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 exhaust the subject. Twenty-four years afterwards it was completed by Laplace. Later astronomical investigations of Lagrange are on cometary perturbations (1778 and 1783), on Kepler's problem, and on a new method of solving the problem of three bodies.

Being anxious to make the personal acquaintance of leading mathematicians, Lagrange visited Paris, where he enjoyed the stimulating delight of conversing with Clairaut, D'Alembert, Condorcet, the Abbé Marie, 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 staidstayed [sic] 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