The Historical Development of the Calculus

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Springer Science & Business Media, 6 dic 2012 - 368 pagine
The calculus has served for three centuries as the principal quantitative language of Western science. In the course of its genesis and evolution some of the most fundamental problems of mathematics were first con fronted and, through the persistent labors of successive generations, finally resolved. Therefore, the historical development of the calculus holds a special interest for anyone who appreciates the value of a historical perspective in teaching, learning, and enjoying mathematics and its ap plications. My goal in writing this book was to present an account of this development that is accessible, not solely to students of the history of mathematics, but to the wider mathematical community for which my exposition is more specifically intended, including those who study, teach, and use calculus. The scope of this account can be delineated partly by comparison with previous works in the same general area. M. E. Baron's The Origins of the Infinitesimal Calculus (1969) provides an informative and reliable treat ment of the precalculus period up to, but not including (in any detail), the time of Newton and Leibniz, just when the interest and pace of the story begin to quicken and intensify. C. B. Boyer's well-known book (1949, 1959 reprint) met well the goals its author set for it, but it was more ap propriately titled in its original edition-The Concepts of the Calculus than in its reprinting.
 

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Indice

Area Number and Limit Concepts in Antiquity
1
Area and the Method of Exhaustion
16
Archimedes
29
The Volume and Surface Area of a Sphere
42
Solids of Revolution
62
References
75
The Arab Connection
81
The Analytic Geometry of Descartes and Fermat
95
The Reversion of Series
204
Applications of Integration by Substitution
210
Arclength Computations
217
The Calculus and the Principia Mathematica
224
References
230
The Characteristic Triangle
239
Transmutation and the Arithmetical Quadrature
245
The Invention of the Analytical Calculus
252

Arithmetical Quadratures
109
Early Tangent Constructions
122
The Relationship Between Quadratures and Tangents
138
Arithmetic and Geometric Progressions
151
References
164
The Calculus According to Newton
189
The Chain Rule and Integration by Substitution
196
The First Publication of the Calculus
258
The Meaning of Leibniz Infinitesimals
264
The Calculus According to Cauchy Riemann
301
The Twentieth Century
335
Index
347
Copyright

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