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I. 47 that the square on the hypotenuse of any right-angled triangle is equal to the sum of the squares on the other two sides. But Proclus also attributes to him, besides the theory of proportionals, the construction of the 'cosmic figures', the five regular solids. This method was fully worked out by the Pythagoreans and proved one of the most powerful in all Greek geometry.
The Just, then, is a certain proportionable thing. So then the joining together of the first and third and second and fourth proportionals is the Just in the distribution, and this Just is the mean relatively to that which violates the proportionate, for the proportionate is a mean and the Just is proportionate.
Theodorus's proof was evidently not general; and it was reserved for Theaetetus to comprehend all these irrationals in one definition, and to prove the property generally as it is proved in Eucl. The solution, attributed to Plato, of the problem of the two mean proportionals by means of a frame resembling that which a shoemaker uses to measure a foot, can hardly be his.
Of his skill as a geometer, his solution of the problem of two mean proportionals, still extant, offers ample evidence; and it is only of late years that the fragments remaining of his Chronicles of the Theban Kings have been properly appreciated.
He also discovered that of all figures having the same boundary, the circle among plane figures and the sphere among solids, are the most capacious. The theory of the regular solids was taught in his school, and his disciple, Archytas, was the author of a solution of the problem of two mean proportionals.
Hippocrates of Chios reduced the problem of duplicating the cube to that of finding two mean proportionals in continued proportion between two straight lines, that is, he showed that, if the latter problem could be solved, the former was thereby solved also; and it is probable that there were still earlier cases in the Pythagorean geometry. Next there is the method of mathematical analysis.
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