^1 >.. The movement of translation is ascerlation " tainable only by asti'onomical proofs, for the difference in velocity of the various parts of the globe, in virtue of this motion, is too slight to be sen- sible to tis, or to produce any effect on terrestrial pheno- mena. 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 appear- ances must evidently go for nothing in this case; as ap- pearances 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.
-r,. ^ The phenomenon called the precession of the equinoxes """^ equinoxes was observed by Hipparcnus, who was struck by the difference of two de- grees which he observed between the longitudes of stars in liis time and those which had been recorded a century and a half before. To account for such a phenomenon, suc- cessive astronomers imagined other heavens; a process that they repeated with regard to nutations, 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. ISTewton 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 TROOFS OF THE EARTH'S TRANSLATION. 181 bulge, — precisely account for the perturbations whicli 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 retro- Retrogradations gradations and stations of the planets, which and stations of are perfectly explained by the annual circuit ^'^^ planets, of our globe, and are otherwise quite incomprehensible. If two boats 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 move- ment. The appearances answering to our scientific expec- tation, the i^roof is practically complete. If the earth moves, the retrogradatiou 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 conjunc- tion. 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 trom tlie various ^^ Uo-ht phenomena of the aberration of light, which are quite incompatible with the stability of the globe. Eoemer's observations of the satellites of Jupiter suggested to him the use of light as a measure of distance. Know- ing what changes must be taking place at various distances from ns 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 propor- tion as we are removed to the one side or the other of our mean distance from Jupiter, on account of the smaller or 182 POSITIVE PHILOSOPHY.
greater space that the light will have to travel thi'ough. By extending our observation, not only to the other satel- lites of Jupiter, but to those of Saturn and Uranus, we have obtained further A^erifications 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, we 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 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 mathe- matical 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 eai'th, 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 comj^els us to add another correction to those of refraction and parallax, and the same is the case with regard to the pre- cession of the equinoxes and nutation. Thus, as science advances, the j^rei^aration of a phenomenon, observed with the best instruments, for scientific use, becomes a delicate and laborious ojDoration.
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 estab- lished upon the plainest and commonest appearances, inti- mately 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 re- ligious 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 with- out 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 in- tellectual 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 dis- covering, 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 facul- ties? And which is the most auimatino- to oiu" feeliusrsP 184 POSITIVE PHILOSOPHY, 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 pheno- mena being essentially independent of the interior pheno- mena 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 doc- trine: 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 our- selves 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 dis- covering 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 addi- tions 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 — • 186 POSITIVE PHILOSOPHY.
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 ap- proaches 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 corre- sponding 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 astro- nomers 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.
188 POSITIVE PHILOSOPHY.
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 applica- tion 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 geo- metrical 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 con- siderations, 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 phe- nomena, 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 conse- quently 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 condi- tions; 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 be- comes 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 vari- able and irregular, — at least virtually. The Positive phi- losophy, 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.
190 POSITIVE PHILOSOPHY.
astronomical phenomena. By tliis ulterior study, we obtain new determinations; but a more important office of the Mechanical theory is to perfect celestial geometry it- self, by giving more precision to its theories, and establish- ing a sublime connection among all the parts of our solar system, without exception. The laws of Kepler, inesti- mable 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 altera- tion. 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, uni- versal and positive in character; and of course it is the same with their ajiplication. Every curvilinear displace- ment 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 hence- forth be know d priori. These inquiries are evidently equally positive, and in like manner founded upon phe- nomena. 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 compen- sates 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 observa- tion 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, dis- covery 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 pur- sued 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 pre- cursors 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.