University of Chicago Press, 2007 Paper: 978-0-226-80668-6 | eISBN: 978-0-226-80669-3 Library of Congress Classification QA303.T6413 2007 Dewey Decimal Classification 515

ABOUT THIS BOOK | AUTHOR BIOGRAPHY | REVIEWS | TOC | REQUEST ACCESSIBLE FILE

ABOUT THIS BOOK

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.

AUTHOR BIOGRAPHY

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.

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."

— Scitech Book News

TABLE OF CONTENTS

I. The Nature of the Infinite Process 1
1.The Beginnings of Greek Speculation on Infinitesimals 1
2.The Greek Theory of Proportions 9
3.The Exhaustion Method of the Greeks 11
4.The Modern Number Concept 14
5.Archimedes' Measurements of the Circle and the Sine Tables 18
6.The Infinite Geometric Series 22
7.Continuous Compound Interest 24
8.Periodic Decimal Fractions 28
9.Convergence and Limit 33
10.Infinite Series 39
II. The Definite Integral 43
11.The Quadrature of the Parabola by Archimedes 43
12.Continuation after 1,880 Years 52
13.Area and Definite Integral 58
14.Non-rigorous Infinitesimal Methods 60
15.The Concept of the Definite Integral 62
16.Some Theorems on Definite Integrals 69
17.Questions of Principle 70
III. Differential and Integral Calculus 77
18.Tangent Problems 77
19.Inverse Tangent Problems 80
20.Maxima and Minima 80
21.Velocity 84
22.Napier 86
23.The Fundamental Theorem 95
24.The Product Rule 99
25.Integration by Parts 103
26.Functions of Functions 104
27.Transformation of Integrals 105
28.The Inverse Function 106
29.Trigonometric Functions 113
30.Inverse Trigonometric Functions 116
31.Functions of Several Functions 119
32.Integration of Rational Functions 121
33.Integration of Trigonometric Expressions 123
34.Integration of Expressions Involving Radicals 124
35.Limitations of Explicit Integration 126
IV. Applications to Problems of Motion 133
36.Velocity and Acceleration 133
37.The Pendulum 138
38.Coordinate Transformations 144
39.Elastic Vibrations 147
40.Kepler's First Two Laws 150
41.Derivation of Kepler's First Two Laws from Newton's Law 156
42.Kepler's Third Law 161
Exercises 173
Bibliography 183
Works on the History of Mathematics 183
Special Works on the History of the Infinitesimal Calculus 183
Biographical Notes 185
Index 191

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University of Chicago Press, 2007 Paper: 978-0-226-80668-6 eISBN: 978-0-226-80669-3

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.

AUTHOR BIOGRAPHY

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.

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."

— Scitech Book News

TABLE OF CONTENTS

I. The Nature of the Infinite Process 1
1.The Beginnings of Greek Speculation on Infinitesimals 1
2.The Greek Theory of Proportions 9
3.The Exhaustion Method of the Greeks 11
4.The Modern Number Concept 14
5.Archimedes' Measurements of the Circle and the Sine Tables 18
6.The Infinite Geometric Series 22
7.Continuous Compound Interest 24
8.Periodic Decimal Fractions 28
9.Convergence and Limit 33
10.Infinite Series 39
II. The Definite Integral 43
11.The Quadrature of the Parabola by Archimedes 43
12.Continuation after 1,880 Years 52
13.Area and Definite Integral 58
14.Non-rigorous Infinitesimal Methods 60
15.The Concept of the Definite Integral 62
16.Some Theorems on Definite Integrals 69
17.Questions of Principle 70
III. Differential and Integral Calculus 77
18.Tangent Problems 77
19.Inverse Tangent Problems 80
20.Maxima and Minima 80
21.Velocity 84
22.Napier 86
23.The Fundamental Theorem 95
24.The Product Rule 99
25.Integration by Parts 103
26.Functions of Functions 104
27.Transformation of Integrals 105
28.The Inverse Function 106
29.Trigonometric Functions 113
30.Inverse Trigonometric Functions 116
31.Functions of Several Functions 119
32.Integration of Rational Functions 121
33.Integration of Trigonometric Expressions 123
34.Integration of Expressions Involving Radicals 124
35.Limitations of Explicit Integration 126
IV. Applications to Problems of Motion 133
36.Velocity and Acceleration 133
37.The Pendulum 138
38.Coordinate Transformations 144
39.Elastic Vibrations 147
40.Kepler's First Two Laws 150
41.Derivation of Kepler's First Two Laws from Newton's Law 156
42.Kepler's Third Law 161
Exercises 173
Bibliography 183
Works on the History of Mathematics 183
Special Works on the History of the Infinitesimal Calculus 183
Biographical Notes 185
Index 191

REQUEST ACCESSIBLE FILE

If you are a student who has a disability that prevents you
from using this book in printed form, BiblioVault may be able to supply you
with an electronic file for alternative access.

Please have the disability coordinator at your school fill out this form.

It can take 2-3 weeks for requests to be filled.

ABOUT THIS BOOK | AUTHOR BIOGRAPHY | REVIEWS | TOC | REQUEST ACCESSIBLE FILE