History of mathematics Books

Book SynopsisAny reader who aspires to be scientifically literate will find this a good starting place.-Publishers WeeklyTrade Review"More than just a celebration of the great equations…[Crease] shows how an equation not only affects science and math but also transforms the thinking of all people." -- Dick Teresi"Wry, probing, philosophically inclined." -- Charles C. Mann, author of 1491: New Revelations of the Americas Before Columbus

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Book SynopsisThe updated new edition of the classic and comprehensive guide to the history of mathematics For more than forty years, A History of Mathematics has been the reference of choice for those looking to learn about the fascinating history of humankind s relationship with numbers, shapes, and patterns.Trade Review"... the book is an essential reference that will help you explore the incredible history of mathematics and the men and women who created it." (Zentralblatt MATH, 2016) "... an 'engaging' read for the mathematically minded." (Inside OR, June 2011)Table of ContentsForeword by Isaac Asimov xi Preface to the Third Edition xiii Preface to the Second Edition xv Preface to the First Edition xvii 1 Traces 1 Concepts and Relationships 1 Early Number Bases 3 Number Language and Counting 5 Spatial Relationships 6 2 Ancient Egypt 8 The Era and the Sources 8 Numbers and Fractions 10 Arithmetic Operations 12 “Heap” Problems 13 Geometric Problems 14 Slope Problems 18 Arithmetic Pragmatism 19 3 Mesopotamia 21 The Era and the Sources 21 Cuneiform Writing 22 Numbers and Fractions: Sexagesimals 23 Positional Numeration 23 Sexagesimal Fractions 25 Approximations 25 Tables 26 Equations 28 Measurements: Pythagorean Triads 31 Polygonal Areas 35 Geometry as Applied Arithmetic 36 4 Hellenic Traditions 40 The Era and the Sources 40 Thales and Pythagoras 42 Numeration 52 Arithmetic and Logistic 55 Fifth-Century Athens 56 Three Classical Problems 57 Quadrature of Lunes 58 Hippias of Elis 61 Philolaus and Archytas of Tarentum 63 Incommensurability 65 Paradoxes of Zeno 67 Deductive Reasoning 70 Democritus of Abdera 72 Mathematics and the Liberal Arts 74 The Academy 74 Aristotle 88 5 Euclid of Alexandria 90 Alexandria 90 Lost Works 91 Extant Works 91 The Elements 93 6 Archimedes of Syracuse 109 The Siege of Syracuse 109 On the Equilibriums of Planes 110 On Floating Bodies 111 The Sand-Reckoner 112 Measurement of the Circle 113 On Spirals 113 Quadrature of the Parabola 115 On Conoids and Spheroids 116 On the Sphere and Cylinder 118 Book of Lemmas 120 Semiregular Solids and Trigonometry 121 The Method 122 7 Apollonius of Perge 127 Works and Tradition 127 Lost Works 128 Cycles and Epicycles 129 The Conics 130 8 Crosscurrents 142 Changing Trends 142 Eratosthenes 143 Angles and Chords 144 Ptolemy’s Almagest 149 Heron of Alexandria 156 The Decline of Greek Mathematics 159 Nicomachus of Gerasa 159 Diophantus of Alexandria 160 Pappus of Alexandria 164 The End of Alexandrian Dominance 170 Proclus of Alexandria 171 Boethius 171 Athenian Fragments 172 Byzantine Mathematicians 173 9 Ancient and Medieval China 175 The Oldest Known Texts 175 The Nine Chapters 176 Rod Numerals 177 The Abacus and Decimal Fractions 178 Values of Pi 180 Thirteenth-Century Mathematics 182 10 Ancient and Medieval India 186 Early Mathematics in India 186 The Sulbasutras 187 The Siddhantas 188 Aryabhata 189 Numerals 191 Trigonometry 193 Multiplication 194 Long Division 195 Brahmagupta 197 Indeterminate Equations 199 Bhaskara 200 Madhava and the Keralese School 202 11 The Islamic Hegemony 203 Arabic Conquests 203 The House of Wisdom 205 Al-Khwarizmi 206 ‘Abd Al-Hamid ibn-Turk 212 Thabit ibn-Qurra 213 Numerals 214 Trigonometry 216 Tenth- and Eleventh-Century Highlights 216 Omar Khayyam 218 The Parallel Postulate 220 Nasir al-Din al-Tusi 220 Al-Kashi 221 12 The Latin West 223 Introduction 223 Compendia of the Dark Ages 224 Gerbert 224 The Century of Translation 226 Abacists and Algorists 227 Fibonacci 229 Jordanus Nemorarius 232 Campanus of Novara 233 Learning in the Thirteenth Century 235 Archimedes Revived 235 Medieval Kinematics 236 Thomas Bradwardine 236 Nicole Oresme 238 The Latitude of Forms 239 Infinite Series 241 Levi ben Gerson 242 Nicholas of Cusa 243 The Decline of Medieval Learning 243 13 The European Renaissance 245 Overview 245 Regiomontanus 246 Nicolas Chuquet’s Triparty 249 Luca Pacioli’s Summa 251 German Algebras and Arithmetics 253 Cardan’s Ars Magna 255 Rafael Bombelli 260 Robert Recorde 262 Trigonometry 263 Geometry 264 Renaissance Trends 271 François Viète 273 14 Early Modern Problem Solvers 282 Accessibility of Computation 282 Decimal Fractions 283 Notation 285 Logarithms 286 Mathematical Instruments 290 Infinitesimal Methods: Stevin 296 Johannes Kepler 296 15 Analysis, Synthesis, the Infinite, and Numbers 300 Galileo’s Two New Sciences 300 Bonaventura Cavalieri 303 Evangelista Torricelli 306 Mersenne’s Communicants 308 René Descartes 309 Fermat’s Loci 320 Gregory of St. Vincent 325 The Theory of Numbers 326 Gilles Persone de Roberval 329 Girard Desargues and Projective Geometry 330 Blaise Pascal 332 Philippe de Lahire 337 Georg Mohr 338 Pietro Mengoli 338 Frans van Schooten 339 Jan de Witt 340 Johann Hudde 341 René François de Sluse 342 Christiaan Huygens 342 16 British Techniques and Continental Methods 348 John Wallis 348 James Gregory 353 Nicolaus Mercator and William Brouncker 355 Barrow’s Method of Tangents 356 Newton 358 Abraham De Moivre 372 Roger Cotes 375 James Stirling 376 Colin Maclaurin 376 Textbooks 380 Rigor and Progress 381 Leibniz 382 The Bernoulli Family 390 Tschirnhaus Transformations 398 Solid Analytic Geometry 399 Michel Rolle and Pierre Varignon 400 The Clairauts 401 Mathematics in Italy 402 The Parallel Postulate 403 Divergent Series 404 17 Euler 406 The Life of Euler 406 Notation 408 Foundation of Analysis 409 Logarithms and the Euler Identities 413 Differential Equations 414 Probability 416 The Theory of Numbers 417 Textbooks 418 Analytic Geometry 419 The Parallel Postulate: Lambert 420 18 Pre- to Postrevolutionary France 423 Men and Institutions 423 The Committee on Weights and Measures 424 D’Alembert 425 Bézout 427 Condorcet 429 Lagrange 430 Monge 433 Carnot 438 Laplace 443 Legendre 446 Aspects of Abstraction 449 Paris in the 1820s 449 Fourier 450 Cauchy 452 Diffusion 460 19 Gauss 464 Nineteenth-Century Overview 464 Gauss: Early Work 465 Number Theory 466 Reception of the Disquisitiones Arithmeticae 469 Astronomy 470 Gauss’s Middle Years 471 Differential Geometry 472 Gauss’s Later Work 473 Gauss’s Influence 474 20 Geometry 483 The School of Monge 483 Projective Geometry: Poncelet and Chasles 485 Synthetic Metric Geometry: Steiner 487 Synthetic Nonmetric Geometry: von Staudt 489 Analytic Geometry 489 Non-Euclidean Geometry 494 Riemannian Geometry 496 Spaces of Higher Dimensions 498 Felix Klein 499 Post-Riemannian Algebraic Geometry 501 21 Algebra 504 Introduction 504 British Algebra and the Operational Calculus of Functions 505 Boole and the Algebra of Logic 506 Augustus De Morgan 509 William Rowan Hamilton 510 Grassmann and Ausdehnungslehre 512 Cayley and Sylvester 515 Linear Associative Algebras 519 Algebraic Geometry 520 Algebraic and Arithmetic Integers 520 Axioms of Arithmetic 522 22 Analysis 526 Berlin and Göttingen at Midcentury 526 Riemann in Göttingen 527 Mathematical Physics in Germany 528 Mathematical Physics in English-Speaking Countries 529 Weierstrass and Students 531 The Arithmetization of Analysis 533 Dedekind 536 Cantor and Kronecker 538 Analysis in France 543 23 Twentieth-Century Legacies 548 Overview 548 Henri Poincaré 549 David Hilbert 555 Integration and Measure 564 Functional Analysis and General Topology 568 Algebra 570 Differential Geometry and Tensor Analysis 572 Probability 573 Bounds and Approximations 575 The 1930s and World War II 577 Nicolas Bourbaki 578 Homological Algebra and Category Theory 580 Algebraic Geometry 581 Logic and Computing 582 The Fields Medals 584 24 Recent Trends 586 Overview 586 The Four-Color Conjecture 587 Classification of Finite Simple Groups 591 Fermat’s Last Theorem 593 Poincaré’s Query 596 Future Outlook 599 References 601 General Bibliography 633 Index 647

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Book SynopsisStatistics have helped shape every area of science. Without the means to analyze critical data, none of the great disoveries of the past would be possible. This paperback reprint of a Wiley bestseller shows the development of these data analysis tools and the manner in which they aided technological development prior to 1750.Trade Review"...the account goes into great detail...very accessible...useful for teachers..." (Short Book Reviews, Vol 24(1), 2004)Table of Contents1. The Book and Its Relation to Other Works. 2. A Sketch of the Background in Mathematics and Natural Philosophy. 3. Early Concepts of Probability and Chance. 4. Cardano and Liber de Ludo Aleae, c. 1565. 5. The Foundation of Probability Theory by Pascal and Fermat in 1654. 6. Huygens and De Ratiociniis in Ludo Aleae, 1657. 7. John Graunt and the Observations Made upon the Bills of Mortality, 1662. 8. The Probabilistic Interpretation of Graunt's Life Table. 9. The Early History of Life Insurance Mathematics. 10. Mathematical Models and Statistical Methods in Astronomy from Hipparchus to Kepler and Galileo. 11. The Newtonian Revolution in Mathematics and Science. 12. Miscellaneous Contributions Between 1657 and 1708. 13. The Great Leap Forward, 1708 - 1718: A Survey. 14. New Solutions to Old Problems, 1708 - 1718. 15. James Bernoulli and Ars Conjectandi, 1713. 16. Bernoulli's Theorem. 17. Tests of Significance Based on the Sex Ratio at Birth and the Binomial Distribution, 1712 - 1713. 18. Montmort and the Essay d'Analyse sur les Jeux de Hazard, 1708 and 1713. 19. The Problem of Coincidences and the Compound Probability Theorem. 20. The Problems of the Duration of Play, 1708–1718. 21. Nicholas Bernoulli. 22. De Moivre and the Doctrine of Chances, 1718, 1738, and 1756. 23. The Problem of the Duration of Play and the Method of Difference Equations. 24. De Moivre's Normal Approximation to the Binomial Distribution, 1733. 25. The Insurance Mathematics of de Moivre and Simpson, 1725-1756. References. Index.

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Book SynopsisThis original anthology collects 10 of Weyl''s less-technical writings that address the broader scope and implications of mathematics. Most have been long unavailable or not previously published in book form. Subjects include logic, topology, abstract algebra, relativity theory, and reflections on the work of Weyl''s mentor, David Hilbert. 2012 edition.

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Book SynopsisThis book is at once an analytical study of one of the most important mathematical texts of antiquity, the Mathematical Collection of the fourth-century AD mathematician Pappus of Alexandria, and also an examination of the work's wider cultural setting.Trade Review'[Cuomo] takes a refreshing approach to the history of mathematics.' Journal of Roman Studies'… Cuomo does an admirable job in hopefully tempting more students and scholars from different fields to tackle these themes and, even more importantly, to cooperate and cross the lines between disciplines.' De novis libris iudiciaTable of ContentsAcknowledgements; Introduction; 1. The outside world; 2. Bees and philosophers; 3. Inclined planes and architects; 4. Altars and strange curves; 5. The inside story; Bibliography; General index; Index locorum.

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Book SynopsisThis book analyzes the historical transformation of early mathematics, from a Greek practice based on the localized solution to an Islamic practice based on the systematic approach. The transformation is accounted for in terms of changing social practices, thereby offering an alternate interpretation of the historical trajectory of mathematics.Trade Review"For the true mathematics historian, this is a fascinating exploration, perhaps different from one's previous ideas of this time period. Highly recommended." M.D. Sanford, Felician College"...engaging, provocative, and definitely worth reading and thinking about." MAA Reviews, Fernando Q. Gouvea"...recommended reading--for its thought-provoking ideas and lively writing--for those with a serious interest in the mathematics of ancient Greece and medieval Islam." - Mathematical Reviews, J.L. BerggrenTable of ContentsAcknowledgements; Introduction; 1. The problem in the world of Archimedes; 2. From Archimedes to Eutocius; 3. From Archimedes to Khayyam; Conclusion; References; Index.

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Book SynopsisThe main part of the third volume of Dr Whiteside's annotated and critical edition of all the known mathematical papers of Isaac Newton reproduces, from the original autograph, Newton's elaborate tract on infinite series and fluxions (the so-called Methodus Fluxionum), including a formerly unpublished appendix on geometrical fluxions. Ancillary documents include, in Part 1, papers on the integration of algebraic functions and, in Part 2, short texts dealing with geometry and simple harmonic motion in a cycloidal arc. Part 3 reproduces, from both manuscript versions of Newton's Lectiones Opticae and from his Waste Book, mathematical excerpts from his researches into light and the theory of lenses at this period. An appendix summarizes mathematical highlights in his contemporary correspondence.Table of ContentsPart I. Researches into Fluxions and Infinate Series: 1. Preliminary Scheme for a Treatsie on Fluxions; 2. The Tract '[De Methodis Serierum et Fluxionum]'; 3. The Quadrature of Curves Defined by Polynomials; Part II. Miscellaneous Researches: 1. The Second Book of Euclid's 'Elements' Reworked; 2. Research into the Elementary Geometry of Curved Surfaces; 3. Harmonic Motion in a Cyclodial Arc; Part III. Researches in Geometrical Optics: 1. Extracts from Newton's Lectures on Optics; 2. Miscellaneous Researches into Refraction at a Curved Interface

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Book SynopsisThis volume reproduces the texts of a number of important, yet relatively minor papers, many written during a period of Newton's life (1677â84) which has been regarded as mathematically barren except for his Lucasian lectures on algebra (which appear in Volume V). Part 1 concerns itself with his growing mastery of interpolation by finite differences, culminating in his rule for divided differences. Part 2 deals with his contemporary advances in the pure and analytical geometry of curves. Part 3 contains the extant text of two intended treatises on fluxions and infinite series: the Geometria Curvilinea (c. 1680), and his Matheseos Universalis Specimina (1684). A general introduction summarizes the sparse details of Newton's personal life during the period, one â from 1677 onwards â of almost total isolation from his contemporaries. A concluding appendix surveys highlights in his mathematical correspondence during 1674â6 with Collins, Dary, John Smith and above all Leibniz.Table of ContentsPart I. Researches in Algebra, Number Theory and Trigonometry: 1. Approaches to a General Theory of Finite Differences; 2. Problems in Elementary Number Theory; 3. Codifications of Elementary Plane and Spherical Trigonometry; 4. Miscellaneous Notes on Annuities and Algebraic Factorization; Part II. Researches in Pure and Analytical Geometry: 1. Miscellaneous Problems in Elementary Geometry; 2. Researches into the Greek 'Solid Locus'; 3. Miscellaneous Topics in Analytical Geometry; Part III. The 'Geometria Curvilinea' and 'Matheseos Universalis Specimina': 1. The 'Geometry of Curved Lines'; 2. Specimens of a Universal System of Mathematics; Appendix.

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Book SynopsisThe fifth volume of this definitive edition centres around Newton's Lucasian lectures on algebra, purportedly delivered during 1673â83, and subsequently prepared for publication under the title Arithmetica Universalis many years later. Dr Whiteside first reproduces the text of the lectures deposited by Newton in the Cambridge University Library about 1684. In these much reworked, not quite finished, professional lectiones, Newton builds upon his earlier studies of the fundamentals of algebra and its application to the theory and construction of equations, developing new techniques for the factorizing of algebraic quantities and the delimitation of bounds to the number and location of roots, with a wealth of worked arithmetical, geometrical, mechanical and astronomical problems. An historical introduction traces what is known of the background to the parent manuscript and assesses the subsequent impact of the edition prepared by Whiston about 1705 and the revised version published by NeTable of ContentsPart I. The Deposited Lucasian Lectures on Algebra (Winter 1683–1684): Introduction; 1. Preliminary notes and drafts for the 'Arithmetica'; 2. The copy deposited in the Cambridge archives; Part II. The 'Arithmeticæ Universalis Liber Primus' (1684): Introduction; Index of Names

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Book SynopsisWhen Newton left Cambridge in April 1696 to take up, at the age of 53, a new career at the London Mint, he did not entirely 'leave off Mathematicks' as he so often publicly declared. This last volume of his mathematical papers presents the extant record of the investigations which for one reason and another he pursued during the last quarter of his life. In January 1697 Newton was tempted to respond to two challenges issued by Johann Bernoulli to the international community of mathematicians, one the celebrated problem of identifying the brachistochrone; both he resolved within the space of an evening, producing an elegant construction of the cycloid which he identified to be the curve of fall in least time. In the autumn of 1703, the appearance of work on 'inverse fluxions' by George Cheyne similarly provoked him to prepare his own ten-year-old treatise De Quadratura Curvarum for publication, and more importantly to write a long introduction to it where he set down what became his besTable of ContentsPart I. Solutions to Challenge-Problems, Revisions of Earlier Researches, and General Retrospections: 1. The Twin Problems of Bernoulli's 1697 'Programma' solved; 2. The 'De Quadratura Curvarum' Revised for Publication; 3. Miscellaneous Writings on Mathematics; 4. The 'Method of [Finite] Differences'; 5. The 'De Quadratura' Amplified as an 'Analysis per Quantitates Fluentes et Earum Momenta'; 6. Proposition X of the Principa's Second Book Reworked; 7. Response to Bernoulli's Second Problem; 8. Analysis and Synthsis: Newton's Declaration of the Manner of their Application in the 'Principia'; 9. Minor Compliments to the 'Arithemetica Universalis'; Part II. Newton's Varied Efforts to Substantiate His Claims to Calculus Priority: Appendix 1; Appendix 2; Appendix 3; Appendix 4; Appendix 5; Appendix 6; Appendix 7; Appendix 8; Appendix 9; Appendix 10; Index of Names

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Book SynopsisThe bringing together, in an annotated and critical edition, of all the known mathematical papers of Isaac Newton marks a step forward in the publication of the works of this great natural philosopher. In all, there are eight volumes in this present edition. Translations of papers in Latin face the original text and notes are printed on the page-openings to which they refer, so far as possible. Each volume contains a short index of names only and an analytical table of contents; a comprehensive index to the complete work is included in Volume VIII. Volume I covers three exceptionally productive years: Newton's final year as an undergraduate at Trinity College, Cambridge, and the two following years, part of which were spent at his home in Lincolnshire on account of the closure of the university during an outbreak of bubonic plague.Table of ContentsPart I. The First Mathematical Annotations 1664–1665: 1. Annotations from Oughtred, Descartes, Schooten and Huygens; 2. Annotations from Viete and Oughtred; 3. Annotations from Wallis; Part II. Researches in Analytical Geometry and Calculus 1664–1666: 1. Early notes on Analytical Geometry; 2. Work on the Cartesian Subnormal; 3. Miscellaneous Problems in Analytical Geometry and Calculus; 4. Normals, Curvature and the Resolution of the General Problem of Tangents; 5. The Calculus Becomes an Algorithm; 6. The General Problems of Tangents, Curvature and Limit-Motion Analysed by the Method of Fluxions; 7. The October 1966 Tract of Fluxions; Part III. Miscellaneous Early Mathematical Researches 1664–1666: 1. Early Scraps in Newton's Waste Book; 2. Early Work in Trigonometry; 3. The Theory and Construction of Equations; 4. Miscellaneous Researches in Arithmetic, Number Theory and Geometry; Appendix

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Book SynopsisProbably the most celebrated controversy in all of the history of science was that between Newton and Leibniz over the invention of the calculus. Philosophers at War reveals how the dispute arose and became embittered, the dispositions of the chief actors, and the shifts in their opinions of each other.Table of ContentsPreface; Chronological outline; 1. Introduction; 2. Beginnings in Cambridge; 3. Newton states his claim: 1685; 4. Leibniz encounters Newton: 1672–1676; 5. The emergence of the calculus: 1677–1699; 6. The outbreak: 1693–1700; 7. Open warfare: 1700–1710; 8. The philosophical debate; 9. Thrust and parry: 1710–1713; 10. The dogs of war: 1713–1715; 11. War beyond death: 1715–1722; Appendix; Notes; Index.

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Book SynopsisHistorical records show that there was no real concept of probability in Europe before the mid-seventeenth century, although the use of dice and other randomizing objects was commonplace. First published in 1975, this edition includes an introduction that contextualizes his book in light of developing philosophical trends.Trade Review"A fascinating in-depth study of the philosophical aspects of the concept of probability during its founding days." Andreas Karlsson, Uppsala University"[Hacking's] knowledge of the pertinent literature is considerable and the vigorous style of writing makes for enjoyable reading. Hacking states that his book was not written as history: be that as it may, but anyone who is interested in the history of probability and statistics, either as a philosopher or as a statistician, will find much here to think about." A.I. Dale, Mathematical ReviewsTable of ContentsIntroduction; 1. An absent family of ideas; 2. Duality; 3. Opinion; 4. Evidence; 5. Signs; 6. The first calculations; 7. The Roannez circle; 8. The great decision; 9. The art of thinking; 10. Probability and the law; 11. Expectation; 12. Political arithmetic; 13. Annuities; 14. Equipossibility; 15. Inductive logic; 16. The art of conjecturing; 17. The first limit theorem; 18. Design; 19. Induction.

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Book SynopsisThis book analyzes the historical transformation of early mathematics, from a Greek practice based on the localized solution to an Islamic practice based on the systematic approach. The transformation is accounted for in terms of changing social practices, thereby offering an alternate interpretation of the historical trajectory of mathematics.Trade Review"For the true mathematics historian, this is a fascinating exploration, perhaps different from one's previous ideas of this time period. Highly recommended." M.D. Sanford, Felician College"...engaging, provocative, and definitely worth reading and thinking about." MAA Reviews, Fernando Q. Gouvea"...recommended reading--for its thought-provoking ideas and lively writing--for those with a serious interest in the mathematics of ancient Greece and medieval Islam." - Mathematical Reviews, J.L. BerggrenTable of ContentsAcknowledgements; Introduction; 1. The problem in the world of Archimedes; 2. From Archimedes to Eutocius; 3. From Archimedes to Khayyam; Conclusion; References; Index.

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Book SynopsisHistorical records show that there was no real concept of probability in Europe before the mid-seventeenth century, although the use of dice and other randomizing objects was commonplace. First published in 1975, this edition includes an introduction that contextualizes his book in light of developing philosophical trends.Trade Review"A fascinating in-depth study of the philosophical aspects of the concept of probability during its founding days." Andreas Karlsson, Uppsala University"[Hacking's] knowledge of the pertinent literature is considerable and the vigorous style of writing makes for enjoyable reading. Hacking states that his book was not written as history: be that as it may, but anyone who is interested in the history of probability and statistics, either as a philosopher or as a statistician, will find much here to think about." A.I. Dale, Mathematical ReviewsTable of ContentsIntroduction; 1. An absent family of ideas; 2. Duality; 3. Opinion; 4. Evidence; 5. Signs; 6. The first calculations; 7. The Roannez circle; 8. The great decision; 9. The art of thinking; 10. Probability and the law; 11. Expectation; 12. Political arithmetic; 13. Annuities; 14. Equipossibility; 15. Inductive logic; 16. The art of conjecturing; 17. The first limit theorem; 18. Design; 19. Induction.

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Book SynopsisIn these selected essays, Charles Parsons surveys the contributions of philosophers and mathematicians who shaped the philosophy of mathematics over the past century: Brouwer, Hilbert, Bernays, Weyl, Gödel, Russell, Quine, Putnam, Wang, and Tait.Trade ReviewParsons is a much admired and highly respected philosopher of mathematics and logic, well-known for his thoughtful and careful reflections on both the great historical figures and on work of the previous century. He is also an astute commentator on the current literature, engaging the contemporary debates and offering illuminating insights about its content and direction. This volume offers a unique opportunity for those not fortunate enough to have attended classes of Parsons’s to form some idea of what such an experience would be like. -- William Demopoulos, University of Western OntarioThis is a truly superb book. Parsons is quite possibly the most distinguished writer on philosophy of mathematics now working and certainly the most careful and probing. These essays examine a rather wide range of historical opinion on mathematical matters, both with an eye to demanding more careful interpretations and formulations from important writers such as Kant or Gödel while remaining sympathetic to their overall philosophical ambitions. Parsons’s treatments are unsurpassed. -- Mark Wilson, University of Pittsburgh

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Book SynopsisSurveys the assortment of methods put forth for fixing Frege's system, in an attempt to determine just how much of mathematics can be reconstructed in each. This work considers every proposed fix, each with its distinctive philosophical advantages and drawbacks.Trade ReviewCo-Winner of the 2007 Shoenfield Prize, Association for Symbolic Logic "Fixing Frege fills a serious gap in the Frege's literature (always increasing but perhaps with an excessive attention paid to semantics and the philosophy of language) and should remain for a long time a necessary reference for scholars in the field."--Ignacio Angelelli, Review of Modern LogicTable of ContentsAcknowledgments ix CHAPTER 1: Frege, Russell, and After 1 CHAPTER 2: Predicative Theories 86 CHAPTER 3: Impredicative Theories 146 Tables 215 Notes 227 References 241 Index 249

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Book SynopsisConcentrates on thirty highlights of pure and applied mathematics. This book opens by discussing the four main philosophical foundations of mathematics of the nineteenth century and ends by describing the four important open mathematical problems of the twenty-first century.Trade Review"Odifreddi's overview is of course a personal one, but it is hard to argue with either his choices or his organization. This is a perfect handle on an otherwise bewildering proliferation of ideas."--Ben Longstaff, New Scientist "Odifreddi clearly and concisely describes important 20th-century developments in pure and applied mathematics... Unlike similar volumes, this book keeps descriptions general and contains a short section on the philosophical foundations of mathematics to help non-mathematicians easily navigate the material."--Library Journal "This is an astonishingly readable, succinct, and wonderful account of twentieth-century mathematics! It is a great book for mathematics majors, students in liberal-arts courses in mathematics, and the general public. I am amazed at how easily the author has set out the achievements in a broad array of mathematical fields. The writing appears effortless."--Paul Campbell, Mathematics Magazine "Piergiogio Odifreddi's book successfully portrays the major developments in 20th century mathematics by an examination of the mathematical problems that have gained prominence during the past 100 years... [T]he literary style is such that the contents are made accessible to a very wide readership, but with no hint of oversimplification."--P.N. Ruane, MathDL "Odifreddi ... has an engaging and effective style and a knack for compact but comprehensible summaries, making his presentation seem effortless. The Mathematical Century can be dabbled in, read through, or perhaps even used as a quick reference."--Danny Yee, Danny ReviewsTable of ContentsForeword xi Acknowledgments xvii Introduction 1 CHAPTER 1: THE FOUNDATIONS 8 1.1. The 1920s: Sets 10 1.2. The 1940s: Structures 14 1.3. The 1960s: Categories 17 1.4. The 1980s: Functions 21 CHAPTER TWO: PURE MATHEMATICS 25 2.1. Mathematical Analysis: Lebesgue Measure (1902) 29 2.2. Algebra: Steinitz Classification of Fields (1910) 33 2.3. Topology: Brouwer's Fixed-Point Theorem (1910) 37 2.4. Number Theory: Gelfand Transcendental Numbers (1929) 39 2.5. Logic: Godel's Incompleteness Theorem (1931) 43 2.6. The Calculus of Variations: Douglas's Minimal Surfaces (1931) 47 2.7. Mathematical Analysis: Schwartz's Theory of Distributions (1945) 52 2.8. Differential Topology: Milnor's Exotic Structures (1956) 56 2.9. Model Theory: Robinson's Hyperreal Numbers (1961) 59 2.10. Set Theory: Cohen's Independence Theorem (1963) 63 2.11. Singularity Theory: Thom's Classification of Catastrophes (1964) 66 2.12. Algebra: Gorenstein's Classification of Finite Groups (1972) 71 2.13. Topology: Thurston's Classification of 3-Dimensional Surfaces (1982) 78 2.14. Number Theory: Wiles's Proof of Fermat's Last Theorem (1995) 82 2.15. Discrete Geometry: Hales's Solution of Kepler's Problem (1998) 87 CHAPTER THREE: APPLIED MATHEMATICS 92 3.1. Crystallography: Bieberbach's Symmetry Groups (1910) 98 3.2. Tensor Calculus: Einstein's General Theory of Relativity (1915) 104 3.3. Game Theory: Von Neumann's Minimax Theorem (1928) 108 3.4. Functional Analysis: Von Neumann's Axiomatization of Quantum Mechanics (1932) 112 3.5. Probability Theory: Kolmogorov's Axiomatization (1933) 116 3.6. Optimization Theory: Dantzig's Simplex Method (1947) 120 3.7. General Equilibrium Theory: The Arrow-Debreu Existence Theorem (1954) 122 3.8. The Theory of Formal Languages: Chomsky's Classification (1957) 125 3.9. Dynamical Systems Theory: The KAM Theorem (1962) 128 3.10. Knot Theory: Jones Invariants (1984) 132 CHAPTER FOUR: MATHEMATICS AND THE COMPUTER 139 4.1. The Theory of Algorithms: Turing's Characterization (1936) 145 4.2. Artificial Intelligence: Shannon's Analysis of the Game of Chess (1950) 148 4.3. Chaos Theory: Lorenz's Strange Attractor (1963) 151 4.4. Computer-Assisted Proofs: The Four-Color Theorem of Appel and Haken (1976) 154 4.5. Fractals: The Mandelbrot Set (1980) 159 CHAPTER FIVE: OPEN PROBLEMS 165 5.1. Arithmetic: The Perfect Numbers Problem (300 BC) 166 5.2. Complex Analysis: The Riemann Hypothesis (1859) 168 5.3. Algebraic Topology: The Poincare Conjecture (1904) 172 5.4. Complexity Theory: The P=NP Problem (1972) 176 Conclusion 181 References and Further Reading 187 Index 189

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Book SynopsisA student in class asks the math teacher: "Shouldn't minus times minus make minus?" Teachers soon convince most students that it does not. Yet the innocent question brings with it a germ of mathematical creativity. What happens if we encourage that thought, odd and ungrounded though it may seem? Few books in the field of mathematics encourage suchTrade Review"Alberto A. Martinez ... shows that the concept of negative numbers has perplexed not just young students but also quite a few notable mathematicians... The rule that minus times minus makes plus is not in fact grounded in some deep and immutable law of nature. Martinez shows that it's possible to construct a fully consistent system of arithmetic in which minus times minus makes minus. It's a wonderful vindication for the obstinate smart-aleck kid in the back of the class."--Greg Ross, American Scientist "Alberto Martinez ... has written an entire book about the fact that the product of two negative numbers is considered positive. He begins by reminding his readers that it need not be so... The book is written in a relaxed, conversational manner... It can be recommended to anyone with an interest in the way algebra was developed behind the scenes, at a time when calculus and analytic geometry were the main focus of mathematical interest."--James Case, SIAM News "[Negative Math] is very readable and the style is entertaining. Much is done through examples rather than formal proofs. The writer avoids formal mathematical logic and the more esoteric abstract algebras such as group theory."--Mathematics MagazineTable of ContentsFigures ix Chapter 1: Introduction 1 Chapter 2: The Problem 10 Chapter 3: History: Much Ado About Less than Nothing 18 The Search for Evident Meaning 36 Chapter 4: History: Meaningful and Meaningless Expressions 43 Impossible Numbers? 66 Chapter 5: History: Making Radically New Mathematics 80 From Hindsight to Creativity 104 Chapter 6: Math Is Rather Flexible 110 Sometimes -1 Is Greater than Zero 112 Traditional Complications 115 Can Minus Times Minus Be Minus? 131 Unity in Mathematics 166 Chapter 7: Making a Meaningful Math 174 Finding Meaning 175 Designing Numbers and Operations 186 Physical Mathematics? 220 Notes 235 Further Reading 249 Acknowledgments 259 Index 261

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Book SynopsisBy any measure, the Pythagorean theorem is the most famous statement in all of mathematics. In this book, the author reveals the full story of this ubiquitous geometric theorem. It shows that the theorem, although attributed to Pythagoras, was known to the Babylonians more than a thousand years earlier.Trade ReviewHonorable Mention for the 2007 Best Professional/Scholarly Book in Mathematics, Association of American Publishers "This excellent biography of the theorem is like a history of thought written in lines and circles, moving from ancient clay tablets to Einstein's blackboards... There is something intoxicating about seeing one truth revealed in so many ways. It all makes for hours of glorious mathematical distraction."--Ben Longstaff, New Scientist "[The Pythagorean Theorem] is aimed at the reader with an interest in the history of mathematics. It should also appeal to most well-educated people...It is a story based on a theme and guided by a timeline...As a popular account of important ideas and their development, the book should be read by anyone with a good education. It deserves to succeed."--Peter M. Neumann, Times Higher Education Supplement "Based on this recent book, Maor just keeps getting better. Already recognized for his excellent books on infinity, the number e, and trigonometry, Maor offers this new work as a comprehensive overview of the Pythagorean Theorem...If one has never read a book by Eli Maor, this book is a great place to start."--J. Johnson, Choice "Maor expertly tells the story of how this simple theorem known to schoolchildren is part and parcel of much of mathematics itself... Even mathematically savvy readers will gain insights into the inner workings and beauty of mathematics."--Amy Shell-Gellasch, MAA Reviews "Maor's book is a concise history of the Pythagorean theorem, including the mathematicians, cultures, and people influenced by it. The work is well written and supported by several proofs and exampled from Chinese, Arabic, and European sources the document how these unique cultures came to understand and apply the Pythagorean theorem. [The book] provides thoughtful commentary on the historical connections this fascinating theorem has to many cultures and people."--Michael C. Fish, Mathematics Teacher "This book will make for good supplementary reading for high school students, high school teachers, and those with a general interest in mathematics... The author's enthusiasm for his subject is evident throughout the book."--James J. Tattersull, Mathematical Reviews "This book goes beyond the theorem and its proofs to set it beautifully in the context of its time and subsequent history."--Eric S. Rosenthal, Mathematics Magazine "This is an excellent book on the history of the Pythagorean Theorem... This book is suitable to any student who has basic knowledge of calculus but the layperson will also find it interesting... Maor has an exceptional method of writing very technical mathematics in a seamlessly way."--Kuldeep, Mathematics and My Diary "All in all, this affordable book, as with Maor's previous titles, is rollicking good fun and highly recommended to anyone with even the slightest interest in the history of mathematics."--Francis A, Grabowski, European Legacy "The Pythagorean Theorem is rich in information, careful in its presentation, and at times personal in its approach... The variety of its topics and the engaging way they are presented make The Pythagorean Theorem a pleasure to read."--Cecil Rousseau, College Math JournalTable of ContentsList of Color Plates ix Preface xi Prologue: Cambridge, England, 1993 1 Chapter 1: Mesopotamia, 1800 bce 4 Sidebar 1: Did the Egyptians Know It? 13 Chapter 2: Pythagoras 17 Chapter 3: Euclid's Elements 32 Sidebar 2: The Pythagorean Theorem in Art, Poetry, and Prose 45 Chapter 4: Archimedes 50 Chapter 5: Translators and Commentators, 500-1500 ce 57 Chapter 6: Francois Viete Makes History 76 Chapter 7: From the Infinite to the Infinitesimal 82 Sidebar 3: A Remarkable Formula by Euler 94 Chapter 8: 371 Proofs, and Then Some 98 Sidebar 4: The Folding Bag 115 Sidebar 5: Einstein Meets Pythagoras 117 Sidebar 6: A Most Unusual Proof 119 Chapter 9: A Theme and Variations 123 Sidebar 7: A Pythagorean Curiosity 140 Sidebar 8: A Case of Overuse 142 Chapter 10: Strange Coordinates 145 Chapter 11: Notation, Notation, Notation 158 Chapter 12: From Flat Space to Curved Spacetime 168 Sidebar 9: A Case of Misuse 177 Chapter 13: Prelude to Relativity 181 Chapter 14: From Bern to Berlin, 1905-1915 188 Sidebar 10: Four Pythagorean Brainteasers 197 Chapter 15: But Is It Universal? 201 Chapter 16: Afterthoughts 208 Epilogue: Samos, 2005 213 Appendixes A. How did the Babylonians Approximate? 219 B. Pythagorean Triples 221 C. Sums of Two Squares 223 D. A Proof that is Irrational 227 E. Archimedes' Formula for Circumscribing Polygons 229 F. Proof of some Formulas from Chapter 7 231 G. Deriving the Equation x2/3 ??y2/3 ??1 235 H. Solutions to Brainteasers 237 Chronology 241 Bibliography 247 Illustrations Credits 251 Index 253

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Book SynopsisRecalls the last words of the great Greek mathematician Archimedes before he was slain by a Roman soldier - "Don't disturb my circles" - words that seem to refer to two radically different concerns: that of the practical person living in the concrete world of reality, and that of the theoretician lost in a world of abstraction.Trade Review"Editors Doxiadis and Mazur have compiled a collection of 15 essays that look at the many possible roles narrative can play in mathematics, which is usually considered far removed from storytelling... Circles Disturbed will be of special value to collections in history of mathematics, philosophy of mathematics, and mathematical pedagogy."--Choice "Circles Disturbed presents a cohesive narrative whose strength lies in helping each side to understand the other. It should encourage scientists to grasp the logic behind storytelling and literary critics to sense the allure of mathematics."--Mel Bayley, British Society for the History of Mathematics Bulletin "Well thought and well written and with a careful balance between erudition and down-to-earthness all through it, Circles Disturbed is a highly recommended reading for mathematicians and students of mathematics, as well as for anyone who wishes to better understand what it is to do mathematics and why they are done the way they are done."--Capi Corrales Rodriganez, European Mathematical Society "Circles Disturbed will spark interest in younger readers in the commonalities among these three disciplines as well as engage other readers. Further, readers with greater background in one or more topics can see the intra- and the intersections rather naturally and inquisitively. The diverse perspectives represented by the various authors are quite refreshing."--Farshid Safi, Mathematics TeacherTable of ContentsIntroduction vii Chapter 1: From Voyagers to Martyrs: Toward a Storied History of Mathematics 1 By AMIR ALEXANDER Chapter 2 Structure of Crystal, Bucket of Dust 52 By PETER GALISON Chapter 3: Deductive Narrative and the Epistemological Function of Belief in Mathematics: On Bombelli and Imaginary Numbers 79 By FEDERICA LANAVE Chapater 4: Hilbert on Theology and Its Discontents: The Origin Myth of Modern Mathematics 105 By COLIN MCLARTY Chapter 5: Do Androids Prove Theorems in Their Sleep? 130 By MICHAEL HARRIS Chapter 6: Visions, Dreams, and Mathematics 183 By BARRY MAZUR Chapter 7: Vividness in Mathematics and Narrative 211 By TIMOTHY GOWERS Chapter 8: Mathematics and Narrative: Why Are Stories and Proofs Interesting? 232 By BERNARD TEISSIER Chapter 9: Narrative and the Rationality of Mathematical Practice 244 By DAVID CORFIELD Chapter 10: A Streetcar Named (among Other Things) Proof: From Storytelling to Geometry, via Poetry and Rhetoric 281 By APOSTOLOS DOXIADIS Chapter 11: Mathematics and Narrative: An Aristotelian Perspective 389 By G .E .R . LLOYD Chapter 12: Adventures of the Diagonal: Non-Euclidean Mathematics and Narrative 407 By ARADY PLOTNITSKY Chapter 13: Formal Models in Narrative Analysis 447 By DAVID HERMAN Chapter 14: Mathematics and Narrative: A Narratological Perspective 481 By URI MARGOL N Chapter 15: Tales of Contingency, Contingencies of Telling: Toward an Algorithm of Narrative Subjectivity 508 By JAN CHRISTOPH MEISTER Contributors 541 Index 545

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Book SynopsisTrade ReviewWinner of the 2016 PROSE Award in Mathematics, Association of American Publishers One of Choice's Outstanding Academic Titles for 2015 "Mathematics without Apologies ... provide[s] an unmatched perspective on life in this 'problematic vocation' ... a kaleidoscope of philosophical, sociological, historical and literary perspectives on what mathematicians do, and why."--Amir Alexander, Nature "Harris is the kind of mathematician one hopes to meet at an intimate dinner party... Recommended for curious readers in any subject wishing to answer problems in creative ways."--Library Journal "If you are interested at all in what mathematics really is and what the best mathematicians really do (and you're up for an intellectual challenge), I highly recommend that you get a copy and set some time aside for delving into this unusual book... Harris manages to move back and forth between the deepest ideas about mathematics at the frontiers of the subject, insightful takes on the sociology of mathematical research, and a variety of topics pursued in a sometimes gonzo version of post-modern academic style. You will surely sometimes be baffled, but definitely will come away knowing about many things you'd never heard of before, and with a lot of new ideas to think about."--Peter Woit, Not Even Wrong "[A] wry and insightful look at what being a pure mathematician is all about, as seen from the inside."--Steven Strogatz, Physics Today "This extraordinary, extravagant Apologia pro Vita Sua--the title more deliberately echoes G. H. Hardy's renowned 1940 memoir A Mathematician's Apology--heads off in many directions and is all the more admirable for it. The book is part memoir, part account of the arcane research that brought number theorist Harris a measure of fame, and part sociological/economic study of academic mathematics. Together with interspersed chapters amusingly titled 'How to Explain Number Theory at a Dinner Party,' the work offers erudition, panache, and an intriguing authorial voice... A book to be read and then read again."--Choice "The erudition displayed by Harris in this book is amazing... The satisfaction it gives is more than rewarding."--A. Bultheel, Adhemar Bultheel Blog "This book is a rich tapestry interweaving various aspects of culture and tradition--social, economic, religious, aesthetic--in an attempt to explicate the three basic philosophical questions underlying mathematics as a human endeavor: the What, Why and How of it."--Swami Vidyanathananda, Prabuddha Bharata "Michael Harris is more than a mathematician; he is a Parisian intellectual."--Brendan Larvor, London Mathematical Society Newsletter "Even apprentice number theorists can understand and enjoy this well-written book. Harris's theories are coherent and rational, and he provides lay readers clarity into what contemporary mathematicians really do."--Bernadette Trainer, Mathematics TeacherTable of ContentsPreface ix Acknowledgments xix Part I 1 Chapter 1. Introduction: The Veil 3 Chapter 2. How I Acquired Charisma 7 Chapter alpha. How to Explain Number Theory at a Dinner Party 41 (First Session: Primes) 43 Chapter 3. Not Merely Good, True, and Beautiful 54 Chapter 4. Megaloprepeia 80 Chapter ss. How to Explain Number Theory at a Dinner Party 109 (Second Session: Equations) 109 Bonus Chapter 5. An Automorphic Reading of Thomas Pynchon's Against the Day (Interrupted by Elliptical Reflections on Mason & Dixon) 128 Part II 139 Chapter 6. Further Investigations of the Mind-Body Problem 141 Chapter ss.5. How to Explain Number Theory at a Dinner Party 175 (Impromptu Minisession: Transcendental Numbers) 175 Chapter 7. The Habit of Clinging to an Ultimate Ground 181 Chapter 8. The Science of Tricks 222 Part III 257 Chapter gamma. How to Explain Number Theory at a Dinner Party 259 (Third Session: Congruences) 259 Chapter 9. A Mathematical Dream and Its Interpretation 265 Chapter 10. No Apologies 279 Chapter delta. How to Explain Number Theory at a Dinner Party 311 (Fourth Session: Order and Randomness) 311 Afterword: The Veil of Maya 321 Notes 327 Bibliography 397 Index of Mathematicians 423 Subject Index 427

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Book SynopsisThe history of mathematics is filled with major breakthroughs resulting from solutions to recreational problems. Problems of interest to gamblers led to the modern theory of probability, for example, and surreal numbers were inspired by the game of Go. Yet even with such groundbreaking findings and a wealth of popular-level books exploring puzzlesTrade ReviewOne of Choice's Outstanding Academic Titles for 2016 "Beineke and Rosenhouse have compiled and edited a fantastic collection of essays dealing with popular mathematics... Anybody who enjoys reading about recreation mathematics should definitely explore these writings."--ChoiceTable of ContentsForeword by Raymond Smullyan vii Preface and Acknowledgments x PART I VIGNETTES 1 Should You Be Happy? 3 Peter Winkler 2 One-Move Puzzles with Mathematical Content 11 Anany Levitin 3 Minimalist Approaches to Figurative Maze Design 29 Robert Bosch, Tim Chartier, and Michael Rowan 4 Some ABCs of Graphs and Games 43 Jennifer Beineke and Lowell Beineke PART II PROBLEMS INSPIRED BY CLASSIC PUZZLES 5 Solving the Tower of Hanoi with Random Moves 65 Max A. Alekseyev and Toby Berger 6 Groups Associated to Tetraflexagons 81 Julie Beier and Carolyn Yackel 7 Parallel Weighings of Coins 95 Tanya Khovanova 8 Analysis of Crossword Puzzle Difficulty Using a Random Graph Process 105 John K. McSweeney 9 From the Outside In: Solving Generalizations of the Slothouber-Graatsma-Conway Puzzle 127 Derek Smith PART III PLAYING CARDS 10 Gallia Est Omnis Divisa in Partes Quattuor 139 Neil Calkin and Colm Mulcahy 11 Heartless Poker 149 Dominic Lanphier and Laura Taalman 12 An Introduction to Gilbreath Numbers 163 Robert W. Vallin PART IV GAMES 13 Tic-tac-toe on Affine Planes 175 Maureen T. Carroll and Steven T. Dougherty 14 Error Detection and Correction Using SET 199 Gary Gordon and Elizabeth McMahon 15 Connection Games and Sperner's Lemma 213 David Molnar PART V FIBONACCI NUMBERS 16 The Cookie Monster Problem 231 Leigh Marie Braswell and Tanya Khovanova 17 Representing Numbers Using Fibonacci Variants 245 Stephen K. Lucas About the Editors 261 About the Contributors 263 Index 269

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Book SynopsisAn anthology of the year's finest writing on mathematics from around the world, featuring promising new voices as well as some of the foremost names in mathematics.Trade Review"[The] essays cover a broad swath of mathematics that include entertaining puzzles, complicated proofs, pedagogical philosophy, and technical discussions of mathematical problems. The pedagogical entries are both serious and light... Many of the technical articles are difficult and demand a mathematical background, other entries are well suited for readers more casual readers; the volume is intended to capture both audiences and does it well."--Publishers Weekly "Abundant diversity and some truly exceptional writing make this collection stand out."--Gretchen Kolderup, Library Journal "I would characterize the articles in the book as extreme in terms of several value functions: clarity, lucidity, instructiveness, wittiness, modern day pertinency, broad accessibility... On the whole, the book is informative and thoroughly entertaining."--Alexander Bogomolny, Cut the Knot "Written in a pleasant and alive style, with suggestive quotations and witty comments of the author (also many photos illustrating the text are made by the author), the book will be of great help for students in computer science specializing in computer vision and computer graphics. Other students who use mathematics in their disciplines (physics, chemistry, biology, economics) will find the book as a good source of rapid and reliable information."--Dana Cobza, Studia Mathematica "For those looking to broaden their knowledge of mathematics, including recent mathematical developments, this is a good option and an enjoyable read."--Frannie Worek, Math Teacher "[Pitici's] work fills a gap between expository mathematics and popular explanation. It is a welcome contribution to improving public perception of our discipline."--Phill Schultz, Australian Mathematical Society GazetteTable of ContentsIntroduction, Mircea Pitici ix Mathematics and the Good Life, Stephen Pollard 1 The Rise of Big Data: How It's Changing the Way We Think about the World, Kenneth Cukier and Viktor Mayer-Schonberger 20 Conway's Wizards, Tanya Khovanova 33 On Unsettleable Arithmetical Problems, John H. Conway 39 Color illustration section follows page 48 Crinkly Curves, Brian Hayes 49 Why Do We Perceive Logarithmically? Lav R. Varshney and John Z. Sun 64 The Music of Math Games, Keith Devlin 74 The Fundamental Theorem of Algebra for Artists, Bahman Kalantari and Bruce Torrence 87 The Arts-Digitized, Quantified, and Analyzed, Nicole Lazar 96 On the Number of Klein Bottle Types, Carlo H. Sequin 105 Adventures in Mathematical Knitting, Sarah-Marie Belcastro 128 The Mathematics of Fountain Design: A Multiple-Centers Activity, Marshall Gordon 144 Food for (Mathematical) Thought, Penelope Dunham 156 Wondering about Wonder in Mathematics, Dov Zazkis and Rina Zazkis 165 The Lesson of Grace in Teaching, Francis Edward Su 188 Generic Proving: Reflections on Scope and Method, Uri Leron and Orit Zaslavsky 198 Extreme Proofs I: The Irrationality of 2, John H. Conway and Joseph Shipman 216 Stuck in the Middle: Cauchy's Intermediate Value Theorem and the History of Analytic Rigor, Michael J. Barany 228 Plato, Poincare, and the Enchanted Dodecahedron: Is the Universe Shaped Like the Poincare Homology Sphere? Lawrence Brenton 239 Computing with Real Numbers, from Archimedes to Turing and Beyond, Mark Braverman 251 Chaos at Fifty, Adilson E. Motter and David K. Campbell 270 Twenty-Five Analogies for Explaining Statistical Concepts, Roberto Behar, Pere Grima, and Lluis Marco-Almagro 288 College Admissions and the Stability of Marriage, David Gale and Lloyd S. Shapley 299 The Beauty of Bounded Gaps, Jordan Ellenberg 308 Contributors 315 Notable Writings 325 Acknowledgments 333 Credits 335

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Book Synopsis"The book that inspired the film The imitation game."Trade ReviewA New York Times Bestseller The Imitation Game, Winner of the 2015 Academy Award for Best Adapted Screenplay Winner of the 2015 (27th) USC Libraries Scripter Award, University of Southern California Libraries One of The Guardian's Best Popular Physical Science Books of 2014, chosen by GrrlScientist "Scrupulous and enthralling."--A. O. Scott, New York Times "One of the finest scientific biographies ever written."--Jim Holt, New Yorker "Andrew Hodges' 1983 book Alan Turing: The Enigma, is the indispensable guide to Turing's life and work and one of the finest biographies of a scientific genius ever written."--Michael Hiltzik, Los Angeles Times "Turing's rehabilitation from over a quarter-century's embarrassed silence was largely the result of Andrew Hodges's superb biography, Alan Turing: The Enigma (1983; reissued with a new introduction in 2012). Hodges examined available primary sources and interviewed surviving witnesses to elucidate Turing's multiple dimensions. A mathematician, Hodges ably explained Turing's intellectual accomplishments with insight, and situated them within their wider historical contexts. He also empathetically explored the centrality of Turing's sexual identity to his thought and life in a persuasive rather than reductive way."--Michael Saler, Times Literary Supplement "On the face of it, a richly detailed 500-page biography of a mathematical genius and analysis of his ideas, might seem a daunting proposition. But fellow mathematician and author Hodges has acutely clear and often extremely moving insight into the humanity behind the leaping genius that helped to crack the Germans' Enigma codes during World War II and bring about the dawn of the computer age... This melancholy story is transfigured into something else: an exploration of the relationship between machines and the soul and a full-throated celebration of Turing's brilliance, unselfconscious quirkiness and bravery in a hostile age."--Sinclair McKay, Wall Street Journal "A first-class contribution to history and an exemplary work of biography."--I. J. Good, Nature "An almost perfect match of biographer and subject... [A] great book."--Ray Monk, Guardian "A superb biography... Written by a mathematician, it describes in plain language Turing's work on the foundations of computer science and how he broke the Germans' Enigma code in the Second World War. The subtle depiction of class rivalries, personal relationships, and Turing's tragic end are worthy of a novel. But this was a real person. Hodges describes the man, and the science that fascinated him--which once saved, and still influences, our lives."--Margaret Boden, New Scientist "Andrew Hodges's magisterial Alan Turing: The Enigma ... is still the definitive text."--Joshua Cohen, Harper's "Andrew Hodges's biography is a meticulously researched and written account detailing every aspect of Turing's life... This account of Turing's life is a definitive scholarly work, rich in primary source documentation and small-grained historical detail."--Mathematics Teacher "Tells a powerful story that combines professional success and personal tragedy."--Nancy Szokan, Washington Post "[A] really excellent biography... The great thing about this book is that the author is a mathematician and can explain the details of Turing's work--as a scientist, mathematician, and a code breaker--in a way that is easy to understand. He is also wonderful at the emotional nuance of Alan's life, who was a somewhat odd--a student was assigned to him in school to help him maintain a semblance of tidiness in his appearance, rooms and school work and at Bletchley Park he was known for chaining his tea mug to a pipe--but he was also charming and intelligent and Hodges brings all the aspects of his personality and life into sharp focus."--Off the Shelf "This book is an incredibly detailed and meticulously researched biography of Alan Turing. Reading it is a melancholy experience, since you know from the outset that the ending is a tragic one and that knowledge overshadows you throughout. While the author divides the text into two parts, it actually reads like a play in four acts... This book is Turing's memorial, and one that does justice to the subject."--Katherine Safford-Ramus, MAA Reviews "The new paperback edition of the 1983 book that inspired the film, with an updated introduction by Oxford mathematics professor Andrew Hodges, is an exhilarating, compassionate and detailed biography of a complicated man."--Jane Ciabattari, BBC "If [The Imitation Game] does nothing else but send you, as it did me, to Alan Hodges's Alan Turing: The Enigma (1983, newly prefaced in the 2014 Princeton University Press edition) it more than justifies its existence. A great read, Hodges's intellectual biography depicts Turing as a brilliant mathematician; a crucial pioneering figure in the theorization and engineering of digital computing; and the biggest brain in Bletchley Park's Hut #8."--Amy Taubin, Artforum "It is indeed the ultimate biography of Alan Turing. It will bring you as close as possible to his enigmatic personality."--Adhemar Bultheel, European Mathematical Society "A book whose time has finally come. I found it to be a page-turner in spite of the occasionally esoteric explanations of mathematical theories that reminded of why Brooklyn Technical High School was not the wisest choice for me."--Terrance, Paris Readers Circle "Thanks to the movie The Imitation Game, Alan Turing has emerged from history's shadows, where his memory had languished for decades. For anyone whose interest in the pioneering computer scientist, mathematician, and logician was piqued by the film, the book that served as the film's source material, Andrew Hodges's exhaustive biography Alan Turing: The Enigma, has the answers."--Frank Caso, Simply CharlyTable of ContentsList of Plates ix Foreword by Douglas Hofstadter xi Preface xv PART ONE: THE LOGICAL 1 Esprit de Corps to 13 February 1930 3 2 The Spirit of Truth to 14 April 1936 60 3 New Men to 3 September 1939 141 4 The Relay Race to 10 November 1942 202 BRIDGE PASSAGE to 1 April 1943 305 PART TWO: THE PHYSICAL 5 Running Up to 2 September 1945 325 6 Mercury Delayed to 2 October 1948 394 7 The Greenwood Tree to 7 February 1952 491 8 On the Beach to 7 June 1954 574 Postscript 665 Author's Note 666 Notes 680 Acknowledgements 714 Index 715

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Book SynopsisThis book presents a new approach to the epistemology of mathematics by viewing mathematics as a human activity whose knowledge is intimately linked with practice. Charting an exciting new direction in the philosophy of mathematics, Jose Ferreiros uses the crucial idea of a continuum to provide an account of the development of mathematical knowledgTrade Review"Both philosophers and mathematicians can find ample food for thought in this study."--Choice "Ferreiros has published a fascinating book which consists of an impressive combination of thought-provoking philosophical ideas and mathematical material. As such, it can be interesting for philosophers of mathematics, mathematicians, and other people interested in the topics of mathematical knowledge and mathematical practice."--Joachim Frans, MathScieNetTable of ContentsList of Illustrations ix Foreword xi 1 On Knowledge and Practices: A Manifesto 1 2 The Web of Practices 17 2.1. Historical Work on Practices 18 2.2. Philosophers Working on Practices 22 2.3. What Is Mathematical Practice, Then? 28 2.4. The Multiplicity of Practices 34 2.5. The Interplay of Practices and Its Basis 39 3 Agents and Frameworks 44 3.1. Frameworks and Related Matters 45 3.2. Interlude on Examplars 55 3.3. On Agents 59 3.4. Counting Practices and Cognitive Abilities 65 3.5. Further Remarks on Mathematics and Cognition 74 3.6. Agents and "Metamathematical" Views 79 3.7. On Systematic Links 83 4 Complementarity in Mathematics 89 4.1. Formula and Meaning 89 4.2. Formal Systems and Intended Models 94 4.3. Meaning in Mathematics: A Tentative Approach 99 4.4. The Case of Complex Numbers 104 5 Ancient Greek Mathematics: A Role for Diagrams 112 5.1. From the Technical to the Mathematical 113 5.2. The Elements: Getting Started 117 5.3. On the Euclidean Postulates: Ruling Diagrams (and Their Reading) 127 5.4. Diagram-Based Mathematics and Proofs 131 5.5. Agents, Idealization, and Abstractness 137 5.6. A Look at the Future-Our Past 147 6 Advanced Math: The Hypothetical Conception 153 6.1. The Hypothetical Conception: An Introduction 154 6.2. On Certainty and Objectivity 159 6.3. Elementary vs. Advanced: Geometry and the Continuum 163 6.4. Talking about Objects 170 6.5. Working with Hypotheses: AC and the Riemann Conjecture 176 7 Arithmetic Certainty 182 7.1. Basic Arithmetic 182 7.2. Counting Practices, Again 184 7.3. The Certainty of Basic Arithmetic 189 7.4. Further Clarifications 195 7.5. Model Theory of Arithmetic 198 7.6. Logical Issues: Classical or Intuitionistic Math? 200 8 Mathematics Developed: The Case of the Reals 206 8.1. Inventing the Reals 207 8.2. "Tenths" to the Infinite: Lambert and Newton 215 8.3. The Number Continuum 221 8.4. The Reinvention of the Reals 227 8.5. Simple Infinity and Arbitrary Infinity 231 8.6. Developing Mathematics 236 8.7. Mathematical Hypotheses and Scientific Practices 241 9 Objectivity in Mathematical Knowledge 247 9.1. Objectivity and Mathematical Hypotheses: A Simple Case 249 9.2. Cantor's "Purely Arithmetical" Proofs 253 9.3. Objectivity and Hypotheses, II: The Case of p() 257 9.4. Arbitrary Sets and Choice 261 9.5. What about Cantor's Ordinal Numbers? 265 9.6. Objectivity and the Continuum Problem 273 10 The Problem of Conceptual Understanding 281 10.1. The Universe of Sets 283 10.2. A "Web-of- Practices" Look at the Cumulative Picture 290 10.3. Conceptual Understanding 296 10.4. Justifying Set Theory: Arguments Based on the Real-Number Continuum 305 10.5. By Way of Conclusion 310 References 315 Index 331

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Book SynopsisThis is a book for those who enjoy thinking about how and why Nature can be described using mathematical tools. Approximating Perfection considers the background behind mechanics as well as the mathematical ideas that play key roles in mechanical applications. Concentrating on the models of applied mechanics, the book engages the reader in the typeTrade Review"A well-written general-interest introduction to classical mechanics."--ChoiceTable of ContentsPreface vii Chapter 1. The Tools of Calculus 1 1.1 Is Mathematical Proof Necessary? 2 1.2 Abstraction, Understanding, Infinity 6 1.3 Irrational Numbers 8 1.4 What Is a Limit? 11 1.5 Series 15 1.6 Function Continuity 19 1.7 How to Measure Length 21 1.8 Antiderivatives 33 1.9 Definite Integral 35 1.10 The Length of a Curve 42 1.11 Multidimensional Integrals 44 1.12 Approximate Integration 47 1.13 On the Notion of a Function 52 1.14 Differential Equations 53 1.15 Optimization 59 1.16 Petroleum Exploration and Recovery 61 1.17 Complex Variables 63 1.18 Moving On 65 Chapter 2. The Mechanics of Continua 67 2.1 Why Do Ships Float? 67 2.2 The Main Notions of Classical Mechanics 71 2.3 Forces, Vectors, and Objectivity 74 2.4 More on Forces; Statics 76 2.5 Hooke's Law 80 2.6 Bending of a Beam 84 2.7 Stress Tensor 94 2.8 Principal Axes and Invariants of the Stress Tensor 100 2.9 On the Continuum Model and Limit Passages 102 2.10 Equilibrium Equations 104 2.11 The Strain Tensor 108 2.12 Generalized Hooke's Law 113 2.13 Constitutive Laws 114 2.14 Boundary Value Problems 115 2.15 Setup of Boundary Value Problems of Elasticity 118 2.16 Existence and Uniqueness of Solution 120 2.17 Energy; Minimal Principle for a Spring 126 2.18 Energy in Linear Elasticity 128 2.19 Dynamic Problems of Elasticity 132 2.20 Oscillations of a String 134 2.21 Lagrangian and Eulerian Descriptions of Continuum Media 137 2.22 The Equations of Hydrodynamics 140 2.23 D'Alembert-Euler Equation of Continuity 142 2.24 Some Other Models of Hydrodynamics 144 2.25 Equilibrium of an Ideal Incompressible Liquid 145 2.26 Force on an Obstacle 148 Chapter 3. Elements of the Strength of Materials 151 3.1 What Are the Problems of the Strength of Materials? 151 3.2 Hooke's Law Revisited 152 3.3 Objectiveness of Quantities in Mechanics Revisited 157 3.4 Plane Elasticity 159 3.5 Saint-Venant's Principle 161 3.6 Stress Concentration 163 3.7 Linearity vs. Nonlinearity 165 3.8 Dislocations, Plasticity, Creep, and Fatigue 166 3.9 Heat Transfer 172 3.10 Thermoelasticity 175 3.11 Thermal Expansion 177 3.12 A Few Words on the History of Thermodynamics 178 3.13 Thermodynamics of an Ideal Gas 180 3.14 Thermodynamics of a Linearly Elastic Rod 182 3.15 Stability 186 3.16 Static Stability of a Straight Beam 188 3.17 Dynamical Tools for Studying Stability 193 3.18 Additional Remarks on Stability 195 3.19 Leak Prevention 198 Chapter 4. Some Questions of Modeling in the Natural Sciences 201 4.1 Modeling and Simulation 201 4.2 Computerization and Modeling 203 4.3 Numerical Methods and Modeling in Mechanics 206 4.4 Complexity in the Real World 208 4.5 The Role of the Cosine in Everyday Measurements 209 4.6 Accuracy and Precision 211 4.7 How Trees Stand Up against the Wind 213 4.8 Why King Kong Cannot Be as Terrible as in the Movies 216 Afterword 219 Recommended Reading 221 Index 223

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Book SynopsisCopyright 2017 by Princeton University Press.Trade Review"[This book] is beautiful in that just about every problem could be explained to anybody with almost no mathematics background at all, but the methods of solving them take you deeply into many complex areas of mathematics. The books gathers together problems which pop up through what one might consider 'silly' or 'frivolous' questions, but which lead to new ways of thinking and have applications in enormously wide-ranging areas of mathematics."---Jonathan Shock, Mathemafrica"The editors once again have brought together an extraordinary list of authors to produce nineteen engaging papers, split into five groups: puzzles and brainteasers, geometry and topology, graph theory, games of chance, and computational complexity. . . . It is often deeply challenging mathematically and, as a result, all the more fun. Each reader will find chapters that appeal to them." * MAA Reviews *"In the second volume of this engaging series, Beineke . . . and Rosenhouse . . . deliver another fantastic collection of essays dealing with popular mathematics. . . . Anyone who enjoys reading about recreational mathematics will find plenty to enjoy and discover in this second volume." * Choice *

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Book SynopsisTrade Review"This short book on a vast subject is the work of a master. With a few sure and authoritative words [Weyl] gives us the heart of the matter. There is no book ... quite like this one on the subject of symmetry and I doubt if any book will be written in the future that will not in some way lean upon this one... [I]t contains so much besides mathematics that it can still be read with profit and enjoyed by someone who has not advanced beyond long division."--John Tyler Bonner, Science "Dr. Weyl presents a masterful and fascinating survey of the applications of the principle of symmetry in sculpture, painting, architecture, ornament, and design; its manifestations in organic and inorganic nature; and its philosophical and mathematical significance."--Scientific American "Weyl offers deep insight into [the concept of symmetry], its foundations in group theory, its applications in physics, chemistry, and biology, and its role in art."--Manfred Eigen and Ruthild Winkler in Laws of the Game "Vivid and picturesque... [Weyl is] an outstanding thinker."--Wolfgang Yourgrau, Philosophy and Phenomenological ResearchTable of ContentsBilateral symmetry 3 Translatory, rotational, and related symmetries 41 Ornamental Symmetry 83 Crystals. The General mathmatical idea of symmetry 119 Appendices A. Determination of all finite groups of proper rotations in 3-space 149 B. Inclusion of improper rotations 155 Acknowledgements 157 Index 161

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Book SynopsisTrade ReviewOne of Amazon.com's 2013 Best Science Books One of Choice's Outstanding Academic Titles for 2013 Honorable Mention for the 2013 PROSE Award in Popular Science & Mathematics, Association of American Publishers "As Fortnow describes... P versus NP is 'one of the great open problems in all of mathematics' not only because it is extremely difficult to solve but because it has such obvious practical applications. It is the dream of total ease, of the confidence that there is an efficient way to calculate nearly everything, 'from cures to deadly diseases to the nature of the universe,' even 'an algorithmic process to recognize greatness.'... To postulate that P ? NP, as Fortnow does, is to allow for a world of mystery, difficulty, and frustration--but also of discovery and inquiry, of pleasures pleasingly delayed."--Alexander Nazaryan, New Yorker "Fortnow effectively initiates readers into the seductive mystery and importance of P and NP problems."--Publishers Weekly "Fortnow's book is just the ticket for bringing one of the major theoretical problems of our time to the level of the average citizen--and yes, that includes elected officials."--Veit Elser, Science "Without bringing formulas or computer code into the narrative, Fortnow sketches the history of this class of questions, convincingly demonstrates their surprising equivalence, and reveals some of the most far-reaching implications that a proof of P = NP would bring about. These might include tremendous advances in biotechnology (for instance, more cures for cancer), information technology, and even the arts. Verdict: Through story and analogy, this relatively slim volume manages to provide a thorough, accessible explanation of a deep mathematical question and its myriad consequences. An engaging, informative read for a broad audience."--J.J.S. Boyce, Library Journal "A provocative reminder of the real-world consequences of a theoretical enigma."--Booklist "The definition of this problem is tricky and technical, but in The Golden Ticket, Lance Fortnow cleverly sidesteps the issue with a boiled-down version. P is the collection of problems we can solve quickly, NP is the collection of problems we would like to solve. If P = NP, computers can answer all the questions we pose and our world is changed forever. It is an oversimplification, but Fortnow, a computer scientist at Georgia Institute of Technology, Atlanta, knows his stuff and aptly illustrates why NP problems are so important."--Jacob Aron, New Scientist "Fortnow's book does a fine job of showing why the tantalizing question is an important one, with implications far beyond just computer science."--Rob Hardy, Commercial Dispatch "A great book... [Lance Fortnow] has written precisely the book about P vs. NP that the interested layperson or IT professional wants and needs."--Scott Aaronson, Shtetl-Optimized blog "[The Golden Ticket] is a book on a technical subject aimed at a general audience... Lance's mix of technical accuracy with evocative story telling works."--Michael Trick, Michael Trick's Operations Research Blog "Thoroughly researched and reviewed. Anyone from a smart high school student to a computer scientist is sure to get a lot of this book. The presentation is beautiful. There are few formulas but lots of facts."--Daniel Lemire's Blog "An entertaining discussion of the P versus NP problem."--Andrew Binstock, Dr. Dobb's "The Golden Ticketis an extremely accessible and enjoyable treatment of the most important question of theoretical computer science, namely whether P is equal to NP."--Choice "The book is accessible and useful for practically anyone from smart high school students to specialists... [P]erhaps the interest sparked by this book will be the 'Golden Ticket' for further accessible work in this area. And perhaps P=NP will start to become as famous as E=mc2."--Michael Trick, INFORMS Journal of Computing "In any case, it is excellent to have a nontechnical book about the P versus NP question. The Golden Ticket offers an inspiring introduction for nontechnical readers to what is surely the most important open problem in computer science."--Leslie Ann Goldberg, LMS Newsletter "The Golden Ticket does a good job of explaining a complex concept in terms that a secondary-school student will understand--a hard problem in its own right, even if not quite NP."--Physics World "[The Golden Ticket] is fun to read and can be fully appreciated without any knowledge in (theoretical) computer science. Fortnow's efforts to make the difficult material accessible to non-experts should be commended."--Andreas Maletti, Zentralblatt MATH "This is a fabulous book for both educators and students at the secondary school level and above. It does not require any particular mathematical knowledge but, rather, the ability to think. Enjoy the world of abstract ideas as you experience an intriguing journey through mathematical thinking."--Gail Kaplan, Mathematics Teacher "Fortnow's book provides much of the background and personal information on the main characters involved in this problem--notably Steven Cook, with a cameo appearance by Kurt Godel--that one does not get in the more technical treatments. There is a lot of information in this book, and the serious computer science student is sure to learn from it."--James M. Cargal, UMAP JournalTable of ContentsPreface ix Chapter 1 The Golden Ticket 1 Chapter 2 The Beautiful World 11 Chapter 3 P and NP 29 Chapter 4 The Hardest Problems in NP 51 Chapter 5 The Prehistory of P versus NP 71 Chapter 6 Dealing with Hardness 89 Chapter 7 Proving P <> NP 109 Chapter 8 Secrets 123 Chapter 9 Quantum 143 Chapter 10 The Future 155 Acknowledgments 163 Chapter Notes and Sources 165 Index 171

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Book SynopsisTrade ReviewHonorable Mention for the 2015 PROSE Award in Popular Science & Popular Mathematics, Association of American Publishers "A book that stimulates the mind as well as the eye."--Scientific American "The combination of art and exposition was quite effective. The writing is accessible to most reasonably well-educated laypeople, and I imagine that many such people would derive considerable pleasure dipping into this attractive and interesting book."--Mark Hunacek, MAA Reviews "Eli Maor's lively writing benefits in equal parts from the geometry of ancient Greece and the eye-popping images conjured by artist Eugen Jost."--Bill Cannon, Scientist's Bookshelf "Graphic illustrations serve as both beautiful abstract art and helpful explanations in this overview of geometric theorems and patterns."--Science News "[Beautiful Geometry] achieves its aim to demonstrate that there is visual beauty in Mathematics. I heartily recommend it."--LSE Review of Books "The explanations are clear, and cover the background to the paintings in a manner that will be appreciated by readers whatever their level of mathematical knowledge... Anyone with any interest in visual mathematics will love this book."--Times Higher Education "A good-looking, large-format book suitable for the coffee table, but with lots of mathematical ideas packed in among the colorful illustrations... [A] handsome book for browsing and for some deep thought, and would be a superb gift for anyone (especially a young person) who has interest in mathematics."--Rob Hardy, Columbus Dispatch "It is a handsome book for browsing and for some deep thought, and would be a superb gift for anyone (especially a young person) who has interest in mathematics."--Rob Hardy, Dispatch "The book by Maor and Jost should be given to everyone--young or old--embarking on the study of mathematics or anyone teaching mathematics. The book will act as a source of inspiration and as a reminder of why it is that mathematics has fascinated the human race for millennia."--Henrik Jeldtoft Jensen, LMS Newsletter "The content is accessible to anyone with even a high school course in geometry. The writing is very clear."--Choice "Clear and lively... The mathematics in this book is first-rate, but the real surprise is how well the art reflects and illuminates the topic at hand... All of it is lovely to look at... [Beautiful Geometry] rises to the level of a coffee-table art book, only with a lot more depth."--Mathematical Reviews "[E]erily captivating book... Maor's style of writing is conversational, and the writing is engaging."--Annalisa Crannell, Journal of Mathematics and the Arts "At a very reasonable price, this is a book which would grace the coffee-table of any mathematics department, and many of the ideas in it will stimulate valuable discussions in the classroom."--Gerry Leversha, Mathematical Gazette "It presents as a coffee-table book for mathematicians and would be a good addition to a classroom library, available for students of all ages to explore."--Susan Mielechowsky, Mathematics Teaching in the Middle School "Visually stunning... [Beautiful Geometry] raises fundamental questions, answered thousands of years later and evidencing the progress made... This is an engaging book of broad appeal and a colourful approach to the history of geometry."--Mathematics TodayTable of ContentsPrefaces ix 1.Thales of Miletus 1 2.Triangles of Equal Area 3 3.Quadrilaterals 6 4.Perfect Numbers and Triangular Numbers 9 5.The Pythagorean Theorem I

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Book SynopsisTrade Review"Nahin includes gems from all over mathematics, ranging from engineering applications to beautiful pure-mathematical identities... It would be good to have more books like this."--Timothy Gowers, Nature "Nahin's tale of the formula e[pi] i+1=0, which links five of the most important numbers in mathematics, is remarkable. With a plethora of historical and anecdotal material and a knack for linking events and facts, he gives the reader a strong sense of what drove mathematicians like Euler."--Matthew Killeya, New Scientist "It is very difficult to sum up the greatness of Euler... This excellent book goes a long way to explaining the kind of mathematician he really was."--Steve Humble, Mathematics Today "What a treasure of a book this is! This is the fourth enthusiastic, informative, and delightful book Paul Nahin has written about the beauties of various areas of mathematics... This book is a marvelous tribute to Euler's genius and those who built upon it and would make a great present for students of mathematics, physics, and engineering and their professors."--Henry Ricardo, MAA Reviews "The heart and soul of the book are the final three chapters on Fourier series, Fourier integrals, and related engineering. One can recommend them to all applied math students for their historical development and sensible content."--Robert E. O'Malley, Jr., SIAM Review "This is a book for mathematicians who enjoy historically motivated mathematical explanations on a high mathematical level."--Eberhard Knobloch, Mathematical Reviews "It is a 'popular' book, written for a general reader with some mathematical background equivalent to a first-year undergraduate course in the UK."--Robin Wilson, London Mathematical Society NewsletterTable of Contents*FrontMatter, pg. i*Contents, pg. ix*Preface to the Paperback Edition, pg. xiii*Preface, pg. xxix*Introduction, pg. 1*Chapter 1. Complex Numbers, pg. 13*Chapter 2. Vector Trips, pg. 68*Chapter 3. The Irrationality of pi2, pg. 92*Chapter 4. Fourier Series, pg. 114*Chapter 5. Fourier Integrals, pg. 188*Chapter 6. Electronics and -1, pg. 275*Euler: The Man and the Mathematical Physicist, pg. 324*Notes, pg. 347*Acknowledgments, pg. 375*Index, pg. 377

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Book SynopsisTrade Review"Glen van Brummelen has prepared a highly recommended, accessible and definitive history of the subject that will serve as a resource for scholars for decades to come."---Daniel Otero, MAA Reviews"The Doctrine of Triangles is an informative and valuable reference work."---Wallace A Ferguson, Institute of Mathematics and its Applications"A guided tour through the museum of mathematics. . . . [The Doctrine of Triangles] takes the history of trigonometry, which is a formidable subject in its scope and size, and transforms it into something readable."---Daniel Mansfield, The Mathematical Intelligencer"Very easy to read, and there are lots of helpful diagrams, especially for the spherical trigonometry . . . [The Doctrine of Triangles] is deeply enriched by extracts from contemporary texts, given first in fairly literal English translations, often accompanied by the original diagrams, and then explained in modern terms. So mathematical readers (and, I hope, their students) can experience a little of what trigonometry was actually like at each stage in its history."---John Hannah, Aestimatio

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Book SynopsisTrade Review"In an orderly sense, the writer introduces the context and then proceeds to state exactly what was the major draw back in the context during the relevant period of time. . . The progression, as well as the way in which he uses simple techniques to demolish towers of problems in the same sense as it was done back in the day is certainly worth appreciation." * Mathemafrica *"Any lover of mathematics will appreciate the time spent among these pages."---A. Misseldine, Choice"A great companion for students studying analysis, and calculus instructors will find it an enriching experience." * Mathematics Magazine *"I thoroughly enjoyed reading this accessible, insightful and well-written book."---Nick Lord, Mathematical Gazette

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Book SynopsisThis volume brings together authors from a variety of specialties to present fascinating problems and solutions in recreational mathematics.Trade Review"This is a fantastic (and entertaining) book on various aspects of recreational mathematics which are also at the forefront of research level mathematics."---Manjil Pratim Saikia, Zentralblatt MATH

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Book SynopsisTrade Review"In The Mathematics of Secrets, Joshua Holden takes the reader on a chronological journey from Julius Caesar’s substitution cipher to modern day public-key algorithms and beyond. . . . Written for anyone with an interest in cryptography." —Noel-Ann Bradshaw, Times Higher Education "Complete in surveying cryptography. . . . This is a marvelous way of illustrating the use of simple mathematics in an important application that has triggered the wit of the designers and the ingenuity of the attackers since antiquity." —Adhemar Bultheel, European Mathematical Society "The best book I have seen on this subject." —Phil Dyke, Leonardo Reviews "This is a fascinating tour of the mathematics behind cryptography, showing how its principles underpin the ways that different codes and ciphers operate. . . . While it’s all about maths, the book is accessible—basic high school algebra is all that’s needed to understand and enjoy it." —Cosmos Magazine

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