**Spectrum**

Volume: 64;
2010;
287 pp;
Hardcover

**Print ISBN: 978-0-88385-572-0
Product Code: SPEC/64**

List Price: $64.00

AMS Member Price: $48.00

MAA Member Price: $48.00

**Electronic ISBN: 978-1-61444-506-7
Product Code: SPEC/64.E**

List Price: $64.00

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# A Historian Looks Back: The Calculus as Algebra and Selected Writings

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*Judith V. Grabiner*

MAA Press: An Imprint of the American Mathematical Society

Recipient of the Mathematical Association of America's
Beckenbach Book Prize in 2014!

Judith Grabiner, the author of A Historian
Looks Back, has long been interested in investigating what
mathematicians actually do, and how mathematics actually has
developed. She addresses the results of her investigations not
principally to other historians, but to mathematicians and teachers of
mathematics. This book brings together much of what she has had to say
to this audience. The centerpiece of the book is The Calculus as
Algebra: J.-L. Lagrange, 1736–1813. The book describes the
achievements, setbacks, and influence of Lagrange's pioneering attempt
to reduce the calculus to algebra. Nine additional articles round out
the book describing the history of the derivative; the origin of
delta-epsilon proofs; Descartes and problem solving; the contrast
between the calculus of Newton and Maclaurin, and that of Lagrange;
Maclaurin's way of doing mathematics and science and his surprisingly
important influence; some widely held myths about the history of
mathematics; Lagrange's attempt to prove Euclid's parallel postulate;
and the central role that mathematics has played throughout the
history of western civilization. The development of mathematics cannot
be programmed or predicted. Still, seeing how ideas have been formed
over time and what the difficulties were can help teachers find new
ways to explain mathematics. Appreciating its cultural background can
humanize mathematics for students. And famous mathematicians' struggles
and successes should interest—and perhaps
inspire—researchers. Readers will see not only what the
mathematical past was like, but also how important parts of the
mathematical present came to be.

# Table of Contents

## A Historian Looks Back: The Calculus as Algebra and Selected Writings

- Cover cover11
- Copyright page ii3
- Title page iii4
- Contents ix10
- Introduction xi12
- I The Calculus as Algebra: J.-L. Lagrange, 1736–1813 118
- Preface to the Garland Edition 320
- Acknowledgements 724
- Introduction 926
- 1 The Development of Lagrange’s Ideas on the Calculus: 1754–1797 1734
- 2 The Algebraic Background of the Theory of Analytic Functions 3754
- Introduction 3754
- The Attractiveness of Algebra: Certainty 3855
- The Attractiveness of Algebra: Methods 4057
- The Algebraic Character of the Taylor Series 4663
- Origins of the “Proof ” that Every Function has a Taylor Series 4865
- The Algebraic Background of the Lagrange Remainder: Approximations 5168
- The Algebraic Background of the Lagrange Remainder: Error-EstimatesBefore Lagrange 5269
- The Algebraic Background of the Lagrange Remainder: Lagrange and Bounds on Error 5774

- 3 The Contents of the Fonctions Analytiques 6380
- Introduction 6380
- Lagrange’s Critique of Earlier Methods 6481
- The Results of the Calculus: By Means of Formal Power Series 6582
- The Results of the Calculus: Those Needed to Derive the Remainder Term of the Taylor Series 6784
- Results of the Calculus: Derivation of the Remainder Term 6986
- Results of the Calculus: Application of the Remainder Term 7491
- Refinements of Lagrange’s Ideas: 1799–1813 7895
- Impact of the Fonctions Analytiques 7996
- Conclusion 8097

- 4 From Proof-Technique to Definition: The Pre-History of Delta-Epsilon Methods 8198
- Conclusion 101118
- Appendix 103120
- Bibliography 105122
- Analytical Bibliography: 1966 105122
- Bibliography: 1966–Present 119136

- II Selected Writings 125142
- 1 The Mathematician, the Historian, and the History of Mathematics 127144
- 2 Who Gave You the Epsilon? Cauchy and the Origins of Rigorous Calculus 135152
- 3 The Changing Concept of Change: The Derivative from Fermat to Weierstrass 147164
- The Seventeenth-Century Background 148165
- Finding Maxima, Minima, and Tangents 149166
- Tangents, Areas, and Rates of Change 152169
- Differential Equations, Taylor Series, and Functions 154171
- Lagrange and the Derivative as a Function 156173
- Definitions, Rigor, and Proofs 157174
- Historical Development Versus Textbook Exposition 159176
- References 160177

- 4 The Centrality of Mathematics in the History of Western Thought 163180
- 5 Descartes and Problem-Solving 175192
- 6 The Calculus as Algebra, the Calculus as Geometry: Lagrange, Maclaurin, and their Legacy 191208
- 7 Was Newton’s Calculus a Dead End? The Continental Influence of Maclaurin’s Treatise of Fluxions 209226
- 1. Introduction 209226
- 2. The Standard Picture 210227
- 3. The Nature of Maclaurin’s Treatise of Fluxions 211228
- 4. The Social Context: The Scottish Enlightenment 211228
- 5. Maclaurin’s Continental Reputation 213230
- 6. Maclaurin’s Mathematics and Its Importance 214231
- 7. Other Examples of Maclaurin’s Mathematical Influence 219236
- 8. Why a Treatise of Fluxions? 220237
- 9. Why the Traditional View? 221238
- 10. Some Final Reflections 223240
- References 224241

- 8 Newton, Maclaurin, and the Authority of Mathematics 229246
- 1. Introduction: Maclaurin, the Scottish Enlightenment, and the “Newtonian Style” 229246
- 2. What is the “Newtonian Style”? 230247
- 3. Maclaurin’s First Use of the Newtonian Style 231248
- 4. Religion, Authority, and Mathematics for Newton and Maclaurin 232249
- 5. Maclaurin’s Mature Use of the Newtonian Style 233250
- 6. Religious Authority Revisited 238255
- 7. Conclusion 239256
- References 240257

- 9 Why Should Historical Truth Matter to Mathematicians? Dispelling Myths while Promoting Maths 243260
- A. Myth: The social history of mathematics is easy; just determine what nation or group your mathematician comes from and generalize. 243260
- B. Second Myth: All Modern Mathematics Comes from Men, Mostly White Christian Men in the Graeco-European Tradition 244261
- C. Third Myth: There Was No Real Mathematics in the European Middle Ages. After the Decline of Greek Mathematics, Nothing Much Happened Mathematically in Europe Until the Renaissance 247264
- D. Fourth Myth: Newton Invented the Calculus Just to Do His Mathematical Physics 249266
- E. Fifth Myth: Colin Maclaurin, Because of His Old-Fashioned Geometrical Approach to the Calculus, Halted Mathematical Progress in 18th-Century Britain 250267
- F. Sixth Myth: Lagrange was a Formalist. He Tried to Rigorize the Calculus, But Failed Because of His Unreflective Reliance on Formal Power Series 251268
- G. Last Myth, Held by a Number of Past and Present Mathematicians: The Mathematical Approach Can be Applied to Solve Almost Any Major Question 254271
- H. A conclusion in four parts 255272

- 10 Why Did Lagrange “Prove” the Parallel Postulate? 257274
- 1. Introduction 257274
- 2. The Contents of Lagrange’s 1806 Paper 258275
- 3. Why Did He Attack the Problem This Way? 260277
- 4. The Crucial Argument: Newtonian Physics 264281
- 5. The Argument from Eighteenth-Century Mathematics and Science 267284
- 6. Why Did It Matter so Much? 269286
- 7. Conclusion 271288
- References 272289

- Index 275292
- About the Author 287304