Shifting the foundations: Descartes's transformation of ancient geometry

The aim of this paper is to analyse how the bases of Descartes's geometry differed from those of ancient geometry. Particular attention is paid to modes of specifying curves of which two types are distinguished — “Specification by genesis” and “Specification by property”. For both Descartes and most of Greek geometry the former was fundamental, but Descartes diverged from ancient pure geometry by according an essential place to the imagination of mechanical instruments. As regards specification by property, Descartes's interpretation of the multiplication of (segments of) straight lines as giving rise to a straight line (segment), together with newer methods of articifical symbolism, led to more concise and suggestive modes of representation. Descartes's account of ancient procedures is historically very misleading, but it allowed him to introduce his own ideas more naturally.