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 in Greek. In it he proves the theorems on the congruence of spherical triangles, and describes their properties in much the same way as Euclid treats plane triangles. In it are also found the theorems that the sum of the three sides of a spherical triangle is less than a great circle, and that the sum of the three angles exceeds two right angles. Celebrated are two theorems of his on plane and spherical triangles. The one on plane triangles is that, "if the three sides be cut by a straight line, the product of the three segments which have no common extremity is equal to the product of the other three." The illustrious Carnot makes this proposition, known as the 'lemma of Menelaus,' the base of his theory of transversals. The corresponding theorem for spherical triangles, the so-called 'regula sex quantitatum,' is obtained from the above by reading "chords of three segments doubled," in place of "three segments."

Claudius Ptolemæus, a celebrated astronomer, was a native of Egypt. Nothing is known of his personal history except that he flourished in Alexandria in 139 A.D. and that he made the earliest astronomical observations recorded in his works, in 125 A.D., the latest in 151 A.D. The chief of his works are the Syntaxis Mathematica (or the Almagest, as the Arabs call it) and the Geographica, both of which are extant. The former work is based partly on his own researches, but mainly on those of Hipparchus. Ptolemy seems to have been not so much of an independent investigator, as a corrector and improver of the work of his great predecessors. The Almagest forms the foundation of all astronomical science down to Copernicus. The fundamental idea of his system, the "Ptolemaic System," is that the earth is in the centre of the universe, and that the sun and planets revolve around the earth. Ptolemy did considerable for mathematics. He created, for astronomical use, a trigonometry remarkably perfect in form.