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A Historian Looks Back

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

Judith V. Grabiner
Series: Spectrum
Copyright Date: 2010
Edition: 1
Pages: 304
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  • Book Info
    A Historian Looks Back
    Book Description:

    The centerpiece of the book is The Calculus as Algebra: J.-L. Lagrange, 1736–1813. This section 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.

    eISBN: 978-1-61444-506-7
    Subjects: Mathematics

Table of Contents

  1. Front Matter
    (pp. i-viii)
  2. Table of Contents
    (pp. ix-x)
  3. Introduction
    (pp. xi-xvi)

    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 … it has frequent...

  4. Part I. The Calculus as Algebra

    • Preface to the Garland Edition
      (pp. 3-6)
    • Acknowledgements
      (pp. 7-8)
    • Introduction
      (pp. 9-16)

      The eighteenth century was an age in which the power of the human mind seemed unlimited. Philosophers believed that they had discovered the nature of human understanding; scientists or “natural philosophers,” that they had found the basic laws of the cosmos; observers of society thought that they knew the principles on which government was based. Human history was expected to show perpetual progress in the future.

      Such predictions were not made merely out of optimistic desire; they were based on the real success that men saw had been achieved by the sciences. The science whose success most captured the imagination...

    • 1 The Development of Lagrange’s Ideas on the Calculus: 1754–1797
      (pp. 17-36)

      The evolution of Lagrange’s ideas on the calculus can be traced through his writings from 1754—the year of his first paper—to 1797, when he published the first edition ofFA. It is not generally appreciated how much Lagrange’s views on the calculus changed throughout his career. In particular, the point of view of his brief early writings has usually—and, in my view, mistakenly—been identified with the conclusions of his more mature thought.¹

      Lagrange’s early work on the calculus (1754–1761) presented investigations of formal relationships which hold between derivatives, differentials, and integrals, but these were not...

    • 2 The Algebraic Background of the Theory of Analytic Functions
      (pp. 37-62)

      TheThéorie des fonctions analytiquesof Joseph-Louis Lagrange has always appeared puzzling to historians of the calculus. Viewing the work from the point of view of the history of the concepts of the calculus, historians have had difficulty in understanding why Lagrange should have begun with the assertion that all functions had Taylor series; they have attributed it to a formalist tendency. Looking at Lagrange’s definition of thenth derivative as thenth coefficient in the Taylor series multiplied byn!, it has been hard to see why the work commanded any attention at all since it defines a relatively...

    • 3 The Contents of the Fonctions Analytiques
      (pp. 63-80)

      In 1797, Lagrange published theThéorie des fonctions analytiques, the expression of his mature views on the nature of the calculus. These views were, as we have seen, the product of a long period of concern with the foundations of the calculus. He had concluded that algebra would provide the only satisfactory foundation; as we have seen, the algebra of the eighteenth century was sufficiently rich to provide a basis for the many diverse results of theFonctions analytiques.

      TheThéorie des fonctions analytiques, in this first edition, is not an especially attractive work, particularly in comparison with the second...

    • 4 From Proof-Technique to Definition: The Pre-History of Delta-Epsilon Methods
      (pp. 81-100)

      The nineteenth century is often called “the Age of Rigor in Analysis.” This name can be understood as “the age of delta and epsilon.” In defining limits, in proving theorems, in exemplifying clear thinking, the rigor known as “Weierstrassian” represents to us the greatest achievement of the nineteenth-century analysts.

      The development of these principles from the less adequate foundations for the calculus of the eighteenth century illustrates a trend common to many branches of nineteenth century mathematics, a trend toward increasing generality and abstraction. In the area of the foundations of analysis, this trend can be viewed as a retreating...

    • Conclusion
      (pp. 101-102)

      Lagrange’s commitment to the necessity of an algebraic foundation for the calculus led him to the major accomplishments of theFAandCF: the sharply argued critique of the prevailing eighteenth century foundations for the calculus, the study of functions by means of their power series expansions, the derivation and use of the remainder term of the Taylor series, and the development of what are essentially delta-epsilon proofs. His influence on the development of analysis in the nineteenth century rests on these accomplishments.¹

      The development of the foundations of the calculus from Newton and Leibniz to Weierstrass, and the length...

    • Appendix
      (pp. 103-104)
    • Bibliography
      (pp. 105-124)
  5. Part II. Selected Writings

    • 1 The Mathematician, the Historian, and the History of Mathematics
      (pp. 127-134)

      The historian’s basic questions, whether he is a historian of mathematics or of political institutions, are: what was the past like? and how did the present come to be? The second question—how did the present come to be?—is the central one in the history of mathematics, whether done by historian or mathematician. But the historian’s view of both past and present is quite different from that of the mathematician. The historian is interested in the past in its full richness, and sees any present fact as conditioned by a complex chain of causes in an almost unlimited past....

    • 2 Who Gave You the Epsilon? Cauchy and the Origins of Rigorous Calculus
      (pp. 135-146)

      Perhaps this exchange will remind us that the rigorous basis for the calculus is not at all intuitive—in fact, quite the contrary. The calculus is a subject dealing with speeds and distances, with tangents and areas—not inequalities. When Newton and Leibniz invented the calculus in the late seventeenth century, they did not use delta-epsilon proofs. It took a hundred and fifty years to develop them. This means that it was probably very hard, and it is no wonder that a modern student finds the rigorous basis of the calculus difficult. How, then, did the calculus get a rigorous...

    • 3 The Changing Concept of Change: The Derivative from Fermat to Weierstrass
      (pp. 147-162)

      Some years ago while teaching the history of mathematics, I asked my students to read a discussion of maxima and minima by the seventeenth-century mathematician, Pierre Fermat. To start the discussion, I asked them, “Would you please define a relative maximum?” They told me it was a place where the derivative was zero. “If that’s so,” I asked, “then what is the definition of a relative minimum?” They told me,that’sa place where the derivative is zero. “Well, in that case,” I asked, “what is the difference between a maximum and a minimum?” They replied that in the case...

    • 4 The Centrality of Mathematics in the History of Western Thought
      (pp. 163-174)

      Since this paper was first given to educators, let me start with a classroom experience. It happened in a course in which my students had read some of Euclid’sElements of Geometry. A student, a social science major, said to me, “I never realized mathematics was like this. Why, it’s like philosophy!” That is no accident, for philosophy is like mathematics. When I speak of the centrality of mathematics in western thought, it is this student’s experience I want to recapture—to reclaim the context of mathematics from the hardware store with the rest of the tools and bring it...

    • 5 Descartes and Problem-Solving
      (pp. 175-190)

      What does Descartes have to teach us about solving problems? At first glance it seems easy to reply. Descartes says a lot about problem-solving. So we could just quote what he says in theDiscourse on Method[12] and in hisRules for Direction of the Mind([2], pp. 9–11). Then we could illustrate these methodological rules from Descartes’ major mathematical work,La Géométrie[13]. After all, Descartes claimed he did his mathematical work by following his “method.” And the most influential works in modern mathematics—calculus textbooks—all contain sets of rules for solving word problems, rather like...

    • 6 The Calculus as Algebra, the Calculus as Geometry: Lagrange, Maclaurin, and their Legacy
      (pp. 191-208)

      Given a regular hexagon and a point in its plane: draw a straight line through the given point that divides the given hexagon into two parts of equal area.¹ Please stop for a few moments, solve the problem, and think about the way you solved it.

      Did you draw an actual diagram? Did you draw a mental diagram? Were you able to solve the problem without drawing one at all? If you had a diagram, do you find it hard to understand how others could proceed without one? Were you motivated at all by analytic considerations? If you were not,...

    • 7 Was Newton’s Calculus a Dead End? The Continental Influence of Maclaurin’s Treatise of Fluxions
      (pp. 209-228)

      Eighteenth-century Scotland was an internationally-recognized center of knowledge, “a modern Athens in the eyes of an enlightened world.” [74, p. 40] [81] The importance of science, of the city of Edinburgh, and of the universities in the Scottish Enlightenment has often been recounted. Yet a key figure, Colin Maclaurin (1698–1746), has not been highly rated. It has become a commonplace not only that Maclaurin did little to advance the calculus, but that he did much to retard mathematics in Britain—although he had (fortunately) no influence on the Continent. Standard histories have viewed Maclaurin’s major mathematical work, the two-volume...

    • 8 Newton, Maclaurin, and the Authority of Mathematics
      (pp. 229-242)

      Sir Isaac Newton revolutionized physics and astronomy in his bookMathematical Principles of Natural Philosophy[27]. This book of 1687, better known by its abbreviated Latin title as thePrincipia, contains Newton’s three laws of motion, the law of universal gravitation, and the basis of all of classical mechanics. As one approaches this great work, a key question is: How did Newton do all of this? An equally important question is: Can Newton’s methods work on any area of inquiry? Newton’s contemporaries hoped that the answer to the second question was yes: that his methods would be universally effective, whether...

    • 9 Why Should Historical Truth Matter to Mathematicians? Dispelling Myths while Promoting Maths
      (pp. 243-256)

      Who is the audience for scholarship in the history of mathematics? Historians of mathematics, perhaps, or maybe some historians of science. But a much larger group is mathematicians and students of mathematics. I talk to mathematicians a lot, and I believe that we are in fact doing something of interest to mathematicians. We have an advantage over historians of other subjects. A distinguished mathematician asked me recently what Euclid had said about a particular result. I doubt that many historians of physics are asked by physicists, “Tell me, how did Aristotle prove that the earth can’t move?” Mathematicians use history...

    • 10 Why Did Lagrange “Prove” the Parallel Postulate?
      (pp. 257-274)

      We begin with an often-told story from theBudget of Paradoxesby Augustus de Morgan: “Lagrange, in one of the later years of his life, imagined” that he had solved the problem of proving Euclid’s parallel postulate. “He went so far as to write a paper, which he took with him to the [Institut de France], and began to read it.”

      But, De Morgan continues, “something struck him which he had not observed: he muttered ‘Il faut que j’y songe encore’ [I’ve got to think about this some more] and put the paper in his pocket” [8, p. 288].


  6. Index
    (pp. 275-286)
  7. About the Author
    (pp. 287-287)