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

Henk J. M. Bos (Author)

Book Description

Series: Sources and Studies in the History of Mathematics and Physical Sciences | Publication Date: 2001

In his "Géométrie" of 1637 Descartes achieved a monumental innovation of mathematical techniques by introducing what is now called analytic geometry. Yet the key question of the book was foundational rather than technical: When are geometrical objects known with such clarity and distinctness as befits the exact science of geometry? Classically, the answer was sought in procedures of geometrical construction, in particular by ruler and compass, but the introduction of new algebraic techniques made these procedures insufficient. In this detailed study, spanning essentially the period from the first printed edition of Pappus' "Collection" (1588, in Latin translation) and Descartes' death in 1650, Bos explores the current ideas about construction and geometrical exactness, noting that by the time Descartes entered the field the incursion of algebraic techniques, combined with an increasing uncertainty about the proper means of geometrical problem solving, had produced a certain impasse. He then analyses how Descartes transformed geometry by a redefinition of exactness and by a demarcation of geometry's proper subject and procedures in such a way as to incorporate the use of algebraic methods without destroying the true nature of geometry. Although mathematicians later essentially discarded Descartes' methodological convictions, his influence was profound and pervasive. Bos' insistence on the foundational aspects of the "Géométrie" provides new insights both in the genesis of Descartes' masterpiece and in its significance for the development of the conceptions of mathematical exactness.

Editorial Reviews

Review

"This is a brilliant book and the most satisfactory account of Descartes' mathematical career that I have come across. Although the author does not claim to provide an exhaustive study of Descartes' geometry, he illuminates it in an original way by focusing on the conception and practice of construction. The book covers essentially the period from 1588, when Commandino's Latin translation of the surviving books of Pappus' Collections appeared, to 1650 when Descartes met his untimely death in Sweden. Bos discusses the opinions of five mathematicians who stated their position on construction explicitly, namely Clavius, Viète, Kepler, Molther and Fermat. He then sketches out the background to Descartes' program for restructuring the art of geometrical problem solving that led to the Geometry of 1637, a work that he analyses in detail. The concluding two chapters summarise the dynamics of Descartes' geometrical thinking and sketch the development that followed upon Descartes' achievement."--MATHEMATICAL REVIEWS

Product Details

• Hardcover: 470 pages
• Publisher: Springer; 1 edition (2001)
• Language: English
• ISBN-10: 0387950907
• ISBN-13: 978-0387950907
• Product Dimensions: 9.3 x 7.1 x 1.1 inches
• Shipping Weight: 2.2 pounds
• Average Customer Review: 5.0 out of 5 stars  See all reviews (1 customer review)
• Amazon Best Sellers Rank: #2,880,586 in Books (See Top 100 in Books)

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5 of 5 people found the following review helpful:

5.0 out of 5 stars Don't read Descartes without it, November 24, 2005

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Viktor Blasjo - See all my reviews
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This review is from: Redefining Geometrical Exactness: Descartes' Transformation of the Early Modern Concept of Construction (Sources and Studies in the History of Mathematics and Physical Sciences) (Hardcover)

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.