De Fermat Research Articles

Pascal's legacy.

The musings of a Belgian monk 300 years ago on the consequences of believing in God, or not, have influenced our Western life probably more than any other scientific or technological invention

Fermat’s principle and acoustic tomography in a time-dependent ocean

Fermat’s principle states that, for nondispersive waves propagating between two given points, travel time is stationary with respect to small variations of the wave propagation path around the actual path (i.e., ray) taken by the wave. This fundamental principle governing short wave kinematics was first established by seventeenth century French mathematician Pierre de Fermat. The principle is of particular importance for acoustic tomography because it allows one to neglect perturbations in ray geometry when performing a linear inversion of acoustic travel times for sound speed variations. Derivations of Fermat’s principle available in the literature does not apply to acoustic waves in media with time-dependent sound speed. Moreover, the very formulation of the principle in this case is not obvious as travel time along a trial ray ceases to be a single-valued functional of the trial ray geometry. In this paper, the Fermat’s principle is extended to media with time-dependent parameters. We consider acoustic waves in the inhomogeneous, moving or quiescent fluid. Formulations of the principle in terms of travel time and eikonal are compared. Applications of the established principle to ocean acoustic tomography and modeling wave propagation in weakly nonstationary, inhomogeneous ocean are discussed. [Work supported by ONR.]

Marginalia: readers writing in books

From Pierre de Fermat to Samuel Taylor Coleridge to Graham Greene, readers have related to books through the notes they write in the margins. In this pioneering book-the first to examine the phenomenon of marginalia-H.J. Jackson surveys an extraordinary range of annotated books to explore the history of marginalia, the forms they take, the psychology that underlies them, and the reactions they provoke. Based on a study of thousands of books annotated by readers both famous and obscure over the last three centuries, this book reveals the intensity of emotion that characterizes the process of reading. For hundreds of years, readers have talked to other people in the margins of their books-not only to authors, but also to friends, lovers, and future generations. With an infectious enthusiasm for her subject, Jackson reflects on the cultural and historical value of writing in the margins, examines works that have invited passionate annotation, and presents examples of some of the most provocative marginalia. Imaginative, amusing, and poignant, this book will be treasured by-and maybe even annotated by-anyone who cares about reading.

Real analysis: a historical approach

Preface to the Second Edition Acknowledgments 1. Archimedes and the 1.1 The Area of the Parabolic Segment 1.2 The Geometry of the 2. Fermat, Differentiation, and Integration 2.1 Fermat s Calculus 3. Newton s Calculus (Part 1) 3.1 The Fractional Binomial Theorem 3.2 Areas and Infinite Series 3.3 Newton s Proofs 4. Newton s Calculus (Part 2) 4.1 The Solution of Differential Equations 4.2 The Solution of Algebraic Equations Chapter Appendix. Mathematica implementations of Newton s algorithm 5. Euler 5.1 Trigonometric Series 6. The Real Numbers 6.1 An Informal Introduction 6.2 Ordered Fields 6.3 Completeness and Irrational Numbers 6.4 The Euclidean Process 6.5 Functions 7. Sequences and Their Limits 7.1 The Definitions 7.2 Limit Theorems 8. The Cauchy Property 8.1 Limits of Monotone Sequences 8.2 The Cauchy Property 9. The Convergence of Infinite Series 9.1 Stock Series 9.2 Series of Positive 9.3 Series of Arbitrary 9.4 The Most Celebrated Problem 10. Series of Functions 10.1 Power Series 10.2 Trigonometric Series 11. Continuity 11.1 An Informal Introduction 11.2 The Limit of a Function 11.3 Continuity 11.4 Properties of Continuous Functions 12. Differentiability 12.1 An Informal Introduction to Differentiation 12.2 The Derivative 12.3 The Consequences of Differentiability 12.4 Integrability 13. Uniform Convergence 13.1 Uniform and Non-Uniform Convergence 13.2 Consequences of Uniform Convergence 14. The Vindication 14.1 Trigonometric Series 14.2 Power Series 15. The Riemann Integral 15.1 Continuity Revisited 15.2 Lower and Upper Sums 15.3 Integrability Appendix A. Excerpts from Quadrature of the Parabola by Archimedes Appendix B. On a Method for Evaluation of Maxima and Minima by Pierre de Fermat Appendix C. From a Letter to Henry Oldenburg on the Binomial Series (June 13, 1676) by Isaac Newton Appendix D. From a Letter to Henry Oldenburg on the Binomial Series (October 24, 1676) by Isaac Newton Appendix E. Excerpts from Of Analysis by Equations of an Infinite Number of Terms by Isaac Newton Appendix F. Excerpts from Subsiduum Calculi Sinuum by Leonhard Euler) Solutions to Selected Exercises Bibliography Index

Exploring Moduli Spaces

There's more to math than just numbers and equations. In fact, much of mathematics concerns space?not outer space, but mental representations of space and spatial relationships. While many people think of Descartes as the first person to describe points in a plane by means of a pair of coordi nates, the idea of coordinatizing planar space was known to Hipparchus in 150 B.C. And by 1640, Pierre de Fermat had already developed the idea of expressing a curve in the plane as the solution set of an equation in x and);. Rene Descartes, however, was the first to systemically use the coordinate plane to turn problems of geometry into problems of algebra, and vice versa. Today, at the very center of modern research in number theory and geometry lies a much-generalized form of Descartes's idea?that of moduli spaces.

Fermat's last theorem: unlocking the secret of an ancient mathematical problem

Around 1637, the French mathematician Pierre de Fermat wrote that he had found a way to prove a seemingly simple statement: while many square numbers can be broken down into the sum of two other squares - for example, 25 (five squared) equals nine (three squared) plus 16 (four squared) - the same can never be done for cubes or any higher powers. This book provides an account of how Fermat's solution was lost, the consequent struggle by mathematicians to solve this scientific mystery and how the solution was finally found in the 1990s.

Fermat's Last Theorem, the Four Color Conjecture, and Bill Clinton for April Fools' Day

1. INTRODUCTION. We often fool our students. We fool them by stating and proving theorems that are in a polished and final form. As a result, students are often unaware of the evolution of ideas, strategies, statements, and even important mistakes that inevitably lead to the beautiful theorems and they encounter. There is much value in understanding early ideas and mistakes. In some cases, mistakes lead to the discovery of powerful and deep new mathematical truths. Thus we believe that it is occasionally appropriate to celebrate mathematical mistakes and we cannot imagine a better time for such a celebration than on April Fools' Day. Here we describe and outline three erroneous from the l9th century. We provide proofs of Fermat's Last Theorem, the Four Color Conjecture, and the fact that one of us is Bill Clinton. This paper is based upon a special undergraduate mathematics colloquium the authors gave on April 1, 1996. 2. FERMAT'S LAST THEOREM. The problem that was to become one of the most infamous open problems in mathematics had a most inauspicious beginning. Circa 1637, Pierre de Fermat (Figure la), while studying Bachet's Latin translation of Arithmetica by Diophantus, came upon a discussion of the Pythagorean theorem. This inspired Fermat to write the following, now famous, lines in the margin:

Generalization of a problem of Diophantus

Greek mathematician Diophantus of Alexandria noted that the numbers 1 16 , 33 16 , 68 16 and 105 16 have the property that the product of any two of them when increased by 1 is a square of a rational number. Let n be an integer. We say the set of natural numbers {a1, a2, . . . , am} has the property of Diophantus of order n, in brief D(n), if for all i, j = 1, 2, . . . ,m, i 6= j, the following holds: ai · aj + n = bij , where bij is an integer. The first set of four natural numbers with property D(1) was found by French mathematician Pierre de Fermat (1601 1665). That set is {1, 3, 8, 120}. Davenport and Baker [3] show that a fifth integer r cannot be added to that set and maintain the same property unless r = 0. For the rational number r = 777480 8288641 the product of any two different members of that set increased by 1 is the square of a rational number (see [1]). In this paper we consider some problems of existence of sets of four natural numbers with property D(n), for any integer n. We prove that, for all e ∈ Z, there exist infinite numbers of sets of four natural numbers with property D(e2). Indeed, we show how a set {a, b} with property D(e2) can be extended to a set {a, b, c, d} with the same property, if a · b is not a perfect square. That construction is applied to the identities

Theory development processes in the social sciences: The case of stochastic choice theory

Almost three hundred years ago, Pascal and de Fermat produced a quantitative model for choice under uncertainty, based on expected payoffs. This event triggered a theory development process fueled by contributions like the logarithmic utility model of Bernoulli, the expected utility theorem (EUT) of von Neumann and Morgenstern, the non-expected utility (NEU) theories of Allais, Bell, and Machina, as well as a long series of empirical studies. This article uses the paradigm-oriented framework of Kuhn and Lakatos to describe and evaluate the history of ideas in stochastic choice theory. Particular attention is paid to the rivalry between the EUT and various NEU models, which has been called one of this century's most important disputes in the social sciences. Specific issues include the quick acceptance of the EUT, the large export of the EUT to several decision-oriented disciplines, and the periods of progress, stagnation and revitalization of NEU research. The article also contrasts the EU and NEU schools by analyzing the remarkable consensus on subject and methodology, the ambiguous use of the rationality concept, the difference in basic implicit assumptions, and the heterogeneous response to given empirical findings.

Studies in the history of probability and statistics XLII. Further details of contacts between Boscovich and Simpson in June 1760

SUMMARY In a recent paper Stigler (1984) described a manuscript fragment giving Thomas Simpson's solution to a least absolute deviations regression problem which Ruggiero (Roger) Boscovich had posed to him. In this paper we present evidence that Boscovich received Simpson's solution in person in June 1760. We also note that the second problem discussed by Stigler is closely related to a two-dimensional generalization of the least absolute deviations problem and that two instances of this third problem may be found in the work of Pierre de Fermat.

Fermat's Last Theorem: 1637—1988

Around 1637 the French jurist and amateur mathematician Pierre de Fermat wrote in the margin of his copy of Diophantus's Arithmetic that he had a “truly marvelous” proof that the equation xn + yn = zn has no solution in positive integers if n > 2. Unfortunately the margin was too narrow to contain it. In 1988 the world thought that the Japanese mathematician Yoichi Miyaoka, working at the Max Planck Institute in Bonn, West Germany, might have discovered a proof of this theorem. Such a proof would be of considerable interest because no evidence has been found that Fermat ever wrote one down, and no one has been able to find one in the 350 years since. In fact Miyaoka's announcement turned out to be premature, and a few weeks later articles reported holes in his argument that could not be repaired.

Pour l'histoire des sept premiers nombres parfaits

Until the end of the last century Pietro Cataldi was considered the first mathematician to calculate the 5th, 6th, and 7th perfect numbers. But in two memoirs (1895 and 1899) M. Curtze pointed out that he had found the 5th in Codex Latinus Monacensis 14908, dated 1461, and in Codex Vindobonensis 5203 written by Regiomontanus (1436–1476) during his stay at the University of Vienna. In recent times (1977 and 1980) M. Folkerts pointed out that the young mathematician had calculated the 5th perfect number and also that the 6th is mentioned in the comments of J. Scheybl in his partial edition of Euclid's Elements. We will show that the 5th perfect number was calculated in 1458 by the author of the Codice Palatino 573 of the Biblioteca Nazionale of Florence, which the author himself declares is a copy of a treatise written in his own hand some years before, and that both the 5th and 6th perfect numbers can be found in the Codex Ottobonianus Latinus 3307 of the Biblioteca Apostolica Vaticana composed by the same author “l'Allievo del Vaiaio” in 1460. We will also explain the criteria followed by P. Cataldi (1552–1626) in calculating the first seven perfect numbers in 1603 and show, step by step, how he demonstrated the first of the three well-known propositions on perfect numbers communicated in 1640 to P. Mersenne by P. de Fermat.

Cartesian geometry: The Dutch contribution

At the beginning of the seven teen th century , as Morris Kline has remarked, mathematics was still essentially a body of geometry with algebraic appendages. Geometry was beginning to take on a new significance, however, particularly in the light of recent developments in science and technology. The conic sections so beloved of the Greeks described the orbits of celestial bodies and the trajectories of projectiles; a study of curves in general was essential for work on optics, astronomy, and mechanics. The mathematical techniques for handling curves were still rudimentary, however, being based on classical methods of Euclid and Apollonius. What was needed, and what was first developed by Ren6 Descartes (1596-1650) and Pierre de Fermat (1601-1655), was a radical new approach to the mathematical description of curves. In essence this was the now familiar algebraic description by which curves are characterised by equations. Fermat published little during his lifetime, but Descartes set ou t his n e w approach to geometr ica l problems in La G~omdtrie, one of three important appendices to his philosophical magnum opus Discours de la mdthode pour bien conduire sa raison et chercher la vdritd dans les sciences (published in 1637). The study of curves was now transformed into the study of equations, and a geometrical problem could be attacked by converting it into an algebraic one, solving the algebraic problem, and finally interpreting the algebraic solution in the context of the original geometrical problem. Descartes and Fermat laid the basis of the modern, algebraic approach to g e o m e t r y a n approach that forms a leitmotiv running through Newtonian mechanics, n ine teen th-cen tury science, and even the Einsteinian revo!ut ion--but the term Cartesian geometry in its m o d e r n sense is someth ing of an anachronism: Descartes never expressed his ideas in the way now so familiar. For example, he continued to use

The Dice Problem-Then and Now

Is it probable that probability gives assurance? Blaise Pascal (1623-1662) asked this question more than three centuries ago. His answer, it appears, was in the negative: Nothing gives certainty but truth [7, p. 268]. This question is not asked much anymore. And if it is, the answer is vastly different, embedded as it must be in definitions and theorems. Nowadays the answer to this question is a qualified yes. Despite his apparent lack of confidence in probability, Pascal was intensely interested in problems involving probability, and in fact is regarded by many as the inventor of probability theory (see [3] and [5]). An oft-repeated story goes like this. In 1654, Antoine Chevalier de Mere (1607-1684) presented Pascal with several gambling problems, among them the so-called dice problem. Pascal, in turn, commu? nicated these problems to Pierre de Fermat (1601-1665), and in the ensuing correspondence between Pascal and Fermat the theory of probability was born. Initially, probability was practical and experiential, rooted in games of chance and in insurance and annuities. Gradually the emphasis became more abstract and analytical, especially after the formulation of the Central Limit Theorem by Pierre-Simon Laplace (1749-1827). It was not until the 1930's, however, with the presentation by Jerzy Neyman (1894-1981) ([4] and [6, p. 310]) of a clear logic for the confidence interval approach to statistical inference that probability realized its potential for giving assurance to the man in the street. More recently, with the prevalence of survey sampling and the extensive use of computers in statistical research, probability has taken a turn again toward the empirical. My purpose in this paper is to look at the dice problem from several points of view: experimentally, analytically, and via the computer. In so doing, I intend to retrace in a way the various phases in the evolution of probability. First, a statement of the dice problem: In throwing two perfectly balanced dice, how many tosses are needed to have at least an even chance of getting a pair of sixes at least once!

The Mathematical Career of Pierre de Fermat (1601-1665)

The Mathematical Career of Pierre de Fermat (1601-1665)

Book Review: The mathematical career of Pierre de Fermat

Book Review: The mathematical career of Pierre de Fermat

Analytic Geometry: The Discovery of Fermat and Descartes

Until a century ago the discovery of analytic geometry was ascribed categorically to the father of modern philosophy— René Descartes. Proles sine matre creata, Chasles called it, but this famous characterization is now quoted only to be refuted. The offspring is now recognized not only as having arrived as twins—for Pierre de Fermat had developed substantially the same method at about the same time—but also as having descended from a long line of legitimate antecedents.

Pierre de Fermat

Throughout his lifetime, Pierre de Fermat was associated with the country in and about Toulouse. He studied law in the local university and later became a member of the local parliament. His leisure time was given over to the study of mathematics and he made contributions to most if not to all of the branches of mathematics then known. Fermat's discoveries were preserved in his Varia Opera Mathematica edited by his son Samuel de Fermat and published at Toulouse in 1679, in his notes to Bachet's translation of Diophantus which were also published by the younger Fermat in 1670, and in his correspondence vvith other mathematicians among whom were Roberval and Pascal. It is difficult, accordingly, to give the precise dates of Fermat's various discoveries, but it is probable that his work on maxima and minima was done by 1629. Professor Cajori says of this that “He substituted x + e for x in the given function of x and then equated to each other the successive values of the function and divided the equation by e. If e be taken 0, then the roots of this equation are the values of x, making the function a maximum or a minimum…. The main difference between it and the rule of the differential calculus is that it introduces the indefinite quantity e instead of the infinitely small dx.” This method was attacked by Descartes and the dispute that followed involved many of the important French mathematicians.

The character, lately published beyond the seas, of an eminent person, not long since dead at Tholouse, where he was a councellor of Parliament

It is the deservedly famous Mounsieur de Fermat, who was, (faith the author of the letter) one of the most excellent men of this age, a genius fo universal, and of so vast an extent, that if very knowing and learned men had not given testimony of his extraordinary merit, what with truth can be said of him, would hardly be believed.