certain corrections to the angles, and on the method of least squares, published for the first time by him without demonstration in 1806.
Legendre wrote an Élements de Géométrie, 1794, which enjoyed great popularity, being generally adopted on the Continent and in the United States as a substitute for Euclid. This great modern rival of Euclid passed through numerous editions; the later ones containing the elements of trigonometry and a proof of the irrationality of and . Much attention was given by Legendre to the subject of parallel lines. In the earlier editions of the Élements, he made direct appeal to the senses for the correctness of the "parallel-axiom." He then attempted to demonstrate that "axiom," but his proofs did not satisfy even himself. In Vol. XII. of the Memoirs of the Institute is a paper by Legendre, containing his last attempt at a solution of the problem. Assuming space to be infinite, he proved satisfactorily that it is impossible for the sum of the three angles of a triangle to exceed two right angles; and that if there be any triangle the sum of whose angles is two right angles, then the same must be true of all triangles. But in the next step, to show that this sum cannot be less than two right angles, his demonstration necessarily failed. If it could be granted that the sum of the three angles is always equal to two right angles, then the theory of parallels could be strictly deduced.
Joseph Fourier (1768–1830) was born at Auxerre, in central France. He became an orphan in his eighth year. Through the influence of friends he was admitted into the military school in his native place, then conducted by the Benedictines of the Convent of St. Mark. He there prosecuted his studies, particularly mathematics, with surprising success. He wished to enter the artillery, but, being of low birth (the son of a tailor), his application was answered thus: "Fourier, not
