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 to mathematicians, in which each vied to outdo all others. The problem of two bodies, requiring the determination of their motion when they attract each other with forces inversely proportional to the square of the distance between them, had been completely solved by Newton. The "problem of three bodies" asks for the motion of three bodies attracting each other according to the law of gravitation. Thus far, the complete solution of this has transcended the power of analysis. The general differential equations of motion were stated by Laplace, but the difficulty arises in their integration. The "solutions" hitherto given are merely convenient methods of approximation in special cases when one body is the sun, disturbing the motion of the moon around the earth, or where a planet moves under the influence of the sun and another planet.

In the discussion of the meaning of negative quantities, of the fundamental processes of the calculus, and of the theory of probability, D'Alembert paid some attention to the philosophy of mathematics. His criticisms were not always happy. In 1754 he was made permanent secretary of the French Academy. During the last years of his life he was mainly occupied with the great French encyclopædia, which was begun by Diderot and himself. D'Alembert declined, in 1762, an invitation of Catharine II. to undertake the education of her son. Frederick the Great pressed him to go to Berlin. He made a visit, but declined a permanent residence there.

Alexis Claude Clairaut (1713–1765) was a youthful prodigy. He read l'Hospital's works on the infinitesimal calculus and on conic sections at the age of ten. In 1731 was published his Recherches sur les courbes à double courbure, which he had ready for the press when he was sixteen. It was a work of remarkable elegance and secured his admission to the Academy of Sciences when still under legal age. In 1731 he gave a proof of