Newton had undertaken to submit the motions of fluids to calculation. He had made researches on the propaga tion of sound ; but his principles were insufficient, and his suppositions inconsistent with each other. La Grange de monstrates this. He founds his new researches on the known laws of dynamics, and, by considering only in the air the particles which arc in a straight line, he reduces the problem to that of vibrating cords, respecting which the greatest mathematicians differed in opinion. He shows that their calculations are insufficient to decide the ques tion. He undertakes a general solution by an analysis equally new and interesting, which enables him to resolve at once an indefinite number of equations, and which em braces even discontinued functions. He establishes on more solid grounds the theory of the mixture of simple and regular vibrations of Daniel Bernoulli. He shows the limits within which this theory is exact, and beyond which it becomes faulty. Then he comes to the construction giv en by Euler, a construction true in itself, although its first author had arrived at it by calculations which were not quite rigorous. He answers the objections of D'Alembert. He demonstrates that whatever figure is given to the cord, the duration of the oscillations is always the same: a truth derived from experiment, which D'Alembert considered as very difficult, if not impossible, to demonstrate. He passes to the propagation of sound, treats of simple and compound echos, of the mixture of sounds, of the possi bility of their spreading in the same space without interfe ring with each other. He demonstrates rigorously the generation of harmonious tounds. Finally, he announces that his intention is to destroy the prejudices of those who still doubt whether the mathematics can ever throw a real light upon physics.
We have given this long account of that memoir, be cause it is the first by which M. la Grange became known. If the analytical reasoning in it be of the most transcendent kind, the object at least has something sensible. He re cals names and facts which are well known to most people. What is surprising is, that such a first essay should be the production of a young man, who took possession of a subject treated by Newton, Taylor, Bernoulli, D'Alem bert, and Euler. He appears all at once in the midst of these great mathematicians as their equal, as a judge, who, in order to put an end to a difficult dispute, points out how far each of them is in the right, and how far they have deceived themselves ; determines the dispute between them, corrects their errors, and gives them the true solu tion, which they had perceived without knowing it to be so.
Euler saw the merit of the new method, and took it for the object of his profoundest meditations. D'Alembert did not yield the point in dispute. In his private letters, as well as in his printed memoirs, he proposed numerous objections, to which La Grange afterwards answered. But these objections may give rise to this question: How comes it that, in a science in which every one admits the merit of exactness, geniuses of the first order take different sides, and continue to dispute for a long time ? The reason is, that in problems of this kind, the solutions of which can not be subjected to the proof of experiment, besides the part of the calculation which is subjected to rigorous laws, and respecting which it is not possible to entertain two opinions, there is always a metaphysical part which leaves doubt and obscurity. It is because in the calculations
themselves, mathematicians are often content with point ing out the way in which the demonstration may be made; they suppress the developments, tvhich are not always so superfluous as they think. The care of filling up these blanks would require a labour which the author alone would have the courage to accomplish. Even he himself, drawn on by his subject and by the habits which he has acquired, allows himself to leap over the intermediate ideas. He defines his definitive equation, instead of ar riving at it step by step with an attention that would pre vent every mistake. Hence it happens that more timid calculators sometimes point out mistakes in the calcula tions of an Euler, a D'Alembert, a La Grange. Hence it happens that men of very great genius do not at first agree, from not having studied each other with sufficient attention to understand each other's meaning.
The first answer of Euler was to make La Grange an associate of the Berlin academy. When he announced to him this nomination on the 20th of October, 1759, he said, " Your solution of the problem of isoperimeters leaves no thing to desire ; and I am happy that this subject, with which I was almost alone occupied since the first attempts, has been carried by you to the highest degree of perfec tion. The importance of the matter has induced me to draw up, with your assistance, an analytical solution of it. But I shall not publish it till you yourself have published the sequel of your researches, that I may not deprive you of any part of the glory which is your due." If these delicate proceedings, and tho testimonies of the highest esteem, were very flattering to a young man of 24 years of age, they do no less honour to the great man, who at that time swayed the sceptre of mathematics, and who thus accurately estimated the merit of a work that announc ed to him a successor.
But these praises are to be found in a letter. It may be supposed that the great and good Euler has indulged in some of those exaggerations which the epistolary style permits. Let us see then how he has expressed himself in the dissertation which his letter announced. It begins as follows : " After having fatigued myself for a long time and to no purpose, in endeavouring to find this integral, what was my astonishment when I learnt that in the Turin Memoirs the problem was resolved with as much facility as felicity ! This fine discovery produced in me so much the more ad miration, as it is very different from the methods which I had given, and far surpasses them all in simplicity." It is thus that Euler begins the memoir, in which he explains with his usual clearness the foundation of the me thod of his young rival, and the theory of the new calculus, which he called the calculus of variations.