Euler, Lagrange, and Laplace

filled by his writings. He had engaged to furnish the Petersburg Academy with memoirs in sufficient number to enrich its acts for twenty years–-a promise more than fulfilled, for down to 1818 the volumes usually contained one or more papers of his. It has

been said that an edition of Euler's complete works would fill $16, 000$ quarto pages. His mode of working was, first to concentrate his powers upon a special problem, then to solve separately all problems growing out of the first. No one excelled him in dexterity of accommodating methods to special problems. It is easy to see that mathematicians could not long continue in Euler's habit of writing and publishing. The material would soon grow to such enormous proportions as to be unmanageable. We are not surprised to see almost the opposite in Lagrange, his great successor. The great Frenchman

delighted in the general and abstract, rather than, like Euler, in the special and concrete. His writings are condensed and give in a nutshell what Euler narrates at great

length.

Jean-le-Rond D'Alembert (1717–1783) was exposed, when

an infant, by his mother in a market by the church of St. Jean-le-Rond, near the Nôtre-Dame in Paris, from which he derived his Christian name. He was brought up by the wife of a poor glazier. It is said that when he began to show signs of great talent, his mother sent for him, but received the reply, "You are only my step-mother; the glazier's wife is my mother." His father provided him with a yearly income. D'Alembert entered upon the study of law, but such was his love for mathematics, that law was soon abandoned. At the age of twenty-four his reputation as a mathematician secured for him admission to the Academy of Sciences. In 1743 appeared his Traité de dynamique, founded upon the important general principle bearing his name: The impressed forces are equivalent to the effective forces. D'Alembert's principle seems to have been recognised before him by Fontaine, and

in some measure by John Bernoulli and Newton. D'Alembert

gave it a clear mathematical form and made numerous applications of it. It enabled the laws of motion and the reasonings

depending on them to be represented in the most general form, in analytical language. D'Alembert applied it in 1744 in a treatise on the equilibrium and motion of fluids, in 1746 to a treatise on the general causes of winds, which obtained a prize from the Berlin Academy. In both these treatises, as also in one of 1747, discussing the famous problem of vibrating chords, he was led to partial differential equations. He was

a leader among the pioneers in the study of such equations. To the equation $\frac{\partial^{2} y}{\partial t^{2}} = a^{2} \frac{\partial^{2} y}{\partial x^{2}}$ , arising in the problem of vibrating chords, he gave as the general solution, $y = f (x + a t) + ϕ (x - a t),$ and showed that there is only one arbitrary function, if $y$ be supposed to vanish for $x = 0$ and $x = l$ . Daniel Bernoulli,

starting with a particular integral given by Brook Taylor,

showed that this differential equation is satisfied by the trigonometric series y = α sin π x l · cos π t l + β sin 2 π x l · cos 2 π t l + ⋯ , and claimed this expression to be the most general solution.