used by him in his tract on the squaring of lunes, which was intended to lead up to the squaring of the circle. The latter problem is one which must have exercised practical geometers from time immemorial. Anaxagoras for instance (about 500-428 B.C.) is said to have worked at the problem while in prison. The essential portions of Hippocrates’s tract are preserved in a passage of Simplicius (on Aristotle’s Physics), which contains substantial fragments from Eudemus’s History of Geometry. Hippocrates showed how to square three particular lunes of different forms, and then, lastly, he squared the sum of a certain circle and a certain lune. Unfortunately, however, the last-mentioned lune was not one of those which can be squared, and so the attempt to square the circle in this way failed after all.

Hippocrates also attacked the problem of doubling the cube. There are two versions of the origin of this famous problem. According to one of them, an old tragic poet represented Minos as having been dissatisfied with the size of a tomb erected for his son Glaucus, and having told the architect to make it double the size, retaining, however, the cubical form. According to the other, the Delians, suffering from a pestilence, were told by the oracle to double a certain cubical altar as a means of staying the plague. Hippocrates did not, indeed, solve the problem, but he succeeded in reducing it to another, namely, the problem of finding two mean proportionals in continued proportion between two given straight lines, i.e. finding x , y such that a : x = x : y = y : b , where a , b are the two given straight lines. It is easy to see that, if a : x = x : y = y : b , then b / a = ( x / a )³, and, as a particular case, if b = 2 a , x ³ = 2