  One may wonder why the proponents of the anthyphairetic definition felt that it was more natural than Euclid’s approach. There is no doubt, however, that it was preferred; Khayyam even refers to the anthyphairetic definition as the “true” nature of proportionality. Part of the explanation might be simply that the Euclidean algorithm applied to geometric quantities was much more familiar to medieval mathematicians than to us. It has also been suggested that Khayyam’s preference is due to the fact that the anthyphairetic definition allows a ratio to be considered on its own, rather than always in equality to some other ratio. Khayyam’s achievement in this topic was not to invent a new definition, but rather to demonstrate that each of the existing definitions logically implies the other. Thus Islamic mathematicians could continue to use ratio theorems from the Elements without having to prove them again according to the anthyphairetic definition.