1 Introduction

From Pythagoras to the mid-nineteenth century, the fundamental problem of geometry was to relate numbers to geometry. It played a key role in the creation of field theory (via the classic construction problems), and, quite differently, in the creation of linear algebra. To resolve the problem, it was necessary to have the modern concept of real number; this was essentially achieved by Simon Stevin, around 1600, andwas thoroughly assimilated into mathematics in the following two centuries. The integration of real numbers into geometry began with Descartes and Fermat in the 1630s, and achieved an interim success at the end of the eighteenth century with the introduction into the mathematics curriculum of the traditional course in analytic geometry. Fromthe point of view of analysis, with its focus on functions, this was entirely satisfactory; but from the point of view of geometry, it was not: the method of attaching numbers to geometric entities is too clumsy, the choice of origin and axes irrelevant and (in view of Euclid) unnecessary.

Leibniz, in 1679, had mused upon the possibility of a universal algebra, an algebra with which one would deal directly and simply with geometric entities. The possibility is already suggested by a perusal of Euclid.