By D'Ooge, Martin Luther; of Gerasa. Nicomachus; Robbins, Frank Egleston; Karpinski, Louis Charles

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6, etc. 11 Republic, 546 B. II! 1 from UNrY£RSlTY Of MICHIGAN GREEK ARITHMETIC BEFORE NICOMACHUS '5 ",p&afhcr«)! ' Finally, if we consider that all mention of them is purely incidental, the treatment of proportion (avaAOY'a) is especially complete. Definitions of the arithmetic and harmonic types occur in the form found also in Archytas and Nicomachus, 8 and the geometric is seen in the simile of the divided line 9 and elsewhere. We have already noted that the theorem concerning the number of means necessary between plane and solid numbers as extremes may well be a contribution of Plato himself; and furthermore the use of the so-called 'musical' proportion by Plato in Timaeus 36 A, though it had doubtless been employed by the Pythagoreans before him, led the authors of aries, like Nicomachus in Introduction, n.

65-66. 1 from UNrY£RSlTY Of MICHIGAN SOURCES OF GREEK MATHEMATICS IS sion to Greece from Babylon, but no Babylonian record confirming the statement has yet been found. The squares and cubes of numbers also received particular attention in Babylon, and tables of squares and cubes have been found. In Egypt the relation 3' + 4' = 5' appears to have been used in the laying out of right angles by means of a stretched rope. " 1 Michael Psellus of the eleventh century mentions the Greek equivalents of the Egyptian names of the higher powers, first power to twelfth power; it is supposed that the statement is based upon the lost commentary upon the Arithmetic of Diophantus by Hypatia.

354-357. ' The name of the book in which this occurred, however, is unknown, and likewise we have no information upon a treatise on arithmetic which he is reported to have written. Euclid, Eudoxus, Hypsicles and Eratosthenes, who has been mentioned in another connection, are practically the only mathematicians whom we can specify as having very probably been concerned in the development of the ars arithmetica from the · fourth century before Christ to the time of Nicomachus. Yet in this age the Greek science of mathematics was in its most flourishing state, centering especially about Alexandria, and it is hardly to be imagined that the Elements of Euclid was the only book of the type of the artes which was written in this period.