Sherlock Holmes in Babylon

Sherlock Holmes in Babylon: and Other Tales of Mathematical History

Marlow Anderson
Victor Katz
Robin Wilson
Series: Spectrum
Copyright Date: 2004
Edition: 1
Pages: 398
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  • Book Info
    Sherlock Holmes in Babylon
    Book Description:

    This book is a collection of 44 articles on the history of mathematics, published in MAA journals over the past 100 years. Covering a span of almost 4000 years, from the ancient Babylonians to the eighteenth century, it chronicles the enormous changes in mathematical thinking over this time, as viewed by distinguished historians of mathematics from the past (Florian Cajori, Max Dehn, David Eugene Smith, Julian Lowell Coolidge, and Carl Boyer etc.) and the present. Each of the four sections of the book (Ancient Mathematics, Medieval and Renaissance Mathematics, The Seventeenth Century, The Eighteenth Century) is preceded by a Foreword, in which the articles are put into historical context, and followed by an Afterword, in which they are reviewed in the light of current historical scholarship. In more than one case, two articles on the same topic are included, to show how knowledge and views about the topic changed over the years. This book will be enjoyed by anyone interested in mathematics and its history—and in particular by mathematics teachers at secondary, college, and university levels.

    eISBN: 978-1-61444-503-6
    Subjects: Mathematics

Table of Contents

  1. Front Matter
    (pp. i-vi)
  2. Introduction
    (pp. vii-viii)

    For the past one hundred years, the Mathematical Association of America has been publishing high-quality articles on the history of mathematics, some written by distinguished historians such as Florian Cajori, Julian Lowell Coolidge, Max Dehn, David Eugene Smith, Carl Boyer, and others. Many well-known historians of the present day also contribute to the MAA’s journals. Some years ago, Robin Wilson and Marlow Anderson, along with the late John Fauvel, a distinguished and sorely missed historian of mathematics, decided that it would be useful to reprint a selection of these papers and to set them in the context of modern historical...

  3. Table of Contents
    (pp. ix-x)
  4. Ancient Mathematics
    • Foreword
      (pp. 3-4)

      The twentieth century saw great strides in our understanding of the mathematics of ancient times. This was often achieved through the combined work of archaeologists, philologists, and historians of mathematics.

      We especially see how this understanding has grown in the study of the mathematics of Mesopotamia. Although the clay tablets on which this mathematics was written were excavated beginning in the nineteenth century, it was not until early in the twentieth century that a careful study of the mathematics on some of these tablets was undertaken. In particular, the tablet known as Plimpton 322 was first published by Neugebauer and...

    • Sherlock Holmes in Babylon
      (pp. 5-13)

      Let me begin by clarifying the title “Sherlock Holmes in Babylon.” Lest some members of the Baker Street Irregulars be misled, my topic is the archaeology of mathematics, and my objective is to retrace a small portion of the research of two scholars: Otto Neugebauer, who is a recipient of the Distinguished Service Award, given to him by the Mathematical Association of America in 1979, and his colleague and long-time collaborator, Abraham Sachs. It is also a chance for me to repay both of them a personal debt. I went to Brown University in 1947, and as a new Assistant...

    • Words and Pictures: New Light on Plimpton 322
      (pp. 14-26)

      In this paper I shall discuss Plimpton 322, one of the world’s most famous ancient mathematical artefacts [Figure 1]. But I also want to explore the ways in which studying ancient mathematics is, or should be, different from researching modern mathematics. One of the most cited analyses of Plimpton 322, published some twenty years ago, was called “Sherlock Holmes in Babylon” [4]. This enticing title gave out the message that deciphering historical documents was rather like solving a fictional murder mystery: the amateur detective-historian need only pit his razor-sharp intellect against the clues provided by the self-contained story that is...

    • Mathematics, 600 B.C.–600 A.D.
      (pp. 27-40)
      MAX DEHN

      Isolated arithmetical and geometrical facts were, without doubt, known in prehistoric times much as such facts are now known among the most primitive tribes. Rather advanced mathematical knowledge appears in ancient Egyptian papyri (for instance in the Rhind Papyrus of the 14th century b.c.) and on numerous Babylonian cuneiform texts dating from 2000 b.c. onwards. Certainly the Greeks learned many of the algebraic methods and the techniques of geometric measurements from these ancient peoples through the lively commerce of the Eastern Mediterranean. Our reports begin with Greek mathematics after 600 b.c.

      The sources for the history of mathematics in Greece...

    • Diophantus of Alexandria
      (pp. 41-46)
      J. D. SWIFT

      The name of Diophantus of Alexandria is immortalized in the designation of indeterminate equations and the theory of approximation. As is perhaps more often the rule than the exception in such cases, the attribution of the name may readily be questioned. Diophantus certainly did not invent indeterminate equations. Pythagoras was credited with the solution

      (2n+ 1, 2n2+ 2n, 2n2+ 2n+ 1)

      of the equationx2+y2=z2; the famous Cattle Problem of Archimedes is far more difficult than anything in Diophantus, and a large number of other ancient indeterminate problems are known. Further, Diophantus...

    • Hypatia of Alexandria
      (pp. 47-51)
      A. W. RICHESON

      The first woman mathematician regarding whom we have positive knowledge is the celebrated mathematician-philosopher Hypatia. The exact date of her birth is not known, but recent studies indicate that she was born about a.d. 370 in Alexandria. This would make her about 45 years of age at her death. Hypatia, it seems, was known by two different names, or at least by two different spellings of the same name; the one, Hypatia; the other, Hyptachia. According to Meyer [6], there were two women with the same name living at about this time; Hypatia, the daughter of Theon of Alexandria; the...

    • Hypatia and Her Mathematics
      (pp. 52-59)

      The first woman mathematician of whom we have reasonably secure and detailed knowledge is Hypatia of Alexandria. Although there is a considerable amount of material available about her, very much of that is fanciful, tendentious, unreferenced or plain wrong. These limitations are to be found even in works that we might hope to be authoritative; for example, the entry in theDictionary of Scientific Biography(DSB) [11]. Even where the account given is more careful and accurate [14, 19, 20], one is disappointed to be told so little of Hypatia’sMathematics.

      This article will direct the reader’s attention to the...

    • The Evolution of Mathematics in Ancient China
      (pp. 60-68)

      A popular survey book on the development of mathematics has its text prefaced by the following remarks:

      Only a few ancient civilizations, Egypt, Babylonia, India and China, possessed what may be called the rudiments of mathematics. The history of mathematics and indeed the history of western civilization begins with what occurred in the first of these civilizations. The role of India will emerge later, whereas that of China may be ignored because it was not extensive and moreover has no influence on the subsequent development of mathematics [1].

      Even most contemporary works on the history of mathematics reinforce this impression,...

    • Liu Hui and the First Golden Age of Chinese Mathematics
      (pp. 69-82)

      Very little is known of the life of Liu Hui, except that he lived in the Kingdom of Wei in the third century a.d., when China was divided into three kingdoms at continual war with one another. What is known is that Liu was a mathematician of great power and creativity. Liu’s ideas are preserved in two works which survived and became classics in Chinese mathematics. The most important of these is his commentary, dated 263 a.d., on theJiuzhang suanshu, the great problem book known in the West as theNine Chapters on the Mathematical Art. The second is...

    • Number Systems of the North American Indians
      (pp. 83-93)
      W. C. EELLS

      The linguistic diversity of the Indians inhabiting the North American continent is one of the most remarkable features of world ethnology. The late director of the Bureau of American Ethnology says: “In philology, North America presents the richest field in the world, for here is found the greatest number of languages distributed among the greatest number of stocks.” [16, p. 78] The Bureau recognizes almost three score distinct linguistic families having no lexical resemblance, no apparent unity of origin, no relation to European or Asiatic languages. These “families” are further subdivided linguistically into 750 “tribes” or languages. [17, p. 1]...

    • The Number System of the Mayas
      (pp. 94-97)
      A. W. RICHESON

      The number systems of the North American Indians have recently been discussed in detail in two papers in thisMonthly[2]. The system of numbers developed by the semi-civilized Maya Indians of Central America is probably the most interesting of all systems developed by the early inhabitants of this continent.

      The examples of the number system of the Mayas that have been found, or at least that have been deciphered, deal with the counting of time events or periods, and many authorities are of the opinion that the recording of time series was the sole purpose of their numbers. The...

    • Before The Conquest
      (pp. 98-104)

      In the late 15th century, through their explorers, Europeans “discovered” the New World. Although the discovery would cause drastic change, the New World was, of course, not new to its inhabitants. When the Europeans arrived, there were at least 9 million people in about 800 different cultures living in the Western Hemisphere. Because of the vast disruptions that eventually took place, what we know about them and their mathematical ideas is limited. Most of the cultures had no writing as we commonly use the term and so there are no writings by them in their own words. For the cultures...

    • Afterword
      (pp. 105-106)

      The two standard accounts of Mesopotamian mathematics (as well as the mathematics of other ancient civilizations) are Otto Neugebauer’sThe Exact Sciences in Antiquity[14] and B. L. Van der Waerden’sScience Awakening I[16]. Although they are both still useful, they have been superseded in some of their technical accounts of the mathematics by the results of new research. Among the newer surveys of Mesopotamian mathematics are articles by Jens Høyrup [7] and Jöoran Friberg [5]. Høyrup also has a book-length treatment of the technical aspects of the Mesopotamian tablets:Lengths, Widths, Surfaces: A Portrait of Old Babylonian Algebra...

  5. Medieval and Renaissance Mathematics
    • Foreword
      (pp. 109-110)

      Although the Middle Ages are often thought of as a period of little progress in mathematics, the statement is true only of Europe; much progress was made in other parts of the world. The first three papers in this section deal with the contributions of medieval south Indian mathematicians to the development of the power series representation of the sine, cosine, and arctangent series; these power series first occur in a work by Nilakantha in the early sixteenth century. A detailed derivation of the series appeared later in that century in a work of Jyesthadeva, who attributed the series to...

    • The Discovery of the Series Formula for π by Leibniz, Gregory and Nilakantha
      (pp. 111-121)

      The formula for π mentioned in the title of this article is\[\frac{\pi }{2}=1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\cdots .\caption {(1)}\]

      One simple and well-known modern proof goes as follows:\[\begin{array}{*{35}{l}} \arctan x & =\int_{0}^{x}{\frac{1}{1+{{t}^{2}}}dt} \\ {} & =x-\frac{{{x}^{3}}}{3}+\frac{{{x}^{5}}}{5}-\cdots +{{(-1)}^{n}}\frac{{{x}^{2n+1}}}{2n+1}+{{(-1)}^{n+1}}\int_{0}^{x}{\frac{{{t}^{2n+2}}}{1+{{t}^{2}}}dt}. \\ \end{array}\]

      The last integral tends to zero if |x| ≤ 1, for\[\begin{array}{*{35}{l}} \left| \int_{0}^{x}{\frac{{{t}^{2n+2}}}{1+{{t}^{2}}}dt} \right| & \le \left| \int_{0}^{x}{{{t}^{2n+2}}dt} \right| \\ {} & =\frac{{{\left| x \right|}^{2n+3}}}{2n+3}\to 0\ \text{as}\ n\to \infty . \\ \end{array}\]

      Thus, arctanxhas an infinite series representation for |x| ≤ 1:\[\arctan x=x-\frac{x}{3}+\frac{x}{5}-\frac{x}{7}+\cdots \caption {(2)}\]

      The series for π/4 is obtained by settingx= 1 in (2). The series (2) was obtained independently by Gottfried Wilhelm Leibniz (1646–1716), James Gregory (1638–1675) and an Indian mathematician of the fourteenth century or probably the fifteenth century whose identity is not definitely known. Usually ascribed to Nilakantha,...

    • Ideas of Calculus in Islam and India
      (pp. 122-130)

      Isaac Newton created his version of the calculus during the years from about 1665 to 1670. One of Newton’s central ideas was that of a power series, an idea he believed he had invented out of the analogy with the infinite decimal expansions of arithmetic [9, Vol. III, p. 33]. Newton, of course, was aware of earlier work done in solving the area problem, one of the central ideas of what was to be the calculus, and he knew well that the area under the curvey=xnbetweenx= 0 andx=bwas given by...

    • Was Calculus Invented in India?
      (pp. 131-137)

      No. Calculus was not invented in India. But two hundred years before Newton or Leibniz, Indian astronomers came very close to creating what we would call calculus. Sometime before 1500, they had advanced to the point where they could apply ideas from both integral and differential calculus to derive the infinite series expansions of the sine, cosine, and arctangent functions:\[\begin{array}{*{35}{r}} \sin x & =x-\frac{{{x}^{3}}}{3!}+\frac{{{x}^{5}}}{5!}-\frac{{{x}^{7}}}{7!}-\cdots , \\ \cos x & =1-\frac{{{x}^{2}}}{2!}+\frac{{{x}^{4}}}{4!}-\frac{{{x}^{6}}}{6!}-\cdots , \\ \arctan x & =x-\frac{{{x}^{2}}}{2}+\frac{{{x}^{3}}}{3}-\frac{{{x}^{4}}}{4}-\cdots . \\ \end{array}\]

      Roy [13] and Katz [7, 8] have given excellent expositions of the Indian derivation of these infinite summations. I will give a slightly different explanation of how Indian astronomers obtained the sine and cosine expansions, with an emphasis on...

    • An Early Iterative Method for the Determination of sin 1°
      (pp. 138-142)

      In his popularHistory of Mathematics, Carl B. Boyer [5] dated the medieval period in Europe from 529 a.d. to 1436. It was in 529 that the Byzantine emperor Justinian, fearing a threat to orthodox Christianity, ordered all pagan philosophical schools at Athens to be closed and the scholars dispersed. Rome, then ruled by the Goths, was hardly a hospitable home for the learned, but many found a haven in Sassanide Persia. To Boyer the year 1436 marked the dawn of a new mathematical era in Christian Europe for two reasons. It saw the birth of the most influential European...

    • Leonardo of Pisa and his Liber Quadratorum
      (pp. 143-147)
      R.B. McCLENON

      The thirteenth century is a period of great fascination for the historian, whether his chief interest is in political, social, or intellectual movements. During this century great and far-reaching changes were taking place in all lines of human activity. It was the century in which culminated the long struggle between the Papacy and the Empire; it brought the beginnings of civil liberty in England; it saw the building of the great Gothic cathedrals, and the establishment and rapid growth of universities in Paris, Bologna, Naples, Oxford, and many other centers. The crusades had awakened the European peoples out of their...

    • The Algorists vs. the Abacists: An Ancient Controversy on the Use of Calculators
      (pp. 148-152)

      In 1299 the bankers of Florence were forbidden to use Arabic numerals and were obliged instead to use Roman numerals. And in 1348 the University of Padua directed that a list of books for sale should have the prices marked “non per cifras, sed per literas clara” (not by figures, but by clear letters). [1], [11], [15]

      Our “modern” decimal system of notation actually comes to us from ancient India. Some of the symbols in use today were used as early as the third century b.c. (The zero, however, did not appear until much later—about a.d. 376.) The Arabs...

    • Sidelights on the Cardan-Tartaglia Controversy
      (pp. 153-163)

      There is quite a difference in the frame of mind which comes with the answer to a problem only vaguely defined and lying in an uncharted field, like the invention of the differential calculus, or with a discovery that comes undivined like a flash of lightning from some human mind, like the invention of logarithms,—and the reaction that greets the answer to a problem posed to the world for centuries when that answer arrives, two thousand years in the coming.

      The solution of the cubic had presented itself to the human mind as an intellectual problem already in the...

    • Reading Bombelli’s x-purgated Algebra
      (pp. 164-168)

      Reading mathematics is hard work and reading a four hundred year old mathematics text is four hundred times harder. The language, notation and also the spirit are different from ours. If the reader is not already convinced from past experience, the following extract should prove the point.

      Let us first assume that if we wish to find the approximate root of13that this will be 3 with 4 left over. This remainder should be divided by6(double the 3 given above) which gives⅔.This is the first fraction which is to be added to the3, making...

    • The First Work on Mathematics Printed in the New World
      (pp. 169-172)

      If the student of the history of education were asked to name the earliest work on mathematics published by an American press, he might, after a little investigation, mention the anonymous arithmetic that was printed in Boston in the year 1729. It is now known that this was the work of Isaac Greenwood who held for some years the chair of mathematics in what was then Harvard College. If he should search the records still further back, he might come upon the American reprint of Hodder’s well-known English arithmetic, the first textbook on the subject, so far as known, to...

    • Afterword
      (pp. 173-174)

      Although there is much current research on medieval Indian mathematics, there are few books or articles accessible to the non-specialist. The classic history of Indian mathematics is B. Datta and A. N. Singh’sHistory of Hindu Mathematics[2], but this book, written in the 1930s, does not at all reflect modern research, such as the material on power series in the three articles on India in this section. A more recent book, which has two chapters devoted to ancient and medieval Indian mathematics, isCrest of the Peacock[6], by George Gheverghese Joseph. We understand, however, that there are other...

  6. The Seventeenth Century
    • Foreword
      (pp. 177-178)

      The seventeenth century saw a great acceleration in the development of mathematics. In particular, it witnessed the invention of analytic geometry and the calculus, achievements accomplished through the work of numerous mathematicians. The articles in this section deal with many aspects of these important ideas. In addition, several of the articles emphasize the relationship of history to the teaching of mathematics.

      The age of exploration in Europe required new and better maps. The most famous of these, produced by Gerardus Mercator in 1569, enabled sailors to plot routes of fixed compass directions as straight lines. To accomplish this, Mercator progressively...

    • An Application of Geography to Mathematics: History of the Integral of the Secant
      (pp. 179-182)

      Every student of the integral calculus has done battle with the formula\[\int{\sec \theta d\theta ={\text {ln}}\left| \sec \theta +\tan \theta \right|+c}. \caption {(1)}\]

      This formula can be checked by differentiation or “derived” by using the substitutionu=secθ+tanθ, but thesead hocmethods do not make the formula any more understandable. Experience has taught us that this troublesome integral can be motivated by presenting its history. Perhaps our title seems twisted, but the tale to follow will show that this integral should be presented not as an application of mathematics to geography, but rather as an application of geography to mathematics

      The secant integral arose from cartography and navigation,...

    • Some Historical Notes on the Cycloid
      (pp. 183-187)
      E. A. WHITMAN

      In this paper our interest is not in a renowned mathematician, a celebrated school, or a famous problem, but in a curve, the cycloid. More particularly, our interest is to center around its relation to the mathematics of the seventeenth century, one of the great centuries in the history of the subject. This curve had the good fortune to appear at a time when mathematics was being developed very rapidly and perhaps mathematicians were fortunate that so useful a curve appeared at this time. A new and powerful tool for the study of curves was furnished by the analytic geometry,...

    • Descartes and Problem-Solving
      (pp. 188-198)

      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 text-books—all contain sets of rules for solving word problems, rather like...

    • René Descartes’ Curve-Drawing Devices: Experiments in the Relations Between Mechanical Motion and Symbolic Language
      (pp. 199-207)

      By the beginning of the seventeenth century it had become possible to represent a wide variety of arithmetic concepts and relationships in the newly evolved language of symbolic algebra [19]. Geometry, however, held a preeminent position as an older and far more trusted form of mathematics. Throughout the scientific revolution geometry continued to be thought of as the primary and most reliable form of mathematics, but a continuing series of investigations took place that examined the extent to which algebra and geometry might be compatible. These experiments in compatibility were quite opposite from most of the ancient classics. Euclid, for...

    • Certain Mathematical Achievements of James Gregory
      (pp. 208-217)

      For a long time the light of James Gregory did not shine as brightly as did that of John Wallis, Isaac Barrow and Isaac Newton, the other three great British mathematicians of the seventeenth century. Only recently, through the endeavors of several Scottish mathematicians, especially E. T. Whittaker, G. A. Gibson and H. W. Turnbull, Gregory’s genius is revealed and fills with admiration all those interested in the development of modern mathematics.

      TheJames Gregory Tercentenary Memorial Volume, edited by H. W. Turnbull [1], contains Gregory’s momentous scientific correspondence, mostly with J. Collins. An extremely important supplement is the large...

    • The Changing Concept of Change: The Derivative from Fermat to Weierstrass
      (pp. 218-227)

      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’s a 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...

    • The Crooked Made Straight: Roberval and Newton on Tangents
      (pp. 228-234)

      In October 1665, about two years after he had first read a mathematics book, Isaac Newton began investigating a method for finding the tangents to “mechanical” curves. He can have known only vaguely that he was following a path trod previously by several outstanding mathematicians, Torricelli, Descartes, Roberval, and Barrow among them. In his ignorance of the details of their work, Newton stumbled before setting himself firmly on the way to his calculus. As he progressed, he overcame the inadequate mathematical language that had kept others from expressing|sometimes from even thinking—their ideas clearly.

      Newton’s method found tangents by regarding...

    • On the Discovery of the Logarithmic Series and Its Development in England up to Cotes
      (pp. 235-239)

      To the expert of today the logarithmic series appears to be a very non-essential detail. In its time it was a very notable discovery as regards itself alone, as well as in the framework of the general theory of series. It was discoveredcirca1667 by Newton and independently by Mercator. Huygens and Gregory were close to the same discovery but they were anticipated by the other two. Newton was then 24 years old, Mercator 47. For Newton the logarithmic series was a beginning, for Mercator the climax.

      Mercator’s life work is almost forgotten today, certainly unjustly. Mercator was a...

    • Isaac Newton: Man, Myth, and Mathematics
      (pp. 240-260)

      Three hundred years ago, in 1687, the most famous scientific work of all time, thePhilosophiae Naturalis Principia Mathematicaof Isaac Newton, was published. Fifty years earlier, in 1637, a work which had considerable influence on Newton, theDiscours de la Méthode, with its famous appendix,La Géométrie, was published by René Descartes. It is fitting that we celebrate these anniversaries by sketching the lives and outlining the works of Newton and Descartes.

      In the past several decades, historians of science have arranged the chaotic bulk of Newton manuscripts into a coherent whole and presented it to us in numerous...

    • Reading the Master: Newton and the Birth of Celestial Mechanics
      (pp. 261-273)

      In January of 1684, the young astronomer Edmund Halley travelled from Islington up to London for a meeting of the Royal Society. Later, perhaps over tea and chocolate at a nearby coffee house, he chatted casually about natural philosophy and other topics with Sir Christopher Wren and Robert Hooke. Talk soon turned to celestial motions, and Halley later reconstructed the conversation [22 p. 26]:

      I, having from the consideration of the sesquialter proportion of Kepler concluded that the centripetall force [to the Sun] decreased in the proportion of the squares of the distances reciprocally, came one Wednesday to town, where...

    • Newton as an Originator of Polar Coordinates
      (pp. 274-278)
      C. B. BOYER

      The name of Newton, indissolubly linked with the calculus, seldom is associated with analytic geometry, a field to which he nevertheless made important contributions. Newton’s use of polar coordinates, for example, seems to have been overlooked completely in the historiography of mathematics. The polar coordinate system is ascribed generally [1] to Jacques Bernoulli in 1691 and 1694, although it has been attributed [2] to others as late as Fontana in 1784. It is the purpose here to call attention to an application of polar coordinates made by Newton probably a score of years before the earliest publication of Bernoulli’s work....

    • Newton’s Method for Resolving Affected Equations
      (pp. 279-287)

      During the 300 years since Newton and Leibniz began disputing which of them had discovered the calculus, debates have continued over the credit due to Newton for various scientific and mathematical achievements. Recent research by Nick Kollerstrom [11] has led him to credit Thomas Simpson (1710–1761) with the first discovery and publication in 1740 [18] of what is now called Newton’s method. William Dunham [8] has pointed out the irony that Newton, who “bitterly resented people’s getting credit for results they did notoriginallydiscover,” is credited with a method of approximation that “in its full generality seems to...

    • A Contribution of Leibniz to the History of Complex Numbers
      (pp. 288-291)
      R. B. McCLENON

      One of the most important and fascinating chapters in the history of mathematics is the development of the concept of complex numbers. Certain parts of this development have not yet been adequately treated by writers on the history of mathematics, and among these is to be mentioned the work of Leibniz.

      It may be worth while to recall that neither the Hindu nor the Arabian algebraists, nor the medieval Europeans, had recognized any possibility of attaching a meaning to a square root of a negative number; indeed it was only the exceptional writer who recognized evennegativeroots of equations...

    • Functions of a Curve: Leibniz’s Original Notion of Functions and Its Meaning for the Parabola
      (pp. 292-296)

      When the notion of a function evolved in the mathematics of the late seventeenth century, the meaning of the term was quite different from our modern set theoretic definition, and also different from the algebraic notions of the nineteenth century. The main conceptual difference was that curves were thought of as having a primary existence apart from any analysis of their numeric or algebraic properties. Equations did not create curves, curves gave rise to equations. When Descartes published hisGeometry[10] in 1637, he derived for the first time the algebraic equations of many curves, but never once did he...

    • Afterword
      (pp. 297-298)

      Further information on the development of the calculus can be found in several good books. Margaret Baron’sThe Origins of the Infinitesimal Calculus[2] deals with many of the methods of the calculus up to the time of Newton and Leibniz. C. H. Edwards’The Historical Development of the Calculus[7] also shows how mathematicians calculated solutions to problems, but covers in more detail the work of Newton, Leibniz, and their successors. The classic work by Carl Boyer,The History of the Calculus and its Conceptual Development[4], concentrates more on the central ideas of the calculus rather than the...

  7. The Eighteenth Century
    • Foreword
      (pp. 301-302)

      Newton and Leibniz invented calculus in the late seventeenth century. The following century saw its continued development, so many of the articles in this section deal with aspects of the calculus. But since the towering figure in the eighteenth century is Leonhard Euler, much of this section deals with aspects of his work as well, both in analysis and in number theory.

      The opening article of this section, however, deals with neither of these subjects. Although Brook Taylor is best known for his 1715 workMethodus Incrementorum Directa et Inversa, in which he discusses the Taylor series expansion of a...

    • Brook Taylor and the Mathematical Theory of Linear Perspective
      (pp. 303-309)
      P. S. JONES

      One can distinguish four overlapping and interrelated periods in the development of the mathematical theory of linear perspective:

      (1) the “prehistory” period in which, for example, the Greeks are reported to have made some use of perspective drawing in their theater,

      (2) the 15th and 16th century period of the origin of the theory with the artists-architects-engineers of the Renaissance (Brunelleschi, Franceschi, Alberti, and da Vinci),

      (3) a period of geometrical expositions typified by the works of del Monte and Stevin in the 17th century, and, finally,

      (4) the period of a generalized, complete, and even abstract theory.

      This last...

    • Was Newton’s Calculus a Dead End? The Continental Influence of Maclaurin’s Treatise of Fluxions
      (pp. 310-324)

      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...

    • Discussion of Fluxions: from Berkeley to Woodhouse
      (pp. 325-331)

      The first direct statement of Newton’s method and notation of fluxions was printed in 1693 in Wallis’sAlgebra. Here and in thePrincipiaof 1687 Newton made use of infinitely small quantities, but in hisQuadrature of Curvesof 1704 he declared that “in the method of fluxions there is no necessity of introducing figures infinitely small.” No other publication of Newton, printed either before 1704 or after, equalled theQuadrature of Curvesin mathematical rigor. Here Newton reached his high water mark of rigidity in the exposition of fluxions. By a fluxion, Newton always meant a finite velocity. With...

    • The Bernoullis and the Harmonic Series
      (pp. 332-335)

      Any introduction to the topic of infinite series soon must address that first great counterexample of a divergent series whose general term goes to zero—the harmonic series$\sum\nolimits_{k=1}^{\infty }{1/k}$. Modern texts employ a standard argument, traceable back to the great 14th-century Frenchman Nicole Oresme (see [3], p. 92), which establishes divergence by grouping the partial sums:\[\begin{array}{*{35}{rl}} 1+\frac{1}{2} & >\frac{1}{2}+\frac{1}{2}=\frac{2}{2}, \\ 1+\frac{1}{2}+\left( \frac{1}{3}+\frac{1}{4} \right) & >\frac{2}{2}+\left( \frac{1}{4}+\frac{1}{4} \right)=\frac{3}{2}, \\ 1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4} & +\left( \frac{1}{5}+\frac{1}{6}+\frac{1}{7}+\frac{1}{8} \right) \\ {} & >\frac{3}{2}+\left( \frac{1}{8}+\frac{1}{8}+\frac{1}{8}+\frac{1}{8} \right)=\frac{4}{2}, \\ \end{array}\]and in general\[1+\frac{1}{2}+\frac{1}{3}+\cdots +\frac{1}{{{2}^{n}}}>\frac{n+1}{2},\]from which it follows that the partial sums grow arbitrarily large asngoes to infinity.

      It is possible that seasoned mathematicians tend to forget how surprising this phenomenon appears to the uninitiated student—that, by adding ever...

    • Leonhard Euler 1707–1783
      (pp. 336-345)

      Born in 1707, Leonhard Euler grew up in the town of Riehen, near Basel, Switzerland. Encouraged by his father, Paulus, a minister, young Leonhard received very early instruction from Johann I Bernoulli, who immediately recognized Euler’s talents. Euler completed his work at the University of Basel at age 15, and at age 19 won a prize in the competition organized by the Academy of Sciences in Paris. His paper discussed the optimal arrangement of masts on sailing ships (Meditationes super problemate nautico…). In 1727 Euler attempted unsuccessfully to obtain a professorship of physics in Basel by submitting a dissertation on...

    • The Number e
      (pp. 346-353)
      J. L. COOLIDGE

      The distinguished American mathematician, Benjamin Peirce, was wont to find all of analysis in the equation\[{{i}^{-i}}=\sqrt{{{e}^{\pi }}}.\]

      In fact, he had his picture taken in front of a blackboard on which this mystic formula, in somewhat different shape, was inscribed. He would say to his hearers, “Gentlemen, we have not the slightest idea of what this equation means, but we may be sure that it means something very important.”

      With regard to the symbols which appear in this charm, there is a vast literature connected withπ; andi, when written$\sqrt{-1}$, leads into the broad field of analysis...

    • Euler’s Vision of a General Partial Differential Calculus for a Generalized Kind of Function
      (pp. 354-360)

      The vibrating string controversy involved most of the analysts of the latter half of the 18th century. The dispute concerned the type of functions which could be allowed in analysis, particularly in the new partial differential calculus. Leonhard Euler held the bold opinion that all functions describing any curve, however irregular, ought to be admitted in analysis. He often stressed the importance of such an extended calculus, but did almost nothing to support his point of view mathematically. After having been abandoned during the introduction of rigor in the latter part of the 19th century, Euler’s ideas began to take...

    • Euler and the Fundamental Theorem of Algebra
      (pp. 361-368)

      A watershed event for all students of mathematics is the first course in basic high school algebra. In my case, this provided an initial look at graphs, inequalities, the quadratic formula, and many other critical ideas. Somewhere near the term’s end, as I remember, our teacher mentioned what sounded like the most important result of them all—the fundamental theorem of algebra. Anything with a name like that, I figured, must be (for want of a better term)fundamental. Unfortunately, the teacher informed us that this theorem was much too advanced to state, let alone to investigate, at our current...

    • Euler and Differentials
      (pp. 369-374)

      Two recent articles by Dunham [5] and Flusser [10] have presented examples of Leonhard Euler’s work in algebra. Both papers are a joy to read; watching Euler manipulate and calculate with incredible facility is a pleasure. A modern mathematician can see the logical flaws in some of the arguments, yet at the same time be aware that the mind behind it all is that of a unique master.

      These two articles reminded me how much fun it is to read Euler. In researching the evolution of the differential a few years ago, I found the work of Euler refreshingly different...

    • Euler and Quadratic Reciprocity
      (pp. 375-382)

      In a letter to Goldbach bearing the date 28 August 1742, Euler described a property of positive whole numbers that was to play a central role in the history of the theory of numbers. (The original is a mixture of Latin and German, which I have translated into English as best I can. The letter can be found in [2] or [3].)

      Whether there are series of numbers which either have no divisors of the form 4n+ 1, or which even are prime, I very much doubt. If such series could be found, however, one could use them to...

    • Afterword
      (pp. 383-384)

      For more information on Maclaurin, the reader can consult H. W. Turnbull,Bicentenary of the Death of Colin Maclaurin[10], which contains numerous articles about aspects of his work.

      Florian Cajori expanded his arguments in the article in this section into a book,A History of the Conceptions of Limits and Fluxions in Great Britain from Newton to Woodhouse[3]. A more recent treatment of much of the same material is Niccolò Guicciardini’sThe Development of Newtonian Calculus in Britain, 1700–1800[8], and a good survey article on calculus in the first half of the eighteenth century is by...

  8. Index
    (pp. 385-386)
  9. About the Editors
    (pp. 387-387)