The Ancient Commentators and Their Methods: Pappus and Eutocius

An important stimulus for research in the medieval Arabic geometry, as also in the Western Renaissance, was the perception of gaps in the tradition received from antiquity. The Arabic geometers supposed, quite wrongly as we have seen, that the Greeks had not much advanced the study of the angle trisection, and so pursued this area with zeal and ingenuity. Among the many variant solutions we can detect several of ancient provenance, for which through accidents of transmission the credit of discovery came to be assigned to Arabic geometers. By contrast, the diversity of Greek methods of cube duplication was well known through the compilation by Eutocius and, doubtless, through other channels like Menelaus or even anonymous collections of excerpts. It is thus not surprising that Arabic geometers directed little energy toward the solution of this problem, and that the method favored in many of their treatments of the conics, although ostensibly Arabic in origin (being sometimes attributed to Abû Bakr al-Harawî), is found through points of technical and textual similarity to the conic methods of Menaechmus and to the neusis of Philo and Apollonius arguably to be of Greek provenance, perhaps with Apollonius.1