Introduction

By the beginning of the seventeenth century it had become possible to represent a wide variety of arithmetic concepts and relationships in the newly evolved language of symbolic algebra [19]. Geometry, however, held a preeminent position as an older and far more trusted form of mathematics. Throughout the scientific revolution geometry continued to be thought of as the primary and most reliable form of mathematics, but a continuing series of investigations took place that examined the extent to which algebra and geometry might be compatible. These experiments in compatibility were quite opposite from most of the ancient classics. Euclid, for example, describes in Books 8–10 of the Elements a number of important theorems of number theory cloaked awkwardly in a geometrical representation [16]. The experiments of the seventeenth century, conversely, probed the possibilities of representing geometrical concepts and constructions in the language of symbolic algebra. To what extent could it be done? Would contradictions emerge if one moved freely back and forth between geometric and algebraic representations?

Questions of appropriate forms of representation dominated the intellectual activities of seventeenth century Europe, not just in science and mathematcs but perhaps even more pervasively in religious, political, legal, and philosophical discussions [13, 24, 25]. Seen in the context of this social history it is not surprising that mathematicians like René Descartes and G. W. von Leibniz would have seen their new symbolic mathematical representations in the context of their extensive philosophical works.