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This Day is Publified,
A CATALOGUE OF BOOKS,
CONSISTING OF NEARLY
THIRTY THOUSAND VOLUMES,
Including the valuable CLASSICAL LIBRARY of the late Profeffor HENSLER OF KIEL, in HOLSTEIN.
On fale at the Shop of
ARCHIBALD CONSTABLE & CO.
EDINBURGH.-JAN. 1. 1808.
To be had of the Publifhers, and of JOHN MURRAY, 32, Fleet-Street, London.
ART. I. Traité de Méchanique Céleste. Par P. S. La Place, Mema bre de l'Institut National de France, et du Bureau des Longia tudes. Paris. Vol. 1. An 7. Vol. 3. & 4. 1805.
As STRONOMY is distinguished by several great and striking characters, which place it decidedly at the head of the physical sciences. The objects which it treats of, cannot fail to impart to it a degree of their own magnificence and splendour; while their distance, their magnitude, the steadiness and regularity of their movements, deeply impress the imagination, and afford a noble exercise to the understanding. Add to this, that the history of astronomy is that which is best marked out in the progress of human knowledge. Through the darkness of the early ages, we perceive the truths of this science shining as it were by their own light, and scattering some rays around them, that serve to discover a few definite objects amid the confusion of ancient tra dition, a few fixed points amid the uncertainty of Greek, Egyp tian, or even Hindoo mythology. But what distinguishes astro nomy the most, is the perfect explanation which it gives of the celestial phenomena. This explanation is so complete, that there is not any fact concerning the motions of the heavenly bodies, from the greatest to the least, which is not reducible to one single law-the mutual gravitation of all bodies to one another, with forces that are directly as the masses of the bodies, and inversely as the squares of their distances. On this principle Sir Isaac Newton long ago accounted for all the great motions in our system; and, on the same principle, his successors, after near a century of the most ingenious and elaborate investigation, have explained all the rest. The work before us brings those explanations into one view, and deduces them from the first principles of meVOL. XI. NO. 22.
chanics. It is not willingly that we have suffered so much time to elapse without laying before our readers an analysis of a work the most important, without doubt, that has distinguished the conclusion of the last or the commencement of the present century. But the book is still, in some respects, incomplete, and a historical volume is yet wanting, which, had we been in possession of it, would have very much facilitated the task that we have now undertaken to perform. We know not whether this volume is actually published. In the present state of Europe it may be a long time before it can find its way to this country; and, in the meantime, our duty seems to require that an account of the fourvolumes, which we possess, should no longer be withheld from the public.
Though the integral calculus, as it was left by the first invent-> ors and their contemporaries, was a very powerful instrument of investigation, it required many improvements to fit it for extending the philosophy of Newton to its utmost limits." A brief enumeration of the principal improvements which it has actually received in the last seventy or eighty years, will very much assist us in appreciating the merit of the work which is now before
1. Descartes is celebrated for having applied algebra to geometry; and Euler hardly deserves less credit for having applied the same science to trigonometry. Though we ascribe the invention of this calculus to Euler, we are aware that the first attempt toward it was made by a mathematician of far inferior note, Christian Mayer, who, in the Petersburgh Commentaries for 1727, published a paper on analytical trigonometry. In that memoir, the geometrical theorems, which serve as the basis of this new species of arithmetic, are pointed out; but the extension of the method, the introduction of a convenient notation, and of a peculiar algorithm, are the work of Euler. By means of these, the sines and cosines of arches are multiplied into one another, and raised to any power, with a simplicity unknown in any other part of algebra, being expressed by the sines and cosines of multiple arches, of one dimension only, or of no higher power than the first. It is incredible of how great advantage this method has proved in all the parts of the higher geometry, but more especially in the researches of physical astronomy. As what we observe in the heavens is nothing but angular position, so if we would compare the result of our reasonings concerning the action of the heavenly bodies, with observations made on the surface of the earth, we must express those results in terms of the angles observed, or the quantities dependent on them, such as sines, tangents, &c. It is evident that a calculus which teaches how this is to be accomplished, must be of the greatest value to the as
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 fluents and the summing of series, makes it one of the most valuable resources in mathematical science. It is a method continually em ployed in the Méchanique Céleste.
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 Auxions. It was discovered by the geometer just named, when he was inquiring into the nature of the figures successively assumed 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 partial 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 advantage. The first idea of this new method, and the first appli cation of it, are due to D'Alembert: it is from Euler, however, that we derive the form and notation that have been generally adopted.
3. Another great addition made to the integral calculus, 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,
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 fluxions 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 determined 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 investigation 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. this work, though a great number of ingenious men have been concerned, yet more is due to Euler than to any other individual. 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 dif 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