Exceeding and Falling Short: Elliptical and Hyperbolical Application of Areas

Abstract

Argument

My purpose in this paper is to venture an explanation for how the peculiar Greek idiom “application of area” came into being, and why Greek mathematicians preferred this idiom to one which seems more straightforward and is undoubtedly older. To those conversant with the Greek theory of , especially Elements VI.28–29, there is little new in my exposition, and they may want to skip sections 1 and 2. I present them nevertheless, partly to initiate newcomers, partly in order to show that Elements II.5 works as a maximum condition (analogous to VI.27) besides being, like II.6, a useful lemma in the solution of the problem. My main point is the shifting of focus from one rectangle to another in problems 3a and 4a, which may have given rise to the esoteric – and more general – wording of problems 3b and 4b.