The Calculus
A Genetic Approach
The Calculus
A Genetic Approach
When first published posthumously in 1963, this bookpresented a radically different approach to the teaching of calculus. In sharp contrast to the methods of his time, Otto Toeplitz did not teach calculus as a static system of techniques and facts to be memorized. Instead, he drew on his knowledge of the history of mathematics and presented calculus as an organic evolution of ideas beginning with the discoveries of Greek scholars, such as Archimedes, Pythagoras, and Euclid, and developing through the centuries in the work of Kepler, Galileo, Fermat, Newton, and Leibniz. Through this unique approach, Toeplitz summarized and elucidated the major mathematical advances that contributed to modern calculus.
Reissued for the first time since 1981 and updated with a new foreword, this classic text in the field of mathematics is experiencing a resurgence of interest among students and educators of calculus today.
Reviews
Table of Contents
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 . Non-rigorous 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