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Yanney and J. Math. Monthly, Vols. 3 and 4, 1896 and 1897. A. Calderhead in the Am. GREECE 51 arithmetical fact has its analogue in geometry, and vice versa. i== the other side, and 5, then the three If n = hypotenuse. sides are 11, 60, 61. This rule yields only triangles whose hypotenuse exceeds one of the sides by unity. 2 n2 + 2 n -f 1 = the ^ Ascribed to the Pythagoreans is matical discoveries of antiquity The discovery is t^JJ^ one of the "greatest mathe that of Irrational Quantities. usually supposed to have gro\vn out of the If each of the equal 1 study of the isosceles right triangle.

Without exhibiting the computation itself. Thus advanced mathematicians frequently had occasion to extract the square root. In his Mensuration of the Circle, Archimedes results states, for instance, that V3 < -^g^- 1- and V3 > fff , but he method of approximation. 2 When sexagesimal numbers (introduced from Babylonia into Greece about the time of the Greek geometer Hypsicles and the Alexandrian astronomer Ptolemseus) were used, then gives no clue to his the mode of root-extraction resembled that of the present time.

CJLNTOB, Yol. I, A HISTOKY OF MATHEMATICS 32 work was brought out in a Latin translation by Appuleius (now lost) and then by Boethius. In Boethius's translation the elementary parts of the work were in high authority in Western Europe until the country was invaded by Hindu Thereupon for several centuries Greek arithmetic bravely but vainly struggled for existence against its immeas arithmetic. urably superior Indian The style of rival. Mcomachus from that of his differs essentially The geo predecessors.