Abstract

Euclid discusses the ex aequali relationship twice in the Elements. The first is in Book V, during his discussion of arithmetical relations between mathematical magnitudes in general. The second is in Books VIIIX, he was not much troubled by the differences between his treatment of ex aequali ratios in these two contexts. Later generations of mathematicians, however, found these differences less acceptable and tried to minimize them in various ways. This paper summarizes Euclid's use of the ex aequali relation in developing his mathematics. The paper then outlines the fate of the post-Theonine Greek attempts to the Euclidean discussion when the Elements entered the Arabic/Islamic intellectual tradition. The study concludes with the attempts by Ibn al-Hayam and Ibn al-Sarī to improve the parallelism between the discussions of ex aequali ratios in Book V and Book VII