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He found this curve to be a very elongated trochoid; and then he assigned the length of the simple pendulum, that performs it's oscillations in the same time, as the vibrating cord. This was a new and original problem at that time, and several other geometricians afterward treated it according to the same data.
The first supposition, that the excursions of the cord on either side of the axis always remain very small, is sufficiently conformable to the nature of things and besides it is the only one, which admits of calculation, even in the present state of analysis. As to the second, that all the points of the cord arrive at the axis in the same time, it is absolutely precarious, and the problem required to be freed from this limitation. D'Alembert invented a solution independant of this. In the Memoirs of the Academy of Berlin for 1747 he has directly and a priori determined the curve, which a vibrating cord forms at every instant, without making any other supposition, but that it moves only a little from the axis in it's greatest deviations. The nature of this curve is expressed at first by an equation of the second order, one member of which is the second differential of the ordinate, taken by making the time alone vary, and supposing it's differential constant: the other member is the second differential of the ordinate, taken by making the absciss alone vary, and supposing it's differential constant. Hence, by satisfying these two conditions successively, we arrive at a finite equation of such a nature, that the ordinate has for it's value the assemblage of two arbitrary functions, one the sum of the absciss and the time, the other their dif
ference. It is obvious, that, by means of this equation, if any two of the three variable quantities, the ordinate, the absciss, and the time, be given, we can determine the third, and all the circumstances of the motion of the cord.
Euler, struck with the beauty of this problem, employed himself on it a very long time, and resumed it more than once in the Memoirs of the Academies of Berlin, Petersburgh, and Turin. Notwithstanding the conformity between the results of the two great geometricians just mentioned, they had a long dispute on the extent that might be given to the arbitrary functions, which enter into the equation of the vibrating cord. D'Alembert would have the initial curvature of the cord subjected to the law of continuity: Euler considered it as altogether arbitrary, and introduced discontinuous functions into the calculation. Other geometricians have thought, that this discontinuity of the functions might be admitted, but that it should be subjected to some law, and that it was necessary, that three consecutive points of the initial curvature should always belong to a continued curve. No person yet, however, appears to have given demonstrative arguments for his opinion, at which we need not be surprised. The question is connected with metaphysical ideas; and the problems of mechanics, or of pure analysis, to which this new kind of calculus has been applied, have not yet furnished any means of discerning which of the opinions affords results conformable, or contrary, to truths already discovered and universally acknowledged.
The celebrated Daniel Bernoulli, without taking any part in the dispute, bestowed the highest encomiums on the calculations of Euler, and d'Alembert: but at the same time he undertook to show, that the vibrating cord always forms either a simple trochoid, such as Taylor's theory gives, or an assemblage of these trochoids; and that all the curves determined by d'Alembert and Euler were inadmissible, and inapplicable to nature, except as far as they were reducible to such a form.
This discussion gave him occasion to investigate the physical formation of sound, which was then very imperfectly known. He explains, for instance, with all the clearness possible, how a cord set in vibration, or generally any sonorous body whatever, may emit at once several different sounds composing one individual whole. But while we admire his address in simplifying this subject, and bringing experience in support of his reasoning, geometricians agree, that his solution is less general and perfect than those of his two rivals. In fact, the latter, whatever extent may be attributed to them, are founded on a kind of calculus, which is beyond the reach of dispute, and contain the general solution of Daniel Bernoulli as a particular case. I say as much with respect to the problem of the propagation of sound, which is of the same nature with that of vibrating cords, and to which Euler and Daniel Bernoulli have equally applied each his particular method.
The different points of view, under which Euler has considered and exhibited the integral calculus with partial differences, have settled it's true nature,
and made known the applications of which it is susceptible in a number of physico-mathematical problems. Lastly, he has thoroughly developed the method, and given it's algorithm, in an excellent paper in the Petersburgh Transactions for 1762, entitled, Investigatio Functionum ex data Differentialium Conditione. Hence some geometricians have considered Euler as the principal, if not the sole inventor of this calculus: but it ought not to be forgotten, that d'Alembert first made an important and original application of it, which furnished Euler with hints, as he himself confesses. Were I allowed to give my opinion, I should say, that these two illustrious men have nearly an equal right to the glory of this beautiful discovery.
In proportion as this new calculus has been investigated, and it's utility perceived, it has been cultivated with so much the more ardour, as the field it offers to research is immense. Some geometricians of our own time have already obtained brilliant success in it; and farther efforts will be crowned with fresh laurels. If the career become daily narrower, and if the steps of the competitors appear shorter and less profound as they advance, those who are real judges of the subject will know how to proportion their esteem to the obstacles surmounted, or the utility of the discovery; and this esteem is the noblest reward, for which he who merits it can contend.
Of some works on analysis.
UNWILLING to interrupt the history of the new calculi, I have abstained from reviewing particular works, great numbers of which appeared in this fourth period, on the analysis of finite quantities as well as of infinites but I shall now take a succinct view of the chief of those, that relate to the analysis of infinites, either as direct treatises on it, or as introductory to the subject, confining myself however to deceased authors.
It has been already observed, that the marquis de l'Hopital's Analysis of Infinites was the first work, in which the differential calculus was explained at large. It was long styled the young geometrician's Breviary. To the general idea before given of it I shall here add, that, independent of the theory of tangents, and of maxima and minima, which constituted the principal object of the differential calculus, the author has resolved a number of other problems, at that time difficult as well as interesting. Some of these problems were new: of others the solutions had been given without analysis, and without demonstration. The marquis unfolded all these mysteries, and thus rendered science one of the most important services it ever received. In sections vi and VII, for instance, he explains in the clearest and most complete