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of the propositions here deduced are already known ; but it is good to have them presented in a new order, and investigated by the same methods that are pursued through the whole of this work, from the most elementary truths to the most remote conclusions.
For the purpose of instructing one in what may be called the Philosophy of Mechanics, that is, in the leading truths in the science of motion, and at the same time, in the way by which those truths are applied to particular investigations, we do not believe any work is better adapted than the first book of the Mécanique Céleste, provided it had a little more expansion given it in particular places, and a little more illustration employed for the sake of those who are not perfectly skilled in the use of the instrument which Laplace himself employs with so much dexterity and ease.
From the differential equations that express the motion of any number of bodies subjected to the mutual attraction of one another, deduced in the second chapter, Laplace proceeds to the integration of these equations by approximation, in the third and the following chapters. The first step in this process gives the integral complete in the case of two bodies, and shows that the curve described by each of them is a conic section. The whole theory of the elliptic motion follows, in which the solution of Kepler's problem, or the expression of the true
anomaly, and of the radius vector of a planet, in terms of the mean anomaly, or of the time, are particularly deserving of attention, as well as the difference between the motion in a parabolic orbit, and in an elliptic orbit of great eccentricity.
In the greater part of this investigation, the theorems are such as have been long since deduced by more ordinary methods; the deduction of them here was however essential, in order to preserve the unity of the work, and to show that the simpler truths, as well as the more difficult, make parts of the same system, and emanate from the same principle. These more elementary investigations have this further advantage, that the knowledge of the calculus, and of the methods peculiar to this work, is thus gradually acquired, by beginning from the more simple cases; and we are prepared, by that means, for the more difficult problems that are to follow.
The general methods of integrating the differential equations above mentioned, are laid down in the fifth chapter, which deserves to be studied with particular attention, whether we would improve in the knowledge of the pure or the mixed mathematics. The calculus of variations is introduced with great effect in the last article of this chapter.
A very curious subject of investigation, and one that we believe to be altogether new, follows in the next chapter. In the general movement of a sys
tem of bodies, such as is here supposed, and such, too, as is actually exemplified in nature, every thing is in motion; not only every body, but the plane of every orbit. The mutual action of the planets changes the positions of the planes in which they revolve; and they are perpetually made to depart, by a small quantity, on one side or another, each from that plane in which it would go on continually, if their mutual action were to cease. The calculus makes it appear, that the inclinations of these orbits in the planetary system is stable, or that the planes of the orbits oscillate a little, backwards and forwards, on each side of a fixed and immoveable plane. This plane is shown to be one, on which, if every one of the bodies of the system be projected by a perpendicular let fall from it, and if the mass of each body be multiplied into the area described in a given time by its projection on the said plane, the sum of all these products shall be a maximum. From this condition, the method of determining the immoveable plane is deduced; and in the progress of science, when observations made at a great distance of time shall be compared together, the reference of them to an immoveable plane must become a matter of great importance to astronomers.
As the great problem resolved in this first book is that which is called the problem of the Three Bodies, it may be proper to give some account of the steps by which mathematicians have been gra
dually conducted to a solution of it so perfect as that which is given by Laplace. The problem is, -Having given the masses of three bodies projected from three points given in position with velocities given in their quantity and direction, and supposing the bodies to gravitate to one another with forces that are as their masses directly and the squares of their distances inversely, to find the lines described by these bodies, and their position, at any given instant.
The problem may be rendered still more general, by supposing the number of bodies to be greater than three.
To resolve the problem in the general form contained in either of these enunciations, very far exceeds the powers even of the most improved analysis. In the cases, however, where it applies to the heavens, that is, when one of the bodies is very great and powerful in respect of the other two, a solution by approximation, and having any required degree of accuracy, may be obtained.
When the number of bodies is only two, the problem admits of a complete solution. Newton had accordingly resolved the problem of two bodies gravitating to one another, in the most perfect manner; and had shown, that when their mutual gravitation is as their masses divided by the squares of their distances, the orbits they describe are conic sections. The application of this theorem and
its corollaries to the motions of the planets round the sun, furnished the most beautiful explanation of natural phenomena that had yet been exhibited to the world; and however excellent, or in some respects superior, the analytical methods may be that have since been applied to this problem, we' hope that the original demonstrations will never be overlooked. When Newton, however, endeavoured to apply the same methods to the case of a planet disturbed in its motion round its primary by the action of a third body, the difficulties were too great to be completely overcome. The efforts, nevertheless, which he made with instruments, that, though powerful, were still inadequate to the work in which they were employed, displayed, in a striking manner, the resources of his genius, and conducted him to many valuable discoveries. Five of the most considerable of the inequalities in the moon's motion were explained in a satisfactory manner, and referred to the sun's action; but beyond this, though there is some reason to think that Newton attempted to proceed, he has not made us acquainted with the route which he pursued. It was evident, however, that besides these five inequalities, there were many more, of less magnitude indeed, but of an amount that was often considerable, though the laws which they were subject to were unknown, and were never likely to be discovered by observation alone.