Redefining Geometrical Exactness: Descartes’ Transformation of the Early Modern Concept of ConstructionSpringer Science & Business Media, 6 dec 2012 - 472 pagina's 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. |
Inhoudsopgave
1 | |
Construction and the interpretation of exactness in Descartes | 16 |
7 | 53 |
Early modern methods of analysis | 99 |
Arithmetic geometry algebra and analysis | 119 |
Using numbers in geometry | 135 |
1 | 144 |
Clavius | 159 |
25 | 289 |
Algebraic operations in geometry | 293 |
The use of algebra in solving plane | 302 |
27 | 304 |
Descartes solution of Pappus problem | 313 |
34 | 330 |
Curves and the demarcation of geometry in the Geometry | 335 |
Simplicity and the classification of curves | 355 |
Viète | 167 |
Kepler | 183 |
Molther | 195 |
Fermat | 205 |
Geometrical problem solving the state of the art c 1635 | 211 |
Introduction to Part II | 225 |
Descartes general construction of solid problems c 1625 | 255 |
11 | 257 |
Problem solving and construction | 261 |
Descartes first studies of Pappus problem early 1632 | 271 |
23 | 272 |
The Geometry introduction and survey 285 | 284 |
Overige edities - Alles bekijken
Redefining Geometrical Exactness: Descartes' Transformation of the Early ... H. J. M. Bos Gedeeltelijke weergave - 2001 |
Redefining Geometrical Exactness: Descartes' Transformation of the Early ... H. J. M. Bos Geen voorbeeld beschikbaar - 2001 |
Veelvoorkomende woorden en zinsdelen
acceptable algebraic analysis algebraic curves algebraic operations analytical angle angular sections Apollonius Archimedes arguments arithmetical Cartesian parabola Ceulen Chapter classical Clavius coefficients conchoid conic sections corresponding cube cubic equations curve tracing degree demarcation Descartes 1637 discussed draw Elements Euclid Euclidean Eutocius Fermat Figure fourth-degree equations geometrical construction geometrical exactness geometrical magnitudes geometrical problem solving Ghetaldi given in position given lines heptagon higher-order hyperbola indeterminate instruments interpretation of exactness irrational Kepler latus latus rectum lignes line segments line-like linea lines and circles loci locus mathematical mathematicians mean proportionals mesolabum method Molther motion neusis construction Nicomedes non-plane Note numbers Pappus parabola plane pointwise construction postulate procedure proof quadratic quadratic equations quadratrix question ratio rectangle reduced Regiomontanus result Roomen root extraction Snellius solid problems solution spiral standard construction Stevin straight lines struction tion tradition of geometrical translation treatise triangle trisection unknown Viète Viète's Vietean