In a way, nothing could be simpler: this Dover paperback contains little more than the text of Descartes’s famous La Géométrie, presented both in the original French (a facsimile of the first printing from 1637) and in English translation. Apart from that, there is a one-page introduction and a one-page (inadequate!) index. The title page says that the text was “translated from the French and Latin,” which the preface makes a bit clearer by saying that one translator worked from the French text and one from the Latin text, in order to adequately convey “the meaning which Descartes had in mind.” Originally published in 1925 by Open Court, this edition has been kept in print by Dover since 1954.
Given the historical importance of the book, this one is a no-brainer: anyone interested in the history of mathematics will want to take a look. Nevertheless, there is a caveat: one can argue that La Géométrie was never intended to stand alone, and historically it was not presented that way.
A reader might be suspicious after looking at the first facsimile page, which is numbered 297. Surely something is missing! In fact, if our reader ever gets to the end and reads the colophon on page 243 of the Dover edition, it will become clear that La Géométrie is in fact part of another famous book: Discours de la Méthode, “Discourse on the Method,” in which Descartes presents his approach to achieving certainty in philosophical reflection. The book contains an introduction describing that method, followed by essays on dioptrics, meteors, and geometry. One of the English translators of the Method, Desmond Clarke, thinks of the essay on method as the preface to the main book, observing that
Since then, [the Discourse on Method] has often been detached from its original context — so much so that some modern commentaries refer to the essays as appendices to the Discourse, as if the main contents of a book could be appendices to its preface. (Discourse on Method and Related Writings, Penguin Books, 2003)
Nevertheless, the Penguin edition includes only the “preface”, continuing the tradition of cutting up the book into little pieces, with philosophers reading only the overall description of the method, mathematicians reading only the geometry, and not many people reading the other parts:
Something similar happened to the Latin translation by Van Schooten: the original appendix appears there supplemented by other texts that explain and extend Descartes’s ideas. This was necessary because Descartes’s presentation is very dense, insufficiently detailed, too concerned to get to results and not concerned enough about making things clear to the reader. The original Latin edition of 1649 was still fairly small at 118 pages, but expansion and explanation were much needed. Over time, the Latin edition grew to almost 1000 pages. These expanded Latin editions were the ones that other mathematicians (for example, Isaac Newton) read and learned from. This is why in Landmark Writings in Western Mathematics the geometry is dated to 1649: that was the important edition for mathematicians.
From a historian’s perspective, then, what we have here is a starting point. For serious engagement with how geometry fit into the thought-world of René Descartes, one needs a full translation of the Method with the accompanying essays. (Such an edition in English does exist.) For full appraisal of the mathematical impact, one needs a full translation of the Latin Geometry. Unfortunately, there is no such translation, though one can easily find the full Latin text online.
As a starting point, however, this is excellent. The English translation is accurate and easy to read. The mathematics is definitely not easy to understand, but that too is a significant fact about this book. The translators offer lots of helpful footnotes.
Fernando Q. Gouvêa wishes his Latin were better.