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Redefining Geometrical Exactness: Descartes' Transformation of the Early Modern Concept of Construction (Sources and Studies in the History of Mathematics and Physical Sciences)

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This is the theme of the book: Euclid built up the whole Elements with just ruler and compass, but, as the Greeks knew perfectly well, for many problems it is necessary to allow higher curves or curve-tracing devices, or other suspicious tricks. This is very bothersome; we don't want things to get out of hand. What methods should be allowed in true geometry? The thoughts on these things before Descartes (comprising the first half of the book) were inconclusive and not terribly interesting. One does not regret that the subject is extinct. In fact, Descartes wrote: "through the dark confusion of this science I have seen some kind of light, and I believe that by its help I can dispel darkness however dense" (p. 232). But we cannot afford the luxury of forgetting all about these matters. Many great mathematicians clearly regarded it as an important issue, and it shaped their work. In particular--and this is the main point of the book--it is absolutely essential for a proper understanding of Descartes' Geometrie, which is in a sense a culmination of this whole tradition (and which of course many of us have tried to read for very different reasons). The high-point in this argument is chapter 19, where Bos argues that the catalyst for Descartes' breakthrough was his study of "Pappus' problem", where the harmony between geometric constructions and algebraic equations unfolds perfectly. This also beautifully closes the circle, since Pappus' Collection was the main ancient source for the whole discussion on geometrical exactness in the first place.