Euler, Lagrange, and Laplace

involve the co-ordinates $x$, $y$, $z$, of the different particles $m$ or $dm$ of the system. But $x$, $y$, $z$, are in general not independent, and Lagrange introduced in place of them any variables $\xi$, $\psi$, $\varphi$, whatever, determining the position of the point at the time. These may be taken to be independent. The equations of motion may now assume the form $\frac{d}{dt}\frac{dT}{d{\xi }^{\prime }}-\frac{dT}{d\xi }+\Xi =0;$ or when $\Xi$, $\Psi$, $\Phi ,\dots$ are the partial differential coefficients with respect to $\xi$, $\psi$, $\varphi ,\dots$ of one and the same function $V$, then the form $\frac{d}{dt}\frac{dT}{d{\xi }^{\prime }}-\frac{dT}{d\xi }+\frac{dV}{d\xi }=0.$ The latter is par excellence the Lagrangian form of the equations of motion. With Lagrange originated the remark that mechanics may be regarded as a geometry of four dimensions. To him falls the honour of the introduction of the potential into dynamics.49 Lagrange was anxious to have his Mécanique Analytique published in Paris. The work was ready for print in 1786, but not till 1788 could he find a publisher, and then only with the condition that after a few years he would purchase all the unsold copies. The work was edited by Legendre.

After the death of Frederick the Great, men of science were no longer respected in Germany, and Lagrange accepted an invitation of Louis XVI. to migrate to Paris. The French queen treated him with regard, and lodging was procured for him in the Louvre. But he was seized with a long attack of melancholy which destroyed his taste for mathematics. For two years his printed copy of the Mécanique, fresh from the press,–-the work of a quarter of a century,–-lay unopened on his desk. Through Lavoisier he became interested in chemistry, which he found "as easy as algebra." The disastrous

crisis of the French Revolution aroused him again to activity. About this time the young and accomplished daughter of the astronomer Lemonnier took compassion on the sad, lonely

Lagrange, and insisted upon marrying him. Her devotion to him constituted the one tie to life which at the approach of death he found it hard to break.

He was made one of the commissioners to establish weights and measures having units founded on nature. Lagrange strongly favoured the decimal subdivision, the general idea of which was obtained from a work of Thomas Williams, London,

1788. Such was the moderation of Lagrange's character, and such the universal respect for him, that he was retained as president of the commission on weights and measures even after it had been purified by the Jacobins by striking out the names of Lavoisier, Laplace, and others. Lagrange took alarm at the fate of Lavoisier, and planned to return to Berlin, but at the establishment of the École Normale in 1795 in Paris, he was induced to accept a professorship. Scarcely had he time to elucidate the foundations of arithmetic and algebra to young pupils, when the school was closed. His additions to the algebra of Euler were prepared at this time. In 1797 the

École Polytechnique was founded, with Lagrange as one of the professors. The earliest triumph of this institution was

the restoration of Lagrange to analysis. His mathematical activity burst out anew. He brought forth the Théorie des fonctions analytiques (1797), Leçons sur le calcul des fonctions, a treatise on the same lines as the preceding (1801), and the Résolution des équations numériques (1798). In 1810 he began a thorough revision of his Mécanique analytique, but he died before its completion.

The Théorie des fonctions, the germ of which is found in a memoir of his of 1772, aimed to place the principles of the calculus upon a sound foundation by relieving the mind of the difficult conception of a limit or infinitesimal. John Landen's