The Calculus: A Genetic Approach

by Otto Toeplitz, David Bressoud

Overview

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.

Product Details

ISBN-13:	9780226806686
Publisher:	University of Chicago Press
Publication date:	06/01/2007
Edition description:	Reprint
Pages:	201
Product dimensions:	5.50(w) x 8.50(h) x 0.70(d)

About the Author

Otto Toeplitz (1881-1940) was a leading scholar in linear algebra and functional analysis. He was the author of many scholarly articles and coauthor of The Enjoyment of Mathematics.

Table of 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 . 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

Editorial Reviews

"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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