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    The Calculus Gallery: Masterpieces from Newton to Lebesgue
    The Calculus Gallery: Masterpieces from Newton to Lebesgue
    The Calculus Gallery: Masterpieces from Newton to Lebesgue
    Ebook352 pages3 hours

    The Calculus Gallery: Masterpieces from Newton to Lebesgue

    By William Dunham

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    More than three centuries after its creation, calculus remains a dazzling intellectual achievement and the gateway to higher mathematics. This book charts its growth and development by sampling from the work of some of its foremost practitioners, beginning with Isaac Newton and Gottfried Wilhelm Leibniz in the late seventeenth century and continuing to Henri Lebesgue at the dawn of the twentieth. Now with a new preface by the author, this book documents the evolution of calculus from a powerful but logically chaotic subject into one whose foundations are thorough, rigorous, and unflinching—a story of genius triumphing over some of the toughest, subtlest problems imaginable. In touring The Calculus Gallery, we can see how it all came to be.
    LanguageEnglish
    PublisherPrinceton University Press
    Release dateNov 13, 2018
    ISBN9780691184548
    The Calculus Gallery: Masterpieces from Newton to Lebesgue

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      Book preview

      The Calculus Gallery - William Dunham

      INTRODUCTION

      The calculus, wrote John von Neumann (1903–1957), was the first achievement of modern mathematics, and it is difficult to overestimate its importance [1].

      Today, more than three centuries after its appearance, calculus continues to warrant such praise. It is the bridge that carries students from the basics of elementary mathematics to the challenges of higher mathematics and, as such, provides a dazzling transition from the finite to the infinite, from the discrete to the continuous, from the superficial to the profound. So esteemed is calculus that its name is often preceded by the, as in von Neumann’s observation above. This gives "the calculus a status akin to the law"—that is, a subject vast, self-contained, and awesome.

      Like any great intellectual pursuit, the calculus has a rich history and a rich prehistory. Archimedes of Syracuse (ca. 287–212 BCE) found certain areas, volumes, and surfaces with a technique we now recognize as proto-integration. Much later, Pierre de Fermat (1601–1665) determined slopes of tangents and areas under curves in a remarkably modern fashion. These and many other illustrious predecessors brought calculus to the threshold of existence.

      Nevertheless, this book is not about forerunners. It goes without saying that calculus owes much to those who came before, just as modern art owes much to the artists of the past. But a specialized museum—the Museum of Modern Art, for instance—need not devote room after room to premodern influences. Such an institution can, so to speak, start in the middle. And so, I think, can I.

      Thus I shall begin with the two seventeenth-century scholars, Isaac Newton (1642–1727) and Gottfried Wilhelm Leibniz (1646–1716), who gave birth to the calculus. The latter was first to publish his work in a 1684 paper whose title contained the Latin word calculi (a system of calculation) that would attach itself to this new branch of mathematics. The first textbook appeared a dozen years later, and the calculus was here to stay.

      As the decades passed, others took up the challenge. Prominent among these pioneers were the Bernoulli brothers, Jakob (1654–1705) and Johann (1667–1748), and the incomparable Leonhard Euler (1707–1783), whose research filled many thousands of pages with mathematics of the highest quality. Topics under consideration expanded to include limits, derivatives, integrals, infinite sequences, infinite series, and more. This extended body of material has come to be known under the general rubric of analysis.

      With increased sophistication came troubling questions about the underlying logic. Despite the power and utility of calculus, it rested upon a less-than-certain foundation, and mathematicians recognized the need to recast the subject in a precise, rigorous fashion after the model of Euclid’s geometry. Such needs were addressed by nineteenth-century analysts like Augustin-Louis Cauchy (1789–1857), Georg Friedrich Bernhard Riemann (1826–1866), Joseph Liouville (1809–1882), and Karl Weierstrass (1815–1897). These individuals worked with unprecedented care, taking pains to define their terms exactly and to prove results that had hitherto been accepted uncritically.

      But, as often happens in science, the resolution of one problem opened the door to others. Over the last half of the nineteenth century, mathematicians employed these logically rigorous tools in concocting a host of strange counterexamples, the understanding of which pushed analysis ever further toward generality and abstraction. This trend was evident in the set theory of Georg Cantor (1845–1918) and in the subsequent achievements of scholars like Vito Volterra (1860–1940), René Baire (1874–1932), and Henri Lebesgue (1875–1941).

      By the early twentieth century, analysis had grown into an enormous collection of ideas, definitions, theorems, and examples—and had developed a characteristic manner of thinking—that established it as a mathematical enterprise of the highest rank.

      What follows is a sampler from that collection. My goal is to examine the handiwork of those individuals mentioned above and to do so in a manner faithful to the originals yet comprehensible to a modern reader. I shall discuss theorems illustrating the development of calculus over its formative years and the genius of its most illustrious practitioners. The book will be, in short, a great theorems approach to this fascinating story.

      To this end I have restricted myself to the work of a few representative mathematicians. At the outset I make a full disclosure: my cast of characters was dictated by personal taste. Some whom I have included, like Newton, Cauchy, Weierstrass, would appear in any book with similar objectives. Some, like Liouville, Volterra and Baire, are more idiosyncratic. And others, like Gauss, Bolzano, and Abel, failed to make my cut.

      Likewise, some of the theorems I discuss are known to any mathematically literate reader, although their original , for example, is included simply as a demonstration of his analytic wizardry.

      Each result, from Newton’s derivation of the sine series to the appearance of the gamma function to the Baire category theorem, stood at the research frontier of its day. Collectively, they document the evolution of analysis over time, with the attendant changes in style and substance. This evolution is striking, for the difference between a theorem from Lebesgue in 1904 and one from Leibniz in 1690 can be likened to the difference between modern literature and Beowulf. Nonetheless—and this is critical—I believe that each theorem reveals an ingenuity worthy of our attention and, even more, of our admiration.

      Of course, trying to characterize analysis by examining a few theorems is like trying to characterize a thunderstorm by collecting a few raindrops. The impression conveyed will be hopelessly incomplete. To undertake such a project, an author must adopt some fairly restrictive guidelines.

      One of mine was to resist writing a comprehensive history of analysis. That is far too broad a mission, and, in any case, there are many works that describe the development of calculus. Some of my favorites are mentioned explicitly in the text or appear as sources in the notes at the end of the book.

      A second decision was to exclude topics from both multivariate calculus and complex analysis. This may be a regrettable choice, but I believe it is a defensible one. It has imposed some manageable boundaries upon the contents of the book and thereby has added coherence to the tale. Simultaneously, this restriction should minimize demands upon the reader’s background, for a volume limited to topics from univariate, real analysis should be understandable to the widest possible audience.

      This raises the issue of prerequisites. The book’s objectives dictate that I include much technical detail, so the mathematics necessary to follow these theorems is substantial. Some of the early results require considerable algebraic stamina in chasing formulas across the page. Some of the later ones demand a refined sense of abstraction. All in all, I would not recommend this for the mathematically faint-hearted.

      At the same time, in an attempt to favor clarity over conciseness, I have adopted a more conversational style than one would find in a standard text. I intend that the book be accessible to those who have majored or minored in college mathematics and who are not put off by an integral here or an epsilon there. My goal is to keep the prerequisites as modest as the topics permit, but no less so. To do otherwise, to water down the content, would defeat my broader purpose.

      So, this is not primarily a biography of mathematicians, nor a history of calculus, nor a textbook. I say this despite the fact that at times I provide biographical information, at times I discuss the history that ties one topic to another, and at times I introduce unfamiliar (or perhaps long forgotten) ideas in a manner reminiscent of a textbook. But my foremost motivation is simple: to share some favorite results from the rich history of analysis.

      And this brings me to a final observation.

      In most disciplines there is a tradition of studying the major works of illustrious predecessors, the so-called masters of the field. Students of literature read Shakespeare; students of music listen to Bach. In mathematics such a tradition is, if not entirely absent, at least fairly uncommon. This book is meant to address that situation. Although it is not intended as a history of the calculus, I have come to regard it as a gallery of the calculus.

      To this end, I have assembled a number of masterpieces, although these are not the paintings of Rembrandt or Van Gogh but the theorems of Euler or Riemann. Such a gallery may be a bit unusual, but its objective is that of all worthy museums: to serve as a repository of excellence.

      Like any gallery, this one has gaps in its collection. Like any gallery, there is not space enough to display all that one might wish. These limitations notwithstanding, a visitor should come away enriched by an appreciation of genius. And, in the final analysis, those who stroll among the exhibits should experience the mathematical imagination at its most profound.

      CHAPTER 1

      Newton

      Isaac Newton

      Isaac Newton (1642–1727) stands as a seminal figure not just in mathematics but in all of Western intellectual history. He was born into a world where science had yet to establish a clear supremacy over medieval superstition. By the time of his death, the Age of Reason was in full bloom. This remarkable transition was due in no small part to his own contributions.

      For mathematicians, Isaac Newton is revered as the creator of calculus, or, to use his name for it, of fluxions. Its origin dates to the mid-1660s when he was a young scholar at Trinity College, Cambridge. There he had absorbed the work of such predecessors as René Descartes (1596–1650), John Wallis (1616–1703), and Trinity’s own Isaac Barrow (1630–1677), but he soon found himself moving into uncharted territory. During the next few years, a period his biographer Richard Westfall characterized as one of incandescent activity, Newton changed forever the mathematical landscape [1]. By 1669, Barrow himself was describing his colleague as a fellow of our College and very young … but of an extraordinary genius and proficiency [2].

      In this chapter, we look at a few of Newton’s early achievements: his generalized binomial expansion for turning certain expressions into infinite series, his technique for finding inverses of such series, and his quadrature rule for determining areas under curves. We conclude with a spectacular consequence of these: the series expansion for the sine of an angle. Newton’s account of the binomial expansion appears in his epistola prior, a letter he sent to Leibniz in the summer of 1676 long after he had done the original work. The other discussions come from Newton’s 1669 treatise De analysi per aequationes numero terminorum infinitas, usually called simply the De analysi.

      Although this chapter is restricted to Newton’s early work, we note that early Newton tends to surpass the mature work of just about anyone else.

      GENERALIZED BINOMIAL EXPANSION

      By 1665, Isaac Newton had found a simple way to expand—his word was reduce—binomial expressions into series. For him, such reductions would be a means of recasting binomials in alternate form as well as an entryway into the method of fluxions. This theorem was the starting point for much of Newton’s mathematical innovation.

      As described in the epistola prior, the issue at hand was to reduce the binomial (P + PQ)m/n and to do so whether m/n is integral or (so to speak) fractional, whether positive or negative [, etc. I write a¹/², a¹/³, a⁵/³, and instead of 1/a, 1/aa, 1/a³, I write a−1, a−2, a−3" [4]. Apparently readers of the day needed a gentle reminder.

      Newton discovered a pattern for expanding not only elementary binomials like (1 + x. The reduction, as Newton explained to Leibniz, obeyed the rule

      where each of A, B, C, … represents the previous term, as will be illustrated below. This is his famous binomial expansion, although perhaps in an unfamiliar guise.

      , m = 1, and n = 2. Thus,

      To identify A, B, C, and the rest, we recall that each is the immediately preceding term. Thus, A = (c²)¹/² = c, giving us

      Likewise B —so at this stage we have

      . Working from left to right in this fashion, Newton arrived at

      Obviously, the technique has a recursive flavor: one finds the coefficient of x⁸ from the coefficient of x⁶, which in turn requires the coefficient of x⁴, and so on. Although the modern reader is probably accustomed to a direct statement of the binomial theorem, Newton’s recursion has an undeniable appeal, for it streamlines the arithmetic when calculating a numerical coefficient from its predecessor.

      For the record, it is a simple matter to replace A, B, C, … by their equivalent expressions in terms of P and Q, then factor the common Pm/n from both sides of (1), and so arrive at the result found in today’s texts:

      Newton likened such reductions to the conversion of square roots into infinite decimals, and he was not shy in touting the benefits of the operation. It is a convenience attending infinite series, he wrote in 1671,

      that all kinds of complicated terms … may be reduced to the class of simple quantities, i.e., to an infinite series of fractions whose numerators and denominators are simple terms, which will thus be freed from those difficulties that in their original form seem’d almost insuperable. [5]

      To be sure, freeing mathematics from insuperable difficulties is a worthy undertaking.

      , which Newton put to good use in a result we shall discuss later in the chapter. We first write this as (1 − x²)−1/2, identify m = − 1, n = 2, and Q = − x², and apply (2):

      Newton would check an expansion like (3) by squaring the series and examining the answer. If we do the same, restricting our attention to terms of degree no higher than x⁸, we get

      where all of the coefficients miraculously turn out to be 1 (try it!). The resulting product, of course, is an infinite geometric series with common ratio

      x. But if the square of the series in (. Voila!

      Newton regarded such calculations as compelling evidence for his general result. He asserted that the common analysis performed by means of equations of a finite number of terms may be extended to such infinite expressions albeit we mortals whose reasoning powers are confined within narrow limits, can neither express nor so conceive all the terms of these equations, as to know exactly from thence the quantities we want [6].

      INVERTING SERIES

      Having described a method for reducing certain binomials to infinite series of the form z = A + Bx + Cx² + Dx³ + ⋅ ⋅ ⋅, Newton next sought a way of finding the series for x in terms of z. In modern terminology, he was seeking the inverse relationship. The resulting technique involves a bit of heavy algebraic lifting, but it warrants our attention for it too will appear later on. As Newton did, we describe the inversion procedure by means of a specific example.

      Beginning with the series z = x − x² + x³ − x⁴ + ⋅ ⋅ ⋅, we rewrite it as

      and discard all powers of x greater than or equal to the quadratic. This, of course, leaves x − z = 0, and so the inverted series begins as x = z.

      Newton was aware that discarding all those higher degree terms rendered the solution inexact. The exact answer would have the form x = z + p, where p is a series yet to be determined. Substituting z + p for x in (4) gives

      which we then expand and rearrange to get

      Next, jettison the quadratic, cubic, and higher degree terms in p and solve to get

      Newton now did a second round of weeding, as he tossed out all but the lowest power of z in numerator and denominator. Hence p , so the inverted series at this stage looks like x = z + p = z + z².

      But p is not exactly z². Rather, we say p = z² + q, where q is a series to be determined. To do so, we substitute into (5) to get

      We expand and collect terms by powers of q:

      As before, discard terms involving powers of q . At this point, the series looks like x = z + z² + q = z + z² + z³.

      The process would be continued by substituting q = z³ + r into (6). Newton, who had a remarkable tolerance for algebraic monotony, seemed able to continue such calculations ad infinitum (almost). But eventually even he was ready to step back, examine the output, and seek a pattern. Newton put it this way: Let it be observed here, by the bye, that when 5 or 6 terms … are known, they may be continued at pleasure for most part, by observing the analogy of the progression [7].

      For our example, such an examination suggests that x = z + z² + z³ + z⁴ + z⁵ + ⋅ ⋅ ⋅ is the inverse of the series z = x − x² + x³ − x⁴ + ⋅ ⋅ ⋅ with which we began.

      In what sense can this be trusted? After all, Newton discarded most of his terms most of the time, so what confidence remains that the answer is correct?

      Again, we take comfort in the following check. The original series z = x − x² + x³ − x⁴ + ⋅ ⋅ ⋅ is geometric with common ratio − x, which we recognize to be the sum of the geometric series z + z² + z³ + z⁴ + z⁵ + ⋅ ⋅ ⋅ . This is precisely the result to which Newton’s procedure had led us. Everything seems to be in working order.

      The techniques encountered thus far—the generalized binomial expansion and the inversion of series—would be powerful tools in Newton’s hands. There remains one last prerequisite, however, before we can truly appreciate the master at work.

      QUADRATURE RULES FROM THE DE ANALYSI

      In his De analysi of 1669, Newton promised to describe the method "which I had devised some considerable time ago, for measuring

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