# 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

**FOREWORD**

PREFACE TO THE GERMAN EDITION

PREFACE TO THE ENGLISH EDITION

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

BIBLIOGRAPHY

Works on the History of Mathematics

Special Works on the History of the Infinitesimal Calculus

**BIBLIOGRAPHICAL NOTES**

INDEX

INDEX

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