Who Gave You the Epsilon?

Who Gave You the Epsilon?: and Other Tales of Mathematical History

Marlow Anderson
Victor Katz
Robin Wilson
Series: Spectrum
Copyright Date: 2009
Edition: 1
Pages: 442
https://www.jstor.org/stable/10.4169/j.ctt13x0n5b
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  • Book Info
    Who Gave You the Epsilon?
    Book Description:

    Who Gave You the Epsilon? is a sequel to the MAA bestselling book, Sherlock Holmes in Babylon. Like its predecessor, this book is a collection of articles on the history of mathematics from the MAA journals, in many cases written by distinguished mathematicians (such as G H Hardy and B.van der Waerden), with commentary by the editors. Whereas the former book covered the history of mathematics from earliest times up to the eighteenth century and was organized chronologically, the 40 articles in this book are organized thematically and continue the story into the nineteenth and twentieth centuries. The topics covered in the book are analysis and applied mathematics, Geometry, topology and foundations, Algebra and number theory, and Surveys. Each chapter is preceded by a Foreword, giving the historical background and setting and the scene, and is followed by an Afterword, reporting on advances in our historical knowledge and understanding since the articles first appeared.

    eISBN: 978-1-61444-504-3
    Subjects: Mathematics

Table of Contents

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

    For over one hundred years, the Mathematical Association of America has been publishing high-quality articles on the history of mathematics, some written by distinguished historians and mathematicians such as J. L. Coolidge, B. L. van der Waerden, Hermann Weyl and G. H. Hardy. Many well-known historians of the present day also contribute to the MAA’s journals, such as Ivor Grattan-Guinness, Judith Grabiner, Israel Kleiner and Karen Parshall.

    Some years ago, we decided that it would be useful to reprint a selection of these papers and to set them in the context of modern historical research, so that current mathematicians can...

  3. Table of Contents
    (pp. ix-x)
  4. Analysis

    • Foreword
      (pp. 3-4)

      In this chapter, we look at some critical ideas in the history of analysis in the nineteenth and twentieth centuries and also consider some of the people involved in their creation.

      Certainly, the most important idea in the rigorous study of calculus is the idea of alimit. Although Newton and Leibniz both understood this idea heuristically and used it in their discussions of fluxions and differentials, respectively, it was not until the nineteenth century that our modern treatment in terms of epsilons and deltas was created, mostly by Augustin-Louis Cauchy. Judy Grabiner explores this creation in her article, showing...

    • Who Gave You the Epsilon? Cauchy and the Origins of Rigorous Calculus
      (pp. 5-13)
      JUDITH V. GRABINER

      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...

    • Evolution of the Function Concept: A Brief Survey
      (pp. 14-26)
      ISRAEL KLEINER

      The evolution of the concept of function goes back 4000 years; 3700 of these consist of anticipations. The idea evolved for close to 300 years in intimate connection with problems in calculus and analysis. (A one-sentence definition of analysis as the study of properties of various classes of functions would not be far off the mark.) In fact, the concept of function is one of the distinguishing features of ‘modern’ as against ‘classical’ mathematics. W. L. Schaaf ([24], p. 500) goes a step further:

      The keynote of Western culture is the function concept, a notion not even remotely hinted at...

    • S. Kovalevsky: A Mathematical Lesson
      (pp. 27-35)
      KAREN D. RAPPAPORT

      Sofya Kovalevsky was a noted writer whose works include both fiction and non-fiction. She was also a political activist and a public advocate of feminism. In addition, she was a brilliant mathematician who made significant contributions despite the enormous educational and political obstacles that she had to overcome. Somehow her many achievements have been forgotten. In those few instances where her work has not been lost it has been denigrated by such studies as Felix Klein’s history of nineteenth-century mathematics. Klein dismisses Kovalevsky’s work in the following manner ([3], p. 294): “Her works are done in the style of Weierstrass...

    • Highlights in the History of Spectral Theory
      (pp. 36-51)
      L. A. STEEN

      Not least because such different objects as atoms, operators and algebras all possess spectra, the evolution of spectral theory is one of the most informative chapters in the history of contemporary mathematics. The central thrust of the modern spectral theorem is that certain linear operators on infinite dimensional spaces can be represented in a “diagonal” form. At the beginning of the twentieth century neither this spectral theorem nor the word “spectrum” itself had entered the mathematician’s repertoire. Thus, although it has deep roots in the past, the mathematical theory of spectra is a distinctly twentieth-century phenomenon.

      Today every student of...

    • Alan Turing and the Central Limit Theorem
      (pp. 52-60)
      S. L. ZABELL

      Because the English mathematician Alan Mathison Turing (1912–1954) is remembered today primarily for his work in mathematical logic (Turing machines and the “Entscheidungsproblem”), machine computation, and artificial intelligence (the “Turing test”), his name is not usually thought of in connection with either probability or statistics. One of the basic tools in both of these subjects is the use of the normal or Gaussian distribution as an approximation, one basic result being the Lindeberg-Feller central limit theorem taught in first-year graduate courses in mathematical probability. No-one associates Turing with the central limit theorem, but in 1934 Turing, while still an...

    • Why did George Green Write his Essay of 1828 on Electricity and Magnetism?
      (pp. 61-68)
      I. GRATTAN-GUINNESS

      Among the centenaries of mathematicians and scientists celebrated in 1993, perhaps the most remarkable was the bicentenary of the birth of a professional miller and part-time mathematician, one George Green (1793–1841) of Sneinton, then near Nottingham. Among other achievements, he was the creator of theorems and functions now named after him which make him a principal contributor to potential theory and to its applications in mechanics and mathematical physics.

      During the week corresponding to that of his birth (which occurred on 14 July) various events took place. A three-day conference was held at the University of Nottingham, mainly on...

    • Connectivity and Smoke-Rings: Green’s Second Identity in its First Fifty Years
      (pp. 69-77)
      THOMAS ARCHIBALD

      James Clerk Maxwell, in his review of Thomson and Tait’sTreatise on Natural Philosophy, noted an important innovation in the authors’ approach to mathematics ([8], Vol. 2, p. 777):

      The first thing which we observe in the arrangement of the work is the prominence given to kinematics, … and the large space devoted under this heading to what has been hitherto considered part of pure geometry. The theory of curvature of lines and surfaces, for example, has long been recognized as an important branch of geometry, but in treatises on motion it was regarded as lying as much outside of...

    • The History of Stokes’s Theorem
      (pp. 78-87)
      VICTOR J. KATZ

      Most current American textbooks in advanced calculus devote several sections to the theorems of Green, Gauss, and Stokes. Unfortunately, the theorems referred to were not original to these men. It is the purpose of this paper to present a detailed history of these results from their origins to their generalization and unification into what is today called the generalized Stokes’ theorem.

      The three theorems in question each relate ak-dimensional integral to a (k− 1)-dimensional integral; since the proof of each depends on the fundamental theorem of calculus, it is clear that their origins can be traced back to...

    • The Mathematical Collaboration of M. L. Cartwright and J. E. Littlewood
      (pp. 88-97)
      SHAWNEE L. McMURRAN and JAMES J. TATTERSALL

      Balthasar van der Pol’s experiments on electrical circuits during the 1920s and 1930s opened an interesting chapter in the history of dynamics. The need for advancements in radio technology made van der Pol’s work pertinent and his research stimulated mathematical interest in non-linear oscillators. In particular, van der Pol’s work caught the attention of Cambridge mathematicians M. L. Cartwright and J. E. Littlewood. Topology and Poincaré’s transformation theory provided a key to analyzing behavior of non-linear oscillators and dissipative systems. Resulting mathematical techniques have played a significant role in the development of the modern theory of dynamical systems and chaos....

    • Dr. David Harold Blackwell, African-American Pioneer
      (pp. 98-108)
      NKECHI AGWU, LUELLA SMITH and AISSATOU BARRY

      Dr. David Blackwell is an African-American educational pioneer and eminent scholar in the fields of mathematics and statistics, whose contributions to our society extend beyond these fields. This paper highlights his significant contributions and the personal, educational, and professional experiences that groomed and nurtured him for leadership as a civic scientist. We hope this account of Dr. Blackwell’s life will enhance the literature on African-American achievers, and motivate students majoring in, or considering careers in mathematics and statistics, particularly those from under-represented groups.

      It is April 24, 1919, an era of heightened segregation and racial discrimination in the United States....

    • Afterword
      (pp. 109-110)

      For more details on the work of Cauchy, the best book is Judith Grabiner’s own work,The Origins of Cauchy’s Rigorous Calculus[8]. A slightly more recent work on nineteenth-century analysis in general is Umberto Bottazzini,The Higher Calculus: A History of Real and Complex Analysis from Euler to Weierstrass[1].

      It is interesting to compare Cauchy’s work with the contemporary work of Bolzano. Bolzano’s important paper on the intermediate value theorem is available in [17], while [18] is the complete collection of his mathematical works. For more discussion of the work of Bolzano, see I. Grattan-Guinness [9]. Grattan-Guinness claims...

  5. Geometry, Topology and Foundations

    • Foreword
      (pp. 113-114)

      One of the most important aspects of geometry in the nineteenth century was the development of non-Euclidean geometry, and this chapter begins with two brief studies of aspects of its development. In the first, George Bruce Halsted reviews volume VII of Gauss’sWerkeand concludes from a study of many of Gauss’s letters first published in that volume that Gauss’s ideas on the subject had no influence on the independent discoveries of János Bolyai and Nikolai Lobachevsky, or on the earlier publication by Ferdinand Karl Schweikart (1780–1859). In the second article, Florence P. Lewis gives us a whirlwind tour...

    • Gauss and the Non-Euclidean Geometry
      (pp. 115-119)
      GEORGE BRUCE HALSTED

      We are so accustomed to the German professor who does, we hardly expect the German professor who does not. Such, however, was Schering of Göttingen, who so long held possession of the papers left by Gauss.

      Schering had Planned and promised to publish a supplementary volume, but never did, and only left behind him at his death certain preparatory attempts thereto, consisting chiefly of excerpts copied from the manuscripts and letters left by Gauss. Meantime these papers for all these years were kept secret and even the learned denied all access to them.

      Schering dead, his work has been quickly...

    • History of the Parallel Postulate
      (pp. 120-124)
      FLORENCE P. LEWIS

      Like the famous problems of construction, Euclid’s postulate concerning parallels is a thought that links the ages. Its history is a long story with a dramatic climax and far-reaching influence on modern mathematical and general scientific thought. I wish to recall briefly the salient features of the story, and to state what seem to me its suggestions in regard to the teaching of elementary geometry.

      Euclid’s fifth postulate (called also the eleventh or twelfth axiom) states: “If a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two...

    • The Rise and Fall of Projective Geometry
      (pp. 125-132)
      J. L. COOLIDGE

      The subject of projective geometry occupied an important position on the mathematical stage during a large portion of the nineteenth century. In recent years it has moved considerably towards the wings. Why did it appear? Why was it prominent? Why is it now moving aside? These are pertinent questions which perhaps it is worth our while to consider.

      First of all, what is projective geometry anyway? It is sometimes defined as that branch of geometrical science which deals with those properties of figures which are unaltered by radial projection from plane to plane or space to space, no matter what...

    • Notes on the History of Geometrical Ideas
      (pp. 133-136)
      DAN PEDOE

      It is agreed, even by those who disparage them (see [6], p. 712) that barycentric coordinates, first introduced by August Ferdinand Möbius [4] in 1827, were the first homogeneous coordinates systematically used in geometry. The Möbius idea, in plane geometry for example, is to attach massesp,qandr, respectively, to three non-collinear pointsA,BandCin the plane under consideration, and then to consider the centroidP=pA+qB+rCof the three masses. The pointPnecessarily lies in the plane, and varies as the ratiosp:q:rvary....

    • A note on the history of the Cantor Set and Cantor function
      (pp. 137-141)
      JULIAN F. FLERON

      A search through the primary and secondary literature on Cantor yields little about the history of the Cantor set and Cantor function. In this note, we would like to give some of that history, a sketch of the ideas under consideration at the time of their discovery, and a hypothesis regarding how Cantor came upon them. In particular, Cantor was not the first to discover “Cantor sets”. Moreover, although the original discovery of Cantor sets had a decidedly geometric flavor, Cantor’s discovery of the Cantor set and Cantor function was neither motivated by geometry nor did it involve geometry, even...

    • Evolution of the Topological Concept of “Connected”
      (pp. 142-147)
      R. L. WILDER

      The purpose of this paper is to trace the evolution of one of the most basic concepts in topology, viz., that ofconnected(not to be confused withsimply connected). Like many other mathematical concepts of a fundamental nature (e.g., continuous function), it had only an intuitive meaning (such asconnected figurein geometry) until the increasingly subtle demands of analysis and topology forced formulation of a satisfactory definition. The latter was not achieved, as one might expect, until a number of definitions had been proposed—each sufficient within its mathematical context, but quite insufficient as the configurations studied became...

    • A Brief, Subjective History of Homology and Homotopy Theory in this Century
      (pp. 148-156)
      PETER HILTON

      I have recently been recalling that about twenty-five years ago, when I first came to settle in this country, I was invited to participate in the celebration of the opening of the Mathematics Building, Van Vleck Hall, at the University ofWisconsin. On that occasion I learned a new American word, namely “banquet”, which has a totally different meaning in the United States from the meaning that it has in Britain. But more importantly, I must recall the immense respect I felt for some of the after-dinner speakers who were able to make the recounting of an event last much longer...

    • The Origins of Modern Axiomatics: Pasch to Peano
      (pp. 157-160)
      H. C. KENNEDY

      The modern attitude toward the undefined terms of an axiomatic mathematical system is that popularized by Hilbert’s remark: “One must be able to say at all times—instead of points, straight lines, and planes—tables, chairs, and beer mugs” ([20], p. 57). This view was not widely accepted before the twentieth century, and even in 1959 the well-known James and JamesMathematics Dictionarygave “A self-evident and generally accepted principle” as first meaning of the term “axiom”, although this may only be meant as a reflection of the view universally accepted before the developments in geometry in the nineteenth century....

    • C. S. Peirce’s Philosophy of Infinite Sets
      (pp. 161-171)
      JOSEPH W. DAUBEN

      American mathematics, like American science generally in the nineteenth century, remained underdeveloped, depended heavily upon European models, and made few independent and recognized contributions of its own. Though presidents like Jefferson might take a pedagogical interest in mathematics and its teaching, and while Garfield, in fact, discovered an interesting variation on the many proofs of the Pythagorean theorem, American mathematics generally remained without support, either institutional or financial, until late in the century (see [28] and [37]).

      Despite the lack of incentives to pursue a mathematical career in America, there were nevertheless some who made important contributions to mathematics in...

    • On the Development of Logics between the two World Wars
      (pp. 172-184)
      I. GRATTAN-GUINNESS

      Logic is a disparate topic, occurring in almost any field of human activity without appearing to have much character of its own. Traditionally it was associated largely with methods of reasoning and regarded as encapsulated in the principles of syllogistic logic. It was the concern mostly of philosophers and developed in the context of rather general questions. During the second half of the nineteenth century there was especial concern with connections with psychology. For example, some authors maintained that psychology is a descriptive theory concerned with how we think, while logic is a normative discipline about how we ought to...

    • Dedekind’s Theorem: $\sqrt{2}\times \sqrt{3}=\sqrt{6}$
      (pp. 185-191)
      DAVID FOWLER

      When the young Richard Dedekind, newly arrived at the Zurich Polytechnik (now the ETH), had to give for the first time the introductory calculus course, it had repercussions that were eventually to spread far beyond his class of students. He tells us in the introduction to hisStetigkeit und irrationale Zahlen[3] how his search for a satisfactory foundation for the calculus led him, on November 24, 1858, to his construction of the real numbers. (It was a Wednesday.) His immediate objective was to make precise and therefore, he argued, arithmetical the previously vague geometrical appeals to what we now...

    • Afterword
      (pp. 192-194)

      Halsted’s review is of Volume VII of Gauss’sWerke, but in Volume VIII, the editors published Gauss’s manuscript material on non-Euclidean geometry. These show that Gauss anticipated much of the work of Bolyai and Lobachevsky, as well as of Schweikart. The existence of these materials does not, however, challenge Halsted’s conclusion that Gauss had no influence on the work of any of these men. Roberto Bonola’sNon-Euclidean Geometry[2] is still a good source for details on Gauss’s work; the English edition also contains Halsted’s translations of the fundamental articles on non-Euclidean geometry by Bolyai and Lobachevsky.

      More recent works...

  6. Algebra and Number Theory

    • Foreword
      (pp. 197-199)

      Algebra in 1800 meant the solving of equations. By 1900, the term was beginning to encompass the study of various mathematical structures—sets of elements with well-defined operations, satisfying certain specified axioms—and by the 1930s, under the influence of Emmy Noether, this change was complete. This chapter explores some aspects of this change in the notion of algebra over that time period.

      The nineteenth century also saw a change in number theory, not so much in the subject matter but in the methods of proof. From the inception of number theory, one proved results about positive integers by using...

    • Hamilton’s Discovery of Quaternions
      (pp. 200-205)
      B. L. VAN DER WAERDEN

      The ordinary complex numbersa+ib(or, as they were formerly written,${a+b}{\sqrt{-1}}$) are added and multiplied according to definite rules. The rule for multiplication reads as follows:

      First multiply according to the rules of high school algebra:

      (a+ib)(c+id) =ac+adi+bci+bdi2

      and then replacei2by −1:

      (a+ib)(c+id) = (acbd) + (ad+bc)i.

      Complex numbers can also be defined as couples (a,b). The product of two couples (a,b) and (c,d) is defined as the couple (acbd,...

    • Hamilton, Rodrigues, and the Quaternion Scandal
      (pp. 206-219)
      SIMON L. ALTMANN

      Some of the best minds of the nineteenth century—and this was the century that saw the birth of modern mathematical physics—hailed the discovery of quaternions as just about the best thing since the invention of sliced bread. Thus James Clerk Maxwell, the discoverer of electromagnetic theory, wrote ([31], p. 226):

      The invention of the calculus of quaternions is a step towards the knowledge of quantities related to space which can only be compared, for its importance, with the invention of triple coordinates by Descartes. The ideas of this calculus, as distinguished from its operations and symbols, are fitted...

    • Building an International Reputation: The Case of J. J. Sylvester (1814–1897)
      (pp. 220-229)
      KAREN HUNGER PARSHALL and EUGENE SENETA

      James Joseph Sylvester—prolific, gifted, flamboyant, egocentric, cantankerous. At the time of his death in London on 15 March 1897, Sylvester’s reputation internationally as one of the nineteenth century’s principal mathematical figures had long been secure. He had worked hard to assure this. Obviously, he had done much seminal work in building the theory of invariants, and this had contributed to his renown. Yet, Sylvester had felt compelled to establish ties directly with mathematicians at home—but more importantly abroad—in order to make his name known. Was this just a matter of egocentrism, or did other factors contribute to...

    • The Foundation Period in the History of Group Theory
      (pp. 230-236)
      JOSEPHINE E. BURNS

      Henri Poincaré has pointed out that the fundamental conception of a group is evident in Euclid’s work; in fact, that the foundation of Euclid’s demonstrations is the group idea. Poincaré establishes this assertion by showing that such operations as successive superposition and rotation about a fixed axis presuppose the displacements of a group. However much the fundamental group notions were unconsciously used in the work of early mathematicians, it was not until the latter part of the eighteenth century that these notions began to take life and develop.

      The period of foundation of group theory as a distinct science extends...

    • The Evolution of Group Theory: A Brief Survey
      (pp. 237-253)
      ISRAEL KLEINER

      This article gives a brief sketch of the evolution of group theory. It derives from a firm conviction that the history of mathematics can be a useful and important integrating component in the teaching of mathematics. This is not the place to elaborate on the role of history in teaching, other than perhaps to give one relevant quotation (C. H. Edwards [11]):

      Although the study of the history of mathematics has an intrinsic appeal of its own, its chief raison d’ˆetre is surely the illumination of mathematics itself. For example the gradual unfolding of the integral concept from the volume...

    • The Search for Finite Simple Groups
      (pp. 254-270)
      JOSEPH A. GALLIAN

      At present, simple group theory is the most active and glamorous area of research in the theory of groups and it seems certain that this will remain the case for many years to come. Roughly speaking, the central problem is to find some reasonable description of all finite simple groups. A number of expository papers [36], [42], [45], [47], [49], [79] and books [21], [46], [67] detailing progress on this problem have been written for professional group theorists, but very little has appeared which is accessible to undergraduates. (Only Goldschmidt’s proof of the Brauer-Suzuki-Wall theorem [44] comes to mind.) This...

    • Genius and Biographers: The Fictionalization of Evariste Galois
      (pp. 271-290)
      TONY ROTHMAN

      In Paris, on the obscure morning of May 30, 1832, near a pond not far from the pension Sieur Faultrier, Evariste Galois confronted Pescheux d’Herbinville in a duel to be fought with pistols, and was shot through the stomach. Hours later, lying wounded and alone, Galois was found by a passing peasant. He was taken to the Hospital Cochin where he died the following day in the arms of his brother Alfred, after having refused the services of a priest. Had Galois lived another five months, until October 25, he would have attained the age of twenty-one.

      The legend of...

    • Hermann Grassmann and the Creation of Linear Algebra
      (pp. 291-298)
      DESMOND FEARNLEY-SANDER

      From Pythagoras to the mid-nineteenth century, the fundamental problem of geometry was to relate numbers to geometry. It played a key role in the creation of field theory (via the classic construction problems), and, quite differently, in the creation of linear algebra. To resolve the problem, it was necessary to have the modern concept of real number; this was essentially achieved by Simon Stevin, around 1600, and was thoroughly assimilated into mathematics in the following two centuries. The integration of real numbers into geometry began with Descartes and Fermat in the 1630s, and achieved an interim success at the end...

    • The Roots of Commutative Algebra in Algebraic Number Theory
      (pp. 299-308)
      ISRAEL KLEINER

      The concepts of field, commutative ring, ideal, and unique factorization are among the fundamental notions of commutative algebra. How did they arise? In large measure, from three central problems in number theory:Fermat’s Last Theorem,reciprocity laws, andbinary quadratic forms. In this paper we will describe how this happened.

      To put the issues in a broader context, these three number-theoretic problems were instrumental in the emergence of algebraic number theory—one of the two main sources of the modern discipline of commutative algebra. The other source was algebraic geometry. It was in the setting of these two subjects that...

    • Eisenstein’s Misunderstood Geometric Proof of the Quadratic Reciprocity Theorem
      (pp. 309-312)
      REINHARD C. LAUBENBACHER and DAVID J. PENGELLEY

      Thequadratic reciprocity theoremhas played a central role in the development of number theory, and formed the first deep law governing prime numbers. Its numerous proofs from many distinct points of view testify to its position at the heart of the subject. The theorem was discovered by Euler, and restated by Legendre in terms of the symbol now bearing his name, but was first proven by Gauss. The eight different proofs Gauss published in the early 1800s, for what he called thefundamental theorem, were followed by dozens more before the century was over, including four given by Gotthold...

    • Waring’s Problem
      (pp. 313-317)
      CHARLES SMALL

      What follows is a non-scholarly survey of the history of Waring’s problem. Although a few easy things are proved along the way, the paper is mostly concerned with telling stories—in other words, quoting many beautiful theorems without proof. The proofs, for the most part, involve hard-core analysis, and are difficult. Anyone wishing to pursue the subject should examine chapters 20 and 21 of Hardy and Wright [4] and then [1] and [2]. Ellison’s paper [2] provides a much more scholarly and detailed version of the story, with many proofs and an extensive bibliography; the present informal version should serve...

    • A History of the Prime Number Theorem
      (pp. 318-327)
      L. J. GOLDSTEIN

      The sequence of prime numbers, which begins

      2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, ···,

      has held untold fascination for mathematicians, both professionals and amateurs alike. The basic theorem which we shall discuss in this lecture is known as theprime number theoremand allows one to predict, at least in gross terms, the way in which the primes are distributed. Letxbe a positive real number, and let π(x) = the number of primes ≤x. Then the prime number theorem asserts that\[\underset{x\to \infty }{\mathop{\lim }}\,\frac{\pi (x)}{x/\log (x)}=1,\caption {(1)}\]

      where logxdenotes the natural log ofx....

    • A Hundred Years of Prime Numbers
      (pp. 328-336)
      PAUL T. BATEMAN and HAROLD G. DIAMOND

      This year marks the hundredth anniversary of the proof of the Prime Number Theorem (PNT), one of the most celebrated results in mathematics. The theorem is an asymptotic formula for the counting function of primes π(x) := #{px:pprime} asserting that\[\pi (x)\sim \frac{x}{\log x}.\quad (\text{PNT})\]

      The twiddle notation is shorthand for the statement\[\underset{x\to \infty }{\mathop{\lim }}\,\frac{\pi (x)}{(x/\log x)}=1.\]

      Here we shall survey early work on the distribution of primes, the proof of the PNT, and some later developments.

      Since the time of Euclid, the primes, 2, 3, 5, 7, 11, 13, …, have been known to be infinite in number. They appear to...

    • The Indian Mathematician Ramanujan
      (pp. 337-348)
      G. H. HARDY

      I have set myself a task in these lectures which is genuinely difficult and which, if I were determined to begin by making every excuse for failure, I might represent as almost impossible. I have to form myself, as I have never really formed before, and to try to help you to form, some sort of reasoned estimate of the most romantic figure in the recent history of mathematics; a man whose career seems full of paradoxes and contradictions, who defies almost all the canons by which we are accustomed to judge one another, and about whom all of us...

    • Emmy Noether
      (pp. 349-359)
      CLARK H. KIMBERLING

      The past two years have seen a surge of interest in Emmy Noether and her mathematics. Along with Auguste Dick’s biography of her, listed below, Constance Reid’s biography,Hilbert, frequently mentions Emmy Noether. New mathematics books, such asIntroduction to the Calculus of Variationsby Hans Sagan, andCommutative Ringsby Irving Kaplansky, are spreading anew her methods, and the adjective “noetherian” abounds in titles to papers in mathematics research journals. The State University of New York at Buffalo has just set up a George William Hill–Emmy Noether Fellowship. A high school textbook,Modern Introductory Analysis, by Dolciani, Donnelly,...

    • “A Marvelous Proof”
      (pp. 360-374)
      FERNANDO Q. GOUVÊA

      No one really knows when it was that the story of what came to be known as “Fermat’s Last Theorem” really started. Presumably it was sometime in the late 1630s that Pierre de Fermat made that famous inscription in the margin of Diophantus’sArithmeticaclaiming to have found “a marvelous proof”. It seems now, however, that the story may be coming close to an end. In June 1993, Andrew Wiles announced that he could prove Fermat’s assertion. Since then, difficulties seem to have arisen, but Wiles’s strategy is fundamentally sound and may yet succeed.

      The argument sketched by Wiles is...

    • Afterword
      (pp. 375-378)

      In chapter 10 of hisA History of Algebra[26], B. L. van der Waerden repeats much of what he wrote about Hamilton’s discovery of quaternions. Interestingly, there he mentions Caspar Wessel as one of the originators of the geometric interpretation of complex numbers, while in the current article he ignores Wessel. But he also goes on to discuss Cayley’s own use of quaternions to describe rotations in three-space, meanwhile pointing out the earlier results of Rodrigues. In addition, he deals with some applications of quaternions to the question of representing integers as sums of four squares. He concludes by...

  7. Surveys

    • Foreword
      (pp. 381-382)

      This final chapter contains three survey articles on mathematics, dating from 1900, 1951, and 2000, as well as a brief and subjective account of the Second International Congress of Mathematicians, held in Paris in August 1900.

      George Bruce Halsted was one of the American delegates to the Congress and wrote a report for theMonthlyshortly after he returned. The major part of the paper deals with his reactions to Hilbert’s famous address on the problems of mathematics, an address that set the agenda for twentieth-century work in mathematics. But the other talk that particularly interested Halsted was one on...

    • The International Congress of Mathematicians
      (pp. 383-384)
      GEORGE BRUCE HALSTED

      On the sixth of August at the Palais des Congrès in the Paris Exposition, was held the opening session of the second International Congress of Mathematicians. The president, Poincaré, is regarded as the greatest of living mathematicians. Among the vice presidents in attendance were Gordan, Lindeloef, Lindemann, Mittag-Leffler.

      Representing Japan was Fujisawa; Spain sent Zoel de Galdeano; the United States, Miss Scott. The president of the section of Arithmetic and Algebra was Hilbert; of Geometry was Darboux, of Bibliography and History was Prince Roland Bonaparte. Among the most interesting personalities present may be mentioned Dickstein of Warsaw, Gutzmer of Jena,...

    • A Popular Account of Some New Fields of Thought in Mathematics
      (pp. 385-390)
      G. A. MILLER

      At the beginning of the nineteenth century, elementary arithmetic was a Freshman subject in our best colleges. In 1802 the standard of admission to Harvard College was raised so as to include a knowledge of arithmetic to the ‘Rule of Three’. A boy could enter the oldest college in America prior to 1803 without a knowledge of a multiplication table ([3], p. 60). From that time on the entrance requirements in mathematics were rapidly increased, but it was not until after the founding of Johns Hopkins University that the spirit of mathematical investigation took deep root in this country.

      The...

    • A Half-Century of Mathematics
      (pp. 391-410)
      HERMANN WEYL

      Mathematics, beside astronomy, is the oldest of all sciences. Without the concepts, methods and results found and developed by previous generations right down to Greek antiquity, one cannot understand either the aims or the achievements of mathematics in the last fifty years. Mathematics has been called the science of the infinite; indeed, the mathematician invents finite constructions by which questions are decided that by their very nature refer to the infinite. That is his glory. Kierkegaard once said religion deals with what concerns man unconditionally. In contrast (but with equal exaggeration) one may say that mathematics talks about the things...

    • Mathematics at the Turn of the Millennium
      (pp. 411-422)
      PHILLIP A. GRIFFITHS

      The last century has been a golden age for mathematics. Many important, long-standing problems have been resolved, in large part because of the growing understanding of the complex interactions among the subfields of mathematics. As those relationships continue to expand and deepen, mathematics is beginning to reach out to explore interactions with other areas of science. These interactions, both within diverse areas of mathematics and between mathematics and other fields of science, have led to some great insights and to the broadening and deepening of the field of mathematics. I discuss some of these interactions and insights, describe a few...

    • Afterword
      (pp. 423-426)

      G. B. Halsted names several mathematicians who participated in the Second International Congress, most of whom are not household names today. These include, first, the representatives of the U.S., Charlotte Angas Scott (1858–1931), from Bryn Mawr College [14]; of Japan, Rikitaro Fujisawa (1861–1933); and of Spain, Zoel Garcia de Galdeano y Yanguas (1846–1924). The last of these was the academic advisor of Julio Rey Pastor, who later became the central figure in the development of mathematics in Argentina in the twentieth century. Then there were Halsted’s “interesting personalities”. One of these was Samuel Dickstein (1851–1939), a...

  8. Index
    (pp. 427-430)
  9. About the Editors
    (pp. 431-431)