and others. He had planned a visit to London, but he fell dangerously ill after a dinner in Paris, and was compelled to return to Turin. In 1766 Euler left Berlin for St. Petersburg, and he pointed out Lagrange as the only man capable of filling the place. D'Alembert recommended him at the same time. Frederick the Great thereupon sent a message to Turin, expressing the wish of "the greatest king of Europe" to have "the greatest mathematician" at his court. Lagrange went to Berlin, and staid there twenty years. Finding all his colleagues married, and being assured by their wives that the marital state alone is happy, he married. The union was not a happy one. His wife soon died. Frederick the Great held him in high esteem, and frequently conversed with him on the advantages of perfect regularity of life. This led Lagrange to cultivate regular habits. He worked no longer each day than experience taught him he could without breaking down. His papers were carefully thought out before he began writing, and when he wrote he did so without a single correction.
During the twenty years in Berlin he crowded the transactions of the Berlin Academy with memoirs, and wrote also the epoch-making work called the Mécanique Analytique. He enriched algebra by researches on the solution of equations.
There are two methods of solving directly algebraic equations,–-that of substitution and that of combination. The
former method was developed by Ferrari, Vieta, Tchirnhausen,
Euler, Bézout, and Lagrange; the latter by Vandermonde and
Lagrange.20 In the method of substitution the original forms are so transformed that the determination of the roots is made to depend upon simpler functions (resolvents). In the method of combination auxiliary quantities are substituted for certain simple combinations ("types") of the unknown roots of the equation, and auxiliary equations (resolvents) are obtained for
these quantities with aid of the coefficients of the given equation. Lagrange traced all known algebraic solutions of equations to the uniform principle consisting in the formation and solution of equations of lower degree whose roots are linear functions of the required roots, and of the roots of unity. He showed that the quintic cannot be reduced in this way, its resolvent being of the sixth degree. His researches on the theory of equations were continued after he left Berlin. In the Résolution des équations numériques (1798) he gave a method of approximating to the real roots of numerical equations by continued fractions. Among other things, it contains also a proof that every equation must have a root,–-a theorem which appears before this to have been considered self-evident. Other proofs of this were given by Argand, Gauss, and Cauchy.
In a note to the above work Lagrange uses Fermat's theorem
and certain suggestions of Gauss in effecting a complete algebraic solution of any binomial equation.
While in Berlin Lagrange published several papers on the theory of numbers. In 1769 he gave a solution in integers of
indeterminate equations of the second degree, which resembles the Hindoo cyclic method; he was the first to prove, in 1771, "Wilson's theorem," enunciated by an Englishman, John