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.

Three problems.

Prediction of
Eclipses.

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 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 this kind to the general mind is the prediction of eclipses, absolutely 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 precision, 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 positive 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.

transit of Venus or Mercury, the difference in duration of the phenomenon, observed in 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 us 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 from 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 incompatibility 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- Foundations of tion of the higher conception to which we are Celestial next to pass on; the mechanical theory of Mechanics.

astronomical phenomena. By this 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.

Character of

SECTION II.

DYNAMICAL PHENOMENA.

Gravitation.

The laws of Motion, more difficult to dislaws of Motion. Cover 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 application. 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 trajectory 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 à 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 rules 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 impel 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 remarkable.

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 be 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 in the discovery, by means of Kepler's other two theorems, of the law of the intensity of this action, which we speak of as exercised by the 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 Huyghens 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 square 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 presented two 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 points; 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 law, 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,