The Calculus
A Genetic Approach
201 pages

5 1/2 x 8 1/2

© 1963, 2007
 Contents
 Review Quotes
Table of Contents
Contents
FOREWORD
PREFACE TO THE GERMAN EDITION
PREFACE TO THE ENGLISH EDITION
I . THE NATURE OF THE INFINITE PROCESS
1 . The Beginnings of Greek Speculation on Infinitesimals
2 . The Greek Theory of Proportions
3 . The Exhaustion Method of the Greeks
4 . The Modern Number Concept
5 . Archimedes’ Measurements of the Circle and the Sine Tables
6 . The Infinite Geometric Series
7 . Continuous Compound Interest
8 . Periodic Decimal Fractions
9 . Convergence and Limit
10 . Infinite Series
II . THE DEFINITE INTEGRAL
11 . The Quadrature of the Parabola by Archimedes
12 . Continuation after 1. 880 Years
13 . Area and Definite Integral
14 . Nonrigorous Infinitesimal Methods
15 . The Concept of the Definite Integral
16 . Some Theorems on Definite Integrals
17 . Questions of Principle
III . DIFFERENTIAL AND INTEGRAL CALCULUS
18 . Tangent Problems
19 . Inverse Tangent Problems
20 . Maxima and Minima
21 . Velocity
22 . Napier
23 . The Fundamental Theorem
24 . The Product Rule
25 . Integration by Parts
26 . Functions of Functions
27. Transformation of Integrals
28 . The Inverse Function
29 . Trigonometric Functions
30 . Inverse Trigonometric Functions
31 . Functions of Several Functions
32 . Integration of Rational Functions
33 . Integration of Trigonometric Expressions
34 . Integration of Expressions Involving Radicals
35 . Limitations of Explicit Integration
IV . APPLICATIONS TO PROBLEMS OF MOTION
36 . Velocity and Acceleration
37 . The Pendulum
38 . Coordinate Transformations
39 . Elastic Vibrations
40 . Kepler’s First Two Laws
41 . Derivation of Kepler’s First Two Laws from Newton’s Law
42 . Kepler’s Third Law
EXERCISES
BIBLIOGRAPHY
Works on the History of Mathematics
Special Works on the History of the Infinitesimal Calculus
BIBLIOGRAPHICAL NOTES
INDEX
PREFACE TO THE GERMAN EDITION
PREFACE TO THE ENGLISH EDITION
I . THE NATURE OF THE INFINITE PROCESS
1 . The Beginnings of Greek Speculation on Infinitesimals
2 . The Greek Theory of Proportions
3 . The Exhaustion Method of the Greeks
4 . The Modern Number Concept
5 . Archimedes’ Measurements of the Circle and the Sine Tables
6 . The Infinite Geometric Series
7 . Continuous Compound Interest
8 . Periodic Decimal Fractions
9 . Convergence and Limit
10 . Infinite Series
II . THE DEFINITE INTEGRAL
11 . The Quadrature of the Parabola by Archimedes
12 . Continuation after 1. 880 Years
13 . Area and Definite Integral
14 . Nonrigorous Infinitesimal Methods
15 . The Concept of the Definite Integral
16 . Some Theorems on Definite Integrals
17 . Questions of Principle
III . DIFFERENTIAL AND INTEGRAL CALCULUS
18 . Tangent Problems
19 . Inverse Tangent Problems
20 . Maxima and Minima
21 . Velocity
22 . Napier
23 . The Fundamental Theorem
24 . The Product Rule
25 . Integration by Parts
26 . Functions of Functions
27. Transformation of Integrals
28 . The Inverse Function
29 . Trigonometric Functions
30 . Inverse Trigonometric Functions
31 . Functions of Several Functions
32 . Integration of Rational Functions
33 . Integration of Trigonometric Expressions
34 . Integration of Expressions Involving Radicals
35 . Limitations of Explicit Integration
IV . APPLICATIONS TO PROBLEMS OF MOTION
36 . Velocity and Acceleration
37 . The Pendulum
38 . Coordinate Transformations
39 . Elastic Vibrations
40 . Kepler’s First Two Laws
41 . Derivation of Kepler’s First Two Laws from Newton’s Law
42 . Kepler’s Third Law
EXERCISES
BIBLIOGRAPHY
Works on the History of Mathematics
Special Works on the History of the Infinitesimal Calculus
BIBLIOGRAPHICAL NOTES
INDEX
Review Quotes
Scitech Book News
"Appropriate for students who have completed basic or high school calculus but have not yet stepped up to the rigors of advanced calculus. Here those students will find motivation for understanding techniques in response to the original problems that gave rise to them."
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