Our point of departure is, as everybody knows, Descartes' innovation of reducing geometry to algebra by referring all curves to a standard rectangular coordinate system. Of course, what everybody knows is not necessarily true, and we discover on the one hand that rectangular coordinate systems had been in use in mathematics (not to mention cartography and astronomy) since time immemorial, or in any case since Apollonios ( they may yet turn up in Old Babylonia ( and, on the other hand, that Descartes' did not, in fact, do precisely this. All of which raises the question: what exactly did Descartes do, and why were his contemporaries thrilled by it?
One thing he did, apparently, was to understand Vihte and to appreciate the power of his system ; a rare achievement in itself, though arrived at independently by Fermat. Vihte aimed explicitly at providing a mechanism for reducing geometry to algebra, and for carrying out the necessary algebra : see Orlansky's article for more on this. Descartes carried the project through to the point that his contemporaries were able to see the value of it. It's still not clear to me how exactly they were able to see the value of it, from Descartes' account. Since neither Vihte nor Fermat after him could find the occasion to publish their ideas, the honor of promulgating this idea in a form of some general use is certainly Descartes'.
It is fortunate that you found that the issues that concern us had been ably dealt with by Scott. I have still not had the leisure to read through that volume, but I intend to shortly. What I have read so far puts the issues clearly and is based on a cl ose study of the actual texts, rather than preconceptions as to what must be in them.
All in all, a fortunate topic, and a good way to bring a seminar on the history of algebra to a close. Note: "Descartes seems to say in his fourth rule that the general case is the one of great importance, not the specific problem." My reading of the fourth rule is rather different, and, I'm afraid far less satisfactory. I was particularly impressed by his abuse of the rule in his explanation of meteors. I read it as a catch-all and an invitation to laziness, meaning that when one has run out of plausible solutions one settles for one's best idea.
The references are uneven. The problem one has with Descartes is that often those who write about his mathematics are really more interested in his philosophy, and they seem constitutionally unsuited to dealing with it with any precision. However there a re two real gems: the Scott, which suits our purposes perfectly, and the primary source material in the translation of Smith and Latham, which also gives the original French in facsimile, allowing us among other things to examine the very peculiar conventions of the algebraic notation first hand. I'll be looking at them again.
The Gaukröger looks interesting, or in any case charming, but didn't inspire the same degree of confidence.
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