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Tables of Contents for The Historical Development of the Calculus
Chapter/Section Title
Page #
Page Count
Area, Number, and Limit Concepts in Antiquity
1
28
Babylonian and Egyptian Geometry
1
4
Early Greek Geometry
5
5
Incommensurable Magnitudes and Geometric Algebra
10
2
Eudoxus and Geometric Proportions
12
4
Area and the Method of Exhaustion
16
3
Volumes of Cones and Pyramids
19
5
Volumes of Spheres
24
4
References
28
1
Archimedes
29
48
Introduction
29
2
The Measurement of a Circle
31
4
The Quadrature of the Parabola
35
5
The Area of an Ellipse
40
2
The Volume and Surface Area of a Sphere
42
12
The Method of Compression
54
1
The Archimedean Spiral
54
8
Solids of Revolution
62
6
The Method of Discovery
68
6
Archimedes and Calculus?
74
1
References
75
2
Twilight, Darkness, and Dawn
77
21
Introduction
77
1
The Decline of Greek Mathematics
78
2
Mathematics in the Dark Ages
80
1
The Arab Connection
81
5
Medieval Speculations on Motion and Variability
86
5
Medieval Infinite Series Summations
91
2
The Analytic Art of Viete
93
2
The Analytic Geometry of Descartes and Fermat
95
2
References
97
1
Early Indivisibles and Infinitesimal Techniques
98
24
Introduction
98
1
Johann Kepler (1571-1630)
99
5
Cavalieri's Indivisibles
104
5
Arithmetical Quadratures
109
4
The Integration of Fractional Powers
113
5
The First Rectification of a Curve
118
2
Summary
120
1
References
121
1
Early Tangent Constructions
122
20
Introduction
122
1
Fermat's Pseudo-equality Methods
122
3
Descartes' Circle Method
125
2
The Rules of Hudde and Sluse
127
5
Infinitesimal Tangent Methods
132
2
Composition of Instantaneous Motions
134
4
The Relationship Between Quadratures and Tangents
138
3
References
141
1
Napier's Wonderful Logarithms
142
24
John Napier (1550-1617)
142
1
The Original Motivation
143
5
Napier's Curious Definition
148
3
Arithmetic and Geometric Progressions
151
2
The Introduction of Common Logarithms
153
1
Logarithms and Hyperbolic Areas
154
4
Newton's Logarithmic Computations
158
3
Mercator's Series for the Logarithm
161
3
References
164
2
The Arithmetic of the Infinite
166
23
Introduction
166
4
Wallis' Interpolation Scheme and Infinite Product
170
6
Quadrature of the Cissoid
176
2
The Discovery of the Binomial Series
178
9
References
187
2
The Calculus According to Newton
189
42
The Discovery of the Calculus
189
1
Isaac Newton (1642-1727)
190
1
The Introduction of Fluxions
191
3
The Fundamental Theorem of Calculus
194
2
The Chain Rule and Integration by Substitution
196
4
Applications of Infinite Series
200
1
Newton's Method
201
3
The Reversion of Series
204
1
Discovery of the Sine and Cosine Series
205
4
Methods of Series and Fluxions
209
1
Applications of Integration by Substitution
210
2
Newton's Integral Tables
212
5
Arclength Computations
217
5
The Newton-Leibniz Correspondence
222
2
The Calculus and the Principia Mathematica
224
2
Newton's Final Work on the Calculus
226
4
References
230
1
The Calculus According to Leibniz
231
37
Gottfried Wilhelm Leibniz (1646-1716)
231
3
The Beginning---Sums and Differences
234
5
The Characteristic Triangle
239
6
Transmutation and the Arithmetical Quadrature of the Circle
245
7
The Invention of the Analytical Calculus
252
6
The First Publication of the Calculus
258
2
Higher-Order Differentials
260
4
The Meaning of Leibniz' Infinitesimals
264
1
Leibniz and Newton
265
2
References
267
1
The Age of Euler
268
33
Leonhard Euler (1707-1783)
268
2
The Concept of a Function
270
2
Euler's Exponential and Logarithmic Functions
272
3
Euler's Trigonometric Functions and Expansions
275
2
Differentials of Elementary Functions a la Euler
277
4
Interpolation and Numerical Integration
281
6
Taylor's Series
287
5
Fundamental Concepts in the Eighteenth Century
292
7
References
299
2
The Calculus According to Cauchy, Riemann, and Weierstrass
301
34
Functions and Continuity at the Turn of the Century
301
3
Fourier and Discontinuity
304
4
Bolzano, Cauchy, and Continuity
308
4
Cauchy's Differential Calculus
312
5
The Cauchy Integral
317
5
The Riemann Integral and Its Reformulations
322
7
The Arithmetization of Analysis
329
4
References
333
2
Postscript: The Twentieth Century
335
12
The Lebesgue Integral and the Fundamental Theorem of Calculus
335
6
Non-standard Analysis---The Vindication of Euler?
341
5
References
346
1
Index
347