InHuence of These are the considerations which have scientific fact led men to the knowledge of tlie double upon Opinion, 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 oj)inions; for no other has had such obstacles to surmount. A very small numijer of philosophers, working apart, without any other social superiority than that which attends positive genius and real science, have overthrown, within two ceu- KNOWLEDGE OF THE EARTJl'S MOTION. 183 turies, 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 aiithorities, 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 tlie entire universe was ordained for Man, a notion which apipears 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 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 cliflficulties 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 yjositive exercise of his intelligence, the general laws of the world, so as to be able to modify them, for his own good, Avithin certain limits. Wliich is the nobler lot? Which is most in harmony with our highest instincts? "VVliich is the most stimulating to our faculties? And which is the most auimatino- to oiu" feeliusrsP 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 fovmded on the error of the stability of the earth as its centre. The discovery of the earth's revolution at once transj^orted 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 cii'cumscription is, as elsewhere, to be regarded as real progress. This restriction is further justified by the knowledge we have obtained of all really universal phenomena 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 possessed with an opposite doctrine: so that the positive knowledge which they after-Vi^ards attain commonly does little more than modify and restrain the bad tendencies of their education, instead of rulinu' and o-uidinc: their hig-hest 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. iiaralVax This transformation of our observation is called the annual parallax, and follows the Kepler's laavs. 185 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 j^arallax 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 p., 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 coiild not account. Thus it is that men cleave to old ideas and methods till they are utterly worn out, and proved beyond recai to be ineffectual, under all additions that can be made to them.
Then came Kej^ler, 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 which were the result of the life of his illustrious precursor, Tycho Brahe. Notwithstanding the natural hardihood of his genius, 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 cliose the plauet 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 oiTr system, constitute the founda-TT-,,, ^, tion 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. T^.,, 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 c^uantities, 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, thoiigh 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.
c 1 1 The second law was less diflficult to disfeecontl Jaw., cover, 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 Grreek 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 KEPLER'S LAWS, 187 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 jiriuciples 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.
These two laws determined the course of ti ' • 1 1 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 hai'mony 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 occujDied 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 thi-ee 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 2>lacing 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 j)roblem, 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 bleiiis ' ^^^^ ^^^ system, is naturally divided into three j^roblems; 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 past and all time to come: and thence again, we can exhibit all the secondary phenomena, past and future, which must result from such relative positions. The next striking fact of Eclipses ^^'^'^ '^^ echpses, absoluteiy conclusive as it is, with regard to the accuracy of our geometrical knowledge. This kind of prediction, quite apart from the vague prophesying of ancient times, when eclipses occurred, as they do now, necessarily from the planetary orbits being all closed curves, and which men's experience told them must return, — began in the immortal school of Alexandria; and its degree of j^recision, to the hour, then to the minute, then to the second, faithfully represents the great historical phases of the gradual perfecting of celestial geometry. It is this which will, apart from all other considerations, for ever make the observation of eclipses a spectacle as interesting for philosophers as for the public, and on grounds which the spread of the jjositive spirit will render, we may hope, more and more analogous, though unequally energetic.
We are learning to make more use of this class of phenomena, and to make out new uses from them, as time goes on. Independently of their practical utility in regard to the great problem of the longitudes, they have been found, within a century, very important in determining with more exactitude the distance of the sun from our earth. Whether it be an eclipse by the moon, or the ECLIPSES: THEIR USES. 189 transit of Venus or Mercury, tlie difference in duration of the phenomenon, observed in Vemis ^ different parts of the earth, will furnish the relative parallax of that body and the sun, and consequently the distance of the sun itself. Some bodies are more fit than others for this experiment, certain conditions being necessary, which are not common to all. Of the three known bodies which can pass between us and the sun, two — the Moon and Mercury — are excluded by these conditions; and there remains only Venus. Halley taught lis how to conduct and use the observation. The parallax, in such a position, offers suitable proportions, being nearly three times that of the sun; and the angular velocity is small enough to allow the phenomenon, (lasting fi'om six to eight hours) to present differences of at least twenty minutes between well chosen observatories. I have specified this case, on account of its extreme importance to the whole system of astronomical science; but it would be quitting our object and plan to notice any other secondary cases.
I must remark upon one very striking truth which becomes apparent during the pursuit of astronomical science; — its distinct and ever-increasing opposition as it attains a higher perfection to the theological and metaphysical spirit. Theological philosophy supposes every thing to be governed by will; and that phenomena are therefore eminently variable and irregular, — at least virtually. The Positive philosophy, on the contrary, conceives of them as subjected to invariable laws, which permit us to predict with absolute precision. The radical incomjDatibility of these two views is nowhere more marked than in regard to the phenomena of the heavens; since, in that direction, our prevision is proved to be perfect. The. punctual arrival of comets and eclipses, with all their train of minute incidents, exactly foretold, long before, by the aid of ascertained laws, must lead the common mind to feel that such events must be free from the control of any will, which could not be will if it was thus subordinated to our astronomical decisions.
The three laws of Kepler form the founda- P"'ounflations of tion of the higher conception to which we are Celestial next to pass on; the mechanical theory of Mechanics.
astronomical phenomena. By tliis ulterior study, we obtain new determinations; but a more important office of the Mechanical theory is to perfect celestial geometry itself, by giving more precision to its theories, and establishing a sublime connection among all the parts of our solar system, without exception. The laws of Kepler, inestimable as they are, have come to be regarded as a sort of approximation, — supposing, as they do, various elements to be constant, while they are subject to more or less alteration. The exact knowledge of the laws of these variations constitutes the principal astronomical result of celestial mechanics, independently of its own high philosophical importance.
SECTION 11.
DYNAMICAL PHENOMENA, Gravitation.
The laws of Motion, more difficult to dis- 1 s of Motion ^'^^^^' than those of extension, and later in ' being discovered, are quite as certain, universal and positive in character; and of course it is the same with their ajiplication. Every curvilinear displacement of any kind of body, — of a star as well as a cannon ball, — may be studied under the two points of view which are equally mathematical: geometrically, in determining by direct observation the form of the trajectoiy and the law by which its velocity varies, as Kepler did with the heavenly bodies; and mechanically, by seeking the law of motion which prevents the body from pursuing its natural straight course, and which, combined with its actual velocity, makes it describe its trajectory, which may henceforth be know d priori. These inquiries are evidently equally positive, and in like manner founded upon phenomena. If we find still in use some terms which seem to relate to the nature and cause of motion, they are only vestiges of a mode of thinking long gone by; and they do not affect the positive character of the research.
The two motions which constitute the course of the cannon ball are perfectly known to us beforehand; but we HISTORY OF THE LAWS OF MOTION. 191 have not the geometrical knowledge of its trajectory. With regard to the star, our knowledge of its trajectory compensates exactly for the difficulty of our preliminary ignorance about its elementary motions. If the law of the fall of weights had not been directly established, we should have learned it, indirectly, but no less surely, from the observation of the curvilinear motions produced by weight.
Celestial Mechanics was then founded on a firm basis, when through Kepler's laws, Their history, and by the I'ules of rational Dynamics, discovery was made of the law of direction and intensity of the force which must act upon the planet to divert it from the tangent which it would naturally describe. This fundamental law once discovered, all astronomical researches enter into the domain of Mechanics, in which the motions of bodies are calculated from the forces which imj^el them. This was the course philosophically and perseveringly pursued by Newton.
It does not detract from Newton's merits that Kepler had some foresight of the results of his great laws. He carried their dynamic interpretation as far as the science of his day permitted; and, seeking for what could not yet be found, he wandered off among fantasies. The true precursors of Newton, as founders of dynamics, were Huyghens and Galileo, — especially the last: yet history tells of no such succession of philosophical efforts as in the case of Kepler, who, after constituting celestial geometry, strove to pursue that science of celestial mechanics which was, by its nature, reserved for a future generation. As the means were wanting, he failed; but the example is not the less remarkalde.
The first of Kepler's laws proves that the accelerating force of each planet is constantly directed towards the sun. The accelerating force, however great it may be supposed, does not at all affect the magnitude of the area which would })e described in a given time by the vector radius of the planet, in virtue of its velocity, if its direction passes exactly through the sun, while it would inevitably change it on any other supposition. Thus, the permanence of this area, — the first general datum of observation, — discloses the law of direction. The great difficulty of the problem, gloriously solved by Newton, lies iu the discovery, by means of Kepler's otlier two theorems, of the law of the intensity of this action, which we sj^eak of as exercised by tlie sun on the planets.
When Newton began to work on this conception, he took Kepler's third law as his basis, supposing the orbits, as he might do for such a purpose, to be circular and uniform. The solar action, equal, and opposed to the centrifugal force of the planet, thus became necessarily constant at the different points of the orbit, and could not vary but in passing from one planet to another. This variation between one planet and another was provided for by the theorems of Hiiyghens relating to the centrifugal force in the circle. This force being in proportion to the relation between the radius of the orbit and the square of the periodic time, must vary from one star to another inversely to the square of its distance from the sun, in virtue of the permanence which Kepler showed to exist of the relation between the cube of this distance and this same sqiiare of the periodic time, for all the planets. It was this mathematical consideration which put Newton in the way of his great discovery, and not any metaphysical reasonings, such as prevailed before it, and which probably never entered his mind, one way or another.
There remained the difficulty of explaining how this law of the variation of the solar action agreed with tho geometrical nature of the orbits, as exhibited by Kepler. The elliptical orbit 2:>resented tAvo remarkable points, ^ — the aphelion and the perihelion, in which the centrifugal force was directly opposed to the action of the sun, and consequently equal to it; and the change in this action there must be at the same time more marked. The curve of the orbit was evidently identical at these two j^oints; the action then had simply to be measured, according to Huyghens' theorems, by the square of the corresponding velocity. Thence, it was easily deduced, from Kepler's first kiw, that the decrease of the solar action, from the perihelion to the aphelion, must be inversely to the square of the distance. Here was a full confirmation of the law which related to the different planets by an exact comparison between the two principal positions of each of them. Still, however.
tlie elliptical motion had not been considered. Any other curve would, thus far, have served as well as the ellipse, provided its two extremities had shown an equal curvature. The remaining portion of the demonstration, — the measurement of the solar action throughout the extent of the orbit, — is to be obtained only by transcendental analysis. The process is necessary for carrying on the comparison of the solar action and the centrifugal force; and the theory of the curvature of the ellipse is required. Huyghens made a. near approach to the principle of this great process; but it could not be completed without the aid of the differential analysis, of which Newton was the inventor, as well as Leibnitz. By the aid of this analysis, the force of the solar action in all parts of the orbit is easily ^^, estimated, m various ways; and it is found demonstration to vary inversely to the square of the distance, and that it is indej^endent of the direction. Furthermore, the same method shows, in accordance with Kepler's third law, that the action varies in proportion to distance alone; so that the sun acts upon all the planets alike, whatever may be their dimensions, their distance only being the circumstance to be considered. Thus Newton completed his demonstration of the fundamental law that the solar action is, in every case, proportionate, at the same distance, to the mass of the planet; in the same way that, by the identity of the fall of all terrestrial bodies in a vacuum, or by the precise coincidence of their oscillations, proof had already been obtained of the proportion between their weight and their masses. We thus see how the three laws of Kepler have concurred in establishing, according to the rules of rational mechanics, this fundamental law of nature. The first shows the tendency of all the planets towards the sun; the second shows that this tendency, the same in every direction, changes with the distance from the sun, inversely to its square; and the third teaches that this action is always simply proportionate, the distance beingequal, to the mass of each planet. In accordance with the laws of Kepler, which relate to the whole interior of our system, the same theory applies to tte connection between the satellites and their planets.
Newton thought it necessary to complete his demonstra- I. o 194 rOSITIVE PHILOSOPHY.
"tion by presenting it in an inverse manner; that is, by determining a priori the planetary motions which must result from such a dynamic law. The process brought him back, as it must do, to Kepler's laws. Besides furnishing some means of simplifying the study of these motions, this labour proved that, whereas, by Kepler's laws, the orbit might have had more figures than one, the ellipse was the only one possible under the Newtonian law.
It was once a great jjerplexity to some exnlaiiied peo])le, which others could not satisfactorily explain, that when the planet is travelling towards its aphelion we cannot say that it tends towards the sun. But the difficulty arose out of the use of inappropriate language. The question is, not whether the planet is nearer to the sun than it lately was, but whether it is nearer than it would have been without the force that sends it forward. It is always tending towards the sun to the utmost that is allowed by the other force to which it is subjected. The orbit is always concave towards the sun; and it would evidently have been insurmountable if the trajectory could have been convex. In the same way, when a bomb ascends, its weight is not suspended or reversed: it always tends towards the earth, and is, in fact, falling towards it more rapidly every moment, even if ascending, because it is eveiy moment further below the point at which it would have been but for the action of the earth upon it; and its trajectory is always concave to the ground.
Hqxiw Attrac- I have thus far carefully avoided giving tion inadmis- any name to the tendency of the planets ^^^^®- towards the sun, and of the satellites towards the planets. To call it attraction would be misleading; and we, in triith, can know nothing of its nature. All that we know is that these bodies are conne(^ted, and that their •effect upon each other is mathematically calculable. It is by quite another property of Newton's great discovery that this effect is explained, in the true sense of the word, — that is, comprehended from its conformity with the ordinary phenomena which gravity continually produces on the surface of our globe. Let us now see what this property of the discovery is.
EXTENT OF THE DEMONSTRATION. 195 We owe a great deal to the moou. If the earth had uo satellite, we might calculate the celestial motions by the rules of dynamics, but we could not connect them with those which are under our immediate observation. It is the moon which affords this connection by enabling us to establish the identity of its tendency towards the earth with weight, properly so called; and from this knowledge, we have risen to the view that the mutual action of the heavenly bodies is nothing else than weight properly generalized; or, putting it the other way, that weight is only a particular case of the general action. The case of the moon is susceptible of the most precise testing. The data are known; and by dynamical analysis, the intensity of the action of the earth upon the moon is exactly ascertainable. We have only to suppose the moon close to the earth, with the due increase of this intensity, inversely to the square of the distance, and compare it with the intensity of weight on the earth, as manifest to us by the fall of Ijodies, or by the pendulum. A coincidence between the two amounts to proof; and we have, in fact, mathematical demonstration of it. It was in pursuing this method of proof that Newton evinced that philosophical severity which we find so interesting in the anecdote of his long delay, because he could not establish the coincidence, while confident that he had discovered the fact. He failed for want of an accurate measurement of a degree on the earth's surface; and he put aside this important part of his great conception till Picard's measurement of the earth enabled him to establish his demonstration.
The identity of weight and the moon's ten- + f +i dency towards the earth places the whole of demonstration celestial mechanics in a new light. It shows us the motions of the stars as exactly like that of projectiles which we have under our immediate observation. If we could start our projectiles with a sufficient and continuous force, we should, except for the resistance of the air, find them the models of the planetary system: or, in other words, astronomy has become to us an artillery problem, simplified by the absence of a resisting medium, but complicated by the variety and plurality of weights. — If our observation of weight on our globe has helped us to a