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 a ³, so that the side of the cube which is double of the cube of side a is found.

The problem of doubling the cube was henceforth tried exclusively in the form of the problem of the two mean proportionals. Two significant early solutions are on record.

(1) Archytas of Tarentum (who flourished in first half of fourth century B.C.) found the two mean proportionals by a very striking construction in three dimensions, which shows that solid geometry, in the hands of Archytas at least, was already well advanced. The construction was usually called mechanical, which it no doubt was in form, though in reality it was in the highest degree theoretical. It consisted in determining a point in space as the intersection of three surfaces: ( a ) a cylinder, ( b ) a cone, ( c ) an “anchor-ring” with internal radius = 0. (2) Menæchmus, a pupil of Eudoxus, and a contemporary of Plato, found the two mean proportionals by means of conic sections, in two ways, (α) by the intersection of two parabolas, the equations of which in Cartesian co-ordinates would be x ² =