 It is not clear whether or not the story about the Delians has any basis in fact. Even if it does, it should not be understood to suggest that the problem of doubling the cube first arose in the fourth century with the Delians. We are told that the mathematician, Hippocrates of Chios, who was active in the second half of the fifth century, had already confronted the problem and had reduced it to a slightly different problem (Eutocius, in Archim. sphaer. et cyl. II [III 88.3–96.27 Heiberg/Stamatis]). Hippocrates recognized that if we could find two mean proportionals between the length of the side of the original cube G, and length D, where D = 2G, so that G : x :: x : y :: y : D, then the cube on length x will be double the cube on length G. Exactly how Hippocrates came to see this is conjectural and need not concern us here, but that he was right can be seen relatively easily. Each of the values in the continued proportion G : x :: x : y :: y : D is equal to G : x, so we can set them all equal to G : x. If we do this and multiply the three ratios together we get the value G3 : x3. On the other hand, if we take the same continued proportion and carry out the multiplication in the original terms, then G : x times x : y yields G : y, and G : y times the remaining term gives G : D. Thus G : D = G3 : x3, but D is twice G so x3 is twice G3. Remember that G was the length of the side of the original cube, so the cube that is twice the cube built on G, will be the cube built on x. The Greeks did not think of the problem as a problem in algebra but rather as a problem in geometry. After Hippocrates the problem of doubling the cube was always seen as the problem of finding two lines such that they were mean proportionals between G, the length of the side of the original cube, and D, a length which is double G. It was to this form of the problem that Archytas provided the first solution.