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able.

The duration of the sidereal revolutions may, of course, be directly observed, in the first instance, by looking for the return of the star to the same spot in relation to the centre of its motion. If we suppose its motion to be uniform, which we may for a first approximation, we can estimate its course by observing the time required between any of the three positions, without waiting for the total revolution, which is sometimes very slow. The geometrical law of this motion permits us to determine, from this kind of observation, the exact time of the planetary revolution. The values of these periodic times are not irregularly divided among the bodies of our system, like the other data that we have noticed. The shorter the course, the more rapid the motion; and the duration increases more rapidly than the corresponding distance, so that the mean velocity diminishes in proportion as the distance increases. We owe to Kepler the discovery of the harmony between these two essential elements, and it is one of the most indispensable bases of celestial mechanics.

Such is the spirit of the methods by which celestial geometry is made to yield us the elementary data which characterize the bodies of the solar system. We have still to consider those of our own planet, before we proceed to the geometrical laws of the planetary motions.

Motion of the Earth

We are accustomed to think of the motions of translation and rotation as inseparable; but, in the transition from supposing the earth to be motionless, to the present state of our knowledge, a theory existed that it whirled round its axis, but was stationary in space. We now perceive that, in addition to the general evidence of the double motion of the planetary bodies, we have special evidence about our own globe,—that the annual motion could not exist without the diurnal, though we might logically suppose beforehand that it could.

As the rotation of the earth cannot be absolutely uniform in all parts of its surface, some indications of its course must exist among terrestrial phenomena. We must therefore distinguish between the celestial and terrestrial proofs of our diurnal motion, while the annual motion admits only of the former.

Immediate appearances go for nothing in this case; for it is clear that, to our eyes (as we do not feel the rotation), it must be exactly the same thing whether we move round among the heavenly bodies, or whether they, fixed in a system, move round us in a contrary direction.

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There was nothing absurd in the latter supposition, in the old days when men had no doubt of the stars being very near, and not much larger than they appear to the eye, while they exaggerated the size of the earth. They could not avoid supposing that such a mass must be immovable, while the small stars with their little intervals, were seen moving every day. Even when the Greek astronomers had sketched out the true geometrical theory of the movements of the planets they treated only of the directions, and had no idea of measuring distances; and it required the whole strength of positive evidence of dimensions and distances to uproot men’s strong and natural persuasion of the stability of their globe. From the moment of our obtaining an idea of the proportions of the universe, the old conception became too revolting to reason to be sustained. When it was understood that the earth is a mere point in the midst of prodigious intervals, and that its dimensions are extremely small in comparison with that of the sun, and even of other bodies of our own system, it was absurd to suppose that such a universe could travel round us every day. What velocities would be required to enable the outlying stars to complete such a daily circuit,—making allowance for their being twenty-four thousand times nearer the earth, if the earth describes no orbit,—and how small the movement of the earth, while those prodigious masses were travelling at such speed! On mechanical grounds, the centrifugal force would be seen to be unmanageable. In every way, the supposition was perceived to be monstrous. Again, the passage of stars before each other, and in a contrary direction to that of the general movement of the sky, showed that they were at different distances from each other, and not bound into an unvarying fabric. Hence arose the notion of Aristotle and Ptolemy, of a system of solid and transparent firmaments. But the existence of comets alone was enough to confute this, appearing as they do in all regions of the sky in turn. As Fontenelle said, this theory put the universe in danger of being fractured. It was, curiously enough, Tycho Brahe, the most illustrious opponent of the Copernican system, who provided for the overthrow of his own arguments by first presenting the true geometrical theory of comets. Long before modern precision was attained, men had been prepared by such considerations as the above to conclude upon the rotation of the earth. Long before Copernicus, a rough conception of the truth existed. Even Tycho Brake felt the astronomical superiority of the true theory; but it seemed to be contradicted by what is before our eyes,—the fall of heavy bodies, etc. Copernicus himself could not remove the objections which arose out of

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men’s ignorance of the laws of Mechanics. These objections held their ground for a century, till Galileo established the great law which we have recognized as one of the three on which Rational Mechanics is based,—that the relative motions of different bodies are independent of the common motion of the whole. Till this was established, the supposition of the rotation of the earth was inadmissible. It is a curious fact, casting much light upon the action of the human mind, that the opponents of Galileo taunted him with the so-called fact that a ball let down from the top of the mast of a ship in motion would not fall at the foot of the mast, but some way behind,—neither they nor anybody else having tried the experiment, which would have shown them that their supposed fact was a mistake. The followers of Copernicus did worse,—they admitted the so-called fact, but tried to reason away its bearings with fantastic subtleties. The matter was not settled even by the demonstrations of Galileo, nor till Gassendi compelled observation by a public experiment in the port of Marseilles.

That order of experiments has been carried on, and would be of high value if we could obtain perpendicular stations of sufficient height for the purpose. It is clear that a lofty tower must describe a larger circle in the same time at the top than at the base; and that any body dropped from it must share the higher rate of velocity, having a slight horizontal velocity in the direction of the earth’s rotation,—falling therefore a little to the east of the base of the tower. Omitting the consideration of the resistance of the air, this amount is calculable in the function of the height of the tower and of its latitude, but experiment would also be valuable; and it is to be hoped that it will be tried at the equator, where the deviation must be greater than anywhere else.

The most certain terrestrial proof of the earth’s rotation is found by tracing the influence of the centrifugal force upon the direction and intensity of weight. This has been done by that observation of Richer, on the shortening of the seconds-pendulum at Cayenne, which has been mentioned as having emboldened Newton to declare the true figure of the earth. The deviation from the spherical form is too small to account for more than one-third of the effect observed; and the other two-thirds are precisely what would be required, at the equator, where the centrifugal force is greatest, on the supposition of the earth’s rotation. Wherever the delicate observation can be made with sufficient precision on other points of the globe’s surface, the result answers to the theory. Thus, we should have sufficient assurance, in the absence of the abun-

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dant astronomical proofs that we possess of the rotation of the earth. Probably no one fact has ever, in the history of our race, produced such consequences as that observation of Richer’s,—two-thirds of the estimated effect having established the rotation of our globe, and the other third having led Newton to the ascertainment of its form.

The movement of translation is ascertainable only by astronomical proofs, for the difference in velocity of the various parts of the globe, in virtue of this motion, is too slight to be sensible to us, or to produce any effect on terrestrial phenomena. When the circuits of other planets were known, men’s minds were prepared for that of the earth,—the question then being whether the earth was in analogy with Venus, Mars, Jupiter, etc., or whether, while they continued their courses round the sun, the sun made a yearly circuit round the earth. Reason must declare, in such a case, that any uncertainty must arise simply from the position of the observer, who, placed on any other planet, would have doubted whether he was not the centre of the heavenly motions. Any observation of mere appearances must evidently go for nothing in this case; as appearances must be exactly the same,—the parallelism of its axis of rotation being unaltered,—whether the earth or the sun is in the ecliptic, and the other in the centre. The proofs must be derived from better testimony than mere appearances; and they so abound that we have only to choose among those which are presented by the whole range of the heavens.

The phenomenon called the precession of the equinoxes was observed by Hipparchus, who was struck by the difference of two degrees which he observed between the longitudes of stars in his time and those which had been recorded a century and a half before. To account for such a phenomenon, successive astronomers imagined other heavens; a process that they repeated with regard to notations, which was a phenomenon too minute for their observation. To account for it, on the supposition of the earth being stable, a third general movement of the whole heavens must be supposed. Newton indicated, and Bradley afterwards proved, that very slight alterations in the parallelism of the earth’s axis—such alterations as must result from the influence of the sun, and yet more of the moon, upon the equatorial bulge,—precisely account for the perturbations which create such confusion under the ancient view of the earth’s stability. The most unquestionable proof of all, however is in that class of phenomena called the retrogradations and stations of the planets, which of are perfectly explained by the annual circuit the planets of our globe, and are otherwise quite incomprehensible. If two boats

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are gliding down a river, at different rates of speed, the one must appear to the other advancing, stationary, or retrograde, according as its own speed is smaller, equal, or greater. With regard to the heavenly bodies, their velocities and other circumstances are known to us, so that we can calculate what their courses ought to be to our eyes, on the supposition of our own annual movement. The appearances answering to our scientific expectation, the proof is practically complete. If the earth moves, the retrogradation of the larger planets ought to happen, as it does, when they are in opposition, and that of Venus and Mercury when they are in inferior conjunction. The regular occurrence of this coincidence was not even attempted to be explained by the ancients.

We have called these proofs practically complete, and they were held to be so by Copernican philosophers before the time of Kepler and Galileo: but our age is not satisfied without a more strict mathematical evidence, amounting to demonstration.

The one demonstration on which modern science rests is that derived from the various phenomena of the aberration of light, which are quite incompatible with the stability of the globe. Roemer’s observations of the satellites of Jupiter suggested to him the use of light as a measure of distance. Knowing what changes must be taking place at various distances from us in the heavens, and knowing the velocity of light, the variations in time at which the changes become visible to us will be a measure of our change of place and distance. For instance, the first satellite of Jupiter is eclipsed every forty-two hours and a half. The eclipse will take place in a shorter or a longer time than this to our eyes, in proportion as we are removed to the one side or the other of our mean distance from Jupiter, on account of the smaller or greater space that the light will have to travel through. By extending our observation, not only to the other satellites, if Jupiter, but to those of Saturn and Uranus, we have obtained further verifications of the relation of our orbit to theirs and also, proof of the uniformity in the passage of light,—at least within our own system. If the earth were immovable, vie might have an error of time, with regard to distant stars, but not of place: but, by compounding the velocity of the earth in its orbit with that of light, which is about ten thousand times greater, we can calculate how far any star ought to appear to deviate from its position. This deviation is found not to exceed, at its maximum, twenty seconds in any direction; and therefore forty seconds is the greatest deviation which can appear in the position of any star in the course of the year. It was the striking periodicity of these

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deviations which led Bradley to seek for the true theory in the combination of the motion of the earth with that of light, and to work it out with the mathematical exactness permitted by modern science: and there is nothing in the case to prevent the direct application of the mathematical process to the visible phenomena. The result is an unquestionable demonstration of the annual movement of the earth, with which all the phenomena of the case precisely agree, and without which they could not exist.

It is evident that this knowledge of aberrations compels us to add another correction to those of refraction and parallax, and the same is the case with regard to the precession of the equinoxes and notation. Thus, as science advances, the preparation of a phenomenon, observed with the best instruments, for scientific use, becomes a delicate and laborious operation.

These are the considerations which have led men to the knowledge of the double motion of the planet we inhabit. No other intellectual revolution has ever so thoroughly asserted the natural rectitude of the human mind, or so well shown the action of positive demonstration upon definitive opinions; for no other has had such obstacles to surmount. A very small number of philosophers, working apart, without any other social superiority than that which attends positive genius and real science, have overthrown, within two centuries, a doctrine as old as our intelligence, directly established upon the plainest and commonest appearances, intimately connected with the whole system of existing opinions, general interests, and dominant authorities, and supported moreover by human pride, powerful in the recesses of each individual mind. The whole system of theological belief rested on the notion that the entire universe was ordained for Man, a notion which appears truly absurd the moment it is seen that our globe is only a subaltern star,— not any centre whatever, but circulating in its place and season, among others, round the sun, whose inhabitants might, with more reason, claim the monopoly of a system which is itself scarcely perceptible in the universe. The notion of final causes and providential laws undergoes dissolution at the same time; for, the once clear and reasonable idea of the subordination of all things to the advantage of Man being exploded, no assignable purpose remains for such providential action. As the admission of the motion of the earth overthrows the whole theory founded on the human destination of the universe, it is no wonder that religious minds revolted from the great disclosure, and that the sacerdotal power

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maintained a bitter rage against its illustrious discoverer.

The Positive philosophy never destroys a doctrine without instantly substituting a conviction, adequate to the needs of our human nature. If the vanity of Man was grievously humbled when science disabused him of his notion of his supreme importance in the universe, to this vanity at once succeeded a lofty sentiment of his true intellectual dignity, when he saw what means were in his power, under such difficulties as his position imposed upon him, for the discovery of such a truth as he had attained. Laplace has pointed this out, showing how to the fantastic and enervating notion of a universe arranged for Man has succeeded the sound and vivifying conception of Man discovering, by a positive exercise of his intelligence, the general laws of the world, so as to be able to modify them, for lids own good, within certain limits. Which is the nobler lot? Which is most in harmony with our highest instincts? Which is the most stimulating to our faculties? And which is the most animating to our feelings?

One more remark suggested by these discoveries is that a clear distinction is for ever established between our system and the universe at large. The old notion of the universe as a single system was founded on the error of the stability of the earth as its centre. The discovery of the earth’s revolution at once transported all the external stars to distances infinitely more considerable than the greatest planetary intervals, and has left no place for the idea of system at all, beyond the limits of our sun’s influence. We do not know, more or less, and men will probably never know, whether the innumerable suns that we see compose a general system, or any number, large or small, of partial systems entirely independent of each other. The idea of the universe therefore is excluded from positive philosophy; and that philosophy is, strictly speaking, bounded by the limits of the solar system, in regard to definite results and this circumscription is, as elsewhere, to be regarded as real progress. This restriction is further justified by the knowledge we have obtained of all really universal phenol mend being essentially independent of the interior phenomena of our system, since the astronomical tables of the state of our system, prepared without reference to any other sun than our own, invariably coincide with the minutest direct observations. The theory of the earth’s revolution has not as yet exerted its due influence on our views, and especially in regard to this last consideration. This is doubtless owing to the imperfections of our education which keep back these high philosophical truths till even the best minds have been pos-

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sessed with an opposite doctrine: so that the positive knowledge which they afterwards attain commonly does little more than modify anal restrain the bad tendencies of their education, instead of ruling and guiding their highest faculties.

Kepler’s Laws

The first idea that occurs to us when we are once satisfied of the revolution of the earth is that our point of view ought henceforth to be the centre of the sun. This transformation of our observation is called the annual parallax, and follows the same rules as the diurnal parallax, allowance being made for the much greater distance. Whether our observations of the sidereal heavens are geocentric or heliocentric,— from the middle of the earth or of the sun,—is of no appreciable consequence, but within our system the annual parallax is of sensible importance. When, from the central point of view, the orbits of the planets are determined, we can proceed to that great aim and end of the science,—the prevision of future conditions of the heavens at appointed times.

The earliest supposition was that the motions of the planets were uniform and circular. The ancients had a superstition, as their writings abundantly show, that the circle was the most perfect of all forms, and therefore the most suitable for the motion of such divine existences as the stars. Their choice of the form was wise: they had to suppose some form, while that of the circle answered best to what they saw; and we ourselves now take it provisionally in forming the theory of a new star. But the superstitious attachment of the ancients to this form was a serious impediment to the advance of astronomy. For every deviation and new appearance a new circle was supposed, till all the simplicity of the original hypothesis was lost in a complication of epicycles. By the end of the sixteenth century the number of circles supposed necessary for the seven stars then known amounted to seventy-four, while Tycho Brahe was discovering more and more planetary movements for which these circles could not account. Thus it is that men cleave to old ideas and methods till they are utterly worn out, and proved beyond recal to be ineffectual, under all additions that can be made to them.

Then came Kepler, the first man for twenty centuries who had the courage to go back to the beginning, as if nothing had been done in the way of theory. He took for his materials the complete system of exact observations whit h were the result of the life of his illustrious precursor, Tycho Brahe. Notwithstanding the natural hardihood of his genius,

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his works reveal to us how strenuously he had to maintain his enthusiasm, in order to support the toils of so bold and difficult an enterprise— rational as it was. He chose the planet Mars for study; and it was a happy choice; because the marked eccentricity of that planet was most apt to suggest the true law of irregularity. Mercury is more eccentric still; but it does not admit of continuous observation. He discovered three great laws, which, extended from the case of Mars to that of all the other planets in our system, constitute the foundation of Celestial Mechanics. The first law regulates the velocity, the second determines the figure of the orbit: the third establishes harmony among all the planetary motions.

It had long been remarked that the angular velocity, (that is, the larger or smaller angle described, in a given time, by its vector radius,) of each planet increases constantly in proportion as the body approaches the centre of its motion; but the relation between the distance and the velocity remained wholly unknown. Kepler discovered it by comparing the maximum and minimum of these quantities, by which their relation became more sensible. He found that the angular velocities of Mars at its nearest and furthest distance from the sun were in inverse proportion to the squares of the corresponding distances. Another way of expressing this law is used by himself; that the area described in a given time by the vector radius of the planet is of a constant magnitude, though its form is variable: or, again, in other words, that the areas described increase in proportion to the times. Thus he destroyed the old notion of the uniformity of the planetary motions, and showed that the uniformity was not in the arcs described, but in the areas.

The second law was less difficult to discover, when once Kepler had surrendered his attachment to the circle. The next figure that presented itself must naturally be the ellipse, which is the simplest form of closed curve, after the circle. The Freely geometers had advanced the abstract theory of this curve some way. Kepler could not long hesitate where to place the sun in it: it must be either in the centre or in one of the two foci. No mathematical labour was needed to show him that it could not be in the centre: and thus, in constructing elliptic orbits, Kepler was necessarily led to place the sun in the focus for all the planets at once. His hypothesis once formed, it was easy to verify it by comparison with observations, the first principles of the required calculations being laid down beforehand. The second law of Kepler then is that the planetary orbits are elliptical, having the sun for their common focus.

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These two laws determined the course of each planet; but the movements of all round their common focus seemed to be purely arbitrary, till Kepler discovered his third law. Being distinguished by the most remarkable genius for analogy ever seen in man, Kepler sought, and successfully, to establish some kind of harmony among all these various movements. He spent much time in pursuing the old metaphysical ideas of certain mystic harmonies which must exist in the universe: but, beyond the general conception of harmony, he obtained no assistance from these vague notions. The ground on which he proceeded was, in fact, the observation of astronomers that the planetary revolutions are always slow in proportion to the extent of their orbits. If he had confined himself to this ground, this discovery would certainly not have occupied seventeen years of assiduous toil. At last his labour issued in the discovery that the squares of the times of the planetary revolutions are proportional to the cubes of their mean distances from the sun: a law which all subsequent observations have verified. One important result of this law is that we may determine the periodic times and mean distances of all the planets by any one. By it, for instance, we have determined the duration of the year of Uranus, when once we knew its distance from the sun: and, conversely, if we discovered a new planet very near the sun, we need only observe its short revolution, to be able to calculate its distance, which, in that position, we could not effect by other means. Astronomers are every day using this double facility, afforded them by Kepler’s third law.

These are the three laws which will for ever constitute the basis of celestial geometry, in regard to planetary motions. They answer for all the bodies in our system, regulating the satellites, by placing the origin of areas and the focus of the ellipse in the centre of the respective planets. Since Kepler’s time, the number of bodies in our system has more than trebled; and all have in turn verified these laws By them, motions of translation require for their determination nothing more than a simple geometrical problem, which demands from direct observation only a certain number of data,—six for each planet. And thus is a perfectly logical character given to astronomy.

The application of these laws, restricted to our own system, is naturally divided into three problems; the problem of the planets; that of the satellites; and that of the comets. These are the three general cases of our system; and, by the application to them of Kepler’s laws, we may assign to every body within the system, its precise position, in all time