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tronomer. Besides, the facility which this calculus gives to all the reasonings and computations into which it is introduced, from the elementary problems of geometry to the finding of fuents and the summing of series, makes it one of the most valuable rea' sources in mathematical science. It is a method continually emi ployed in the Méchanique Céleste. A
2. An improvement in the integral calculus, made by M. D'Alembert, has doubled its power, and added to it a territory not inferior in extent to all that it before possessed. This is the method of partial differences, or, as we must call it, of partial fluxions. It was discovered by the geometer just named, when he was inquiring into the nature of the figures successively as= sumed by a musical string during the time of its vibrations. When a variable quantity is a function of other two variable quantities, as the ordinates belonging to the different abscissæ in these curves must necessarily be, (for they are functions both of the abscissæ and of the time counted from the beginning of the vibrations), it becomes convenient to consider how that quantity varies, while each of the other two varies singly, the remaining one being supposed constant. Without this simplification, it would, in most cases, be quite impossible to subject such complicated functions to any rules of reasoning whatsoever. The calculus of para tial differences, therefore, is of great utility in all the more complicated problems both of pure and mixt mathematics ; every thing relating to the motion of fluids that is not purely elementary, falls within its range; and in all the more difficult researches of physical astronomy, it has been introduced with great ada vantage. The first idea of this new method, and the first application of it, are due to D'Alembert : it is from Euler, however, that we derive the form and notation that have been generally adopted.
marco ! 3. Another great addition made to the integral calculús, is the invention of La Grange, and is known by the name of the Calcu. lus variationum. The ordinary problems of determining the greatest and least states of a given function of one or more variable quantities, is easily reduced to the direct method of fluxions, or the differential calculus, and was indeed one of the first classes of questions to which those methods were applied. But when the function that is to be a maximum or a minimum, is not given in its form ; or when the curve, expressing that function, is not : known by any other property, but that, in certain circumstances, it is to be the greatest or least possible, the solution is infinitely mote difficult, and science seems to have no hold of the question by which to reduce it to a mathematical investigation. The problem of the line of swiftest descent is of this nature ; and though, R %
from some facilities which this and other particular instances afforded, they were resolved, by the ingenuity of mathematicians, before any method generally applicable to them was known, yet such a method could not but be regarded as a great desideratum in mathematical science. The genius of Euler had gone far to supply it, when La Grange, taking a view entirely different, fell upon a method extremely convenient, and, considering the difficulty of the problem, the most simple that could be expected. The 'supposition it proceeds on is greatly more general than that of the fluxionary or differential calculus. In this last, the fuxions or changes of the variable quantities are restricted by certain laws. The fluxion of the ordinate; for example, has a relation to the fluxion of the abscissa that is deter' mined by the nature of the curve to which they both belong.. But in the method of variations, the change of the ordinate may be any whatever ; it may no longer be bounded by the original curve, but it may pass into another, having to the former no determinate relation. This is the calculus of La Grange ; and, though it was invented expressly with a view to the problems just mentioned, it has been found of great use in many physical questions with which those problems are not immediately connected. :
4. Among the improvements of the higher geometry, besides those which, like the preceding, consisted of methods entirely new, the extension of the more ordinary methods to the integration of a vast number of formulas, the inyestigation of many, new theorems concerning quadratures, and concerning the solution of fluxionary equations of all orders, had completely changed the appearance of the calculus; so that Newton or Leibnitz, had they returned to the world any time since the middle of the last century, would have been unable, without great study, to follow the discoveries which their disciples had made, by proceeding in the line which they themselves had pointed out. In this work, though a great number of ingenious men have been concerned, yet more is due to Euler than to any other indivi, dual. With indefatigable industry, and the resources of a most inventive mind, he devoted a long life entirely to the pursuits of science. Besides producing many works on all the difa ferent 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 originality 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 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 motion which the particle had in the preceding instant, then the sum of all these third motions must be such, that they are in equilibri. um 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 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 obseryations of astronomy, than the errors unavoidable in the latter were suf. ficient to account for. The time seemed now to be come for reducing the whole theory of astronomy into one work, that should R3
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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 en, tire System of Physical Astronomy, from the first elements to 22 the most remote conclusions of the science. The author has been careful to preserve the same method of investigation through, out; 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 vi rious parts of the higher mathematics, the integral calculus, the m thod of partial differences and of variations, are from the first or set introduced, whenever they can enable the author to abbrevi: or to generalize his investigations. No diagrams or geometri figures are employed ; and the seader must converse with the jects presented to him by the language of arbitrary symbols alc Whether the rejection of figures be in all respects an improvem and whether it may not be in some degree hurtful to the pobara of the imagination, we will not take upon us to decide. It is tain, however, that the perfection of Algebra tends to the þa ment of diagrams, and of all reference to them. La Grangote his treatise of Analytical Mechanics, has no reference to figpose of notwithstanding the great number of mechanical problems 1 * Jlecian, he reşolyes. The resolution of all the forces that act o , 2014 at point, into three forces, in the direction of three axes at leicos angles to one another, enables one to express their re' tidar very distinctly, without representing them by a figure, ledit ad pressing them by any other than algebraic symbols. T i ede thod is accordingly followed in the Méchanique Céleste. Lot Jemi thing of thę same kind, indeed, seems applicable to alm Singers part of the mathematics; -and a very distinct treatise on th sections, we doubt not, might be written, where there we tarantia be a single diagram introduced, and where all the propres zabies the ellipse, the parabola, and the hyperbola would be ease seer either by words or by algebraic characters. Whether these eau nation would lose or gain by this exercise, we shall not a szepters stop to inquire. It is curious, however, to pbşerve, there in the bra, which was first intoduced for the mere purpose of bor each geometry, and supplying its defects, has ended, asma Suptic notion aries have done, with discarding that Kece A culiar methods) almost entirely. WC AKR cause there are, doubtless, a
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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,mwitla 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 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 cle. 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 nléchanique 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 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 anon ther, deduced in the second chapter, La Place proceeds to the in. tegration 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.