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 VII–IX, where he focuses on arithmetical relations in the case of numbers only. Although the distinction between mathematical magnitudes in general and numbers in particular often seems somewhat forced to contemporary philosophers, it was apparently very real to Euclid. Because Euclid seemed so conscious of the differences between the subject matter of Book V and Books VII–IX, 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 “improve” 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.