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 are quadratic residues, or non-residues of odd prime numbers, $\scriptstyle{q}$; he proved in 1770 Méziriac's theorem that every integer is equal to the sum of four, or a less number, of squares. He proved Format's theorem on $\scriptstyle{x^n+y^n=z^n}$, for the case $\scriptstyle{n=4}$, also Fermat's theorem that, if $\scriptstyle{a^2+b^2=c^2}$, then $$\scriptstyle{ab}$$ is not a square.

In his memoir on Pyramids, 1773, Lagrange made considerable use of determinants of the third order, and demonstrated that the square of a determinant is itself a determinant. He never, however, dealt explicitly and directly with determinants; he simply obtained accidentally identities which are now recognised as relations between determinants.

Lagrange wrote much on differential equations. Though the subject of contemplation by the greatest mathematicians (Euler, D'Alembert, Clairaut, Lagrange, Laplace), yet more than other branches of mathematics did they resist the systematic application of fixed methods and principles. Lagrange established criteria for singular solutions (Calcul des Fonctions, Lessons 14–17), which are, however, erroneous. He was the first to point out the geometrical significance of such solutions. He generalised Euler's researches on total differential equations of two variables, and of the ninth order; he gave a solution of partial differential equations of the first order (Berlin Memoirs, 1772 and 1774), and spoke of their singular solutions, extending their solution in Memoirs of 1779 and 1785 to equations of any number of variables. The discussion on partial differential equations of the second order, carried on by D'Alembert, Euler, and Lagrange, has already been referred to in our account of D'Alembert.

While in Berlin, Lagrange wrote the Méchanique Analytique," the greatest of his works (Paris, 1788). From the principle of virtual velocities he deduced, with aid of the calculus of variations, the whole system of mechanics so elegantly and