History of mathematics Books

Book SynopsisThis Element shows that Plato keeps a clear distinction between mathematical and metaphysical realism. It also shows that methodological commitments to mathematical objects are made in light of mathematical practice; foundational considerations; and, mathematical applicability. This title is also available as Open Access on Cambridge Core.Trade Review'… Landry's response to the Platonic call for collaboration with his text opens up the possibility of very fruitful debates.' Susanna Saracco, MetascienceTable of Contents1. Introduction; 2. The interprative lay of the land; 3. The divided line; 4. Book 7; 5. The good in mathematics; 6. Mathematics versus metaphysics; References.

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Book SynopsisThis book introduces prime numbers and explains the celebrated, unsolved Riemann hypothesis in a direct manner. Suitable for both scholars and those with a minimal mathematical background.Trade Review'This is an extraordinary book, really one of a kind. Written by two supreme experts, but aimed at the level of an undergraduate or curious amateur, it emphasizes the really powerful ideas, with the bare minimum of math notation and the maximum number of elegant and suggestive visuals. The authors explain why this legendary problem is so beautiful, why it is difficult, and why you should care.' Will Hearst, Hearst Corporation'This book is a soaring ride, starting from the simplest ideas and ending with one of the deepest unsolved problems of mathematics. Unlike in many popular math books puffed up with anecdotal material, the authors here treat the reader as seriously interested in prime numbers and build up the real math in four stages with compelling graphical demonstrations revealing in deeper and deeper ways the hidden music of the primes. If you have ever wondered why so many mathematicians are obsessed with primes, here's the real deal.' David Mumford, Brown University, Rhode Island'This is a delightful little book, not quite like anything else that I am aware of … a splendid piece of work, informative and valuable. Undergraduate mathematics majors, and the faculty who teach them, should derive considerable benefit from looking at it.' Mark Hunacek, MAA Reviews'This book is divided into four parts, and succeeds beautifully in giving both an overview for the general audience and a sense of the details needed to understand how quickly the number of primes grows. This is accomplished through a very clear exposition and numerous illuminating pictures.' Steven Joel Miller, MathSciNet'Where popularizers of mathematics usually succumb either to a journalist's penchant for 'man bites dog' irony and spectacle or a schoolteacher's iron will to simplify away the terror, one might call the distinctive approach here 'take a lay reader to work'. Computers now provide mathematicians a laboratory, and the authors exploit this modern power to exhibit graphics, making the key equivalence a luminous phenomenon of experimental mathematics … for its clarity and the importance of its topic, this book deserves the same classic status as A Brief History of Time (CH, Jul'88). Summing Up: Essential. All readers.' D. V. Feldman, CHOICE'Prime Numbers and the Riemann Hypothesis is an agile, unusual book written over a decade, one week per year; it can be considered a sort of collaborative work, in that each version was put online with the purpose of getting feedback.' Massimo Nespolo, Acta Crystallographica Section A: Foundations and Advances'… a great gift for a curious student. Using the graphical methods found in calculus reform texts, this beautiful little book allows a patient reader with a good grasp of first-year calculus to explore the most famous unsolved problem in mathematics, the so-called Riemann Hypothesis, and to understand why it points to as yet undiscovered regularities in the distribution of prime numbers.' Donal O'Shea, The Herald Tribune'The book under review succeeds handsomely in making the case for the Riemann Hypothesis to a wide audience … Beginning with the definition of prime numbers, the authors weave their way through concrete and picturesque presentations of elementary techniques and descriptions of unsolved problems connected with the primes. They provide many insightful footnotes, concrete and illuminating figures, pointers to arXiv pages for added information, and a rich set of endnotes that contain further descriptions and details with varying levels of sophistication. After 23 short sections (a few pages each) they have arrived at a formulation of the Riemann Hypothesis in terms of counting primes up to a given size. By this point in their masterful and compelling presentation, the Hypothesis appears to be completely natural and inevitable … I have no doubt that many newcomers to the subject who have read to the end of the book will be eager to learn more and will be drawn into this fertile playground.' Peter Sarnak, Bulletin of the AMS'I really recommend this book if you want to get a feeling for the Riemann hypothesis without sinking into technicalities.' John Baez, The n-Category Café (http://golem.ph.utexas.edu/category)Table of Contents1. Thoughts about numbers; 2. What are prime numbers?; 3. 'Named' prime numbers; 4. Sieves; 5. Questions about primes; 6. Further questions about primes; 7. How many primes are there?; 8. Prime numbers viewed from a distance; 9. Pure and applied mathematics; 10. A probabilistic 'first' guess; 11. What is a 'good approximation'?; 12. Square root error and random walks; 13. What is Riemann's hypothesis?; 14. The mystery moves to the error term; 15. Césaro smoothing; 16. A view of Li(X) - π(X); 17. The prime number theorem; 18. The staircase of primes; 19. Tinkering with the staircase of primes; 20. Computer music files and prime numbers; 21. The word 'spectrum'; 22. Spectra and trigonometric sums; 23. The spectrum and the staircase of primes; 24. To our readers of part I; 25. Slopes and graphs that have no slopes; 26. Distributions; 27. Fourier transforms: second visit; 28. Fourier transform of delta; 29. Trigonometric series; 30. A sneak preview; 31. On losing no information; 32. Going from the primes to the Riemann spectrum; 33. How many θi's are there?; 34. Further questions about the Riemann spectrum; 35. Going from the Riemann spectrum to the primes; 36. Building π(X) knowing the spectrum; 37. As Riemann envisioned it; 38. Companions to the zeta function.

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Book SynopsisPaul Erdos published more papers during his lifetime than any other mathematician, especially in discrete mathematics. He had a nose for beautiful, simply-stated problems with solutions that have far-reaching consequences across mathematics. This captivating book, written for students, provides an easy-to-understand introduction to discrete mathematics by presenting questions that intrigued Erdos, along with his brilliant ways of working toward their answers. It includes young Erdos''s proof of Bertrand''s postulate, the Erdos-Szekeres Happy End Theorem, De Bruijn-Erdos theorem, Erdos-Rado delta-systems, Erdos-Ko-Rado theorem, Erdos-Stone theorem, the Erdos-Rényi-Sós Friendship Theorem, Erdos-Rényi random graphs, the Chvátal-Erdos theorem on Hamilton cycles, and other results of Erdos, as well as results related to his work, such as Ramsey''s theorem or Deza''s theorem on weak delta-systems. Its appendix covers topics normally missing from introductory courses. Filled with personal aneTrade Review'Vašek Chvátal was born to write this one-of-a-kind book. Readers cannot help but be captivated by the evident love with which every page has been written. The human side of mathematics is intertwined beautifully with first-rate exposition of first-rate results.' Donald Knuth, Stanford University'This book is a treasure trove from so many viewpoints. It is a wonderful introduction and an alluring invitation to discrete mathematics - now a central field of mathematics identified mostly with the hero of this book. With lucid, carefully planned chapters on different topics it demonstrates the unique way in which Paul Erdős, one of the most prolific and influential mathematicians of the twentieth century, invented and approached problems. Sprinkled with historical and personal anecdotes and pictures, it opens a window to the unique personality of 'Uncle Paul'. And implicitly, it reveals the charming and candid way in which Vašek Chvátal, an authority in the field and a lifelong friend and collaborator of Erdős, likes to combine teaching and story-telling.' Avi Wigderson, IAS, Princeton'Paul Erdős is one of the founding fathers of modern combinatorics, whose ability to pose beautiful problems greatly determined the development of this field and influenced many other areas of mathematics. This book uses some basic questions, which intrigued Paul Erdős, to give a nice introduction to many topics in discrete mathematics. It contains a collection of beautiful results, covering such diverse subjects as discrete geometry, Ramsey theory, graph colorings, extremal problems for graphs and set systems and some others. It presents many elegant proofs and exposes the reader to various powerful combinatorial techniques.' Benjamin Sudakov, ETH Zurich'This is a brilliant book. It manages in one fell swoop to survey and develop a large part of combinatorial mathematics while at the same time chronicling the work of Paul Erdős. His contributions to different areas of mathematics are seen here to be part of a coherent whole. Chvátal's presentation is particularly appealing and accessible. The wonderful personal recollections add to the mathematical content to provide a portrait of Erdős' mind recognizable to those who knew him.' Bruce Rothschild, University of California, Los Angeles'Vašek Chvátal's book is a gem. Paul Erdős' favorite problems and best work are beautifully laid out. Readers unfamiliar with Erdős' work cannot fail to appreciate its power and elegance, and those who have seen bits and pieces will have the pleasure of seeing it thoughtfully and lovingly presented by a master. It's hard to imagine now, but there was a time when combinatorics was thought to be a jumble of results without depth or coherence. 'Uncle' Paul understood its heart and soul, and nowhere is this more evident than in Chvátal's wonderful compendium. This volume belongs on every math-lover's night-table!' Peter Winkler, Dartmouth College'Beautiful mathematics is presented with great care and clarity in Vašek Chvátal's book, complemented with well-written anecdotes and personal reminiscences about Paul Erdős. This combination makes the book a very enjoyable reading and a lively tribute to the memory of one of the most prolific mathematicians of all time. Studying discrete mathematics from this book is likely to give a great experience to students and established researchers alike.' Gábor Simonyi, Rényi Institute, Budapest'… Chvátal (emer., Concordia Univ.) has created a gem in this work and deserves congratulation … Highly recommended.' J. Johnson, Choice Magazine'This wonderfully written book is undoubtedly a significant contribution to the growing body of literature on the various developments in discrete mathematics over the last several decades. Still, to reduce it to only its mathematical dimension would be an act of injustice not only towards the book but also towards its author. The book is a powerful homage to Paul Erdos as one of the leading mathematicians of the twentieth century as well as a person who, with his unprecedented level of academic generosity and overall human kindness, was one of the pillars of the discrete mathematics community during his lifetime.' Veselin Jungic, MathSciNetTable of ContentsForeword; Preface; Acknowledgments; Introduction; 1. A glorious beginning – Bertrand's postulate; 2. Discrete geometry and spinoffs; 3. Ramsey's theorem; 4. Delta-systems; 5. Extremal set theory; 6. Van der Waerden's theorem; 7. Extremal graph theory; 8. The friendship theorem; 9. Chromatic number; 10. Thresholds of graph properties ; 11. Hamilton cycles; Appendix A. A few tricks of the trade; Appendix B. Definitions, terminology, notation; Appendix C. More on Erdős; References; Index.

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Book SynopsisScience & Scientists in Berlin is a richly illustrated guidebook providing informative biographies of 22 major scientists and 11 mathematicians linked to the metropolis, from polymath Gottfried W. Leibniz (b. 1646) to computer inventor Konrad Zuse (d. 1995). As well as renowned figures like Albert Einstein, the book includes scientists who deserve to be better known, such as flight pioneer Otto Lilienthal. Their world-changing achievements are described in a lively and accessible style. Follow in the footsteps of the protagonists using the comprehensive gazetteer and 18 colour maps which guide you to almost 200 sites associated with their lives: such as plaques, monuments, laboratories, museums, residences & graves. Anyone who is interested in both science and Berlin’s history, and who wants to learn about the people who created this unique past and experience the places where it comes alive, needs a guidebook like this…

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Book SynopsisThis textbook provides an accessible account of the history of abstract algebra, tracing a range of topics in modern algebra and number theory back to their modest presence in the seventeenth and eighteenth centuries, and exploring the impact of ideas on the development of the subject. Beginning with Gauss’s theory of numbers and Galois’s ideas, the book progresses to Dedekind and Kronecker, Jordan and Klein, Steinitz, Hilbert, and Emmy Noether. Approaching mathematical topics from a historical perspective, the author explores quadratic forms, quadratic reciprocity, Fermat’s Last Theorem, cyclotomy, quintic equations, Galois theory, commutative rings, abstract fields, ideal theory, invariant theory, and group theory. Readers will learn what Galois accomplished, how difficult the proofs of his theorems were, and how important Camille Jordan and Felix Klein were in the eventual acceptance of Galois’s approach to the solution of equations. The book also describes the relationship between Kummer’s ideal numbers and Dedekind’s ideals, and discusses why Dedekind felt his solution to the divisor problem was better than Kummer’s. Designed for a course in the history of modern algebra, this book is aimed at undergraduate students with an introductory background in algebra but will also appeal to researchers with a general interest in the topic. With exercises at the end of each chapter and appendices providing material difficult to find elsewhere, this book is self-contained and therefore suitable for self-study.Trade Review“This volume is well written and nicely complements other works on the history of algebra. It can be recommended to all mathematicians and students of mathematics who want to understand how algebra turned into the rather abstract field it is today.” (C. Baxa, Monatshefte für Mathematik, Vol. 201 (4), August, 2023)“The book under review is an excellent contribution to the history of abstract algebra and the beginnings of algebraic number theory. I recommend it to everyone interested in the history of mathematics.” (Franz Lemmermeyer, zbMATH 1411.01005, 2019)“This is a nice book to have around; it reflects careful scholarship and is filled with interesting material. … there is much to like about this book. It is quite detailed, contains a lot of information, is meticulously researched, and has an extensive bibliography. Anyone interested in the history of mathematics, or abstract algebra, will want to make the acquaintance of this book.” (Mark Hunacek, MAA Reviews, June 24, 2019)Table of ContentsIntroduction.- 1 Simple quadratic forms.- 2 Fermat’s Last Theorem.- 3 Lagrange’s theory of quadratic forms.- 4 Gauss’s Disquisitiones Arithmeticae.- 5 Cyclotomy.- 6 Two of Gauss’s proofs of quadratic reciprocity.- 7 Dirichlet’s Lectures.- 8 Is the quintic unsolvable?.- 9 The unsolvability of the quintic.- 10 Galois’s theory.- 11 After Galois – Introduction.- 12 Revision and first assignment.- 13 Jordan’s Traité.- 14 Jordan and Klein.- 15 What is ‘Galois theory’?.- 16 Algebraic number theory: cyclotomy.- 17 Dedekind’s first theory of ideals.- 18 Dedekind’s later theory of ideals.- 19 Quadratic forms and ideals.- 20 Kronecker’s algebraic number theory.- 21 Revision and second assignment.- 22 Algebra at the end of the 19th century.- 23 The concept of an abstract field.- 24 Ideal theory.- 25 Invariant theory.- 26 Hilbert’s Zahlbericht.- 27 The rise of modern algebra – group theory.- 28 Emmy Noether.- 29 From Weber to van der Waerden.- 30 Revision and final assignment.- A Polynomial equations in the 18th Century.- B Gauss and composition of forms.- C Gauss on quadratic reciprocity.- D From Jordan’s Traité.- E Klein’s Erlanger Programm.- F From Dedekind’s 11th supplement.- G Subgroups of S4 and S5.- H Curves.- I Resultants.- Bibliography.- Index.

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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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Book SynopsisThis is the perfect introduction for those who have a lingering fear of maths. If you think that maths is difficult, confusing, dull or just plain scary, then The Maths Handbook is your ideal companion. Covering all the basics including fractions, equations, primes, squares and square roots, geometry and fractals, Dr Richard Elwes will lead you gently towards a greater understanding of this fascinating subject. Even apparently daunting concepts are explained simply, with the assistance of useful diagrams, and with a refreshing lack of jargon. So whether you're an adult or a student, whether you like Sudoku but hate doing sums, or whether you've always been daunted by numbers at work, school or in everyday life, you won't find a better way of overcoming your nervousness about numbers and learning to enjoy making the most of mathematics.Trade Review'Elwes takes the key concepts, perfectly illustrates them with practical examples and easy-to-follow explanations, tests us with quizzes, and applies the principles to everyday situations. The effect is strangely liberating, and you might soon find yourself acquiring a love of logarithms and a respect for reflex quadrilaterals' Good Book Guide. * Good Book Guide *Table of ContentsIntroduction. The language of mathematics. Addition. Subtraction. Multiplication. Division. Primes, factors and multiples. Negative numbers and the number line. Decimals. Fractions. Arithmetic with fractions. Powers. The power of 10. Roots and logs. Percentages and proportions. Algebra. Equations. Angles. Triangles. Circles. Area and volume. Polygons and solids. Pythagoras' theorem. Trigonometry. Coordinates. Graphs. Statistics. Probability. Charts. Answers to quizzes. Index.

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Book SynopsisTrade Review"Everything in the book is very well illustrated with insightful graphics that, together with the text, make results almost like being obvious."---Adhemar Bultheel, European Mathematical Society

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Book SynopsisWhen Kurt Gödel published his celebrated theorem, showing that no axiomatization can determine the whole truth and nothing but the truth concerning arithmetic, it had a profound impact on mathematical ideas and philosophical thought. Adrian Moore places the theorem in its intellectual and historical context, explaining the key concepts and misunderstandings.

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Book SynopsisHow can we be sure that Pythagoras''s theorem is really true? Why is the ''angle in a semicircle'' always 90 degrees? And how can tangents help determine the speed of a bullet?David Acheson takes the reader on a highly illustrated tour through the history of geometry, from ancient Greece to the present day. He emphasizes throughout elegant deduction and practical applications, and argues that geometry can offer the quickest route to the whole spirit of mathematics at its best. Along the way, we encounter the quirky and the unexpected, meet the great personalities involved, and uncover some of the loveliest surprises in mathematics.Trade ReviewWell written, clear and informative. * Edward Rochead, Mathematics today *This delightful book should be available, at the minimum, in every high school library and in every public library. * F. -J. Papp, Mathematical Reviews Clippings *It would make an ideal addition both to readers' bookshelves and for every school library. * GERRY LEVERSHA, The Mathematical Gazette *Everything was explained clearly and concisely so that the wonders of geometry could definitely be seen. * Jasmine Wootten, LMS Newsletter *Don't Miss: The Wonder Book of Geometry is full of pretty surprises... * New Scientist *Give this to a curious teenager and they will fall in love with geometry. * Alex Bellos *David Acheson has set geometry free from the confines of stuffy textbooks and lets loose its potential to surprise and delight. Theres a rich and ancient history to be found in these pages, and a future for the field that extends beyond neat (yet elegant) equations. * BBC Science Focus, Books of the Year *This is by far the most approachable book on geometry I've ever read, and I wish it had been around in my day... if you need to learn the basics of geometry for whatever reason (there must be several reasons, surely) then this blows every known textbook on the topic out of the water... The Wonder Book of Geometry does what it does wonderfully. Acheson has done a remarkable job. * Popular Science *Anyone who has read David's earlier books will instantly recognise his almost playful style... I highly recommend it as a marvellous source book on geometry. * Ray Huntley, Mathematics in Schools *There is no better tour guide to the wonders of geometry than the delightful David Acheson. * Matt Parker, author of Humble Pi: A Comedy of Maths Errors and Things to Make and Do in the Fourth Dimension *Table of Contents1: Introduction 2: Getting Started 3: Euclid's Elements 4: Thales' Theorem 5: Geometry in Action 6: Pythagoras' Theorem 7: 'In Love with Geometry'? 8: 'Imagine my exultation, Watson...' 9: Congruence and Similarity 10: Conversely... 11: Circle Theorems 12: Off at a Tangent 13: From Tangents to Supersonic Flow 14: What is pi, exactly? 15: The Story of the Ellipse 16: Geometry by Coordinates 17: Geometry and Calculus 18: A Royal Road to Geometry? 19: Unexpected Meetings 20: Ceva's Theorem 21: A Kind of Symmetry 22: 'Pyracy' in Woolwich? 23: Fermat's Problem 24: A Soap Solution 25: Geometry in 'The Ladies' Diary' 26: What Euclid Did 27: Euclid on Parallel Lines 28: 'A New Theory of Parallels'? 29: Anti-Euclid? 30: When Geometry Goes Wrong... 31: New Angles on Geometry 32: And Finally...

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Book SynopsisThe Life and Work of James Bradley: The New Foundations of 18th Century Astronomy is the first major work on the life and achievements of James Bradley for 190 years. This book offers a new perspective and new interpretations of previously published materials, together with various insights about recently researched sources.This book is a complete account of the life and work of Bradley as discerned from surviving documents of his working archive, as well as other documents and records. In addition, it offers a new interpretation of Bradley''s work as an astronomer, not merely from his observations of Jupiter and Saturn and their satellites and annual aberration and the nutation of the Earth''s axis, but also his corroborative work with pendulums and other horological work with George Graham. It also explores the little amount documented about his private life including a degree of speculation about his personal relationships.This work on 18th century astronomy is intended for studentsTable of ContentsPreface Table of contents Introduction: Contexts and connections 1: The King's observator 2: May it please your Honours 3: An ingenious young man 4: A new discovered motion 5: And yet it moves 6: The laws of nature 7: On the figure of the Earth 8: The triumph of Themistocles 9: If such a man could have enemies... 10: Observations beyond compare 11: Fundamenta Astronomiae Conclusion: The man who moved the world

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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 SynopsisShe highlights the human side of what many view as that most arcane and otherworldly of intellectual endeavors, mathematics, which indeed answers to such diverse factors as religion, ego, and depression.Trade ReviewA masterful biography. American Scientist 2006 An important and impressively documented contribution to the history of nineteenth-century mathematics. -- Craig G. Fraser Mathematical Reviews 2006 A thoroughly enjoyable read. -- J. W. Anderson The London Mathematical Society Newsletter 2006 This is an exceptional example of scholarly research. -- Gail Kaplan Convergence 2007 Parshall has already established herself as a leading expert on Sylvester and his milieu, carefully reconstructing the trajectory of Sylvester's professional life on the basis of copious documentary evidence, describing Sylvester's more important mathematical results in his career context, and writing for broad audiences with no detailed mathematical exposition or technical analysis of Sylvester's mathematics... Highly recommended. Choice 2007 A well-written and thorough account of its subject... a wealth of useful and well-researched information that is difficult to find elsewhere. -- Robin Wilson Historia Mathematica 2006 This well-written, thoroughly researched biography will become the definitive study of Sylvester. -- Jeremy Gray British Journal for the History of Science 2008Table of ContentsIntroduction: The Myth, the Mathematician, the ManChapter 1. Born to "the Faith in Which the Founder of Christianity Was Educated"Chapter 2. A Price of DissentChapter 3. The Hollow Walls of AcademeChapter 4. Actuary by Day...Mathematician by NightChapter 5. Into the Invariant-Theoretic UnknownChapter 6. A New BeginningChapter 7. At War with the MilitaryChapter 8. The Uneasy YearsChapter 9. Exploring Familiar Ground on Unfamiliar TerritoryChapter 10. Tackling New Challenges in a Home Away from HomeChapter 11. A Bittersweet VictoryChapter 12. The Final TransitionEpilogueNotesReferencesIndex

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Book SynopsisZeta and L-functions have played a major part in the development of number theory. This book for graduate students and researchers presents a big picture of some key results and surrounding theory, whilst taking the reader on a journey through the history of their development.Trade Review'The book will be of interest to both young mathematicians and physicists as well as experienced scholars.' Nikolaj M. Glazunov, zbMATH OpenTable of ContentsIntroduction; 1. The Riemann zeta function; 2. The zeta function of a Z-scheme of finite type; 3. The Weil Conjectures; 4. L-functions from number theory; 5. L-functions from geometry; 6. Motives; Appendix A. Karoubian and monoidal categories; Appendix B. Triangulated categories, derived categories, and perfect complexes; Appendix C. List of exercises; Bibliography; Index.

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Book SynopsisSuitable for both postgraduate students and researchers in the field of operator theory, this book is an excellent resource providing the complete proof of the Brown-Douglas-Fillmore theorem. The book starts with a rapid introduction to the standard preparatory material in basic operator theory taught at the first year graduate level course. To quickly get to the main points of the proof of the theorem, several topics that aid in the understanding of the proof are included in the appendices. These topics serve the purpose of providing familiarity with a large variety of tools used in the proof and adds to the flexibility of reading them independently.Table of ContentsPreface; Overview; 1. Spectral Theory for Hilbert Space Operators; 2. Ext(X) as a Semigroup with Identity; 3. Splitting and the Mayer-Vietoris Sequence; 4. Determination of Ext(X); 5. Applications to Operator Theory; 6. Epilogue; Appendix A. Point Set Topology; Appendix B. Linear Analysis; Appendix C. The Spectral Theorem; Subject Index; Index of Symbols; References.

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Book SynopsisThis book presents a history of differential equations, both ordinary and partial, as well as the calculus of variations, from the origins of the subjects to around 1900. Topics treated include the wave equation in the hands of d’Alembert and Euler; Fourier’s solutions to the heat equation and the contribution of Kovalevskaya; the work of Euler, Gauss, Kummer, Riemann, and Poincaré on the hypergeometric equation; Green’s functions, the Dirichlet principle, and Schwarz’s solution of the Dirichlet problem; minimal surfaces; the telegraphists’ equation and Thomson’s successful design of the trans-Atlantic cable; Riemann’s paper on shock waves; the geometrical interpretation of mechanics; and aspects of the study of the calculus of variations from the problems of the catenary and the brachistochrone to attempts at a rigorous theory by Weierstrass, Kneser, and Hilbert. Three final chapters look at how the theory of partial differential equations stood around 1900, as they were treated by Picard and Hadamard. There are also extensive, new translations of original papers by Cauchy, Riemann, Schwarz, Darboux, and Picard. The first book to cover the history of differential equations and the calculus of variations in such breadth and detail, it will appeal to anyone with an interest in the field. Beyond secondary school mathematics and physics, a course in mathematical analysis is the only prerequisite to fully appreciate its contents. Based on a course for third-year university students, the book contains numerous historical and mathematical exercises, offers extensive advice to the student on how to write essays, and can easily be used in whole or in part as a course in the history of mathematics. Several appendices help make the book self-contained and suitable for self-study.Trade Review“This book is a very good example of a text for a course in the history of mathematics. … the author provides for students and readers a historical overview of how mathematics, physics, celestial mechanics and difficult problems to tackle from differential equations as well as applications were intertwined, and the resulting dialogues between mathematicians, physicists and astronomers. This book is a successful attempt to fill in some of the gaps on the history of differential equations.” (‪Clara Silvia Roero, Mathematical Reviews, September, 2022)Table of Contents1 The First Ordinary Differential Equations.- 2 Variational Problems and the Calculus.- 3 The Partial Differential Calculus.- 4 Rational Mechanics.- 5 Partial Differential Equations.- 6 Lagrange's General Theory.- 7 The Calculus of Variations.- 8 Monge and Solutions to Partial Differential Equations.- 9 Revision.- 10 The Heat Equation.- 11 Gauss and the Hypergeometric Equation.- 12 Existence Theorem.- 13 Riemann and Complex Function Theory.- 14 Riemann and the Hypergeometric Equation.- 15 Schwarz and the Complex Hypergeometric Equation.- 16 Complex Ordinary Differential Equations: Poincaré.- 17 More General Partial Differential Equations.- 18 Green's Functions and Dirichlet's Principle.- 19 Attempts on Laplace's Equation.- 20 Applied Wave Equations.- 21 Revision.- 22 Riemann's Shock Wave Paper.- 23 The Example of Minimal Surfaces.- 24 Partial Differential Equations and Mechanics.- 25 Geometrical Interpretations of Mechanics.- 26 The Calculus of Variations in the 19th Century.- 27 Poincaré and Mathematical Physics.- 28 Elliptic Equations and Regular Variational Problems.- 29 Hyperbolic Equations.- 30 Revision.- 32 Translations.- A Newton's Principia Mathematica.- B Characteristics.- C First-order Non-linear Equations.- D Green's Theorem and Heat Conduction.- E Complex Analysis.- F Möbius Transformations.- G Lipschitz and Picard.- H The Assessment.- Bibliography.- Index.

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Book SynopsisThe book addresses for the first time the dynamics associated with the modernization of mathematics in China from the nineteenth to the mid-twentieth century from a transcultural global historical perspective. Rather than depict the transformations of mathematical knowledge in terms of a process of westernization, the book analyzes the complex interactions between different scientific communities and the ways in which the past, modernity, language, and mathematics were negotiated in a global context. In each chapter, Andrea Bréard provides vivid portraits of a series of go-betweens (such as translators, educators, or state statisticians) based on a vast array of translated primary sources hitherto unavailable to a non-Chinese readership. They not only illustrate how Chinese scholars mediated between new mathematical objects and discursive modes, but also how they instrumentalized their autochthonous scientific roots in specific political and intellectual contexts. While sometimes technical in style, the book addresses all readers who are interested in the global and cultural history of science and the complexities involved in the making of universal mathematics. “While the pursuit of modernity is in the title, entanglement is of as much interest. Using the famous ‘Nine Chapters’ as a framework, Bréard considers a wide range of that entanglement from divination to data management. Bréard’s analysis and thought-provoking insights show once again how much we can learn when two cultures intersect. A fascinating read!” (John Day, Boston University).Trade Review“This collection of essays will make great reading for college students interested in Chinese history or in the history of mathematics and sciences. The topics in many essays are worth further exploration and continue to be a fertile ground for research.” (Jiang-Ping Jeff Chen, Mathematical Reviews, September, 2020)“This book is very useful. It is thoughtful and well researched. The inclusion of many, many translations of original source material, from Chinese into English, makes it a valuable reference in that sense as well.” (Joel Haack, MAA Reviews, January 19, 2020)Table of Contents1 Visions of Antiquity.- 2 The Ellipse Seen from 19th Century China.- 3 Filling Euclid’s Gaps.- 4 Negotiating a Linguistic Space in-between.- 5 Discourse Transformed: Changing Modes of Argumentation.- 6 Fate Calculation : The Mathematics of Divination.- 7 Data Management and Knowledge Production in Late Qing Institutions.- 8 Data Management and Knowledge Production in Late Qing Institutions.- 9 Visions of Modernity.

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Book SynopsisWhenever the topic of mathematics is mentioned, people tend to indicate their weakness in the subject as a result of not having enjoyed its instruction during their school experience. Many students unfortunately do not have very positive experiences when learning mathematics, which can result from teachers who have a tendency 'to teach to the test'. This is truly unfortunate for several reasons. First, basic algebra and geometry, which are taken by almost all students, are not difficult subjects, and all students should be able to master them with the proper motivational instruction. Second, we live in a technical age, and being comfortable with basic mathematics can certainly help you deal with life's daily challenges. Other, less tangible reasons, are the pleasure one can experience from understanding the many intricacies of mathematics and its relation to the real world, experiencing the satisfaction of solving a mathematical problem, and discovering the intrinsic beauty and historical development of many mathematical expressions and relationships. These are some of the experiences that this book is designed to deliver to the reader.The book offers 101 mathematical gems, some of which may require a modicum of high school mathematics and others, just a desire to carefully apply oneself to the ideas. Many folks have spent years encountering mathematical terms, symbols, relationships and other esoteric expressions. Their origins and their meanings may never have been revealed, such as the symbols +, -, =, π. ꝏ, √, ∑, and many others. This book provides a delightful insight into the origin of mathematical symbols and popular theorems such as the Pythagorean Theorem and the Fibonacci Sequence, common mathematical mistakes and curiosities, intriguing number relationships, and some of the different mathematical procedures in various countries. The book uses a historical and cultural approach to the topics, which enhances the subject matter and greatly adds to its appeal. The mathematical material can, therefore, be more fully appreciated and understood by anyone who has a curiosity and interest in mathematics, especially if in their past experience they were expected to simply accept ideas and concepts without a clear understanding of their origins and meaning. It is hoped that this will cast a new and positive picture of mathematics and provide a more favorable impression of this most important subject and be a different experience than what many may have previously encountered. It is also our wish that some of the fascination and beauty of mathematics shines through in these presentations.

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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"This is not your father’s – or grandfather’s – standard collection of conic sections."---Jim Stein, New Books Network"Undoubtedly [this book], written in the same entertaining unmistakable style of the author and containing a lot of information - mathematical, historical and general - will attract, as the previous ones, a large audience."---S. Cobzas, Studia Mathematica"What a beautiful book!"---Jonathan Shock, Mathemafrica.org"A wonderful addition to libraries where the mathematically curious find their reading." * Choice *"Havil’s narrative for each curve is a cornucopia of fun facts and rigorous explanation."---Andrew J. Simoson, Mathematical Intelligencer"Overall, the book was a delight to read. The writing is witty and entertaining, the history is at times peculiar and surprising, and the mathematics is rich and engaging. It would make a fine addition to a classroom bookcase or home coffee table, but while there are plenty of elegant diagrams and intriguing stories to give every curious reader the chance to glimpse mathematical beauty, only those with the ability to dig beneath the surface will understand just how much beauty this book has to offer."---Samuel Hewitt, Mathematical Gazette

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Book SynopsisAlong the way, he tells us what various cultures knew about math and how they came to learn it, providing instructors with a wonderful way to incorporate multicultural mathematics into the middle school, high school, and college classroom.Trade ReviewSwetz has collected word problems, or story problems, used to teach mathematics around the world and throughout history, so mathematics teachers in middle and secondary schools can use them today. University students of mathematics and its history might also find them useful as well as entertaining. Reference and Research Book News Mathematical Expeditions is a wonderful resource for any teacher who would like to use old problems in a course to help students understand the context of mathematical ideas. -- Victor J. Katz Mathematical Reviews The book is well thought-out and is recommended to readers interested in the history of mathematics. -- E. Keith Lloyd London Mathematical Society Newsletter One of my graduate students, who is majoring in mathematics, was excited when I showed her a sample of problems in the book. A month later, she asked whether I had finished my review-she wanted to borrow the book! -- Winifred A. Mallam Mathematics TeacherTable of ContentsPreface1. Word Problems: Footprints from the History of Mathematics2. Problems, Problems: A Resource for Teaching3. Ancient Babylonia (2002–1000 BCE)4. Ancient Egypt5. Ancient Greece6. Ancient China7. India8. Islam9. Medieval Europe10. Renaissance Europe11. Japanese Temple Problems12. The Ladies Diary (1704–1841)13. Nineteenth-Century Victorian Problems14. Eighteenth- and Nineteenth-Century American Problems15. Problems from the Farmer's Almanac16. Nineteenth-Century Calculus Problems17. Some Sample Problem Solution Methods18. Where to from Here? Where Do You Want to Go?AcknowledgmentsAnswers to Numbered ProblemsGlossary of Strange and Exotic Terms: Measurements, Monetary Units, and Culturally Relevant WordsBibliographyIndex

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Book SynopsisThis book shares the life story of William Playfair, the father of statistical graphics, who experienced extreme ups and downs in his various careers, including as a statistician, economist, and fraudster.Table of ContentsPreface: Playfair Is Introduced 1. Playfair Is Sent to Newgate Prison 2. Playfair Goes to Birmingham to Work for Boulton and Watt 3. Playfair Goes to London to Set Up His Own Business 4. Playfair Evolves into a Writer by Profession 5. Playfair Expresses His Early Political Views 6. Playfair Makes His Mark on Statistical Graphics 7. Playfair Goes to Paris 8. Playfair Tries to Take Advantage of the French Revolution 9. Playfair Escapes from France and Returns to England 10. Playfair Becomes an Avid Anti-Jacobin Propagandist 11. Playfair Gets Involved with Forged Assignats 12. Playfair Starts a Bank and Goes Bankrupt 13. Playfair Ekes Out a Living as a Bankrupt 14. Playfair Has a Good Year during 1805 with Hints of Ending Badly 15. Playfair Has Serious Legal and Other Problems 16. Playfair Dabbles Deeply into Family History and Political Biography 17. Playfair Continues Writing and Tries a Few More Scams to Get to Paris 18. Playfair Returns to Paris 19. Playfair Spends His Last Few Years in England in Poverty Afterword: Playfair Avoids a Shakespearean Epitaph Appendix: Assignat Forging by French Emigres in England Notes Index

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Book SynopsisMathematics and Religion: Our Languages of Sign and Symbol is the sixth title published in the Templeton Science and Religion Series, in which scientists from a wide range of fields distill their experience and knowledge into brief tours of their respective specialties. In this volume, Javier Leach, a mathematician and Jesuit priest, leads a fascinating study of the historical development of mathematical language and its influence on the evolution of metaphysical and theological languages.Leach traces three historical moments of change in this evolution: the introduction of the deductive method in Greece, the use of mathematics as a language of science in modern times, and the formalization of mathematical languages in the nineteenth and twentieth centuries. As he unfolds this fascinating history, Leach notes the striking differences and interrelations between the two languages of science and religion. Until now there has been little reflection on these similarities and differences, or about how both languages can complement and enrich each other.Table of ContentsPreface viiChapter 1: Mathematics and Natural Sciences 3Chapter 2: Metaphysical Language 16Chapter 3: Origins of Mathematics 35Chapter 4: Euclid and Beyond 44Chapter 5: Dawn of Science 55Chapter 6: Mathematics Formalized 67Chapter 7: Propositional Logic 93Chapter 8: Language and Meaning 106Chapter 9: Science, Language, and Religion 120Appendix 1: Syntax of Propositional Logic 133Appendix 2: Semantics of Propositional Logic 136Appendix 3: Syntax of First-Order Logic 139Appendix 4: Semantics of First-Order Logic 143Appendix 5: Numerical Systems:Their Role in First-Order Logic 147

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Book SynopsisFrom Ancient Greek times, music has been seen as a mathematical art, and this relationship has fascinated generations. This new in paperback edition of diverse, comprehensive and fully-illustrated papers, authored by leading scholars, links the two fields in a lucid manner that is suitable for students of each subject as well as the general reader.Trade ReviewAn attractive volume that covers almost al of the important aspects of the interplay between mathematics and music. * Ehrhard Behrends, The Mathematical Intelligencer, Vol 28, 3 *Table of ContentsPART I: MUSIC AND MATHEMATICS THROUGH HISTORY; PART II: THE MATHEMATICS OF MUSICAL SOUND; PART III: MATHEMATICAL STRUCTURE IN MUSIC; PART IV: THE COMPOSER SPEAKS

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Book SynopsisTrade ReviewBestselling physicist Carlo Rovelli argues in this enjoyable and provocative little book that a little-known Greek philosopher invented the idea of the cosmos -- Tim Adams * Observer *Carlo Rovelli’s Anaximander is a knockout: there’s nobody like Rovelli for bridging the Two Cultures, and I was enlarged by his lucid, optimistic account, full of fascinating historical nuggets, of what scientists do and why it’s exciting -- Sam Leith * TLS , Best Books of the Year *Rovelli is a very good scientist and a very good writer. He explains some of the most conceptually difficult and densest areas of physics lightly and breezily. Here, he tells the story of an ancient thinker who had a revolutionary idea about the Earth's place in the cosmos -- Tom Whipple * The Times *Anaximander is a delight and so is this book -- James McConnachie * Sunday Times *As Rovelli's fans will expect, this book is excellent. It is never less than engaging, and enviably compendious -- Tim Smith-Laing * The Telegraph *A celebration of the scientific spirit of inquiry and the remarkable achievements of one man more than 2,500 years ago -- John Sellars * TLS *A bold and persuasive case that this ancient Greek philosopher scientist was the founder of critical thinking -- Adam Rutherford * Start the Week, BBC Radio 4 *This is seriously astounding. So lucid, so imaginative, so subtle, and so large in scope. It's like the best primer you can imagine for the non-scientist on why what you think you know about Ptolemy and Copernicus, or Popper and Kuhn, is not quite right -- Sam Leith * Twitter *

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Book SynopsisNearly a century before Mondrian made geometrical red, yellow, and blue lines famous, 19th-century mathematician Oliver Byrne employed the color scheme for his 1847 edition of Euclid’s mathematical and geometric treatise Elements. Byrne’s idea was to use color to make learning easier and “diffuse permanent knowledge.” The result has been described as one of the oddest and most beautiful books of the 19th century. The facsimile of Byrne’s seminal publication is now available in a beautiful new edition. A masterwork of art and science, it is as remarkable in the boldness of its red, yellow, and blue figures and diagrams as it is in the mathematical precision of its theories. In the simplicity of forms and colors, the pages anticipate the vigor of De Stijl and Bauhaus design. In making complex information at once accessible and aesthetically engaging, this work is a forerunner to the information graphics that today define much of our data consumption.Trade Review“Every graphic designer, book lover and math nerd will be awestruck.” * The New York Times *

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Book SynopsisAll the Math Your 4th Grader Needs to SucceedThis book will help your elementary school student develop the math skills needed to succeed in the classroom and on standardized tests. The user-friendly, full-color pages are filled to the brim with engaging activities for maximum educational value. The book includes easy-to-follow instructions, helpful examples, and tons of practice problems to help students master each concept, sharpen their problem-solving skills, and build confidence.Features include:â A guide that outlines national standards for Grade 4â Concise lessons combined with lot of practice that promote better scoresâin class and on achievement testsâ A pretest to help identify areas where students need more workâ End-of-chapter tests to measure studentsâ progressâ A helpful glossary of key terms used in the bookâ More than 1,000 math problems with answersTopics covered:â Adding and

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Book SynopsisAll the Math Your 5th Grader Needs to SucceedThis book will help your elementary school student develop the math skills needed to succeed in the classroom and on standardized tests. The user-friendly, full-color pages are filled to the brim with engaging activities for maximum educational value. The book includes easy-to-follow instructions, helpful examples, and tons of practice problems to help students master each concept, sharpen their problem-solving skills, and build confidence.Features include:â A guide that outlines national standards for Grade 5â Concise lessons combined with lot of practice that promote better scoresâin class and on achievement testsâ A pretest to help identify areas where students need more workâ End-of-chapter tests  to measure studentsâ progressâ A helpful glossary of key terms used in the bookâ More than 1,000 math problems with answersTopics covered:â Opera

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Book SynopsisTrade Review"Excellent. . . . [A]n exceptionally well-informed, very readable and clear introduction to the subject. If you are looking for an entry point into the extensive philosophical literature on the nature of mathematics, look no further."---A. C. Paseau, Mathematical Gazette"Linnebo's slender volume is an admirable addition to the many existing books on the philosophy of mathematics. It is clear, concise, and well written. . . . All in all, this is an excellent introduction to the philosophy of mathematics and should be seriously considered by any individual interested in the subject." * Choice *"This is a thought-provoking book, and is a useful addition to the textbook literature on this subject." * MAA Reviews *"This book provides a nice lay of the land for anyone interested in contemporary philosophy of mathematics."---Gregory Lavers, Philosophia Mathematica"[This book] is very, very good. Superbly clear, concise, well organised, it gives not only a very accessible introduction but also takes the reader all the way to the cutting edge of what philosophers are doing in the philosophy of mathematics. Above all, Linnebo writes as a fully engaged philosopher and makes his preferred choice of philosophical position clear. But this is no mere polemic: I felt he clearly and forcefully presents the strengths and weaknesses of all the philosophical positions he discusses."---Henri Laurie, Mathemafrica"[A] very readable and . . . superb introduction to the philosophy of mathematics."---Jason Wakefield, Avello Publishing JournalTable of ContentsAcknowledgments vii Introduction 1 1 Mathematics as a Philosophical Challenge 4 2 Frege's Logicism 21 3 Formalism and Deductivism 38 4 Hilbert's Program 56 5 Intuitionism 73 6 Empiricism about Mathematics 88 7 Nominalism 101 8 Mathematical Intuition 116 9 Abstraction Reconsidered 126 10 The Iterative Conception of Sets 139 11 Structuralism 154 12 The Quest for New Axioms 170 Concluding Remarks 183 Bibliography 189 Index 199

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Book SynopsisTrade Review"A volume that should be on every scientist's reading list."—Barbara Kiser, Nature"A terrific book."—Mathematics Magazine"Fun and entertaining to read."—MAA Reviews"To anyone with an interest in probability or statistics, this is a book you must read. . . . [It] is far-ranging and can be read at many levels, from the novice to the expert. It is also thoroughly engaging."—David M. Bressoud, UMAP Journal"A very enriching journey. Your vision will be broadened."—Adhemar Bultheel, European Mathematical Society"A great book for anyone who wants to understand some of the central tenets of probability, how they were discovered, and how they can be tamed in our day-to-day lives."—ZME Science

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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 the Year Winner of the 2016 PROSE Award in Mathematics, Association of American Publishers One of Choice's Outstanding Academic Titles for 2015 "Mathematics without Apologies is a kaleidoscope of philosophical, sociological, historical and literary perspectives on what mathematicians do, and why."--Amir Alexander, Nature "A wry and insightful look at what being a pure mathematician is all about, as seen from the inside."--Steven Strogatz, Physics Today "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 "Harris is the kind of mathematician one hopes to meet at an intimate dinner party. By sharing his professional and personal relationship to mathematics, [he] links art, philosophy, music, and literature to academic culture and research problems."--Library Journal "Extraordinary, extravagant... Harris is a polyglot, deeply learned. Threading through his remarkable book, unifying it, is Hardy's lament regarding whether a pure mathematician can make a claim that the vocation has a philosophically 'useful' purpose. Harris's reply is multivalent, persuasive, and profound. 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 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 SynopsisInthe tradition of Fermat’s Enigma and Pi, Marcus du Sautoy tells the illuminating, authoritative, and engagingstory of Bernhard Reimann and the ongoing quest tocapture the holy grail of mathematics—the formula to predict prime numbers.Oliver Sacks, author of The Man Who Mistook His Wife for a Hat, calls TheMusic of the Primes “an amazing book. . . . I could not put it down once Ihad started.” Simon Winchester, author of The Professor and the Madman,writes, “this fascinating account, decoding the inscrutable language of themathematical priesthood, is written like the purest poetry. Marcus du Sautoy''s enthusiasm shines through every line of this hymnto the joy of high intelligence, illuminating as it does so even the darkestcorners of his most arcane universe.”

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Book SynopsisOriginally published between 1880 and 1886, this two-volume work by George Shoobridge Carr (1837–1914) was intended as an aid to students preparing for the Cambridge Mathematical Tripos. Most notably, it played an important part in the mathematical education of the Indian prodigy Srinivasa Ramanujan (1887–1920).Table of ContentsPart I. 1. Mathematical tables; 2. Algebra; 3. Theory of equations; 4. Plane trigonometry; 5. Spherical trigonometry; 6. Elementary geometry; 7. Geometrical conics.

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Book SynopsisPresents the history in English of the origins and early development of trigonometry. This book identifies the earliest known trigonometric precursors in ancient Egypt, Babylon, and Greece, and examines the revolutionary discoveries of Hipparchus. It traces trigonometry's development into a full-fledged mathematical discipline in India and Islam.Trade Review"Fans of the history of mathematics will be richly rewarded by this exhaustively researched book, which focuses on the early development of trigonometry... Finally, the generous and lucid explanations provided throughout the text make Van Brummelen's history a rewarding one for the mathematical tourist."--Mathematics Teacher "[T]his new and comprehensive history of trigonometry is more than welcome--even more so because it is the first in English... [T]his book will be appreciated by many with an interest--general or more specific--in the history of mathematics."--Steven Wepster, Centaurus "[T]his book will have wide appeal, for students, researchers, and teachers of history and/or trigonometry. The excerpts selected are balanced and their significances well articulated... It is a book written by an expert after many years of exposure to individual sources and in this way Van Brummelen uniquely advances the field. The book will no doubt become a necessary addition to the libraries of mathematicians and historians alike."--Clemency Montelle and Kathleen M. Clark, Aestimatio "Van Brummelen's history does far more than simply fill a vacant spot in the historical literature of mathematics. He recounts the history of trigonometry in a way that is both captivating and yet more than satisfying to the crankiest and most demanding of scholars... The Mathematics of the Heavens and the Earth should be a part of every university library's mathematics collection. It's also a book that most mathematicians with an interest in the history of the subject will want to own."--Rob Bradley, MAA Reviews "I highly recommend the book to all those interested in the way in which the ancient people solve their practical problems and hope that the next volume of this interesting history of spherical and plane trigonometry will appear soon."--Cristina Blaga, Studia MathematicaTable of ContentsPreface xi The Ancient Heavens 1 Chapter 1: Precursors 9 What Is Trigonometry? 9 The Seqed in Ancient Egypt 10 * Text 1.1 Finding the Slope of a Pyramid 11 Babylonian Astronomy, Arc Measurement, and the 360 Circle 12 The Geometric Heavens: Spherics in Ancient Greece 18 A Trigonometry of Small Angles? Aristarchus and Archimedes on Astronomical Dimensions 20 * Text 1.2 Aristarchus, the Ratio of the Distances of the Sun and Moon 24 Chapter 2: Alexandrian Greece 33 Convergence 33 Hipparchus 34 A Model for the Motion of the Sun 37 * Text 2.1 Deriving the Eccentricity of the Sun's Orbit 39 Hipparchus's Chord Table 41 The Emergence of Spherical Trigonometry 46 Theodosius of Bithynia 49 Menelaus of Alexandria 53 The Foundations of Spherical Trigonometry: Book III of Menelaus's Spherics 56 * Text 2.2 Menelaus, Demonstrating Menelaus's Theorem 57 Spherical Trigonometry before Menelaus? 63 Claudius Ptolemy 68 Ptolemy's Chord Table 70 Ptolemy's Theorem and the Chord Subtraction/Addition Formulas 74 The Chord of 1 76 The Interpolation Table 77 Chords in Geography: Gnomon Shadow Length Tables 77 * Text 2.3 Ptolemy, Finding Gnomon Shadow Lengths 78 Spherical Astronomy in the Almagest 80 Ptolemy on the Motion of the Sun 82 * Text 2.4 Ptolemy, Determining the Solar Equation 84 The Motions of the Planets 86 Tabulating Astronomical Functions and the Science of Logistics 88 Trigonometry in Ptolemy's Other Works 90 * Text 2.5 Ptolemy, Constructing Latitude Arcs on a Map 91 After Ptolemy 93 Chapter 3: India 94 Transmission from Babylon and Greece 94 The First Sine Tables 95 Aryabhata's Difference Method of Calculating Sines 99 * Text 3.1 Aryabhata, Computing Sines 100 Bhaskara I's Rational Approximation to the Sine 102 Improving Sine Tables 105 Other Trigonometric Identities 107 * Text 3.2 Varahamihira, a Half-angle Formula 108 * Text 3.3 Brahmagupta, the Law of Sines in Planetary Theory? 109 Brahmagupta's Second-order Interpolation Scheme for Approximating Sines 111 * Text 3.4 Brahmagupta, Interpolating Sines 111 Taylor Series for Trigonometric Functions in Madhava's Kerala School 113 Applying Sines and Cosines to Planetary Equations 121 Spherical Astronomy 124 * Text 3.5 Varahamihira, Finding the Right Ascension of a Point on the Ecliptic 125 Using Iterative Schemes to Solve Astronomical Problems 129 * Text 3.6 Paramesvara, Using Fixed-point Iteration to Compute Sines 131 Conclusion 133 Chapter 4: Islam 135 Foreign Junkets: The Arrival of Astronomy from India 135 Basic Plane Trigonometry 137 Building a Better Sine Table 140 * Text 4.1 Al-Samaw'al ibn Yahya al-Maghribi, Why the Circle Should Have 480 Degrees 146 Introducing the Tangent and Other Trigonometric Functions 149 * Text 4.2 Abu'l-Rayhan al-Biruni, Finding the Cardinal Points of the Compass 152 Streamlining Astronomical Calculation 156 * Text 4.3 Kushyar ibn Labban, Finding the Solar Equation 156 Numerical Techniques: Approximation, Iteration, Interpolation 158 * Text .4 Ibn Yunus, Interpolating Sine Values 164 Early Spherical Astronomy: Graphical Methods and Analemmas 166 * Text 4.5 Al-Khwarizmi, Determining the Ortive Amplitude Geometrically 168 Menelaus in Islam 173 * Text 4.6 Al-Kuhi, Finding Rising Times Using the Transversal Theorem 175 Menelaus's Replacements 179 Systematizing Spherical Trigonometry: Ibn Mucadh's Determination of the Magnitudes and Nasir al-Din al-Tusi's Transversal Figure 186 Applications to Religious Practice: The Qibla and Other Ritual Needs 192 * Text 4.7 Al-Battani, a Simple Approximation to the Qibla 195 Astronomical Timekeeping: Approximating the Time of Day Using the Height of the Sun 201 New Functions from Old: Auxiliary Tables 205 * Text 4.8 Al-Khalili, Using Auxiliary Tables to Find the Hour-angle 207 Trigonometric and Astronomical Instruments 209 * Text 4.9 Al-Sijzi (?), On an Application of the Sine Quadrant 213 Trigonometry in Geography 215 Trigonometry in al-Andalus 217 Chapter 5: The West to 1550 223 Transmission from the Arab World 223 An Example of Transmission: Practical Geometry 224 * Text 5.1 Hugh of St. Victor, Using an Astrolabe to Find the Height of an Object 225 * Text 5.2 Finding the Time of Day from the Altitude of the Sun 227 Consolidation and the Beginnings of Innovation: The Trigonometry of Levi ben Gerson, Richard of Wallingford, and John of Murs 230 * Text 5.3 Levi ben Gerson, The Best Step Size for a Sine Table 233 * Text 5.4 Richard of Wallingford, Finding Sin(1 ) with Arbitrary Accuracy 237 Interlude: The Marteloio in Navigation 242 * Text 5.5 Michael of Rhodes, a Navigational Problem from His Manual 244 From Ptolemy to Triangles: John of Gmunden, Peurbach, Regiomontanus 247 * Text 5.6 Regiomontanus, Finding the Side of a Rectangle from Its Area and Another Side 254 * Text 5.7 Regiomontanus, the Angle-angle-angle Case of Solving Right Triangles 255 Successors to Regiomontanus: Werner and Copernicus 264 * Text 5.8 Copernicus, the Angle-angle-angle Case of Solving Triangles 267 * Text 5.9 Copernicus, Determining the Solar Eccentricity 270 Breaking the Circle: Rheticus, Otho, Pitiscus and the Opus Palatinum 273 Concluding Remarks 284 Bibliography 287 Index 323

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Book SynopsisProviding insight into one of the most fascinating and unique subjects in statistics, this book examines classic problems of probability that have both contributed to the field and have been of historical significance, including Parrondo's Amazing Paradox, Laplace's Rule of Succession, and Jacob Bernoulli and His Golden Theorem.Trade Review“Thus, the book can be highly recommend to every lecturer in this field and every student interested in probability and statistics.” (Zentralblatt Math, 1 September 2013)Table of ContentsPreface ix Acknowledgments xi 1 Cardano and Games of Chance (1564) 1 2 Galileo and a Discovery Concerning Dice (1620) 9 3 The Chevalier de Méré Problem I: The Problem of Dice (1654) 13 4 The Chevalier de Méré Problem II: The Problem of Points (1654) 20 5 Huygens and the Gambler’s Ruin (1657) 39 6 The Pepys–Newton Connection (1693) 49 7 Rencontres with Montmort (1708) 54 8 Jacob Bernoulli and his Golden Theorem (1713) 62 9 De Moivre’s Problem (1730) 81 10 De Moivre, Gauss, and the Normal Curve (1730, 1809) 89 11 Daniel Bernoulli and the St. Petersburg Problem (1738) 108 12 d’Alembert and the “Croix ou Pile” Article (1754) 119 13 d’Alembert and the Gambler’s Fallacy (1761) 124 14 Bayes, Laplace, and Philosophies of Probability (1764, 1774) 129 15 Leibniz’s Error (1768) 156 16 The Buffon Needle Problem (1777) 159 17 Bertrand’s Ballot Problem (1887) 169 18 Bertrand’s Strange Three Boxes (1889) 175 19 Bertrand’s Chords (1889) 179 20 Three Coins and a Puzzle from Galton (1894) 186 21 Lewis Carroll’s Pillow Problem No. 72 (1894) 189 22 Borel and a Different Kind of Normality (1909) 194 23 Borel’s Paradox and Kolmogorov’s Axioms (1909, 1933) 199 24 Of Borel, Monkeys, and the New Creationism (1913) 208 25 Kraitchik’s Neckties and Newcomb’s Problem (1930, 1960) 215 26 Fisher and the Lady Tasting Tea (1935) 224 27 Benford and the Peculiar Behavior of the First Significant Digit (1938) 233 28 Coinciding Birthdays (1939) 240 29 Lévy and the Arc Sine Law (1939) 247 30 Simpson’s Paradox (1951) 253 31 Gamow, Stern, and Elevators (1958) 260 32 Monty-Hall, Cars, and Goats (1975) 264 33 Parrondo’s Perplexing Paradox (1996) 271 Bibliography 277 Photo Credits 296 Index 299

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Book SynopsisTrade ReviewPublishers Weekly "Stewart shares his enthusiasm as well as his knowledge in this tour of ground-breaking equations and the research they supported... An entertaining and illuminating collection of curious facts and histories suitable for random dipping-in or reading straight through." Kirkus Reviews "Stewart provides clear, cogent explanations of how the equations work without burdening the reader with cumbersome derivations... He gives a fascinating explanation of how Newton's laws, when extended to three-body problems, are still used by NASA to calculate the best route from Earth to Mars and have laid the basis for chaos theory. Throughout, Stewart's style is felicitous." Discover "Seemingly basic equations have enabled us to predict eclipses, engineer earthquake-proof buildings, and invent the refrigerator. In this lively volume, mathematician Ian Stewart delves into 17 equations that shape our daily existence, including those dreamed up by the likes of Einstein, Newton, and Erwin Schrodinger." Maclean's "Stewart is the finest living math popularizer-a writer who can tackle eye-spraining mathematical topics approachably, and yet dazzle hard-core nerds with new and surprising information. It is hard not to get your money's worth from him, and in a book like this he is at his best because of the very wide ground covered." Library Journal "Stewart's expertise and his well-developed style (enhanced by a nice sense of humor) make for enjoyable reading... [A] worthwhile and entertaining book, accessible to all readers. Recommended for anyone interested in the influence of mathematics on the development of science and on the emergence of our current technology-driven society." Washington Independent Review of Books "Stewart has managed to produce a remarkably readable, informative and entertaining volume on a subject about which few are as well informed as they would like to be." New York Journal of Books "Stewart is a genius in the way he conveys his excitement and sense of wonder... He has that valuable grasp of not only what it takes to make equations interesting, but also to make science cool."Table of ContentsWhy Equations? 1. The squaw on the hippopotamus 2. Shortening the proceedings 3. Ghosts of departed quantities 4. The system of the world 5. Portent of the ideal world 6. Much ado about knotting 7. Patterns of chance 8. Good vibrations 9. Ripples and blips 10. The ascent of humanity 11. Waves in the ether 12. Law and disorder 13. One thing is absolute 14. Quantum weirdness 15. Codes, communications, and computers 16. The imbalance of nature 17. The Midas formula Where Next?

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Book SynopsisA gently guided, profusely illustrated Grand Tour of the world of mathematics.

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