Book II of Euclid's elements in the light of the theory of conic sections

Ken Saito

pp. 139-168

in: Jean Christianidis (ed), Classics in the history of Greek mathematics, Berlin, Springer, 2004

Abstract

This paper proposes an alternative to the prevailing interpretation which regards the second book of the Elements (hereafter Elem. II) as basic part of the "geometric algebra". Chapter I of this paper is dedicated to an examination of the Conics of Apollonius. Though the central part of the "geometric algebra" is usually explained as a translation of the Babylonian algebraic techniques, it is not reasonable to attempt a determination of the nature of Book II by inconfirmable conjectures regarding its origin. The significance of Elem. II should be sought by studying applications of the propositions there. Thus we should examine how Euclid utilizes his propositions in Elem. II, when arguing about this book. The propositions in the Elements have been thoroughly examined by Ian Mueller.¹ But Mueller's study is not sufficient for purposes of the present paper, precisely because he limits his study to the Elements; other works of Euclid as well should be examined. Hence, the study of the Conics is necessary, since compilation of the fundamental part of the theory of conic sections is attributed to Euclid. The examination of the Conics to shed light on Elem. II can also be justified by the fact that the term "geometric algebra" originates in Zeuthen's study of Apollonius. Throughout my examination, geometric intuition in the Conics will be emphasized. In the second chapter, I examine Elem. II itself. Overall, this study is a refutation of the common interpretation of Elem. II, and an attempt to advance Mueller's study a step further.