473 that the planets moved in ellipses having the Sun in one of the foci: the law, therefore, of the force which urged the planets was the inverse square of the distance *. The first two problems of this chapter, solved by M. La PLACE, are the same as the two of Newton above mentioned: but the solution is by a different method; less simple, perhaps, in this particular case, yet to be preferred on account of its connexion with the methods previously used by the author in the preceding parts of his volume. In a systematic work, no demonstrations ought to be admitted but such as pendent from previously established principles, and connected with preceding methods; though independent propositions often present themselves, allure with a specious facility, and tempt the author to depart from the plain and direct road, in the hope of finding a bye-path to truth. In general, the facility and conciseness of such demonstrations, if thoroughly examined, are illusive: but, were such qualities real, they must be sternly sacrificed, for the sake of unity and systematic regularity. According to the method of M. LA PLACE, the position of a body moving in a curve of simple curvature is determined by two rectangular co-ordinates. Let x and y be two co-ordinates, which have their origin at the centre of force; let P and 2 be two forces acting on a body, and parallel to the axis of x and y then the differential equations of the body's motion are but, according to the first law of Kepler, xdy-ydx-edt (e a constant quantity) or, P. 2:x:y; therefore, the resulting force of P and passes through the. origin of the co-ordinates, or through the Sun. To discover the law of the resulting force, let the resulting force, let vangle formed by x, and a radius (r) drawn from the Sun to the planet; then P. cos. v, 2. sin. v: *The law of gravitation cannot, however, be admitted as demon strated on such simple reasoning as this: not but that the reasoning is exact, granting the premises. from 1 from which equations, and those preceding, an expression for may be deduced in terms of r, v, and other constant quantities: but, from the second law of Kepler, the orbits are ellipses; from the equation to an ellipse, find the value of r, substitute it in the equation for 4, and the value of p will appear to be P h h being a constant co-efficient, M. LA PALCE next shews the method of ascertaining the law of gravitation, according to which the satellites of Jupiter, Saturn, and Uranus are attracted towards their primaries; also the method of ascertaining the law of the Moon's tendency to the earth. These methods do not differ in their principle from those given by Newton in his Principia: but they possess the advantage of being fully developed, and of being detailed with perspicuity and exactness. The arguments for the universality of gravitation, and its law, are stated by the author with great clearness, precision, and force. The Sun, and the planets which have satellites, are consequently endowed with an attractive force; which, decreasing to infinity, reciprocally proportional to the squares of distances, embraces all bodies within the sphere of its activity. Analogy leads us to suppose that a like force resides universally in all the planets and comets: but we may be directly assured of this truth thus. It is a constant law of nature, that a body cannot act on another without experiencing an équal and contrary re-action; thus, the planets and comets being attracted towards the Sun, they ought, according to the same law, to attract this star. For the same reason, satellites attract their planets; this attracting property is common then to planets, to comets, and to satellites, and consequently the mutual gravitation of heavenly bodies may be regarded as a general property of the universe. We have just seen that gravitation follows the ratio of the inverse square of distances; in truth, this ratio is given by the laws of the elliptic motion, to which the heavenly motions are not rigorously subjected but we must consider that the most simple laws are always to be preferred, until observation compels us to abandon them. It is natural to suppose, at first, that the law of gravitation is according to a reciprocal power of the distance; and we find, by calculation, that the slightest difference between this power and the square would become extremely sensible in the position of the perihelia of planetary orbits, where observation scarcely discovers certain motions almost insensible, of which we shall develope the cause. In general, we shall see, in the course of this work, that the law of gravitation, according to the inverse square of the distances, represents with extreme precision all the inequalities observable in the heavenly motions; this agreement, joined to the simplicity of this law, authorizes us in the supposition that it is rigorously the law of nature. Gravitation Gravitation is proportional to the masses; for it results from No. 3, that the planets and comets, being supposed at the same distance from the Sun, and abandoned to their tendency towards this star, would fall through an equal height in equal times; so that their gravity would be proportional to their masses. The nearly circular motions of satellites round their planets proved that they gra vitate, as these planets, towards the Sun, in the ratio of their masses; the slightest difference in this respect would be sensible in the motion of the satellites; and observation discovers no inequality dependent on this cause. We perceive, then, that comets, planets, and their satellites, placed at the same distance from the Sun, would gravitate towards this star in proportion to their masses; 'whence it follows, by virtue of the equality of action and re-action, that they would attract the Sun in the same proportion; and that thus their action on this star is proportional to their masses, divided by the square of, their distances from his centre. The same law is observed on the Earth. We are assured by the most precise experiments, made by means of the pendulum, that, without the resistance of the air, all bodies would be precipitated to wards its centre with an equal velocity; terrestrial bodies, then, gravitate on the Earth, in the ratio of their masses; as planets gra vitate towards the Sun, and satellites towards their planets. This conformity and consistency of nature on the Earth, and in the immensity of the heavens, demonstrate to us, in the most striking manner, that the gravity observed here below is only a particular case of the general law which pervades the universe. The attractive property of celestial bodies is not peculiar to them only while they exist in masses; it belongs to each of their individual particles. If the Sun acted only on the centre of the Earth without particularly attracting each of its parts, there would result, in the ocean, oscillations incomparably greater than and very different from the oscillations now observed there. The gravitation of the Earth towards the Sun is, then, the result of the gravitation of all its particles; which, by consequence, attract the Sun in the ratio of their respective masses.-Moreover, each body on the earth gravitates towards its centre, proportionally to its mass;-it re-acts, then, on the Earth, and attracts it according to the same proportion. If this were not the case, and if any part whatever of the Earth, however small we suppose it, did not attract another part, as itself was attracted; the centre of gravity of the Earth would move in space, by virtue of its gravity; which is impossible. The heavenly phænomena, compared with the laws of motion, conduct us then to this grand principle of nature; that all the particles of matter mutually attract in the ratio of their masses, and the inverse ratio of the square of their distances. Already, in this universal gravitation, we have a glimpse of the cause of the perturbations which the heavenly bodies experience; since the planets and comets, being subjected to their reciprocal action, must necessarily deviate a little from the laws of elliptic motion, which they would follow exactly if they obeyed only the action of the Sun. The satellites disturbed in their motions round their planets, by their mutual attraction and by that of the Sun, ought in like manner to deviate from these these laws. We perceive, moreover, that the particles of cach celestial body, re-united by their attraction, ought to form a mass nearly spherical; and that the result of their action, at the surface of bodies, ought there to produce all the phenomena of gravitation. In like manner, we perceive that the motion of rotation of heavenly bodies must necessarily alter in a small degree their spherical figure, and flatten at the poles; and that then the result of their mutual actions, not passing exactly through their centres of gravity, would produce, in their axis of rotation, motions similar to those discovered by observation.-Finally, we discern that the particles of the ocean, anequally attracted by the Sun and Moon, ought to have an oscil latory motion similar to the flux and re-flux of the sea.--The developement of these various effects of universal gravitation requires the most profound analysis. To embrace them in all their generality, we proceed to give the differential equations of the motion of a system of bodies, under the influence of their mutual attraction; and to investigate the rigorous integrals which can thence be obtained. We shall in course avail ourselves of the advantages in point of facility, which the ratios of the masses and distances of heavenly bodies offer to us; in order to deduce the integrals more and more nearly, and so to determine the celestial phenomena, with all the exactness which observations require.' Chapter II. On the differential Equations of the Motion of a System of Bodies subjected to their mutual Attraction.-The form of these differential equations is m. ddx dt the complete integration of which cannot be obtained, except in the case of the system consisting of two bodies only. When the system is composed of more than two bodies, the analysis must have recourse to the methods of approximation. The subject next treated is the attraction of spheroids; and conclusions the same as those of Newton (sect. 12.) are obtained, though by different methods; viz. that, in the law of nature, a particle within a spherical superficies is at rest, and that spherical superficies and spheres attract in the same manner as if their masses were united at their centres. M. LA PLACE investigates, generally, under what laws of attraction the two curious circumstances just mentioned can take place; a problem which M. D'Alembert solved in his Opuscules. It appears from the investigation, that all the laws of attraction, in which a sphere acts on an exterior point placed at the distance r from its centre, are comprehended under this formula: Ar+: ; if B B A=o, the expression becomes, the law of nature *: whence B For, if r be made infinitely great, Ar+ becomes infinitely great; except A=0. it it appears that, in the infinite number of laws which render the attraction very small at great distances, the law of nature is the only one in which spheres have the property of acting in the same manner as if their masses were united at their centres. Chapter III. First Approximation of the Heavenly Motions, or the Theory of the Elliptical Motion.-In this chapter, are given the integration of the differential equations which determine the relative motion of two bodies mutually attracting each other; the curve described is proved to be a conic section, and the time is expressed by means of a converging series. Also are deduced the finite equations of the elliptical motions expressions of the mean anomaly, radius vector, and true anomaly, in functions of the excentric anomaly; and a general method for the reductions of functions into series, &c. M. LA PLACE shews that the law of Kepler, viz. that the squares of the periodic times are as the cubes of the axes, is not rigorous, and only obtains in as much as the mutual action of the planets, and their action on the Sun, are neglected. Chapter IV. Determination of the Elements of the Elliptical Motion.-Newton has shewn, in the 17th proposition of his Principia, that, if the velocity of projection of the heavenly bodies be given, the elements of their orbits may easily be determined. This M. LA PLACE demonstrates conformably to his preceding methods; if V be the velocity of the revolving body, U the velocity in a circle, radius=1, then VU ; which equation gives the semi-axis major a, by means of the body's primitive motion, and primitive distance from the body round which it revolves. The principal part of this chapter is occupied with the theory of Comets. The author observes that the preceding formulas, for determining the elements of the planets' orbits, cannot be applied, since observation does not make known the eircumstances of their primitive motion: but that the elements of their orbits may be determined by means of their oppositions and conjunctions, and easily, since the excentricity of the orbits and their inclination to the ecliptic are very small. Besides, the planets may be continually observed; and, by the com parison of a great number of observations, the elements of their orbits may be corrected. The case is widely different respecting Comets; which can only be observed towards their perihelion; and which, returning after a long succession of ages, cannot be recognized with any certainty. To determine the elements of their orbits by the observations of the appearance of a comet, is beyond the reach and powers of analysis; we must have recourse to the methods of approximation to determine the |