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AMS / MAA SPECTRUM VOL AMS / MAA SPECTRUM VOL 64 64 The Calculus as Algebra and Selected Writings A Historian Looks Back 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 A Historian Looks Back the results of her investigations not principally to other histori- The Calculus as Algebra and Selected Writings ans, 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 Alge- bra: J.-L. Lagrange, 1736–1813. The book describes the achievements, setbacks, and influence of Lagrange’s pioneering attempt Judith V. Grabiner 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 Judith V. Grabiner 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 civiliza- tion. 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. M A PRESS / MAA AMS

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A Historian Looks Back The Calculus as Algebra and Selected Writings Part I was formerly published as The Calculus as Algebra: J.-L. Lagrange, 1736–1813 by Garland Publishing, New York, 1990.

Library of Congress Catalog Card Number 2010927261

Print ISBN 978-0-88385-572-0 Electronic ISBN 978-1-61444-506-7

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10.1090/spec/064

A Historian Looks Back The Calculus as Algebra and Selected Writings

Judith V. Grabiner Pitzer College

Published and Distributed by The Mathematical Association of America P1: JZP MABK013-FM MABK013/Grabiner Trim Size: 7in× 10in July 28, 2010 19:19 SPECTRUM SERIES The Spectrum Series of the Mathematical Association of America was so named to reflect its purpose: to publish a broad range of books including biographies, accessible expositions of old or new mathematical ideas, reprints and revisions of excellent out-of-print books, popular works, and other monographs of high interest that will appeal to a broad range of readers, including students and teachers of mathematics, mathematical amateurs, and researchers.

Council on Publications and Communications Frank Farris, Chair Spectrum Editorial Board Gerald L. Alexanderson, Editor Robert F. Bradley Kenneth A. Ross Susanna S. Epp Sanford Segal Richard K. Guy Franklin F. Sheehan Michael A. Jones Robin Wilson Keith M. Kendig Robert S. Wolf Jeffrey L. Nunemacher

777 Mathematical Conversation Starters, by John de Pillis 99 Points of Intersection: Examples—Pictures—Proofs, by Hans Walser. Translated from the original German by Peter Hilton and Jean Pedersen. Aha Gotcha and Aha Insight, by Martin Gardner All the Math That’s Fit to Print, by Keith Devlin Calculus Gems: Brief Lives and Memorable Mathematics, by George F. Simmons Carl Friedrich Gauss: Titan of Science, by G. Waldo Dunnington, with additional material by Jeremy Gray and Fritz-Egbert Dohse The Changing Space of Geometry, edited by Chris Pritchard Circles: A Mathematical View, by Dan Pedoe Complex Numbers and Geometry, by Liang-shin Hahn Cryptology, by Albrecht Beutelspacher The Early Mathematics of Leonhard Euler, by C. Edward Sandifer The Edge of the Universe: Celebrating 10 Years of Math Horizons, edited by Deanna Haunsperger and Stephen Kennedy Euler and Modern Science, edited by N. N. Bogolyubov, G. K. Mikhailov, and A. P. Yushkevich. Translated from Russian by Robert Burns. Euler at 300: An Appreciation, edited by Robert E. Bradley, Lawrence A. D’Antonio, and C. Edward Sandifer Five Hundred Mathematical Challenges, Edward J. Barbeau, Murray S. Klamkin, and William O. J. Moser The Genius of Euler: Reflections on his Life and Work, edited by William Dunham The Golden Section, by Hans Walser. Translated from the original German by Peter Hilton, with the assistance of Jean Pedersen. The Harmony of the World: 75 Years of Mathematics Magazine, edited by Gerald L. Alexanderson with the assistance of Peter Ross A Historian Looks Back: The Calculus as Algebra and Selected Writings, by Judith Grabiner P1: JZP MABK013-FM MABK013/Grabiner Trim Size: 7in× 10in July 28, 2010 19:19

History of Mathematics: Highways and Byways, Amy Dahan-Dalmedico and Jeanne Peiffer. Translated by Sanford Segal. How Euler Did It, by C. Edward Sandifer Is Mathematics Inevitable? A Miscellany, edited by Underwood Dudley I Want to Be a Mathematician, by Paul R. Halmos Journey into Geometries, by Marta Sved JULIA: a life in mathematics, by Constance Reid The Lighter Side of Mathematics: Proceedings of the Eugene` Strens Memorial Conference on Recreational Mathematics & Its History, edited by Richard K. Guy and Robert E. Woodrow Lure of the Integers, by Joe Roberts Magic Numbers of the Professor, by Owen O’Shea and Underwood Dudley Magic Tricks, Card Shuffling, and Dynamic Computer Memories: The Mathematics of the Perfect Shuffle, by S. Brent Morris Martin Gardner’s Mathematical Games: The entire collection of his Scientific American columns The Math Chat Book, by Frank Morgan Mathematical Adventures for Students and Amateurs, edited by David Hayes and Tatiana Shubin. With the assistance of Gerald L. Alexanderson and Peter Ross. Mathematical Apocrypha,byStevenG.Krantz Mathematical Apocrypha Redux, by Steven G. Krantz Mathematical Carnival, by Martin Gardner Mathematical Circles Vol I: In Mathematical Circles Quadrants I, II, III, IV,byHowardW.Eves Mathematical Circles VolII: Mathematical Circles Revisited and Mathematical Circles Squared,byHoward W. Eves Mathematical Circles Vol III: Mathematical Circles Adieu and Return to Mathematical Circles,byHoward W. Eves Mathematical Circus, by Martin Gardner Mathematical Cranks, by Underwood Dudley Mathematical Evolutions, edited by Abe Shenitzer and John Stillwell Mathematical Fallacies, Flaws, and Flimflam, by Edward J. Barbeau Mathematical Magic Show, by Martin Gardner Mathematical Reminiscences,byHowardEves Mathematical Treks: From Surreal Numbers to Magic Circles, by Ivars Peterson Mathematics in Historical Context, by Jeff Suzuki Mathematics: Queen and Servant of Science,byE.T.Bell Memorabilia Mathematica, by Robert Edouard Moritz Musings of the Masters: An Anthology of Mathematical Reflections, edited by Raymond G. Ayoub New Mathematical Diversions, by Martin Gardner Non-Euclidean Geometry,byH.S.M.Coxeter Numerical Methods That Work, by Forman Acton Numerology or What Pythagoras Wrought, by Underwood Dudley Out of the Mouths of Mathematicians, by Rosemary Schmalz Penrose Tiles to Trapdoor Ciphers . . . and the Return of Dr. Matrix, by Martin Gardner Polyominoes, by George Martin Power Play, by Edward J. Barbeau P1: JZP MABK013-FM MABK013/Grabiner Trim Size: 7in× 10in July 28, 2010 19:19

Proof and Other Dilemmas: Mathematics and Philosophy, edited by Bonnie Gold and Roger Simons The Random Walks of George Polya´ , by Gerald L. Alexanderson Remarkable Mathematicians, from Euler to von Neumann,IoanJames The Search for E.T. Bell, also known as John Taine, by Constance Reid Shaping Space, edited by Marjorie Senechal and George Fleck Sherlock Holmes in Babylon and Other Tales of Mathematical History, edited by Marlow Anderson, Victor Katz, and Robin Wilson Student Research Projects in Calculus, by Marcus Cohen, Arthur Knoebel, Edward D. Gaughan, Douglas S. Kurtz, and David Pengelley Symmetry, by Hans Walser. Translated from the original German by Peter Hilton, with the assistance of Jean Pedersen. The Trisectors, by Underwood Dudley Twenty Years Before the Blackboard, by Michael Stueben with Diane Sandford Who Gave You the Epsilon? and Other Tales of Mathematical History, edited by Marlow Anderson, Victor Katz, and Robin Wilson The Words of Mathematics, by Steven Schwartzman

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Contents

Introduction ...... xi

Part I. The Calculus as Algebra ...... 1 Preface to the Garland Edition ...... 3 Acknowledgements ...... 7 Introduction ...... 9 1. The Development of Lagrange’s Ideas on the Calculus: 1754–1797 ...... 17 2. The Algebraic Background of the Theory of Analytic Functions ...... 37 3. The Contents of the Fonctions Analytiques ...... 63 4. From Proof-Technique to Deﬁnition: The Pre-History of Delta-Epsilon Methods . . 81 Conclusion ...... 101 Appendix ...... 103 Bibliography ...... 105

Part II. Selected Writings ...... 125 1.TheMathematician,theHistorian,andtheHistoryofMathematics...... 127 2. Who Gave You the Epsilon? Cauchy and the Origins of Rigorous Calculus∗ ....135 3. The Changing Concept of Change: The Derivative from Fermat to Weierstrass† . . 147 4. The Centrality of Mathematics in the History of Western Thought† ...... 163 5. Descartes and Problem-Solving† ...... 175 6. The Calculus as Algebra, the Calculus as Geometry: Lagrange, Maclaurin, and Their Legacy ...... 191 7. Was Newton’s Calculus a Dead End? The Continental Inﬂuence of Maclaurin’s Treatise of Fluxions∗ ...... 209 8. Newton, Maclaurin, and the Authority of Mathematics∗ ...... 229 9. Why Should Historical Truth Matter to Mathematicians? Dispelling Myths while PromotingMaths...... 243 10. Why Did Lagrange “Prove” the Parallel Postulate?∗ ...... 257 Index ...... 275 About the Author ...... 287

∗ Recipient of the Lester R. Ford Award † Recipient of the Carl B. Allendoerfer Award ix P1: JZP MABK013-FM MABK013/Grabiner Trim Size: 7in× 10in July 28, 2010 19:19 P1: JZP MABK013-FM MABK013/Grabiner Trim Size: 7in× 10in July 28, 2010 19:19

Introduction

In 1869, Darwin’s champion Thomas Henry Huxley praised scientific experiment and observation over dogmatism. But his criticisms extended also to the textbook view of mathematics. Huxley said, “The mathematician starts with a few simple propositions, the proof of which is so obvious that they are called self-evident, and the rest of his work consists of subtle deductions from them....Mathematics...knows nothing of observation, nothing of experiment.” In reply, the great English algebraist J. J. Sylvester spoke about what he knew from his own work: Mathematics “unceasingly call[s] forth the faculties of observation and comparison ...ithasfrequent recourse to experimental trials and verification ...itaffordsa boundless scope for the exercise of the highest efforts of imagination and invention.” [3, 204] As a historian of mathematics, I’m with Sylvester. I have long been interested in what mathematicians actually do, and how mathematics actually has developed. I have nothing against textbooks and logically structured subjects. It is just that they represent the finished product, not the creativity that produced it. The past, as L. P. Hartley said in opening his novel The Go-Between, “is a foreign country: they do things differently there.” [2, 1] Historians provide guidebooks to that past, although mathematicians tend to be more interested in knowing how that foreign past became transformed into the mathematics we know and teach today. Mathematics is incredibly rich and mathematicians have been unpredictably ingenious. Therefore, the history of mathematics is not rationally reconstructible. It must be the subject of empirical investigation. I address the results of my own investigations not principally to other historians, but to mathematicians and teachers of mathematics. The present volume brings together much of what I have tried to say to this audience. In The Calculus as Algebra: J.-L. Lagrange, 1736–1813, I show what Lagrange’s mathematical practice was like, in order to understand the genesis of the rigorous analysis of Cauchy, Bolzano, and Weierstrass. For Lagrange, the calculus was not about rates of change or ratios of differentials, or even about limits as then understood. Lagrange thought that the calculus should be reduced to “the algebraic analysis of finite quantities.” This sounds as though he was about to introduce deltas and epsilons. But instead he believed that there was an algebra of infinite series, and that every function had a power-series expansion except perhaps at finitely many isolated points. Lagrange defined the derivative as the coefficient of the linear term in the function’s power-series expansion. Why he thought this was justified tells us both about his philosophy of mathematics and about the way many mathematicians practiced their subject in the eighteenth century. Euler, for example, did marvelous things by what we would now call the carefree formal manipulation of infinite series, infinite products, and infinite continued fractions. But Lagrange found something else in infinite series as well. He imported what we now call delta-epsilon techniques from the 18th-century study of approximations into some of his proofs about the

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

concepts of the calculus. He was the first to attempt to prove, let alone to use inequalities in so doing, statements like “a function with a positive derivative on an interval is increasing there.” He justified many results of calculus using inequalities, including the mean-value theorems for derivatives and integrals, and the Lagrange remainder of the Taylor series. This, and much more, helped build Cauchy’s work in the 1820s on the foundations of analysis. The story is instructive in many ways, especially for teachers. Students often have difficulties like those encountered by the mathematicians who invented key ideas. For example, “You say ‘can be made as close as you like’ is the same as ‘equals the limit,’ but how can a bunch of inequalities somehow turn into two things being equal?” is an excellent question. Bishop Berkeley’s criticisms of arguments about the limit of the ratio of two quantities when the quantities vanish—and the inadequacy of eighteenth-century attempts to answer him in his own terms—recur among very good students. Knowledge of the history of the relevant mathematical concepts can help the teacher understand what is troubling the student. And the history also lets those of us who learned a subject with ease appreciate which concepts are inherently hard; it’s no accident that it took a hundred and eighty years from the understanding of the derivative by Newton and Leibniz to the proofs of Weierstrass. Students can see something else as well. Mathematicians solve problems, ask questions, explore fuzzy ideas, make false starts or go in the wrong direction. They can be right for the wrong reasons, like Lagrange. And the “right” definitions often come only at the end. Thus following difficult and crooked paths to problem solution makes the student part of the mathematical community. Historians, mathematicians, and students can all gain from understanding Lagrange’s grappling with the calculus as algebra. Both historians and mathematicians can contribute to the history of mathematics, as I argue in the earliest of the articles reprinted here, “The Mathematician, the Historian, and the History of Mathematics,” but they may do this in different ways. Historians in general, as I said in that paper, ask two types of questions: What was the past like? And, how did the present come to be? Mathematicians more often tend to be interested in the second question, historians in the first. But these two approaches complement one another, and I have tried to use both in my own work. For instance, contrast the older and the modern understandings of the derivative. “The Changing Concept of Change: The Derivative from Fermat to Weierstrass” shows how a key mathematical idea can develop quite differently than in the way we now understand it. Properties of varying quantities reaching their maximum or minimum values were discovered and used by people like Fermat and Descartes, but these men did not see their calculations as examples of calculating rates of change. When Newton and Leibniz instead explained how such problems were special cases of fluxions or of differential quotients, they opened the door for research into previously inconceivable topics like differential equations. Understanding and proving the properties of derivatives in terms of a sufficiently powerful and precise concept of limit came even later, and teasing out the distinction between pointwise and uniform convergence to a limit came later still. Beginning the study of rigorous calculus with the delta-epsilon definitions of limit and derivative turns this history on its head. Recapturing the history helps us motivate our students’ rediscovery of the concepts, and also makes us marvel at the brilliance of our predecessors, who created these concepts out of such chaos. In “Who Gave You the Epsilon?” I do another thing historians do: making something familiar in the present look strange and unexpected, and only then showing the path by which it came P1: JZP MABK013-FM MABK013/Grabiner Trim Size: 7in× 10in July 28, 2010 19:19

Introduction xiii

to be. I imagine a student asking how anybody could ever have imagined explaining “the car is going 50 miles per hour” in delta-epsilon terms. I try to answer this by explaining what the calculus was like before Cauchy, and then how different it looked afterwards. Contrast Euler’s successful discoveries by adding, multiplying, and transforming infinite series, even divergent ones, with Cauchy’s “A divergent series has no sum” and Abel’s lamenting “Can you imagine anything more horrible” than to claim that 1n − 2n + 3n − 4n +···=0? This change in the agreed-on rules of the game is an example of what historians have come to call a paradigm shift. Such a change requires an explanation, which the article presents. The role of approximations in bringing about the change is signaled by the notation “epsilon,” which, I argue, comes from the first letter in the French word for error. Moving from the internal history of mathematics into its cultural setting, in “The Centrality of Mathematics in the History of Western Thought” I observe that, although modern mathematicians may find it hard to explain to outsiders what they do, this has not always been the case. For two thousand years mathematics played a key role in philosophy. Whether it was proof in geometry or the algorithmic power of notation in algebra, mathematics provided a non-material “reality” for idealists, empiricists, and realists to dispute about, as well as a model for methods of demonstration and discovery. Examples abound of mathematics influencing philosophy, from Plato to Spinoza’s Ethics to the Declaration of Independence to Condorcet’s prediction of universal progress. Examples of hostility to using mathematics as a model of all human achievement abound as well, ranging from Pascal to Wordsworth. “Centrality” may be an awkward word, but it is important to establish and document the phenomenon it describes. “Descartes and Problem-Solving” presents a special case of some of the generalizations in “Centrality.” Although it is common to be told that Descartes’ mathematics comes from his general “method,” I explore the possibility that it was the other way around. Observing that Descartes did not have Cartesian coordinates helps understand what he actually did. Descartes developed a new method to solve problems in geometry. He translated them into algebra, then used the algorithmic power of algebra to transform the original question into something more tractable, and finally translated that new “something” back into a geometric construction. His focus on problems and the methods of solving them has influenced mathematicians from Newton to Polya.´ And the power of Descartes’ new problem-solving method reinforced his philosophical commitment to renewing all of science by finding the one true method, and to portraying that method as inspired by mathematics. In my three articles on Colin Maclaurin, I turn to “rehabilitating” a mathematician of lesser reputation than Descartes, Lagrange, and Cauchy. In “Was Newton’s Calculus a Dead End? The Continental Influence of Maclaurin’s Treatise of Fluxions” I ask why Maclaurin’s contributions have so often been undervalued—this is social history—and I describe his considerable impact on 18th- and 19th-century Continental analysis. It is often considered, especially by the French school exemplified by Lagrange, that old- fashioned geometry hindered progress of mathematics. In “The Calculus as Algebra, the Calculus as Geometry,” I argue that Maclaurin’s geometric mind-set helped him formulate results in analysis, and that the decline of such geometric insight was a loss, however much the new analysis was a gain. By contrast, Lagrange’s achievements came from his rejection of geometric intuition in favor of a commitment to the general and to his appreciation of abstract algebraic structures. The success of these two different approaches suggests that a diversity of approaches will best promote mathematical progress, as is clear also from works on the psychology of P1: JZP MABK013-FM MABK013/Grabiner Trim Size: 7in× 10in July 28, 2010 19:19

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invention, like that of Jacques Hadamard [2]. We ought to recognize and nourish this variety of approaches as we teach our subject. In “Newton, Maclaurin, and the Authority of Mathematics,” I show how Maclaurin internal- ized the way of doing mathematical physics exemplified in Newton’s Principia, and then applied it to solve problems ranging from the shape of the earth to the computation of annuities. This article also illustrates the authority mathematicians and mathematics were thought to have in the eighteenth century. The universal agreement and clear reasoning of the mathematical sciences were exploited to promote political and social consensus on controversial issues. This provides an instructive example of the deference to quantitative measures as “objective” and therefore beyond politics, which is, for good or ill, alive and well today. “Why Should Historical Truth Matter to Mathematicians? Dispelling Myths while Promoting Mathematics” began its life as a 20-minute talk at the 2004 Joint Meetings in a session entitled “Truth in Using the History of Mathematics in Teaching Mathematics.” I expanded this talk into a colloquium lecture, and after I gave it to several mathematical audiences, with different examples each time, two conclusions became more and more apparent. First, misconceptions about the history of the calculus were not the only important ones for me to address. Second, mathematicians are especially interested in seeing the record set straight when doing this can promote mathematics and improve its teaching. In the current version of the paper, I address myths like “all important mathematics is done by men and within the European tradition,” “there was no European mathematics in the Middle Ages,” and “the mathematical approach can be applied to settle any major question,” as well as “Newton invented the calculus in order to do physics” and various long-held but inadequate views about Maclaurin and Lagrange. I hope to show how knowing what actually happened is more valuable in understanding the mathematical experience than any plausible but erroneous story could be. Finally, in “How Did Lagrange ‘Prove’ the Parallel Postulate?” I return to Lagrange, this time via an unpublished manuscript. I cannot rehabilitate Lagrange’s “proof ” of Euclid’s Fifth Postulate. And, unlike Lagrange’s power-series definition of the derivative, no “success” arose from Lagrange’s approach to this problem. The episode, though, further illuminates Lagrange’s approach to science and mathematics, as well as casting light on the broader intellectual and social concerns of Continental society. Although of course society cannot call mathematical results into being at will, society can and does influence a great deal: for instance, the choice of problems, the applications considered most useful, and how mathematics is taught. Mathematics is part of human culture. And mathematics answers questions of interest and importance to the society around it. In the case of Lagrange and Euclid’s postulate, we have the mutual influences of geometry and the arts, the professionalization of mathematics in the nineteenth century, and the increasing need for mathematicians to teach, especially from the time of the French Revolution onward. For Enlightenment thinkers, Euclidean geometry was thought to embody the true nature of space, and space was the really existing framework for Newtonian mechanics. The universal agreement claimed for these ideas served as a model for the philosophy of the Enlightenment, and for its search for universally-accepted truths by reason. The history of mathematics, then, has a lot to teach us. Mathematicians’ creative ideas can come from within mathematics or from the wider society. Mathematical progress cannot be programmed. That’s why the history of mathematics is interesting. Mathematics progresses, not only by using and extending the prevailing ideas and techniques, but also by transforming P1: JZP MABK013-FM MABK013/Grabiner Trim Size: 7in× 10in July 28, 2010 19:19

Introduction xv

and transcending them. Progress depends on people being willing to use, discover, explore, and develop concepts and techniques—to risk being incomplete or even wrong. Appreciating all of this can attract students, help teachers, and inspire researchers. I hope the present volume can contribute to these goals.

References 1. Hadamard, Jacques. The Psychology of Invention in the Mathematical Field, Princeton University Press, 1945. Dover reprint, 1954. 2. Hartley, L. P., The Go-Between, London, 1953. 3. Parshall, Karen Hunger, James Joseph Sylvester: Jewish Mathematician in a Victorian World, Johns Hopkins, Baltimore, 2006. P1: JZP MABK013-FM MABK013/Grabiner Trim Size: 7in× 10in July 28, 2010 19:19 P1: JZZ MABK013-IND MABK013/Grabiner Trim Size: 7in× 10in July 30, 2010 9:41

Index

Abel, Neils Henrik, xiii, 22n, 26, 47, 95n, 102, 254 Argand, Jean-Robert, 270 Actuarial science, 229, 236–237 Aristaeus, 178 Adam, Charles, 189 Aristotle, 128n, 151, 164, 168, 170, 243, 247 Agassiz, Louis, 164 Arnold, Matthew, 210, 224 Agnesi, Maria Gaetana, 123–124, 244 Art and architecture, 261–262, 264–265, 267, Algebra of inequalities, 31, 51–52, 57–62, 68, 270–273 83–88, 90–93, 99, 135, 138–139, 141–142, Auchter, Heinrich, 117 145, 205, 207, 214, 216, 250, 252–253 Augustus, Roman Emperor, 255 Algebra of power series, 3, 11, 35n, 37, 38, 65, 205 Authority of mathematics and mathematicians, xiv, Algebra, origin of the word, 245–246 121, 172, 229–230, 232–240, 254, 270 Algebraic notation, heuristic power of, 10, 38, Average man, 169 40–41, 165, 188–189 Algorithm, origin of term, 165, 245 Babbage, Charles, 17n, 79n, 80n, 109, 166–167, Al-Khwarizmi, Mohammed ibn Musa, 165, 245 172–173, 194, 216, 221–222, 224, 226, 251 Al-Samawa’al, 245 Bachmacova, Isabella, 113 Amburger, Erik, 26n, 111 Bacon, Francis, 178, 188–189, 205, 234, 238 Ampere,` A.-M., 81, 83–84, 88–94, 96, 98–99, 117, Baker, Keith Michael, 173, 189, 206n 158 Ball, W. W. Rouse, 109, 222, 224 Ampere’s` definition of derivative, 89 Baron, Margaret, 160 Analyse algebrique´ , 4, 13n, 65n, 80, 89n, 95n Barraclough, June, 173, 189, 206n Analysis, method of, 10n, 166, 171, 177–182, 189, Barrett, William, 172 238 Barroso-Filho, Wilton, 119 Analyst, The, see Berkeley Barrow, Isaac, 150, 187 Analytic geometry, invention of, 10, 148, 166, 176, Barrow, John, 264, 272 186–188 Beaujouan, Guy, 107 Analytical bibliography for The Calculus as Beck, L. J., 189 Algebra (up to 1966), 105–119 Beechler, Barbara, 239 Apollonius, 178 Belhoste, Bruno, 119 Apostol, Tom M., 81n, 110 Bell, Eric Temple, 17n, 22n, 37n, 50n, 110, 127, Approximations in eighteenth-century algebra, xi, 206n, 243–244, 251 xiii, 4–5, 37–38, 41–42, 51–62, 71, 84, 128, Beltrami, Eugenio, 106 130, 138–139, 141–142, 145, 216 Bentham, Jeremy, 254 Arago, François, 88n, 117 Bentley, Richard, 174 Arbogast, L. F. A., 7, 25n, 28–34, 67, 79n, 115–116 Berg, Jan, 118 Arc lengths, 18, 77, 149, 186–187, 249 Berkeley, George, xii, 10, 21n, 26, 34, 64n, 110, “Arnolfini Wedding,” by Jan van Eyck, 265 114–116, 129–130, 139–140, 145, 154, 157, Archibald, R. C., 109 160, 210, 212–213, 219–220, 224, 233, 236, Archimedean axiom, 153, 180 240, 250–252, 254, 263 Archimedes, 75n, 148, 164, 173, 201, 206, 215, Berlin Academy prize competition on foundations 217–218, 225, 259 of calculus, 25–28, 34, 64–65, 124, 140 Areas under curves, 18, 33, 69, 76–77, 79, 135–136, Bernoulli family, 10, 11, 39n, 64, 78, 83, 115, 129 143–144, 148–149, 152–153, 186–187, 200, Bernoulli numbers, 200, 218 204, 214–216, 231, 249 Bernoulli polynomials, 219

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276 Index

Bernoulli, Daniel, 51n, 52–53, 112, 154, 212 Boyer, Carl B., 7, 17n, 20n, 21n, 27n, 29n, 37n, Bernoulli, Jakob, 47n, 90, 116, 129n, 248 41n, 47n, 48n, 82n, 83n, 95n, 111, 113–114, Bernoulli, Johann III, 26–27 120, 131, 145, 160, 172, 189, 224 Bernoulli, Johann, 19, 44n, 45, 83n, 90, 115–116, Bradley, Robert E., 120 129n, 137, 145, 153, 210, 213, 221 Bradwardine, Thomas, 247 Bernoulli, Nikolaus, 205 Briano, Giorgio, 107 Berthollet, Claude-Louis, 257 Bricker, David, 172 Beth, E. W., 117 Brill, A., 113–114 Bezout, Etienne, 112 British Linen Company, 235 Bibliography for TheCalculusasAlgebra Broadie, Alexander, 240 (1966–present), 119–124 Brunelleschi, Filippo, 273 Bibliography in analytical categories for The Bruno, Giordano, 260–261 Calculus as Algebra (up to 1966), 105–119 Buchwald, J. Z., 241 Bibliotheque` de l’Institut de France, 7, 12n, 106, Burtt, E. A., 173, 189 272–273 Butterfield, Herbert, 223 Bieberbach, Ludwig, 244 Biermann, Kurt-R., 26n, 107, 111 Cajori, Florian, 10n, 43n, 64n, 82n, 83n, 110, Binomial theorem, 18, 30, 51 113–115, 146, 161, 174, 206n , 210, 224, 274 Biot, Jean-Baptiste, 107, 257, 270, 272 Calculus, invention of, 114, 136, 147, 152, 187, Birkhoff, Garrett, 119 210, 249 Black, Joseph, 212, 229 Calinger, Ronald, 120, 191n Blanton, John, 120 Camarena,˜ Mar´ıa, 172 Bloom, Paul, 272 Cambridge Analytical Society, 17n, 79, 109, 194n, Boak, Arthur E. R., 174 221–222, 224 Boatner, Charlotte H., 106 Campbell, Reverend Colin, 231 Bohlmann, G., 114 Cantor, Geoffrey N., 224–225 Boissonnade, Auguste Claude, 122, 273 Cantor, Georg, 132, 159, 243–244, 249 Boltzmann, Ludwig, 170, 172 Cantor, Moritz, 26n, 28n, 109–110, 114–115, 223, Bolyai, Janos, 169, 270 225 Bolzano, Bernhard, xi, 3–5, 65, 68, 73n, 74, 81, 95, Caracciolo, Marquis de, 107 102, 117–119, 121, 123, 145–146, 203n, Cardan, Jerome, 41 206–207 Carnap, Rudolf, 272 Bolzano’s 1817 “purely analytic” proof of Carnot, Lazare, 90, 96, 111, 115–116, 121, 124, intermediate-value theorem, 3–4, 65n, 95, 119, 140, 268, 270 145, 206 Carruccio, Ettore, 110 Bolzano’s recognition of Lagrange definition of Cartesian coordinates, xiii, 176, 180, 261 continuity, 5 Cartwright, Dame Mary, 244 Bombelli, Rafael, 59n Cataldi, Pietro, 59n Boncompagni, B., 19n, 105, 118 Cauchy, Augustin-Louis, xi–xiii, 3–6, 11n, 14, 19, Bonnie Prince Charlie, 224 26, 47, 50–51, 61, 63, 65, 67–69, 71, 73n, 74, Bonola, Roberto, 110, 272 77, 80–81, 83–85, 86n, 87–89, 91–99, 102, Bopp, Karl, 107 117–121, 128–129, 132, 135–136, 138, Borel, Emile, 103n, 118 141–146, 157–161, 171–173, 194n, 201n, Borgato, Maria Teresa, 20n, 119, 122, 202n, 224, 202n, 203, 206–207, 210, 214, 216–217, 219, 273 221, 225, 240, 251–256, 273 Boring, E. G., 273 Cours d’analyse, 3–4, 5n, 65n, 80, 89, 94n, 95n, Bortolotti, Ettore, 107 96, 98, 102, 120, 128n, 141, 145, 172, 207n, Bos, H. J. M., 119, 182, 189 255 Bossut, Abbe´ Charles, 39n, 82n, 115, 257 Cours d’analyse, English translation by Bradley Bottazzini, Umberto, 3n, 120 and Sandifer, 120 Bourbaki, Nicolas, 37n, 110, 206n, 224, 251 Definition of continuous function, 95 Boutroux, Emile, 10n, 111 Definition of derivative, 94, 135, 143, 157–158 Boutroux, Pierre, 110, 113–114 Definition of limit, 94–96, 135, 157–158 P1: JZZ MABK013-IND MABK013/Grabiner Trim Size: 7in× 10in July 30, 2010 9:41

Index 277

First delta-epsilon proof, 97–99, 158, 253 Cours d’analyse of Cauchy, English translation by Leçons sur le calcul infinitesimal´ , 3-4, 6n, 14n, Bradley and Sandifer, 120 94n, 95n, 96n, 97n, 98n, 99n, 102, 118, 145, Courtivron, Marquis de, 51n, 112 160, 203n, 225 Craige, John, 232, 241 Cauchy, Catholicism of, 254 Crelle, August Leopold, 79n, 108, 206n Cavalleri, P. Antoine, 212 Cubical parabola of Descartes, 182, 184–185 Chandrasekhar, Subramanyan, 201n, 217, 225, 234, Curves, 32, 45, 75–76, 79, 84n, 95, 115, 120, 143, 240 148–149, 157, 176, 180, 182, 184–187, Change, medieval conceptions of, 151, 247–249 189–190, 195–197, 199, 203–204, 219–220, Charles, Jacques, 257 231, 269 Chasles, Michel, 220 Cuvier, Georges, 257 Chatelet,ˆ Marquise du, 244 Child, J. M., 114n D’Alembert, Jean-le-Rond, 9, 11, 27, 34, 39, 45n, Chinese Remainder Theorem, 245 70n, 82–84, 88, 93–94, 96, 102, 110, 113, 115, Chio,` Felice, 107 120, 129, 130n, 135–136, 139–141, 145–146, Chitnis, Anand, 225 154, 158, 200, 213, 216–220, 222, 225–226, Christie, J. R. R., 224–225 234, 268, 271–272 Church of Scotland, 212, 231, 236 D’Antonio, Lawrence A., 120 Clairaut, Alexis-Claude, 41n, 112, 136, 201, 213, D’Asero, Marcelo, 172 217–218, 224–225, 234, 240, 251, 267, 272 Dahan-Dalmedico, Amy, 119, 201n Clear and distinct ideas, 10, 267 Darwin, Charles, xi, 168, 173, 237, 248 Clifford, W. K., 170, 270 Dauben, Joseph, 120 Cohen, I Bernard, 7, 123, 172, 189, 230, 240–241, Davie, George Elder, 193n, 205n, 222, 225 250n, 257 Davis, P. J., 189 Compensation of errors, 20, 21, 26n, 28, 34, 64–65 De Gua de Malves, Jean Paul, 112 Completeness of real numbers, 81, 96n, 104, 142, De la Chapelle, Abbe´ de, 225 145 De la Lande, J., 39n, 82n, 115 Complex numbers, 49–51 De Moivre, Abraham, 218, 225 Complex variables, 15, 131, 214 De Morgan, Augustus, 257, 272 Comte, Auguste, 79n, 110, 121, 168, 171, 173, De Witt, Jan, 186 254 Debeaune, Florimond, 186 Comte, Claude, 119 Declaration of Independence, xiii, 164–165 Concavity, 195–199, 217, 220, 250 Dedekind, Richard, 39n, 81, 140 Condorcet, Marquis de, xiii, 39n, 82n, 109, 111, Delambre, Jean-Baptiste-Joseph, 10n, 11n, 20n, 115, 165–166, 171–173, 188–189, 206 28n, 35n, 79n, 105, 107–108, 114, 122, 257 Constructability of curves, 182–184 Delta-epsilon techniques, xi, 5, 38n, 67n, 81–83, Continued fractions, xi, 42, 59–62, 139, 156 85–86, 97–99, 101–102, 135, 141, 203, 214 Continuity, uniform, 74n, 94, 99, 102n, 104, 142, Denina, Carlo, 107 144 Deretsky, Tatiana, 189 Continuous function, 3, 5, 67–68, 74, 89, 91–92, 93, Derivative, definitions of, xi, 3, 14, 20, 37, 47, 95, 102, 104, 141–142, 145, 154 65–66, 94, 135, 143, 156–159, 206, 253 Convergence, xii, 4–5, 11, 14, 19, 45, 47, 52, 55, Derivative, history of concept, 147–160 57, 59, 61–62, 81n, 57n, 84, 87, 91, 92n, Desaguliers, John T., 211 98–99, 128–129, 132, 141–142, 145, 159, 203, Desargues, Girard, 262, 272 207, 254 Descartes, Rene,´ xii–xiii, 10, 12, 110, 121, Convergence tests and criteria for convergence, 4, 131–132, 148, 150, 158, 166–167, 170–171, 19, 47n, 141 173, 175–190, 205–206, 221, 223, 238, 249, Convergence, uniform, xii, 84, 87, 91n, 92n, 98–99, 254, 260–261, 265, 269, 271 142, 159, 203, 254 Descartes’ rule of signs, 185 Coolidge, J. L., 130n, 189 Dhombres, Jean, 120, 201n Copenhagen, University of, 240 Dhombres, Nicole, 120 Copernicus, Nicolas, 12, 168, 187 Dickens, Charles, 171–173 Cotes, Roger, 44–45, 200, 218 Dickstein, S., 17n, 63n, 79n, 107 P1: JZZ MABK013-IND MABK013/Grabiner Trim Size: 7in× 10in July 30, 2010 9:41

278 Index

Diderot, Denis, 9n, 11, 110, 129, 140, 146, 225, 272 Euler, Leonhard, xi, xiii, 9–11, 18n, 19, 21n, 26n, Dieudonne,´ Jean, 132, 160n 30–31, 34, 39, 41–43, 45–50, 51n, 53–54, 55n, Dijksterhuis, E. J., 190, 214, 225 56–57, 59n, 60–61, 64, 78, 83, 88, 90, 102, Dini, Ulisse, 141 105, 107–112, 114, 116, 120, 122, 129, Diophantus of Alexandria, 123, 132, 246 136–140, 142–146, 154–157, 159–160, 187, Discovery in mathematics, xii–xiii, xv, 19, 152, 194n, 199–201, 202n, 206 , 210–214, 217–222, 160, 166, 175, 187, 190, 192, 199, 207, 219, 225–227, 239, 251–254, 266, 271–272 238, 249, 264 Algebra, 39n, 51n, 53, 55n, 112 Divergent series, xiii, 129n, 132, 254 Institutiones calculi differentialis, 18n, 21n, 43, Division of labor, 166–167, 172 45–46, 51n, 56–57, 65n, 83n, 107, 116, 120, Domingues, Joao˜ Caramalho, 120 160, 199n, 202n, 217–218, 225 Dou, A., 132 Introductio in analysin infinitorum, 30, 42–50, Dow, J. B., 240 53–54, 56, 59n, 60, 61n, 112, 120, 156, 194n, Dugac, Pierre, 3n, 120, 160 201, 218, 225 Dugas, Rene,´ 111 Zeros as foundation for calculus, 21n, 64, 83, 90, Dunham, William, 120 102 Dunken, Gerhard, 26n, 111 Euler-Maclaurin formula, 199–201, 206, 214, Dunlop, A. I., 240 218–219, 227 Excise, 234–235 Echeverria, Javier, 122 Existence of definite integral, 144 Ecole polytechnique, 4, 34–35, 78, 83, 85n, 97, 111, Existence of solutions to differential equations, 158 130, 140–141, 168, 206 Externalist history, 221, 223 Economy of thought, 12, 102 Eyles, V. A., 225 Edinburgh Philosophical Society, 212, 235 Edinburgh, University of, 80, 205–206, 212, 222, Fagnano, Giulio Carlo, 18n, 105–107 224, 229, 231, 237 Falconer, Etta Zuber, 245 Edwards, H., 132 Fauvel, John, 120, 225 Ellipsoids, gravitational attraction of, 200–201, 206, Favaro, Antonio, 107 217–218, 220, 233–234 Fermat, Pierre de, xii, 10, 41, 132, 147–150, Elliptic integrals, 44, 200, 214, 219 158–161, 185–187, 190 Encyclopedie´ (Diderot-D’Alembert), 9, 39n, 82n, Ferraro, Giovanni, 121 110, 115, 129, 216, 219, 225, 267, 272 Feuerbach, Ludwig, 168 Enestrom,¨ G., 114, 115 Fibonacci, see Leonardo of Pisa Engels, Friedrich, 174 Field, J. V., 272 Engelsman, Stephen B., 120 First and last ratios, 17, 20n, 21–22, 28, 35, 64, 102, Enlightenment, xiv, 3, 9, 11, 124, 171, 188, 136–137, 153 205–206, 209, 212, 223, 226–227, 240, 268, Flett, P. M., 3, 121 272 Fluxions, xii, 11, 13, 17, 20, 21n, 41, 42n, 43n, Enneper, Alfred, 200n, 225 45–46, 51n, 55–56, 64, 68–69, 75n, 78, 80, 82, Enriques, Federigo, 111 101–102, 106, 114, 136, 139, 152–153, 156, Epsilon, introduction by Cauchy, xiii, 5, 97–98, 193–195, 197, 200–201, 209–211, 214–216, 121, 128, 145, 252–253 220, 222–223, 249, 251, 269 Eratosthenes, 178 Folkerts, M., 122 Ernst, Max, 270 Foncenex, Daviet de, 20, 25n, 116 Error bounds, 5, 51, 54, 57–62, 84, 139, 145 Fontaine, Alexis des Bertins, 112 Error estimates, 38, 51–54, 59–62, 88 Forti, Achille, 107 Ersch, Johann Samuel, 108, 118 Fourcroy, Antoine-François, 257 Escher, M. C., 271, 272 Fourier, Joseph, 132, 268, 271 Euclid, xiv, 12n, 32n, 75n, 111, 163–164, 173, 178, Fox, Robert, 121, 273 182, 207, 222, 243. 245–246, 257–260, Franklin, J., 272 262–264, 267–270, 272–273 Fraser, A. C., 10n, 21n, 110, 116 Euclidean geometry, xiv, 12n, 163–165, 167, 170, Fraser, Craig, 121 261–262 Frechet,´ Maurice, 115 P1: JZZ MABK013-IND MABK013/Grabiner Trim Size: 7in× 10in July 30, 2010 9:41

Index 279

Frederick the Great of Prussia, 26n, 34 Green, H. G., 44n, 116 Fredericksen, A., 274 Greenberg, John L., 225, 240 French Revolution, xiv, 10, 28n, 34, 106, 111, 123, Greene, John C., 173 130, 140, 166, 270 Gregory, David, 234 Freudenthal, Hans, 4, 117 Gregory, Duncan, 222 Friedman, Michael, 273 Gregory, James, 155, 186–187, 216 Frisi, Paolo, 107 Grifﬁths, P. L., 218, 226 Froberg, J. P., 109 Grosholz, Emily, 190 Function, concept of, 13, 30, 42–44, 46, 78, 113, Gruber, Johann Gottfried, 108, 118 124, 154, 156, 222 Gruson, Jean Philippe, 109 Fundamental theorem of algebra, 19, 40, 43, 49n Guareschi, Icilio, 107 Fundamental theorem of calculus, 5–6, 74n, 136, Guicciardini, Niccolo,` 122, 211, 226, 250 144–145, 152–154, 158, 204, 210, 214–217, Guitard, Thierry, 122 253 Hadamard, Jacques, xiv, xv, 191–192 Galileo, 12, 151, 168, 173, 247–249 Hahn, Roger, 122 Gambioli, D., 107 Hall, A. Rupert, 173, 226 Garber, Elizabeth, 273 Halley, Edmond, 230, 237 Gardiner, William, 211 Hamburg, Robin, 122 Gassendi, Pierre, 261 Hamilton, William Rowan, 170, 270, 273 Gauging, 234–235, 239 Hamilton, Sir William, 193, 222 Gaukroger, Stephen, 190 Hankel, Hermann, 107–108, 113–114, 118 Gauss, Carl Friedrich, 19, 40n, 112, 118, 145, 169, Hankins, Thomas, 213, 226, 240, 273 217–218, 270 Hardin, Garrett, 173 General Assembly of the Church of Scotland, 236 Hare, D. J. P., 237, 240 Generality of algebra, 38, 140, 180, 207 Harmonic series, 47, 128–129, 248 Genocchi, Angelo, 107 Harnack, Adolf, 26n, 111 Gerdil, Hyacinth Sigismund, 20 Harriot, Thomas, 40n, 112 Gerhardt, C. I., 114–115 Harris, John, 84n, 115 Germain, Sophie, 244 Hartley, L. P., xi, xv Gertner, D., 273 Hauy,¨ Rene-Juste,´ 257 Geymonat, Ludovico, 122 Hawkins, Thomas, 127n, 129, 131–132 Gilain, Christian, 121 Heath, T. L., 75n, 173, 190, 272 Gillespie, Neal C., 173 Heine, Eduard, 103n, 118, 159 Gillies, Donald, 190 Heinekamp, A., 123, 274 Gillispie, Charles C., 121–122, 124, 173, 273 Helmholtz, Hermann von, 170, 173, 270 Girard, Albert, 40n, 112 Henry, Charles, 107, 160 Gmeiner, J. Anton, 119 Hermann, Dieter, 108 Godel,¨ Kurt, 170 Hermite, Charles, 113, 192 Godwin, William, 173 Herrnstein, R. J., 273 Goethe, Johann Wolfgang von, 171 Herschel, Caroline, 244 Goldbach, Christian, 213, 226 Herschel, John Frederick William, 17n, 80n, 109, Goldin-Meadow, S., 273 216, 222, 226 Goldstine, Herman, 3n, 121, 146, 199, 223–225 Hersh, Reuben, 189 Gottlieb, Jason, 172 Heytesbury, William of, 247 Grabiner, Judith V., 3n, 4n, 5n, 6n, 121, 132, 146, Hindu-Arabic numbers, 165, 245 161, 173, 190, 194n, 201n, 202n, 203n, 206n, Hippocrates of Chios, 176 207n, 225, 240, 243n, 251n, 253n, 273 Historians’ basic questions, xii, 127 Grabiner, Sandy, 7, 172 Historical order vs. textbook exposition, xiii–xiv, Graham, George, 234 101, 135, 159–160 Grattan-Guinness, Ivor, 4, 120–123, 142, 146, History, questions about, 127–128 251n, 273–274 Hitler, Adolf, 244 Gray, Jeremy, 120, 122, 272–273 Hobbes, Thomas, 165, 172–173, 239 P1: JZZ MABK013-IND MABK013/Grabiner Trim Size: 7in× 10in July 30, 2010 9:41

280 Index

Hochstetter, E., 114–115 Jushkevich, A. P., see Juskevich Hodge, M. J. S., 224 Juskevic,ˇ A. P., see Juskevich Hodgkin, Luke, 273 Juskevich, A. P., 21n, 46n, 90n, 107–108, 116, 118, Hoene-Wronski, J. M., 109 124, 146, 199n, 226 Hofmann, J. E., 106, 109–110, 114–116, 190 Jussieu, Antoine-Laurent de, 257 Homogeneity of algebraic expressions, 180–181 Kahane, Jean-Pierre, 132 Hopper, Grace, 245 Kant, Immanuel, 167, 170, 173, 266–267, 271, Houzel, C., 123 273 Howell, R., 272 Katz, Victor J., 122, 240, 246n Hudde, Johann, 150–151 Kelland, Phillip, 222 Huggett, Nick, 273 Kelvin, Lord, 218 Hume, David, 10, 110, 167, 205, 212, 229, 239 Kemp, Martin, 264, 273 Hutcheson, Francis, 229, 239, 254 Kempe, A. B., 184, 190 Hutton, James, 212, 225, 229, 239 Kepler, Johann, 120, 168, 230–231, 239, 262 Huxley, Thomas Henry, xi Kiernan, B. M., 122 Huygens, Christiaan, 45n, 151 Kitcher, Philip, 122 Hyman, Anthony, 173, 194n, 224 Klein, Felix, 110 Hypatia of Alexandria, 244 Klein, Jacob, 190 Kline, Morris, 122, 161, 173, 190, 206n, 223, 226 Ibarra, A., 122 Klopstock, Friedrich Gottlieb, 210 Inequality proofs in calculus, 6, 12, 84, 86–87, Klugel,¨ G. S., 129n 90–91, 93–94, 97–99, 101–104, 130, 141–145, Knorr, Wilbur, 178, 190 155, 158, 197–199, 207, 214–216 Konig,¨ G., 120 Infinitesimals, 6, 13, 18, 20–21, 28, 31, 33, 35, 38, Korn, A., 108 44, 64, 80, 83, 86, 93, 101–102, 115, 131, Kossak, Ernst, 118 136, 138, 152–153, 187, 201, 213, 251–252, Kovalevskaya, Sofya (Sonia Kowalevsky), 159, 269 244 Infinity, 13, 21, 26–27, 46n, 50, 90–91, 200, Kowalewski, Gerhard, 90n, 115, 118 247–248, 261–262, 267–268, 272 Koyre,´ Alexandre, 273 Institut de France, 7, 12, 88, 105, 257, 272–273 Kranz, Merle, 272 Integral form of Lagrange remainder, 69–73 Kronecker, Leopold, 243 Integral, definite, 69, 71, 73, 118, 143–144, 160, Krylov, A. N., 108 218 Kuhn, Thomas, 130, 187, 190 Intermediate-value theorem for continuous Kulb,¨ Ph. H., 108 functions, 3–5, 65, 73–74, 81, 88, 91, 95–97, 99, 119, 139, 141–142, 145, 158, 203–204, L’Hospital, Marquis de, 45n, 64, 83n, 95, 116, 206 153 Invention, mathematical, xi–xv, 192, 249, 254 L’Huilier, Simon, 27–28, 31n, 34, 47n, 115–117, Islamic world, mathematics in, 148, 164–165, 140–141, 158, 216, 219, 222 245–247, 249, 258 Lacroix, Silvestre-François, 17n, 32, 69n, 70n, 79, Itard, Jean, 122, 226 80n, 107, 109, 113, 115, 120, 141, 158, 187, Iushkevich, A. P., see Juskevich 206, 211, 214, 216–217, 219, 222, 226, 257, 270 Jacobi, Carl G. J., 206, 219, 226, 234, 251 Ladd-Franklin, Christine, 244 Jaeger, Werner, 173 Ladyzhenskaya, Olga Alexandrovna, 245 Jahnke, Hans Niels, 121–122 Lafleur, L. J., 173, 189 Jefferson, Thomas, 165 Lagrange property of derivative, 5, 85–87, 94, Jessop, T. R., 240 142–143, 157, 202–203, 206, 253 Jesuits, 211, 248 Lagrange remainder of Taylor series, xii, 4–6, 14, Jouguet, Emile, 111 32, 33n, 34, 37–38, 46n, 51–52, 57, 61–62, 67, Jourdain, P. E. B., 17n, 18n, 20, 108, 118 69–79, 84, 87–88, 91, 93, 101, 109, 129, 139, Juschkewitsch, A. P., see Juskevich 143–144, 157, 204, 217, 231, 251, 253, 258 P1: JZZ MABK013-IND MABK013/Grabiner Trim Size: 7in× 10in July 30, 2010 9:41

Index 281

Lagrange, Joseph-Louis, xi–xiv, 3–6, 9–15, 17–35, modern views of his foundations of calculus as 37–51, 52n, 54, 56–62, 63–80, 81–97, 99, overly formalistic or as step backward, 3, 17, 101–103, 105, 111, 122, 129, 130n, 139–145, 62, 69, 251 155–159, 161, 168, 171, 187, 192–194, philosophy of mathematics, 12–13, 34, 38, 40, 199–207, 210–211, 213–214, 216–225, 231, 51, 69, 78, 88, 101, 201–202 234, 239–240, 251–254, 257–260, 266–274, “proof” of the parallel postulate, 257–260, et passim 269–272 applications of calculus to mechanics, 78, proof that a function with positive derivative on 204 an interval increases, xii, 5, 67, 71, 74, 84–87, calculus for the differential operator d (1772), 22, 103–104, 143, 157, 203, 253 23, 25 “proof” that every function has a Taylor series, characterization of continuity of functions, 5, 14, 18, 38, 48–51, 156 67–68, 85, 104 remainder term for Taylor series, xii, 4–6, 14, critique of earlier “foundations” for calculus, 32, 33n, 34, 37–38, 46n, 51–52, 57, 61–62, 26n, 64–65, 101, 156 67, 69–79, 84, 87–88, 91, 93, 101, 109, 129, definition of derivative, xi, 3, 14, 18, 23, 37, 47, 139, 143–144, 157, 204, 217, 231, 251, 253, 65–66, 80, 156–157, 202, 206, 252 258 definition of function, 13, 40n, 43, 129, 202 Resolution´ des equations´ numeriques´ , 5n, 21n, definition of higher-order derivatives, 14, 47, 66, 38, 59–62, 105–106, 129n, 130n, 146, 156, 202, 252 203n derivation of results of calculus from his Taylor series foundation for calculus, xi, 3, 14, power-series foundation, 14, 37, 63, 66–79, 18, 23, 37, 47, 65–66, 80, 156–157, 202, 206, 253 252 development of his ideas on the calculus, Theorie´ des fonctions analytiques, 4, 5n, 6n, 12, 1754–1797, 17–35 13, 17n, 35, 37, 38, 42n, 63–80, 82n, 83n, “Discours sur l’objet de la theorie´ des fonctions 85–86, 101–102, 105–108, 120–121, 123, analytiques,” 78, 82n, 105 129n, 146, 161, 193, 194n, 201n, 202n, 203n, early acceptance of first and last ratios, 17, 21 204n, 205n, 207, 216, 223, 226–227 f ’(x) notation for derivatives of, 4, 14, 24, view of algebra as the study of systems of 25n, 27n, 47, 65–66, 156–157, 197, 202, operations, 39–40, 184, 187, 194, 269 252 work on solvability of algebraic equations, 21, first paper (1754), 18, 19, 22, 105 22n general impact of the Fonctions analytiques, Lakanal, Joseph, 106 79–80, 253 Lamarck, Jean-Baptiste, 257 his foundations of calculus in the 1750s, 17, Lambert, Johann Heinrich, 26n, 52, 113, 20n, 21 267–268 inequality proof-techniques, 12, 84, 86, 97, Lampe, Emil, 119 99, 101, 103–104, 130, 142–143, 158, Landau, David, 173 214, 216 Landau, Lev, 141 influence on Ampere,` 85, 89, 92 Landen, John, 44–46, 116, 220 influence on Cauchy, 65, 85, 96, 99, 141–145, Laplace, Pierre-Simon, 9, 11, 79, 107, 109, 111, 206, 253 121–122, 129, 137, 139, 168–169, 173, 201n, lack of strong political and philosophical views, 210, 217–218, 227, 230, 239, 254, 257, 9–10 268–269, 271, 273 Leçons sur le calcul des fonctions, 5n, 13, 39n, “Last Supper” by Leonardo da Vinci, 262 50, 71, 73, 78–79, 84, 85n, 86, 101, 104–105, Latham, M. L., 173, 189 130n, 146, 202, 203n, 204n, 226 Launay, Louis de, 9n, 80n, 117 letter to Euler of 1759, 19–20 Lavirotte, Louis-Anne, 212 Mechanique analytique, 11–12, 18n, 63–64, 78, Lavoisier, Antoine, 10n, 165 105, 121–123, 168, 194, 213, 221, 226, 254, Lawson, Rachel, 172 258, 268–269, 273–274 Lazarsfeld, Paul, 173 memoir on parallels in the original French, Le Vavasseur, R., 113 published version, 273 (item #32) Lebesgue, Henri, 120, 129, 131n P1: JZZ MABK013-IND MABK013/Grabiner Trim Size: 7in× 10in July 30, 2010 9:41

282 Index

Lebovitz, Norman, 217 Mahoney, Michael, 161, 190, 227, 250 Lee, Desmond, 173 Malthus, Thomas, 171, 173, 237 Leeds, University of, 224, 272 Mancosu, Paolo, 190 Legendre, Adrien-Marie, 111, 200, 217–219, 226, Mandelbaum, Maurice, 174 257–259, 270, 273 Mangione, C., 122 Lehmer, Emma, 245 Manning, Kenneth R., 123 Leibniz, Gottfried Wilhelm, xii, 3, 9–10, 13, 19, Markuschewitsch, A. I., 113–114 21–22, 25, 31, 34, 38, 46, 49n, 64, 69, 78, 83, Marx, Karl, 168, 174 101, 111, 113–115, 123, 132, 135–136, 143, Matthiessen, L., 113 152–154, 156, 158, 160, 165, 167, 173–174, Maupertuis, Pierre Louis Moreau de, 124 187–188, 190, 194, 205–206, 210–211, 221, Maurice, Fred´ eric,´ 107–108 223, 226, 244, 249, 259–260, 263, 265–266, Maxima and minima, 31n, 41, 75, 143, 147–152, 268–269, 271, 274 155–157, 160, 195–199, 202, 204, 206, 214, Leonardo da Vinci, 262 216–217, 220, 231–232, 250 Leonardo of Pisa, 246 Maxwell, James Clerk, 123, 169, 174 Leslie, John, 205 May, Kenneth O., 123, 131n Lever, law of, 164, 259, 267–268 Mazzotti, Massimo, 123 Levinson, Stephen C., 273 McElroy, Davis Dunbar, 227 Limit concept, xii, 4–5, 11, 13, 21, 27–28, 32n, 61, Mean value theorem for derivatives, xii, 5, 6n, 73, 64–65, 77, 80, 81–84, 86, 90, 93–95, 99, 93, 96–99, 121, 143, 158, 203–204, 215, 101–102, 113, 116, 135–137, 140–142, 145, 253 153–154, 156–159, 200, 213, 216, 219–220, Mean value theorem for integrals, xii, 6n, 71–72, 222, 225, 233, 252–253 74n, 144 Lindgren, U., 122 Mecca, direction of, 246, 249 Linnaeus, C., 164 Menabrea, Luigi Federigo, 108 Littlewood, J. E., 127 Mendelsohn, Everett, 121, 146, 207n Lobachevsky, Nikolai Ivanovich, 169, 270 Merton College, Oxford, 247–248 Locke, John, 10, 111, 166 Merton, Robert, 130 Lord Kelvin, see Kelvin Merz, John Theodore, 111, 130n Lorey, Wilhelm, 108 Merzbach, Uta C., 7, 120 Loria, Gino, 107–109, 226 Method of Exhaustion, 27, 83, 214 Louis XIV, 255 Mill, John Stuart, 173, 189 Lovelace, Ada, 244 Mills, Stella, 227, 240 Luce, A. A., 160, 224, 240 Mittag-Lefﬂer, Gosta, 159 Lull, Raymond, 178 Moderate Party of the Church of Scotland, 212 Luneburg, Rudolf, 263 Modern views of Lagrange’s foundations of Lusternik, L. A., 122 calculus as overly formalistic or as a step Lutzen,¨ Jesper, 240 backward, 3, 17, 62, 69, 251 Molasses barrels, 225, 234–235 Mach, Ernst, 12, 111 Molk, Jules, 89, 113–114 Machamer, Peter K., 273 Monge, Gaspard, 9, 111, 187, 257, 270, 273 Mackey, George, 133 Monro, Alexander, 212 Maclaurin series, 70, 216, 231, 250, 255 Moritz, Robert E., 110 Maclaurin, Colin, xiii–xiv, 21n, 41, 52, 55–56, 64n, Mormann, T., 122 70, 75n, 112–113, 117, 121, 123, 135, Motte, Andrew, 146, 161, 174, 274 140–141, 155, 159, 193–202, 204–207, Moufang, Ruth, 245 209–241, 250–251, 253, 255 Murdoch, Patrick, 227, 236, 240 Account of Sir Isaac Newton’s Philosophical Discoveries, 212, 227, 237–240 Nash, Richard, 241 De Viribus Mentium Bonipetis, 231, 240n21 Nasr, Seyyed Hossein, 174 Treatise of Fluxions, xiii, 21n, 64n, 75n, 117, Natural Theology, 168, 171 121, 123, 193, 199, 206, 209–214, 216–224, Neil, William, 187 227, 231, 233, 240, 250, 251n Netto, E., 40n, 112–113 Maclean, L., 123 Neumann, Hanna, 245 P1: JZZ MABK013-IND MABK013/Grabiner Trim Size: 7in× 10in July 30, 2010 9:41

Index 283

Newton, Isaac, xii–xiv, 3, 9–11, 17, 18n, 20–21, 27, Paoli, Pietro, 107 34, 39, 41, 42n, 43n, 45–46, 51n, 54–59, 64, Pappus 4-line problem, 178–181, 187 78, 82n, 84n, 96, 101–102, 105, 111–115, Pappus of Alexandria, 149, 178–181, 184–185, 187 120–123, 130, 135–136, 141, 146, 151–156, Parallel postulate, xiv, 12, 121, 169, 257–259, 158–161, 164, 168, 170–174, 184, 186–190, 267–269, 271–272 194, 200–201, 205, 206n, 210, 212–214, 216, Parallelogram of forces, 268 218, 220–221, 224–227, 229–233, 235–241, Parmigianino, 264 244, 249–251, 261, 264, 266, 269, 271, 274 Parshall, Karen Hunger, xv Approximation method of, 54–59 Partial differential equations, 120, 129, 137, 154, Laws of motion, 154, 164, 168, 188, 213, 229, 221 231, 238, 249, 261 Partial fractions, 219 Method of Fluxions, 42n, 45, 51n, 54n, 55n, 115, Pascal, Blaise, xiii, 171, 174 153, 161 Pasquich, Johann, 109 Principia (Newton’s Mathematical Principles of Pasquier, L. G. du, 116 Natural Philosophy), xiv, 9, 10, 21n, 41, 64n, Peacock, George, 17n, 80n, 109, 216, 222, 226 111, 115, 123, 146, 153, 161, 164, 174, 205, Pearson, E. S., 227 214, 220, 229–230, 232–233, 241, 249, 274 Pearson, Karl, 219, 227 Newton-Cotes numerical integration formula, 200, Peirce, Benjamin, 170, 174 218 Penz, F., 272 Newtonian style, 229–241 Pepe, Luigi, 119, 123, 202n, 224, 227, 273n32 Nicomedes, conchoid of, 182 Perspective in painting, 260–262, 264, 270, 274 Nielsen, Niels, 108 Petrova, S. S., 122 Nieuwentijt, Bernhard, 26, 115, 117 Pez´ enas,´ R. P., 211, 226–227 Nightingale, Florence, 244 Phillipson, Nicholas, 227 Noether, Emmy, 244 Philo of Alexandria, 174 Noether, M., 113–114 Philosophy, xiii–xiv, 38, 111, 121, 132–133, Non-Euclidean geometry, 110, 131, 169–170, 264, 163–168, 170, 172, 174–175, 188–190, 267, 270, 272 205–206, 211–212, 222, 227, 233–234, Normal curve, 169 238–239, 261, 267, 269, 272, 274 Normal to a curve, 184–185 Picon, Antoine, 119 Notation for unknowns in algebra, 178, 180 Piero della Francesca, 262 Notations, choice of, xiii, 20, 24, 25n, 47, 57, 69, Pincherle, S., 119, 159 98, 110, 128, 132–133, 136, 152, 154, Pittarelli, G, 107 157–158, 165, 167, 180–181, 202, 210–211, Plato, xiii, 12, 101, 122, 133, 164, 167–170, 219, 221–222, 252 172–175, 248, 264 Novy, Lubos, 123 Playfair, John, 221, 227, 258–259 Nye, Mary Jo, 274 Poincare,´ Henri, 119, 192, 261, 274 Poisson, Simon-Denis, 136, 143–144, 217, 254, Oldenburg, Henry, 174 257, 268 Oleinik, Olga Arsen’evna, 245 Polya,´ George, xiii, 175, 188, 190–192, 207 Olson, Richard, 222, 227 Poncelet, Victor, 220 Oravas, G., 123 Pope, Alexander, 230 Orders of contact between curves, 32, 75, 143, 157, Popkin, Richard H., 174 199, 204, 220 Porter, Roy, 227 Oresme, Nicole, 129n, 247–248 Porter, Theodore M., 174 Orwell, George, 167, 172 Positivism, 79, 121, 168, 173 Osen, Lynn, 130n Prag, A., 190 Otte, Michael, 121 Prime and ultimate ratios, see First and last ratios Ovaert, J.-L., 123 Principle of sufﬁcient reason, 259–261, 265, Ovals of Descartes, 185 267–269 Pringsheim, Alfred, 70n, 89, 113–114 Pambuccian, Victor, 123 Proclus, 258 Panza, Marco, 121 Professionalization of mathematics, xiv P1: JZZ MABK013-IND MABK013/Grabiner Trim Size: 7in× 10in July 30, 2010 9:41

284 Index

Progress, idea of, xiii, 9–10, 165–166, 168, 171, Sandifer, C. Edward., 120, 123 173, 178, 188–189 Sarton, George, 107–110, 131n Projective geometry, 260, 262 Sartre, Jean-Paul, 166 Prony, G.-F. de, 166–167, 257 Schaaf, Willilam L., 172 Proportional sides of similar ﬁgures, 180–181, 258, “School of Athens,” by Raphael, 263 261, 269 Schroder,¨ Kurt, 116 Psychology, xiii, xv, 168, 170, 191–192, 254, Schubring, Gert, 123 263–264, 273 Schulze, Johann Karl Gottlieb, 26–27 Pulte, Helmut, 123, 269, 274 Schwarz, H. A., 159 Putnam, Hilary, 133 Sclopis, Federigo, 107 Scott, Charlotte Angas, 244 Queen’s College, Oxford, 243n, 254 Scott, J. F., 227 Quetelet, Adolphe, 168–169, 173–174 Scott, W. F., 237, 240 Scottish Enlightenment, 205, 209, 211–212, 225, Radick, Gregory, 272 227, 229, 240–241 Ramus, Petrus, 178 Scottish Excise Commission, 234 Randall, John Herman Jr., 111 Scottish Ministers’ Widows Fund, 236–237, 240 Rankenian Society, 212 Segner, J. A., 113 Raphael, 263 Seidel, V., 159 Rashed, Roshdi, 119, 123 Serret, M. J. A., 38n, 63n, 82n, 105, 122, 161, 193n, Rates of change, xi–xii, 101, 136, 150, 152, 156, 226 214, 269 Servois, Franc¸ois-Joseph, 79n, 109, 270 Redman, Ben Ray, 12n, 274 Seurat, Georges, 273 Reduction, method of, 176–178, 184–185 Shapin, Stephen, 227 Rees, Minna, 245 Simpson, Thomas, 44, 220, 251 Reid, Thomas, 263, 274 Simpson’s rule, 200, 219 Reiff, Richard, 113 Simson, Robert, 205, 222 Religion, 95, 139, 154, 167, 174, 210, 212, 229, Sinaceur, Hourya, 123 232–233, 235, 238–239, 244, 248, 254–255, Sine, origin of the term, 246 267 Sinnigen, William G., 174 Renaissance, 167–168, 245–247, 249, 261, 264, 272 Skinner, Andrew, 174 Residual Analysis of John Landen, 44–46, 116 Sluse, Rene,´ 151 Riccardi, Pietro, 107, 108 Smith, Adam, 166, 174, 205, 212, 229, 239 Richards, Joan L., 123, 270, 274 Smith, David Eugene, 173, 189 Riemann, Bernhard, 122, 138, 146, 170, 192, 210, Smith, G. E., 232, 241 274 Smithies, Frank, 123 Robinson, Abraham, 131 Snell’s law of refraction of light, 150 Robinson, Julia Bowman, 245 Social history, xiii, 225, 243–244 Rolle, Michel, 55n, 113 Socrates, 167 Rootselaar, B. van, 118 Somerville, Martha, 80n, 111 Rosenfeld, B. A., 274 Somerville, Mary, 80n, 111 Rosenkilde, Carl, 217 Space, xiv, 12, 167–168, 170, 213, 260–271, Rossner, Laurence, 217 273–274 Royal Bank of Scotland, 235 Spiess, Otto, 116 Royal Society of London, 216, 222 Spinoza, Baruch, xiii, 164, 174 Russ, S. B., 119, 123, 145 Steps in evolution of concept of derivative, 147 Russell, Bertrand, 165, 174, 190, 274 Stewart, Larry, 241 Rychlik, Karel, 118 Stewart, M. A., 227 Stirling, James, 44, 199n, 200, 213, 217–218, 228, Saccheri, Girolamo, 12n, 111, 169 251 Sadosky, Cora Ratto de, 245 Stirling’s formula, 200, 218 Sageng, Erik, 123, 205, 211, 223, 227, 241, Stokes, G. G., 159 251n Stolz, Otto, 95n, 117–119 P1: JZZ MABK013-IND MABK013/Grabiner Trim Size: 7in× 10in July 30, 2010 9:41

Index 285

Struik, Dirk J., 3n, 7, 11n, 110, 123, 127n, 146, 161, “True metaphysic” of calculus, 19–20, 88 190, 227 Truesdell, Clifford, 111, 124, 129n, 201n, 213, 227 Students, use of history of mathematics in the Turin, 18n, 19, 20n, 34–35, 105, 107–108, 116, education of, xii, xv, 97, 101, 130–131, 133, 119, 140, 202n, 206, 217, 224, 270 147, 160, 163, 172, 187–188, 194, 207, 243, Turnbull, H. W., 117, 222, 228, 250 254–255 Turnbull, Robert G., 273 Subsecant, 82–83 Turner, J. M. W., 264, 266, 274 Subtangent, 82–83 Tweedie, Charles, 117, 199n, 228 Sum of a series, deﬁnition of, 128n, 141, 200, Tymoczko, Thomas, 124 220–221 Swineshead, Richard, 247–248 Ugoni, Camillo, 108 Sylvester, J. J., xi, xv Universal agreement, xiv, 168, 233, 235–236, 267, Symmetry, 233, 259–262, 265, 267–268, 271 271–272 Synthesis, method of, 238 Universal arithmetic, 38–39, 43, 112, 269

Tangents, 18, 32, 69, 75–76, 83n, 101, 135–136, Vacca, Giovanni, 108 148–153, 157, 184–187, 195–199, 213, 216, Vagliente, Victor N., 122, 273 220, 249 Valperga-Caluso, Tommaso, 109 Tanguy, Yves, 270 Valson, C.-A., 4, 96, 118 Tannery, Paul, 189 Van Cleve, James, 274 Taton, Juliette, 227 Van der Warden, B. L., 128n Taton, Rene,´ 107, 109, 115, 120, 123–124, 199n, Van Egmond, Warren, 120 226–227, 274 Van Eyck, Jan, 264–265 Taussky-Todd, Olga, 245 Van Heuraet, Hendrik, 186–187 Taylor series, non-uniqueness of, 14, 157 Van Schooten, Jan, 184, 186–187, 249 Taylor series, xii, 3–6, 14, 15, 17–18, 19n, 20–23, Velocities in conceptualizing calculus, 10n, 11, 64, 25, 30–32, 33n, 34, 37–38, 41, 45–51, 56–60, 101–102, 139, 153, 194–195, 197, 201, 216, 62, 66–74, 76–77, 78n, 79, 83–85, 87–88, 91, 249 93–94, 101, 109, 113, 139, 142–144, 155–157, Vibrating string, 42n, 129, 137, 154 197–200, 202, 204, 207, 216–218, 222, Viete,` Franc¸ois [Franciscus Vieta], 10, 41, 148, 250–253, 258 178, 180 Taylor, Brook, 115, 117, 155 Virey, Julien, 107–108 Teaching and teachers of mathematics, xi–xv, Vivanti, G., 28n, 79n, 113–114 34–35, 79, 97, 101, 130, 140–141, 147, 160, Voltaire, 12n, 167, 172, 174, 267, 274 172, 187–188, 190–191, 207, 221, 243, 272 Volterra, Vito, 107 Teaching as a motivation for work on foundations, Von Humboldt, Alexander, 107 34–35, 97, 140–141 Voss, A., 113–114 Teich, Mikulas, 227 Vuillemin, Jules, 108 Terrall, Mary, 124 Thackray, Arnold, 227 Wagner, Mark, 274 Thayer, H. S., 174 Wallace, Robert, 236–237 Theology, 130, 164, 168, 171, 232–233, 239, 241, Wallace, William, 80 249 Wallis, John, 10, 150, 186–187, 249 Thomas, George B., 71n, 73n, 111 Ward, Seth, 211 Thomson, William, see Kelvin Waring, Edward, 220, 251 Tiulina, I. A., 124 Wave equation, see Vibrating string Todhunter, 201n, 227 Webster, Alexander, 236–237 Toeplitz, Otto, 113–114 Weierstrass, Karl, xi–xii, 3, 15, 51, 63, 74, 80–81, Torino, see Turin 84, 91n, 97, 99, 101–102, 118–120, 122, 138, Torretti, R., 274 140, 159–160, 192, 206–207, 210, 214, 254 Transcendental functions, series development of, Weil, Andre,´ 124 44, 46–47, 245, 252 Weizenbaum, Joseph, 171, 174 Tropfke, Johannes, 113 Westfall, Richard S., 190, 240, 250n P1: JZZ MABK013-IND MABK013/Grabiner Trim Size: 7in× 10in July 30, 2010 9:41

286 Index

Wheeler, Anna Johnson Pell, 244 Women of Mathematics, MAA Poster, 244n Whig interpretation of history, 223 Wood, Paul B., 241 Whiteside, D. T., 84n, 114–115, 161, 190, 227, 249, Woodhouse, Robert, 43n, 82n, 114, 206n, 220, 224 249n, 250, 250n Wordsworth, William, xiii, 171 Whitman, Anne, 123, 241 Wren, Christopher, 186–187 Whitman, Walt, 171 Wronski, see Hoene-Wronski Widder, David V., 75n, 111 Wussing, Hans, 124 Wiener, Philip P., 173, 190 Winter, Eduard, 119, 226 Young, Grace Chisholm, 244 Winter, H. J. J., 44n, 116 Youschkevitch, A. P., see Juskevich, A. P. Withers, Charles W. J., 241 Wolf, J.-C., 19n Zamyatin, Evgeny, 172 Wolf, Rudolf, 117 Zeros as foundation for the calculus, see Euler Wolfson, Harry Austryn, 174 Zimmerman, Karl, 28, 29n, 30n, 31n, 32n, 33, 116 P1: JZP MABK013-ATA MABK013/Grabiner Trim Size: 7in× 10in July 28, 2010 11:24

About the Author

Judith Victor Grabiner received her B. S. in Mathematics from the University of Chicago, M. A. in the History of Science from Radcliffe College, and Ph. D. in the History of Science from Harvard in 1966. She is a member of both Phi Beta Kappa and Sigma Xi. She was a Sigma Xi Society National Lecturer in 1988–90. Among the many awards she has received are three Carl Allendoerfer Awards (for the best article in the Mathematics Magazine, 1984, 1988, 1996), and four Lester Ford Awards (for the best article in the American Mathematical Monthly, 1984, 1998, 2005, 2009). In 2003, she won the national Deborah and Franklin Tepper Haimo Award for Distinguished College or University Teaching of the Mathematical Association of America. Professor Grabiner has held a Woodrow Wilson Fellowship, an American Council of Learned Societies Fellowship, and two National Science Foundation fellowships. Professor Grabiner has had Visiting Scholar positions in England, Scotland, Australia, and Denmark. She currently teaches at Pitzer College, one of the Claremont Colleges in Claremont, California, where she is the Flora Sanborn Pitzer Professor of Mathematics. She is the author of numerous articles in the history of mathematics, and her books include: The Origins of Cauchy’s Rigorous Calculus (MIT Press, 1981), which was reprinted by Dover in 2005. She is also the author of The Calculus as Algebra: J.-L. Lagrange, 1736–1813 (Garland, 1990), which is reprinted as part of the present volume.

287 AMS / MAA SPECTRUM VOL AMS / MAA SPECTRUM VOL 64 64 The Calculus as Algebra and Selected Writings A Historian Looks Back 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 A Historian Looks Back the results of her investigations not principally to other histori- The Calculus as Algebra and Selected Writings ans, 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 Alge- bra: J.-L. Lagrange, 1736–1813. The book describes the achievements, setbacks, and influence of Lagrange’s pioneering attempt Judith V. Grabiner 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 Judith V. Grabiner 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 civiliza- tion. 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

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