Viète, descartes, and the emergence of modern mathematics

Francois Viete (1540-1603) is often regarded as the first modern mathematician on the grounds that he was the first to develop the literal notation, that is, the use of two sorts of letters, one for the unknown and the other for the known parameters of a problem. The fact that he achieved neither a modern conception of quantity nor a modern understanding of curves, both of which are explicit in Descartes’ Geometry (1637), is to be explained on this view “by an incomplete symbolization rather than by any obstacle intrinsic in the system.” Descartes’ Geometry provides only a “clearer expression” of themes already sounded in Viete’s work, one that perfects Viete’s literal calculus and gives it “its modern form”; it merely continues the “‘new’ and ‘pure’ algebra which Viete first established as the ‘general analytic art’.” It can seem, furthermore, that this must be right, that had there been some obstacle intrinsic to Viete’s system that barred the way to a modern conception of quantity and a modern understanding of curves, then Descartes’ Geometry would have had to have taken a very different form than it did. As it was, Descartes had only to improve Viete’s symbolism, free himself of the last vestiges of the ancient view of geometrical and arithmetical objects, and apply the new symbolism to the study of curves in order to achieve what Viete did not but could have. But what, really, is the status of this “could have”; what would it actually have taken for Viete to achieve Descartes’ results in the Geometry? The interest of the question lies in its potential to better our understanding of the nature of (early) modern mathematics.