a long life entirely to the pursuits of science. Besides producing many works on all the different branches of the higher mathematics, he continued, for more than fifty years during his life, and for no less than twenty after his death, to enrich the memoirs of Berlin, or of Petersburgh, with papers that bear, in every page, the marks of origina lity and invention. Such, indeed, has been the industry of this incomparable man, that his works, were they collected into one, notwithstanding that they are full of novelty, and are 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 mixed. 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 motion 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 equations, and the solution of them reduced to the integration of those equations. The full value of the proposition, however, was not understood, till Lagrange published his Mécanique Analytique: the principle is there reduced to still greater simplicity; and the connection between the pure and the mixed 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 distinguished above the rest, by the greatness of their achievements. These are, Clairaut, Euler, D'Alembert, Lagrange, and Laplace 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 embrace the entire compass of that science and its discoveries for the last hundred years: Laplace 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 everywhere algebraical; and the various parts of the higher mathematics, the integral calculus, the method of partial differences and of variations, are from the first outset introduced, whenever they can enable the author to abbreviate or to generalize his investiga tions. 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 fi gures 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 banishment of diagrams, and of all reference to them. Lagrange, 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 method is accordingly followed in the Mécanique 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 introduced for the mere purpose of assisting geometry, and supplying its defects, has ended, as many auxiliaries 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 of geometry, that never can have any but a geometrical, and some of them a synthetical demonstration. The work of Laplace 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 difficult 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 |