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 to wrestle with it was Hippias of Elis, a contemporary of Socrates, and born about 460 B.C. Like all the later geometers, he failed in effecting the trisection by means of a ruler and compass only. Proclus mentions a man, Hippias, presumably Hippias of Elis, as the inventor of a transcendental curve which served to divide an angle not only into three, but into any number of equal parts. This same curve was used later by Deinostratus and others for the quadrature of the circle. On this account it is called the quadratrix.

The Pythagoreans had shown that the diagonal of a square is the side of another square having double the area of the original one. This probably suggested the problem of the duplication of the cube, i.e. to find the edge of a cube having double the volume of a given cube. Eratosthenes ascribes to this problem a different origin. The Delians were once suffering from a pestilence and were ordered by the oracle to double a certain cubical altar. Thoughtless workmen, simply constructed a cube with edges twice as long, but this did not pacify the gods. The error being discovered, Plato was consulted on the matter. He and his disciples searched eagerly for a solution to this "Delian Problem." Hippocrates of Chios (about 430 B.C.), a talented mathematician, but otherwise slow and stupid, was the first to show that the problem could be reduced to finding two mean proportionals between a given line and another twice as long. For, in the proportion $$\scriptstyle{a:x=x:y=y:2a}$$, since $$\scriptstyle{x^2=ay}$$ and $$\scriptstyle{y^2=2 ax}$$ and $$\scriptstyle{x^4=a^2y^2}$$, we have $$\scriptstyle{x^4=2 a^3x}$$ and $$\scriptstyle{x^3= 2 a^3}$$. But he failed to find the two mean proportionals. His attempt to square the circle was also a failure; for though he made himself celebrated by squaring a, he committed an error in attempting to apply this result to the squaring of the circle.

In his study of the quadrature and duplication-problems, Hippocrates contributed much to the geometry of the circle.