Venus is more certain both because it produces larger perturbations of the earth, and because its attraction tends to produce a continual change in the plane of the ecliptic, which in many years amounts to a very sensible quantity. The mass of Mercury is still very uncertain; lately attempts have been made to deduce it from the perturbations which Mercury produces in the motion of one of the comets.
There is, however, one mass which is more important than the others, and that is the mass of the moon. There are several methods by which the mass of the moon is determined. In speaking of the precession of the equinoxes and nutation, I pointed out that lunar nutation is, in fact, an inequality of lunar precession, connected with it by a certain proportion which is known from the theoretical investigation. Therefore, as we can observe the amount of lunar nutation, we can, by taking the proportion backwards, compute the annual amount of lunar precession; and we can observe the whole annual precession produced by both the sun and the moon; and, subtracting the lunar part, there remains the part due to the sun. Thus we have got ths proportion of lunar precession to solar precession. Now, you may remember, that on a former occasion, I went through the steps of a calculation, showing how, if we assumed the proportion of the moon's mass to the sun's mass, we might find the proportion of the lunar precession to the solar precession. By going backward through the same steps, knowing the proportion of lunar precession to solar precession, we may find the proportion of the moon's mass to the sun's mass.
There is a second method by which the mass of the moon may be obtained, from the proportion which its effect (depending upon the difference of its attractions upon different parts of the earth) bears to the sun's effect (depending upon a similar difference); and that is, by comparing the tides at different times, Everybody knows that the tides follow the moon generally but not entirely. They do not follow the time of the moon's meridian passage by the same interval at all times; and they are much larger shortly after new moon and full moon than at other times. From a careful examination of all the phenomena of tides, it appears that they may be most accurately represented by the combination of two independent tides, the larger produced by the moon, and the smaller produced by the sun; that at spring tides these two tides are added together, and make a very large tide; but that at neap-tides the high water produced by the sun is combined with the low water produced by the moon, and the low water produced by the sun is combined with the high water produced by the moon, and thus a small tide is produced.
By comparing the spring-tides with the neap-tides, we can find the proportion of the effect produced by the moon to that produced by the sun. Now, the tides are produced, not by the whole attraction of the moon and the sun upon the water, but by the difference between their attraction upon the water and their attraction upon the mass of the earth, by which difference the moon (and similarly the sun) attracts the water nearest to it from the earth, and attracts the earth from the water which is farthest from the moon.
Still these forces undergo some very peculiar modifications in their actions which produce the lunar and solar tides, which in many cases alter them in proportions slightly different. Thus their tidal effects are nearly but not exactly in the proportion of their difference of attractions, of which I have spoken; but with proper investigation it is possible to find, from their tidal effects, the proportion of their differences of attraction. And when this is found, we have obtained a proportion of differences of attraction which are exactly the same as the differences of attraction concerned in producing precession (of which I have already spoken); and, from knowing this proportion, and knowing the distances of the sun and moon, we can, in the same way, find the proportions of the masses of the sun and moon.
Fig. 65. The third method is this. In Figure 65, suppose C to be the sun, E the earth, M the moon. I have spoken continually of the sun's attraction upon the earth and of the earth's revolution round the sun, as if the sun were the only body whose attraction influenced in a material degree the earth's movement. But in reality the moon also acts in a very sensible degree upon the earth. And the immediate effect upon the motion of the earth is found by proper investigation to be the following. Draw a line from E to M, and in this line take the point G, which is called the "centre of gravity," so that the proportion of EG to GM is the same as the proportion of the weight of M to the weight of E; or so that if E and M were like two balls fastened upon the ends of a rod, they would balance at G. Then investigation shows that the motion of the earth may be almost exactly represented by saying that the point G travels round the sun in an ellipse, describing areas proportional to the times, (according to Kepler's laws), and that the earth E revolves round the point G in a month, being always on the side opposite to the moon.
Consequently, the direction in which the earth would be seen from the sun (and therefore the direction in which the sun is seen from the earth) depends in a certain degree on the distance EG. And therefore, if we observe the sun regularly, and if we compute where we ought to see the sun, according to Kepler's laws, the difference between these two directions will be the angle EGG; and knowing the distance CG, we can then compute the length of EG, and the proportion which it bears to GM; and this proportion, as I have said, is the same as the proportion of the mass of the moon to the mass of the earth.
A fourth method of determining the mass of the moon depends upon an accurate estimation of the force of gravity at the earth's surface. We know the moon's distance very accurately, and we know the earth's attraction at the earth's surface (that is, gravity) very accurately; and therefore we know the earth's attraction on the moon accurately. But the force which thus acts on the moon makes it revolve, not round E, in Figure 65, but round G. Now, in Figure 56, from a knowledge of the distance EM, and taking MN to be that proportion of the orbit which is described in one hour, we found the length of the line mN through which the earth's attraction pulls the moon in one hour. Here in Figure 65, we have the opposite problem; knowing the length of the line through which the earth's attraction pulls the moon in one hour, we have to find what is the length of GM, the semi-diameter of the orbit in which it revolves. Having found this, we find EG; and then, as in the last method, the proportion of EG to GM is the same as the proportion of the mass of the moon to the mass of the earth.
All these different methods agree very well in giving the result that the mass of the moon is about 1 of the earth's mass. And when this mass, and the known mass of the sun, are used in combination with a law of density of the strata of the earth which will well explain the observed ellipticity of the earth, it is found that they explain almost exactly the observed amount of precession.
There remains but one set of bodies whose masses can be determined, namely, Jupiter's satellites. It so happens that these little bodies disturb each other very much. In consequence of the periodic time of Jupiter's second satellite being very nearly double that of the first, and the periodic time of the third being very nearly double that of the second, there is a kind of "inequality of long period" in their motions which admits of tolerably accurate observation, by the observation of their eclipses. From this their masses are computed in the same manner as the masses of the planets from their mutual perturbations. Computations are made of the effect of one satellite upon the others, on the assumption, for instance, that the satellite is 1 part of the mass of Jupiter; if this does not produce, in calculation, the perturbations which are actually observed, the assumed mass must be altered in the proper proportion. For the fourth satellite, there are no perturbations of the nature of "inequalities of long period," but there are others sufficiently sensible, which are treated in just the same way. In this manner the masses of these distant little bodies are ascertained with reasonable accuracy. The largest of them (the third satellite) is about as large as our moon.
I have thus redeemed my pledge of explaining how the weights of the principal bodies of the Solar System are estimated by means of a pound avoirdupois. And I will here briefly recapitulate the principal steps.
First of all, I remarked that this estimation rests absolutely upon the truth of the Theory of Universal Gravitation, and I therefore pointed out the principal evidences of that theory. As these different evidences are nearly independent of each other, I shall not repeat them, but refer you back to what was said upon each of them.
Then the reference of weights to avoirdupois pounds begins with the weighing of the rocks of Schehallien for the Schehallien experiment, and the weighing of the large leaden balls for the Cavendish experiment; or, if you please, by weighing water, because the weights both of rocks and of lead are conveniently expressed by expressing the proportion which their weights bear to the weight of water.
The next step was this: by means of the Schehallien experiment and the Cavendish experiment, as well as by inferences from the ellipticity of the earth, we found that the mean density of the earth is between five and six times the density of water, and from that we were able to compute the weight of the earth.
The next step was this: having the dimensions of the moon's orbit round the earth, we could find how far the earth draws the moon in one hour; and having the dimensions of the earth's orbit round the sun, we could find how far the sun draws the earth in one hour; and comparing these with the proper allowance for the difference of distances, we could find the proportion of the sun's mass to the earth's mass.
The masses of Jupiter and Saturn I explained to be found by ascertaining, from the dimensions of the orbits of their satellites and their periodic times, how far they draw their satellites in one hour; and then comparing this space with the space through which the earth draws the moon, or the sun draws a planet, in one hour, only making the proper allowance for difference of distances.
For the masses of the other planets, I explained that there is no method but by the disturbances which they produce in the Solar System; and that these are made available by computing with an assumed mass what the perturbations would be, and altering the mass till these agree with the observed perturbations. Those of Jupiter's satellites, as I explained, are found in an analogous way.
For our moon, I indicated several different methods. One of these was, to infer (by theoretical considerations) from the observed amount of lunar nutation, what is the amount of lunar precession; to subtract this from the whole observed precession, which leaves solar precession; and thus to obtain the proportion of lunar precession to solar precession, which is the same as the proportion of the force with which the moon tends to pull the earth's surface from its centre to the similar force of the sun. A second method was from the proportion of lunar and solar tides, which is referred to the same proportion of forces as in the first method. A third method was from the circumstance that it is not the earth, but the centre of gravity of the earth and moon, which moves very nearly in an ellipse round the sun. A fourth method was, that knowing the earth's attraction at its surface, and computing from this its attraction at the moon, we could infer from that the distance of the moon from the centre of gravity of the earth and moon. In the two latter methods we are led to an immediate comparison of the weight of the earth with that of the moon.
I shall now repeat what I said in commencing this course of lectures: that I fully believe that there is no part whatever of these subjects of which the principle cannot be well understood by persons of fair intelligence, giving reasonable attention to them; but more especially by persons whose usual occupations lead them to consider measures and forces; not without the exercise of thought, but by the application only of so much thought as is necessary for the understanding of practical problems of measures and forces.
- Perhaps a still more striking proof of the accuracy of the theory is afforded by the discovery of the planet Neptune. It was found that the irregularities of the motion of Uranus could not be completely explained by the action of any planet then known; and the idea suggested itself that they might be due to some undiscovered body. Two astronomers, M. Leverrier, in France, and Mr. Adams, in England, set themselves to calculate what must be the position of a planet whose attraction would account for the deviations of the actual place of Uranus from that which theory assigned. On the 23rd of September, 1846, Leverrier's calculations were communicated to Dr. Galle, of Berlin, who discovered Neptune the same evening within one degree of its predicted position.
- Another determination of the mean density of the earth has been made by the Astronomer Royal since these lectures were delivered. A short account of it is given in Appendix III.