By Averbach B., Chein O.
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Archytas was one of the first to give a solution of the problem to duplicate a cube, that is, to find the side of a cube whose volume is double that of a given cube. 1 The construction given by Archytas is equivalent to the following. On the diameter OA of the base of a right circular cylinder describe a semicircle whose plane is perpendicular to the base of the cylinder. Let the plane containing this semicircle rotate round the generator through O, then the surface traced out by the semicircle will cut the cylinder in a tortuous curve.
I mention the problem and give the construction used by Archytas to illustrate how considerable was the knowledge of the Pythagorean school at the time. Theodorus. Another Pythagorean of about the same date as Archytas was Theodorus of Cyrene, who is√said√to have √ proved √ √geomet√ 3, 5, 6, 7, 8, 10, rically that the numbers represented by √ √ √ √ √ √ 11, 12, 13, 14, 15, and 17 are incommensurable with unity. Theaetetus was one of his pupils. Perhaps Timaeus of Locri and Bryso of Heraclea should be mentioned as other distinguished Pythagoreans of this time.
Eratosthenes gives a somewhat similar account of its origin, but with king Minos as the propounder of the problem. Hippocrates reduced the problem of duplicating the cube to that of finding two means between one straight line (a), and another twice as long (2a). If these means be x and y, we have a : x = x : y = y : 2a, from which it follows that x3 = 2a3 . It is in this form that the problem is usually presented now. Hippocrates did not succeed in finding a construction for these means. CH. III] THE SCHOOLS OF ATHENS AND CYZICUS 35 Plato.