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written in the most concise language by which human thought can be expressed, might vie in magnitude with, the most trite and verbose compilations.

5. The additions we have enumerated were made to the pure mathematics; that which we are going to mention, belongs to the mixt. It is the mechanical principle, discovered by D'Alembert, which reduces every question concerning the motion of bodies, to a case of equilibrium. It consists in this: If the motions, which the particles of a moving body, or a system of moving bodies, have at any instant, be resolved each into two, one of which is the mo tion which the particle had in the preceding instant, then the sum of all these third motions must be such, that they are in equilibrium with one another. Though this principle is, in fact, nothing else than the equality of action and reaction, properly explained, and traced into the secret process which takes place on the communication of motion, it has operated on science like one entirely new, and deserves to be considered as an important discovery. The consequence of it has been, that as the theory of equilibrium is perfectly understood, all problems whatever, concerning the motion of bodies, can be so far subjected to mathematical computation, that they can be expressed in fluxionary or differential equa tions, and the solution of them reduced to the integration of those equations. The full value of the proposition, however, was not understood, till La Grange published his Méchanique Analytique: the principle is there reduced to still greater simplicity; and the connexion between the pure and the mixt mathematics, in this quarter, may be considered as complete.

Furnished with a part, or with the whole of these resources, according to the period at which they arose, the mathematicians who followed Newton in the career of physical astronomy, were enabled to add much to his discoveries, and at last to complete the work which he so happily began. Out of the number who embarked in this undertaking, and to whom science has many great obligations, five may be regarded as the leaders, and as distinguishe ed above the rest, by the greatness of their achievements. These are, Clairaut, Euler, D'Alembert, La Grange, and La Place himself, the author of the work now under consideration. By their efforts, it was found, that, at the close of the last century, there did not remain a single phenomenon in the celestial motions, that was not explained on the principle of Gravitation; nor any greater difference between the conclusions of theory, and the observations of astronomy, than the errors unavoidable in the latter were sufficient to account for. The time seemed now to be come for reducing the whole theory of astronomy into one work, that should

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embrace the entire compass of that science and its discoveries for the last hundred years: La Place was the man in all Europe, whom the voice of the scientific world would have selected for so great an undertaking.

The nature of the work required that it should contain an entire System of Physical Astronomy, from the first elements to the most remote conclusions of the science. The author has been careful to preserve the same method of investigation throughout; so that even where he has to deduce results already known, there is a unity of character and method that presents them under a new aspect.

The reasoning employed is every where algebraical; and the va rious parts of the higher mathematics, the integral calculus, the me thod of partial differences and of variations, are from the first outset introduced, whenever they can enable the author to abbreviate or to generalize his investigations. No diagrams or geometrical figures are employed; and the reader must converse with the objects presented to him by the language of arbitrary symbols alone. Whether the rejection of figures be in all respects an improvement, and whether it may not be in some degree hurtful to the powers of the imagination, we will not take upon us to decide. It is certain, however, that the perfection of Algebra tends to the panishment of diagrams, and of all reference to them. La Grange, in his treatise of Analytical Mechanics, has no reference to figures, notwithstanding the great number of mechanical problems which he resolves. The resolution of all the forces that act on any point, into three forces, in the direction of three axes at right angles to one another, enables one to express their relations very distinctly, without representing them by a figure, or expressing them by any other than algebraic symbols. This me thod is accordingly followed in the Mechanique Céleste. Something of the same kind, indeed, seems applicable to almost any part of the mathematics; and a very distinct treatise on the conic sections, we doubt not, might be written, where there would not be a single diagram introduced, and where all the properties of the ellipse, the parabola, and the hyperbola would be expressed either by words or by algebraic characters. Whether the 'imagination would lose or gain by this exercise, we shall not at present stop to inquire. It is curious, however, to observe, that Algebra, which was first intoduced for the mere purpose of assisting geometry, and supplying its defects, has ended, as many auxili aries have done, with discarding that science (or at least its peculiar methods) almost entirely. We say, almost entirely; because there are, doubtless, a great number of the elementary propositions

propositions of geometry, that never can have any but a geome trical, and some of them a synthetical demonstration.

The work of La Place is divided into two parts, and each of these into five books. The first part lays down the general principles applicable to the whole inquiry, and afterwards deduces from them the motions of the primary planets, as produced by their gravitation to the sun. The second part, treats first of the disturbances of the primary planets, and next of those of the secondary.

In the first book, the theory of motion is explained in a manner very unlike what we meet with in ordinary treatises,-with extreme generality, and with the assistance of the more difficul parts of the mathematics,-but in a way extremely luminous, concise, and readily applicable to the most extensive and arduous researches. This part must be highly gratifying to those who have a pleasure in contemplating the different ways in which the same truths may be established, and in pursuing whatever tends to simplicity and generalization. The greater part 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 ele mentary 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 Mechanique Celeste, provided it had a little more expansion given it in particu lar places, and a little more illustration employed for the sake of those who are not perfectly skilled in the use of the instrument which La Place 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 ano ther, deduced in the second chapter, La Place 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.

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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 mixt 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 system 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 for wards, on each side of a fixt 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 gradually conducted to a solution of it so perfect as that which is given by La Place. The problem is,-Having given the

masses

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 beside 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.

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