Ebook: Redefining Geometrical Exactness: Descartes’ Transformation of the Early Modern Concept of Construction

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.

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Content:
Front Matter....Pages i-xix
Front Matter....Pages 1-1
General introduction....Pages 3-22
The legitimation of geometrical procedures before 1590....Pages 23-36
1588: Pappus’ “Collection”....Pages 37-57
The early modern tradition of geometrical problem solving; survey and examples....Pages 59-94
Early modern methods of analysis....Pages 95-117
Arithmetic, geometry, algebra, and analysis....Pages 119-134
Using numbers in geometry — Regiomontanus and Stevin....Pages 135-143
Using algebra — Vi?te’s analysis....Pages 145-158
Clavius....Pages 159-166
Vi?te....Pages 167-181
Kepler....Pages 183-194
Molther....Pages 195-204
Fermat....Pages 205-210
Geometrical problem solving — the state of the art c. 1635....Pages 211-221
Front Matter....Pages 223-223
Introduction to Part II....Pages 225-229
Construction and the interpretation of exactness in Descartes’ studies of c. 1619....Pages 231-253
Descartes’ general construction of solid problems c. 1625....Pages 255-260
Problem solving and construction in the “Rules for the direction of the mind” (c. 1628)....Pages 261-270
Descartes’ first studies of Pappus’ problem (early 1632)....Pages 271-283
The Geometry, introduction and survey....Pages 285-291
Front Matter....Pages 223-223
Algebraic operations in geometry....Pages 293-301
The use of algebra in solving plane and indeterminate problems....Pages 303-311
Descartes’ solution of Pappus’ problem....Pages 313-334
Simplicity and the classification of curves....Pages 335-354
The canon of geometrical construction....Pages 355-361
Conclusion of Part II....Pages 363-381
Epilogue....Pages 383-397
Back Matter....Pages 399-413
....Pages 415-428