Philosophy of mathematics Books

Book SynopsisAnother terrific book by Rob Eastaway' SIMON SINGHA delightfully accessible guide to how to play with numbers' HANNAH FRYHow many cats are there in the world?What''s the chance of winning the lottery twice?And just how long does it take to count to a million?Learn how to tackle tricky maths problems with nothing but the back of an envelope, a pencil and some good old-fashioned brain power.Join Rob Eastaway as he takes an entertaining look at how to figure without a calculator. Packed with amusing anecdotes, quizzes, and handy calculation tips for every situation, Maths on the Back of an Envelope is an invaluable introduction to the art of estimation, and a welcome reminder that sometimes our own brain is the best tool we have to deal with numbers.Trade Review‘A delightfully accessible guide to how to play with numbers’ – Dr Hannah Fry, author of Hello World and The Mathematics of Love ‘Put aside those calculators and computers, and find a pen and piece of paper! In a collection of riveting tips and examples, Eastaway shows us amazing short-cuts to get rough answers to important questions. I still find it remarkable that 16% of 25 is exactly the same as 25% of 16!’ – Professor Sir David Spiegelhalter, author of The Art of Statistics ‘Another terrific book by Rob Eastaway’ – Simon Singh ‘Packed with fun examples and fresh ideas. I thought I was on top of this subject, but I learned a lot’ – Tim Harford ‘A joyful primer about the lost art of calculating without a calculator’ – Guardian

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Book SynopsisThe Quadrivium consists of the four Liberal Arts of Number, Geometry, Music, and Cosmology, studied from antiquity to the Renaissance as a way of glimpsing the nature of reality. They synthesize number, space, and time. Geometry is number in space, music is number in time, and the cosmos expresses number in space and time. Number, music, and geometry are metaphysical truths, good and beautiful everywhere at all times. Life across the universe investigates them. They foreshadow the physical sciences. This is the first volume to bring together the Quadrivium for many hundreds of years.Trade Review"Music to the eye" - The New York Times. "Fascinating" - Financial Times. "Excellent" - New Scientist. "Genuinely mind-expanding" - Fortean Times. "Engaging and accessible" - Seattle Times. "Beautiful" - London Review of Books. "Rich and artful" - The Lancet.

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Book SynopsisAn inclusive vision of mathematics—its beauty, its humanity, and its power to build virtues that help us all flourishTrade Review“Beautifully written, contains well-chosen and interesting mathematical puzzles, and offers an important viewpoint for mathematicians to consider. . . . The book is aimed at a broader audience and is also a call to being more inclusive, to recognising that there are many paths to success.”—Pamela Gorkin, Mathematical IntelligencerAwarded Book of the Year by Aleo ReviewWinner of the Euler Book Prize, sponsored by the Mathematical Association of AmericaSelected for the 2021 Phi Beta Kappa Award for Science Short List“The ancient Greeks argued that the best life was filled with beauty, truth, justice, play and love. The mathematician Francis Su knows just where to find them.”—Kevin Hartnett, Quanta Magazine “Please read this beautiful, compelling, galvanizing book if you care about mathematics, social justice, or humanity, which I hope is everyone.”—Eugenia Cheng, author of The Art of Logic in an Illogical World “The world desperately needs this all‑embracing and deeply human perspective on what mathematics is and why it matters. The key qualities developed by mathematical thinking are characteristics that we should all value and long for.”—Eddie Woo, author of It’s a Numberful World “I was mesmerized by this unusual, sublime book. Original insights and engaging puzzles made me feel young again, discovering a way to Zen and the Art of Mathematics.”—Nalini Joshi, University of Sydney “Francis Su believes that math can make us better humans—and he leads by example. Every page is a work of generosity and compassion. Plus, the puzzles will haunt you for weeks.”—Ben Orlin, author of Math with Bad Drawings “A celebration of mathematics and the human spirit. Learning mathematics enriches our lives, and Su wants everyone to have a seat at the banquet.”—Edward Scheinerman, author ofvThe Mathematics Lover’s Companion “A delightful mixture of philosophy, mathematical illustrations, and compassion.”—John Cook, Singular Value Consulting “Francis Su has written a lyrical meditation on the beauty of mathematics and how it connects to our common humanity.”—John Urschel, author of Mind and Matter: A Life in Math and Football “Su elegantly uncovers the beauty and power of mathematics as they relate to our desires to be loved, trusted, and accepted. A powerful narrative of mathematical beauty, this book is the antidote for a mathematically fixed mindset.”—Talithia Williams, author of Power in Numbers: The Rebel Women of Mathematics “This is perhaps the most important mathematics book of our time. Francis Su shows mathematics is an experience of the mind and, most important, of the heart.”—James Tanton, Global Math Project“The ancient Greeks argued that the best life was filled with beauty, truth, justice, play and love. The mathematician Francis Su knows just where to find them.”—Kevin Hartnett, Quanta Magazine

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Book SynopsisA theoretical physicist takes readers on an awe-inspiring journey—found in 'no other book' (Science)—to discover how the universe generates everything from nothing at all: 'If you want to know what's really going on in the realms of relativity and particle physics, read this book' (Sean Carroll, author of The Biggest Ideas in the Universe). In Waves in an Impossible Sea, physicist Matt Strassler tells a startling tale of elementary particles, human experience, and empty space. He begins with a simple mystery of motion. When we drive at highway speeds with the windows down, the wind beats against our faces. Yet our planet hurtles through the cosmos at 150 miles per second, and we feel nothing of it. How can our voyage be so tranquil when, as Einstein discovered, matter warps space, and space deflects matter?   The answer, Strassler reveals, is that empty space is a sea, albeit a paradoxically strange one.

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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"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 SynopsisLeonhard Euler's polyhedron formula describes the structure of many objects - from soccer balls and gemstones to Buckminster Fuller's buildings and giant all-carbon molecules. Yet Euler's formula is so simple it can be explained to a child. This title tells the story of this indispensable mathematical idea.Trade ReviewWinner of the 2010 Euler Book Prize, Mathematical Association of America One of Choice's Outstanding Academic Titles for 2009 "The author has achieved a remarkable feat, introducing a naive reader to a rich history without compromising the insights and without leaving out a delicious detail. Furthermore, he describes the development of topology from a suggestion by Gottfried Leibniz to its algebraic formulation by Emmy Noether, relating all to Euler's formula. This book will be valuable to every library with patrons looking for an awe-inspiring experience."--Choice "This is an excellent book about a great man and a timeless formula."--Charles Ashbacher, Journal of Recreational Mathematics "I liked Richeson's style of writing. He is enthusiastic and humorous. It was a pleasure reading this book, and I recommend it to everyone who is not afraid of mathematical arguments and has ever wondered what this field of 'rubbersheet geometry' is about. You will not be disappointed."--Jeanine Daems, Mathematical Intelligencer "The book is a pleasure to read for professional mathematicians, students of mathematicians or anyone with a general interest in mathematics."--European Mathematical Society Newsletter "I found much more to like than to criticize in Euler's Gem. At its best, the book succeeds at showing the reader a lot of attractive mathematics with a well-chosen level of technical detail. I recommend it both to professional mathematicians and to their seatmates."--Jeremy L. Martin, Notices of the AMS "I highly recommend this book for teachers interested in geometry or topology, particularly for university faculty. The examples, proofs, and historical anecdotes are interesting, informative, and useful for encouraging classroom discussions. Advanced students will also glimpse the broad horizons of mathematics by reading (and working through) the book."--Dustin L. Jones, Mathematics Teacher "The book should interest non-mathematicians as well as mathematicians. It is written in a lively way, mathematical properties are explained well and several biographical details are included."--Krzysztof Ciesielski, Mathematical ReviewsTable of ContentsPreface ix Introduction 1 Chapter 1: Leonhard Euler and His Three "Great" Friends 10 Chapter 2: What Is a Polyhedron? 27 Chapter 3: The Five Perfect Bodies 31 Chapter 4: The Pythagorean Brotherhood and Plato's Atomic Theory 36 Chapter 5: Euclid and His Elements 44 Chapter 6: Kepler's Polyhedral Universe 51 Chapter 7: Euler's Gem 63 Chapter 8: Platonic Solids, Golf Balls, Fullerenes, and Geodesic Domes 75 Chapter 9: Scooped by Descartes? 81 Chapter 10: Legendre Gets It Right 87 Chapter 11: A Stroll through Konigsberg 100 Chapter 12: Cauchy's Flattened Polyhedra 112 Chapter 13: Planar Graphs, Geoboards, and Brussels Sprouts 119 Chapter 14: It's a Colorful World 130 Chapter 15: New Problems and New Proofs 145 Chapter 16: Rubber Sheets, Hollow Doughnuts, and Crazy Bottles 156 Chapter 17: Are They the Same, or Are They Different? 173 Chapter 18: A Knotty Problem 186 Chapter 19: Combing the Hair on a Coconut 202 Chapter 20: When Topology Controls Geometry 219 Chapter 21: The Topology of Curvy Surfaces 231 Chapter 22: Navigating in n Dimensions 241 Chapter 23: Henri Poincare and the Ascendance of Topology 253 Epilogue The Million-Dollar Question 265 Acknowledgements 271 Appendix A Build Your Own Polyhedra and Surfaces 273 Appendix B Recommended Readings 283 Notes 287 References 295 Illustration Credits 309 Index 311

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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 SynopsisElliptic Tales describes the latest developments in number theory by looking at one of the most exciting unsolved problems in contemporary mathematics--the Birch and Swinnerton-Dyer Conjecture. In this book, Avner Ash and Robert Gross guide readers through the mathematics they need to understand this captivating problem. The key to the conjectureTrade Review"The authors present their discussion in an informal, sometimes playful manner and with detail that will appeal to an audience with a basic understanding of calculus. This book will captivate math enthusiasts as well as readers curious about an intriguing and still unanswered question."--Margaret Dominy, Library Journal "Minimal prerequisites and its clear writing make this book (which even has a few exercises) a great choice for a seminar for mathematics majors, who at some point should have such an excursion to one of the frontiers of mathematics."--Mathematics Magazine "The authors of Elliptic Tales do a superb job in demonstrating the approach that mathematicians take when they confront unsolved problems involving elliptic curves."--Sungkon Chang, Times Higher Education "One cannot help being impressed, in reading the book and pursuing a few of the references, by the magnitude of the enterprise it chronicles."--James Case, SIAM News "Ash and Gross thoroughly explain the statement and significance of the linchpin Birch and Swinnerton-Dyer conjection... [A]sh and Gross deliver ample and current intellectual and technical substance."--Choice "I would envision this book as an excellent text for an undergraduate 'capstone' course in mathematics; the book lends itself to independent reading, but topics may be explored in much greater depth and rigor in the classroom. Additionally, the book indeed brings together ideas from calculus, complex variables and algebra, showing how a single mathematical research question may require an integrated understanding of the various branches of mathematics. Thus, it encourages students to reinforce their understanding of these various fields, while simultaneously introducing them to an open question in mathematics and a vibrant field of study."--Lisa A. Berger, Mathematical Reviews Clippings "The book is very pleasantly written, and in my opinion, the authors have done an admirable job in giving an idea to non-experts what the Birch-Swinnerton Dyer conjecture is about."--Jan-Hendrik Evertse, Zentralblatt MATH "The book's most important contributions ... are the sense of discovery, invention, and insight into the habits of mind used by mathematicians on this journey. I would recommend this book to anyone who wants to be challenged mathematically or who wants to experience mathematics as creative and exciting."--Jacqueline Coomes, Mathematics Teacher "[T]his book is a wonderful introduction to what is arguably one of the most important mathematical problems of our time and for that reason alone it deserves to be widely read. Another reason to recommend this book is the opportunity to share in the readily apparent joy the authors have for their subject and the beauty they see in it, not least because ... joy and beauty are the most important reasons for doing mathematics, irrespective of its dollar value."--Rob Ashmore, Mathematics Today "This book has many nice aspects. Ash and Gross give a truly stimulating introduction to elliptic curves and the BSD conjecture for undergraduate students. The main achievement is to make a relative easy exposition of these so technical topics."--Jonathan Sanchez-Hernandez, Mathematical SocietyTable of ContentsPreface xiii Acknowledgments xix Prologue 1 PART I. DEGREE Chapter 1. Degree of a Curve 13 1.Greek Mathematics 13 2.Degree 14 3.Parametric Equations 20 4.Our Two Definitions of Degree Clash 23 Chapter 2. Algebraic Closures 26 1.Square Roots of Minus One 26 2.Complex Arithmetic 28 3.Rings and Fields 30 4.Complex Numbers and Solving Equations 32 5.Congruences 34 6.Arithmetic Modulo a Prime 38 7.Algebraic Closure 38 Chapter 3. The Projective Plane 42 1.Points at Infinity 42 2.Projective Coordinates on a Line 46 3.Projective Coordinates on a Plane 50 4.Algebraic Curves and Points at Infinity 54 5.Homogenization of Projective Curves 56 6.Coordinate Patches 61 Chapter 4. Multiplicities and Degree 67 1.Curves as Varieties 67 2.Multiplicities 69 3.Intersection Multiplicities 72 4.Calculus for Dummies 76 Chapter 5. B'ezout's Theorem 82 1.A Sketch of the Proof 82 2.An Illuminating Example 88 PART II. ELLIPTIC CURVES AND ALGEBRA Chapter 6. Transition to Elliptic Curves 95 Chapter 7. Abelian Groups 100 1.How Big Is Infinity? 100 2.What Is an Abelian Group? 101 3.Generations 103 4.Torsion 106 5.Pulling Rank 108 Appendix: An Interesting Example of Rank and Torsion 110 Chapter 8. Nonsingular Cubic Equations 116 1.The Group Law 116 2.Transformations 119 3.The Discriminant 121 4.Algebraic Details of the Group Law 122 5.Numerical Examples 125 6.Topology 127 7.Other Important Facts about Elliptic Curves 131 5.Two Numerical Examples 133 Chapter 9. Singular Cubics 135 1.The Singular Point and the Group Law 135 2.The Coordinates of the Singular Point 136 3.Additive Reduction 137 4.Split Multiplicative Reduction 139 5.Nonsplit Multiplicative Reduction 141 6.Counting Points 145 7.Conclusion 146 Appendix A: Changing the Coordinates of the Singular Point 146 Appendix B: Additive Reduction in Detail 147 Appendix C: Split Multiplicative Reduction in Detail 149 Appendix D: Nonsplit Multiplicative Reduction in Detail 150 Chapter 10. Elliptic Curves over Q 152 1.The Basic Structure of the Group 152 2.Torsion Points 153 3.Points of Infinite Order 155 4.Examples 156 PART III. ELLIPTIC CURVES AND ANALYSIS Chapter 11. Building Functions 161 1.Generating Functions 161 2.Dirichlet Series 167 3.The Riemann Zeta-Function 169 4.Functional Equations 171 5.Euler Products 174 6.Build Your Own Zeta-Function 176 Chapter 12. Analytic Continuation 181 1.A Difference that Makes a Difference 181 2.Taylor Made 185 3.Analytic Functions 187 4.Analytic Continuation 192 5.Zeroes, Poles, and the Leading Coefficient 196 Chapter 13. L-functions 199 1.A Fertile Idea 199 2.The Hasse-Weil Zeta-Function 200 3.The L-Function of a Curve 205 4.The L-Function of an Elliptic Curve 207 5.Other L-Functions 212 Chapter 14. Surprising Properties of L-functions 215 1.Compare and Contrast 215 2.Analytic Continuation 220 3.Functional Equation 221 Chapter 15. The Conjecture of Birch and Swinnerton-Dyer 225 1.How Big Is Big? 225 2.Influences of the Rank on the Np's 228 3.How Small Is Zero? 232 4.The BSD Conjecture 236 5.Computational Evidence for BSD 238 6.The Congruent Number Problem 240 Epilogue 245 Retrospect 245 Where DoWe Go from Here? 247 Bibliography 249 Index 251

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Book SynopsisTrade ReviewOne of Choice's Outstanding Academic Titles for 2013 Shortlisted for the 2013 BSHM Neumann Book Prize, British Society for the History of Mathematics "Once a mainstay of mathematics, spherical trigonometry no longer appears on school curricula. Here, Glen Van Brummelen reasserts the field's importance, sharing in illuminating detail how it figured in astronomy, cartography and our understanding of Earth's rotation."--Rosalind Metcalfe, Nature "The present book is very well written; it leaves a clear impression that the author intended to endear--not merely present and teach--spherical trigonometry to the reader. Although not a history book, there are separate chapters shedding light on the approaches to the subject in the ancient, medieval, and modern times. There are also chapters on spherical geometry, polyhedra, stereographic projection and the art of navigation. The book is thoroughly illustrated and is a pleasant read. Chapters end with exercises; the appendices contain a long list of available and not so available textbooks and recommendations for further reading organized by individual chapters. The book made a valuable addition to my library. I freely recommend it to math teachers and curious high schoolers."--Alexander Bogomolny, CTK Insights "A no-nonsense introduction to spherical trigonometry."--Book News, Inc. "A beautiful popular book."--ThatsMaths.com "Full of academic, textbook content, the book is a delight to math students. So if you are game for a journey into the world of spherical trigonometry, pick up the book. Van Brummelen gives exercises at the end of the chapters that can be fun."--R. Balashankar, Organiser "Heavenly Mathematicsis a truly enjoyable description of the somewhat forgotten science of spherical trigonometry... As readers discover this discipline, they will also appreciate the beauty inherent in the topic."--Choice "Heavenly Mathematics proves the value of bringing a fascinating piece of mathematical history within the grasp of the general reader."--Florin Diacu, Literary Review of Canada "Van Brummelen has written a wonderful introduction ... that draws on the history of [spherical trigonometry] to illuminate the mathematics itself and at the same time gives readers a real sense of what research in the history of early mathematics is all about."--Metascience "[Heavenly Mathematics] is an excellent survey of spherical trigonometry... Simply an appreciation of a beautiful lost subject, with historical overtones... [D]istinguishable for its appealingly fresh style."--Mathematical Reviews "[Heavenly Mathematics] is a lovely book to read... [A] wonderful introduction for anyone who wishes to learn more about this subject... I am in full agreement with the author that spherical trigonometry ought to be brought to a wider audience, and I believe that this is the book to do it."--Mathematics Today "Engaging, clear and not overly technical; you can safely lend this book to your friends in the history department... [Heavenly Mathematics] is excellent."--Zentralblatt MATH "Heavenly Mathematics will be of interest to mathematically inclined historians of science and also to students of mathematics and engineering. Because spherical trigonometry is relevant in applications of modern science, this elegant book may even contribute to a renaissance of the subject."--Jan P. Hogendijk, Isis "This book could serve as an excellent textbook for any secondary school mathematics classroom at or above the level of geometry and certainly trigonometry; as the basis for a high school honors class; or as a textbook and seminar topic for college students."--Teresa Floyd, Mathematics Teacher "Any reader of this book (and there should be many) will see how present day mathematics may be viewed through the kaleidoscope of its historical origins... Glen Van Brummelen has written a beautifully produced book that includes fascinating biographical detail at every stage of his narrative."--P.N. Ruane, Mathematical Gazette "An engaging read that will appeal to historians of science, mathematicians, trigonometry teachers, and anyone interested in the history of mathematics."--Elizabeth Hamm, Aestimatio Critical Reviews in the History of ScienceTable of ContentsPreface vii 1 Heavenly Mathematics 1 2 Exploring the Sphere 23 3 The Ancient Approach 42 4 The Medieval Approach 59 5 The Modern Approach: Right- Angled Triangles 73 6 The Modern Approach: Oblique Triangles 94 7 Areas, Angles, and Polyhedra 110 8 Stereographic Projection 129 9 Navigating by the Stars 151 Appendix A. Ptolemy's Determination of the Sun's Position 173 Appendix B. Textbooks 179 Appendix C. Further Reading 182 Index 189

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Book SynopsisThis is a cultural history of mathematics and art, from antiquity to the present. Mathematicians and artists have long been on a quest to understand the physical world they see before them and the abstract objects they know by thought alone. Taking readers on a tour of the practice of mathematics and the philosophical ideas that drive the disciplinTrade Review"This is a marvelous coffee table book ... very well researched and documented. It touches upon so many fundamental questions that philosophers, scientists, mathematicians and artists have asked since antiquity. But yet it guides the reader smoothly through all these competing visions and theories without becoming dull or getting lost in abstraction. This is the history of Western civilization with particular interest in art and mathematics, illuminating and instructive, and all wrapped up in a rich, colorful, and fancy book."--Adhemar Bultheel, European Mathematical Society "This is the beauty and power of this book: [Mathematics and Art] is an intellectual tour de force of art history and its interaction with mathematics that will draw most readers, including me, back for further reading and study."--Frank Swetz, MAA Reviews "Excellent new book... Overall this is a comprehensive, valuable and detailed book. It is written in an accessible style, with enough mathematics to interest the technical reader without overwhelming one with an arts background... Its rich anthology is particularly relevant today, given the explosion of interest in the digital arts and the need for digital artists to use maths creatively. I will definitely be keeping it close at hand."--William Latham, New Scientist "The author does an artful job in creating a wide-ranging and beautifully illustrated survey that mathematicians and art historians will enjoy."--John Barrow, The Art Newspaper "This sumptuously illustrated book chronicles the history of mathematics through its intersection with the development of visual art... Gamwell articulates the compelling, far-reaching connections within these fields in a way that is rewarding for scholars yet accessible to non-specialists."--Choice "Beautiful books that display the beauty of art are fine additions to many coffee tables; beautiful books that display the beauty of mathematics are fine additions to few coffee tables. Gamwell's impressive work integrates the beauty of these two disciplines to create a work larger than their sum... A book for all ages and of all ages: truly a brilliant 'millennial' composition!"--Sandra L. Arlinghaus, Mathematical Reviews "This splendidly produced volume will appeal to everybody interested in mathematics and art and offers room for agreement and disagreement with the author... This volume stands out by its richness in contents, its wealth of colour reproductions and, last but not least, its very affordable price."--Dirk Werner, Zentralblatt MATH "This wonderful book gives a very thorough overview of the impact of mathematics (and science) of the visual arts (and architecture) over the centuries."--Eos "An interesting read, filled with paradigm-shifting history and art, the book still posits a linear perspective of the relationship of art and mathematics, specifically recounting the ways math has influenced art."--Karie Brown, Mathematics Teacher "A monumental volume... Excellently illustrated by 523 images... Many highlighted quotations from writings of outstanding personalities of the sciences and the arts make the volume more colourful."--Gyorgy Darvas, Symmetry "Mathematics and Art is an enjoyable read accessible to anyone interested in the history of mathematics and art."--Andre Michael Hahn, British Journal for the History of ScienceTable of ContentsFOREWORD by Neil deGrasse Tyson IX PREFACE XI 1 Arithmetic and Geometry 1 2 Proportion 73 3 Infinity 109 4 Formalism 151 5 Logic 197 6 Intuitionism 225 7 Symmetry 249 8 Utopian Visions after World War I 277 9 The Incompleteness of Mathematics 321 10 Computation 355 1 1 Geometric Abstraction after World War II 385 12 Computers in Mathematics and Art 455 13 Platonism in the Postmodern Era 499 NOTES 512 ACKNOWLEDGMENTS 547 CREDITS 548 INDEX 549

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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 SynopsisFrom his return to Cambridge in 1929 to his death in 1951, Wittgenstein influenced philosophy almost exclusively through teaching and discussion. These lecture notes indicate what he considered to be salient features of his thinking in this period of his life.

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Book SynopsisA text on mathematical methods in the life sciences, aimed at advanced undergraduate & graduate students, providing a foundation for understanding the methods used in today''s quantitative biology--

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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 SynopsisThird printing. First paperback printing. Original copyright date: 2013.Trade Review"Meshing logic problems with the stories of two extraordinary men ... Paul Nahin fashions a tale of innovation and discovery. Alongside a gripping account of how Shannon built on Boole's work, Nahin explores others key to the technological revolution, from Georg Cantor to Alan Turing."--Nature "Engaging... Nahin assumes some rudimentary knowledge but expertly explains concepts such as relay circuits, Turing machines, and quantum computing. Reasoning through the problems and diagrams will give persistent readers genuine aha moments and an understanding of the two revolutionaries who helped to lay the foundation of our digital world."--Scientific American "Part biography, part history, and part a review of basic information theory, this book does an excellent job of fitting these interlocking elements together."--Library Journal "The reader is taken on a journey from the development of some abstract mathematical ideas through a nearly ubiquitous application of those ideas within the modern world with so many embedded digital computers... I enjoyed the discussion of Claude Shannon. In the history of the computer and development of the internet and World Wide Web, his ideas and contributions are too often overlooked. He is one of my heroes and I believe that everyone that reads this book will come to the same conclusion."--Charles Ashbacher, MAA Reviews "Paul J. Nahin really knows how to tell a good story... The Logician and the Engineer is truly a gem."--New York Journal of Books "A short but fairly detailed exploration of the genesis of Boolean logic and Shannon's information theory... [G]ood background reading for anyone studying electronics or computer science."--Christine Evans-Pughe, Engineering & Technology "Although the book is technical, it is always easily understandable for anyone (for those who need it, some basic rules for electrical circuits are collected in a short appendix). It is not only understandable but also pleasantly bantering and at occasions even facetious."--A. Bultheel, European Mathematical Society "Most valuable to this reviewer, and likely to many potential readers, is the closing chapter, aptly titled Beyond Boole and Shannon. Here is provided an introduction to quantum computing and its logic, possibly portending the future of computers, yet unmistakably bearing the footprints of the two early pioneers. It is an unexpected yet fitting conclusion to this thoroughly enjoyable read."--Ronald E. Prather, Mathematical Reviews Clippings "Nahin has had the very good idea of connecting the very different worlds and times of Boole, Shannon, and others to demonstrate that a little Victorian algebra can turn out to be very useful."--SIAM Review "The exposition is clear and does not assume any prior knowledge except elementary mathematics and a few basic facts from physics. I recommend this well-written book to all readers interested in the history of computer science, as well as those who are curious about the fundamental principles of digital computing."--Antonin Slavik, Zentralblatt MATH "[T]his is a useful and often interesting introduction to the life and work of two intellectual giants who are largely unknown to the general public."--Gareth and Mary Jones, London Mathematical Society Newsletter "The problems are varied and indeed intriguing, and the solutions are delightful."--Mathematics Magazine "This book is not light reading. It would be excellent for advanced high school juniors or seniors with a strong interest in computer science as well as mathematics."--Tom Ottinger, Mathematics Teacher "Nahin leavens the math and engineering with humor and an infectious intellectual curiosity, and the parallels between Boole and Shannon are convincingly drawn... [The Logician and the Engineer] will give your brain a workout, but an enjoyable one."--San Francisco Book ReviewTable of ContentsPreface xi 1 What You Need to Know to Read This Book 1 Notes and References 5 2 Introduction 6 Notes and References 14 3 George Boole and Claude Shannon: Two Mini-Biographies 17 *3.1 The Mathematician 17 *3.2 The Electrical Engineer 28 * Notes and References 39 4 Boolean Algebra 43 *4.1 Boole's Early Interest in Symbolic Analysis 43 *4.2 Visualizing Sets 44 *4.3 Boole's Algebra of Sets 45 *4.4 Propositional Calculus 48 *4.5 Some Examples of Boolean Analysis 52 *4.6 Visualizing Boolean Functions 59 * Notes and References 65 5 Logical Switching Circuits 67 *5.1 Digital Technology: Relays versus Electronics 67 *5.2 Switches and the Logical Connectives 68 *5.3 A Classic Switching Design Problem 71 *5.4 The Electromagnetic Relay and the Logical NOT 73 *5.5 The Ideal Diode and the Relay Logical AND and OR 76 *5.6 The Bi-Stable Relay Latch 81 * Notes and References 84 6 Boole, Shannon, and Probability 88 *6.1 A Common Mathematical Interest 88 *6.2 Some Fundamental Probability Concepts 89 *6.3 Boole and Conditional Probability 96 *6.4 Shannon, Conditional Probability, and Relay Reliability 99 *6.5 Majority Logic 106 * Notes and References 110 7 Some Combinatorial Logic Examples 114 *7.1 Channel Capacity, Shannon's Theorem, and Error-Detection Theory 114 *7.2 The Exclusive-OR Gate (XOR) 122 *7.3 Error-Detection Logic 127 *7.4 Error-Correction Theory 128 *7.5 Error-Correction Logic 132 * Notes and References 137 8 Sequential-State Digital Circuits 139 *8.1 Two Sequential-State Problems 139 *8.2 The NOR Latch 142 *8.3 The Clocked RS Flip-Flop 146 *8.4 More Flip-Flops 154 *8.5 A Synchronous, Sequential-State Digital Machine Design Example 158 * Notes and References 160 9 Turing Machines 161 *9.1 The First Modern Computer 162 *9.2 Two Turing Machines 164 *9.3 Numbers We Can't Compute 168 * Notes and References 173 10 Beyond Boole and Shannon 176 *10.1 Computation and Fundamental Physics 176 *10.2 Energy and Information 178 *10.3 Logically Reversible Gates 180 *10.4 Thermodynamics of Logic 184 *10.5 A Peek into the Twilight Zone: Quantum Computers 188 *10.6 Quantum Logic--and Time Travel, Too! 197 Notes and References 205 Epilogue For the Future: The Anti-Amphibological Machine 210 Appendix Fundamental Electric Circuit Concepts 219 Acknowledgments 223 Index 225

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Book SynopsisSpherical trigonometry was at the heart of astronomy and ocean-going navigation for two millennia. This title traces the rich history of this forgotten art, revealing how the cultures of classical Greece, medieval Islam, and the modern West used spherical trigonometry to chart the heavens and the Earth.Trade ReviewOne of Choice's Outstanding Academic Titles for 2013 Shortlisted for the 2013 BSHM Neumann Book Prize, British Society for the History of Mathematics "Once a mainstay of mathematics, spherical trigonometry no longer appears on school curricula. Here, Glen Van Brummelen reasserts the field's importance, sharing in illuminating detail how it figured in astronomy, cartography and our understanding of Earth's rotation."--Rosalind Metcalfe, Nature "The present book is very well written; it leaves a clear impression that the author intended to endear--not merely present and teach--spherical trigonometry to the reader. Although not a history book, there are separate chapters shedding light on the approaches to the subject in the ancient, medieval, and modern times. There are also chapters on spherical geometry, polyhedra, stereographic projection and the art of navigation. The book is thoroughly illustrated and is a pleasant read. Chapters end with exercises; the appendices contain a long list of available and not so available textbooks and recommendations for further reading organized by individual chapters. The book made a valuable addition to my library. I freely recommend it to math teachers and curious high schoolers."--Alexander Bogomolny, CTK Insights "A no-nonsense introduction to spherical trigonometry."--Book News, Inc. "A beautiful popular book."--ThatsMaths.com "Full of academic, textbook content, the book is a delight to math students. So if you are game for a journey into the world of spherical trigonometry, pick up the book. Van Brummelen gives exercises at the end of the chapters that can be fun."--R. Balashankar, Organiser "Heavenly Mathematicsis a truly enjoyable description of the somewhat forgotten science of spherical trigonometry... As readers discover this discipline, they will also appreciate the beauty inherent in the topic."--Choice "Heavenly Mathematics proves the value of bringing a fascinating piece of mathematical history within the grasp of the general reader."--Florin Diacu, Literary Review of Canada "Van Brummelen has written a wonderful introduction ... that draws on the history of [spherical trigonometry] to illuminate the mathematics itself and at the same time gives readers a real sense of what research in the history of early mathematics is all about."--Metascience "[Heavenly Mathematics] is an excellent survey of spherical trigonometry... Simply an appreciation of a beautiful lost subject, with historical overtones... [D]istinguishable for its appealingly fresh style."--Mathematical Reviews "[Heavenly Mathematics] is a lovely book to read... [A] wonderful introduction for anyone who wishes to learn more about this subject... I am in full agreement with the author that spherical trigonometry ought to be brought to a wider audience, and I believe that this is the book to do it."--Mathematics Today "Engaging, clear and not overly technical; you can safely lend this book to your friends in the history department... [Heavenly Mathematics] is excellent."--Zentralblatt MATH "Heavenly Mathematics will be of interest to mathematically inclined historians of science and also to students of mathematics and engineering. Because spherical trigonometry is relevant in applications of modern science, this elegant book may even contribute to a renaissance of the subject."--Jan P. Hogendijk, Isis "This book could serve as an excellent textbook for any secondary school mathematics classroom at or above the level of geometry and certainly trigonometry; as the basis for a high school honors class; or as a textbook and seminar topic for college students."--Teresa Floyd, Mathematics Teacher "Any reader of this book (and there should be many) will see how present day mathematics may be viewed through the kaleidoscope of its historical origins... Glen Van Brummelen has written a beautifully produced book that includes fascinating biographical detail at every stage of his narrative."--P.N. Ruane, Mathematical Gazette "An engaging read that will appeal to historians of science, mathematicians, trigonometry teachers, and anyone interested in the history of mathematics."--Elizabeth Hamm, Aestimatio Critical Reviews in the History of ScienceTable of ContentsPreface vii 1 Heavenly Mathematics 1 2 Exploring the Sphere 23 3 The Ancient Approach 42 4 The Medieval Approach 59 5 The Modern Approach: Right- Angled Triangles 73 6 The Modern Approach: Oblique Triangles 94 7 Areas, Angles, and Polyhedra 110 8 Stereographic Projection 129 9 Navigating by the Stars 151 Appendix A. Ptolemy's Determination of the Sun's Position 173 Appendix B. Textbooks 179 Appendix C. Further Reading 182 Index 189

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Book SynopsisIn 1983 Gerry Kennedy set off on a tour through Russia, China, Japan and the USA to visit others involved in the global anti-war movement. Only dimly aware of his Victorian ancestors: George Boole, forefather of the digital revolution and James Hinton, eccentric philosopher and advocate of polygamy, he had directly followed in the footsteps of two dynasties of radical thinkers and doers.Their notable achievements, in which the women were particularly prominent, involved many spheres. Boole's wife, Mary Everest, niece of George Everest, surveyor of the eponymous mountain, was an early advocate of hands-on education. Of the five talented Boole daughters, Ethel Voynich, wife of the discoverer of the enigmatic, still unexplained Voynich Manuscript, campaigned with Russian anarchists to overthrow the Tsar. Her 1897 novel The Gadfly, filmed later with music by Shostakovich, sold in millions behind the Iron Curtain. She was rumoured to have had an affair with the notorious 'Ace of Spies', Sidney Reilly. One of Ethel's sisters married Charles Howard Hinton: a leading exponent of the esoteric realm of the fourth dimension and inventor of the gunpowder baseball-pitcher.Of their descendants, Carmelita Hinton also pioneered progressive education in the USA at her school in Putney, Vermont. Her children dedicated their lives to Mao's China. Appalled by the dropping on Japan of the atomic bomb that she had helped design, Joan Hinton defected to China and actively engaged in the Cultural Revolution. William Hinton wrote the influential documentary Fanshen based on his experience in 1948 of revolutionary change in a Shanxi village. Other members of the clan became renowned in their fields of physics, entomology and botany. Their combined legacy of independent and constructive thinking is perhaps typified by the invention of the Jungle Gym: the climbing-frame now used by children the world over. In The Booles and the Hintons the author embarks on a quest to reveal the stories behind their remarkable lives.Table of ContentsContentsAcknowledgements viiiForeword ixWho's Who xii 1 GRAVY OVER A TABLECLOTH 1The author's own background in West London. 'Discovery' of the Booles, brief outline of the Boole/Hinton major characters.2 THE UGLY DUCKLING 12The Voynich Manuscript in brief: its history and research to date.3 BRINGING STARRY WISDOM DOWN 29The author's visit to Lincoln. George Boole's life and background.4 THE MISSUS 59Mary Everest Boole: her life with George Boole and her own views on life and education.5 TADPOLES INTO FROGS 83The five Boole daughters' early lives.6 THE WIZARD 89James Hinton: his life and philosophy.7 TRAPDOORS AND VELVETEEN 104Tsarist Russia in the late nineteenth century. Russian anarchists: Kropotkin and Stepniak. Charlotte Wilson and Wyldes Farm.Radical politics in England in the 1880s.8 CITIZENS' DIPLOMACY 119The author's peace politics, Moscow dissidents and journey on the Trans-Siberian railway, 1983. Ethel Boole's stay in St Petersburg, 1887-89.9 ROUND AND ROUND THE GARDEN 141James Hinton's comeuppance. Olive Schreiner and Havelock Ellis. Caroline Haddon. The Men and Women's Club.10 THE RIFF-RAFF OF RASCALDOM 153The Society of Friends of Russian Freedom. Russian anarchists and related novels of the period. London arrival of Wilfrid Voynich.11 COMMUNING WITH SPACE 170Charles Howard Hinton and the fourth dimension. His and Mary Ellen Boole's exile to Japan and the USA in 1887. Hinton'swritings and influence in Europe. Peter Ouspensky in Russia.12 THE CITADEL 197The author's visit to Warsaw. Proletariat: early Polish socialists. History of Wilfrid's escape from Siberia.13 DEAD AS MUTTON? 213Ethel Voynich's novel The Gadfly. Wilfrid Voynich establishes his book trade. Jack Raymond, another novel by Ethel.14 THE ENGLISH AUNTS 227The five Boole sisters: Mary Ellen, Margaret, Alice, Lucy and Ethel's later lives. Ethel and Ivor Gurney.15 BRINGING HOME THE BACON 247The Voynich Manuscript again. Ethel and Wilfrid in the USA.16 THE BIGGLES OF COMMUNISM 259Ethel in New York; her last novel. Ethel 're-discovered'. Scandal with the spy, Sidney Reilly.17 OLIVE AND OVOD 277The author's return to Russia via Lithuania searching for The Gadfly in Moscow and St Petersburg.18 A CQR LIFE 302The life of Cambridge scientist, Sir Geoffrey Taylor, grandson of George Boole.19 THE HINTON GENUS 314Hinton descendants in Mexico in the mining industry and their plant collecting. The life of Howard Everest Hinton, entomologist.20 UNCOLLAPSIBLE HINTONS 337The Byrdcliffe arts community, Woodstock USA. Carmelita Hinton and the setting up of Putney School, Vermont USA.21 THE GADGET 356The Manhattan Project and Joan Hinton. Hiroshima.22 PEKING JOAN 374Joan Hinton and Sid Engst in Mongolia, Si'an and Beijing during the Cultural Revolution.23 FANSHEN 398The life of Bill Hinton and the rural revolution in China. Author's visit to Long Bow village.24 THE VERMONTER 417The author returns to the USA to visit Putney School.Afterword 422Notes 427Illustration Credits 453Index 456

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Book SynopsisTrade Review"A historical and philosophical tour of major insights in the development of probability theory."---James Ryerson, New York Times Book Review"A volume that should be on every scientist’s reading list."---Barbara Kiser, Nature"Mathematically rigorous, yet also reasonably accessible; informative, yet fun and entertaining to read. Both students and faculty should find reading this to be a rewarding experience." * MAA Reviews *"The audience is quite specific, but for them it will be a gem. . . . I would recommend this to any student studying or having studied anything statistics related at university."---Jonathan Shock, Mathemafrica"A very enriching journey. Your vision will be broadened assimilating all these issues and solutions as well as open problems from the early history of probability, game theory, financial markets, politics, thermodynamics, quantum theory and much much more."---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 *"This book will not increase your odds of winning at games of chance, but it will give you some greater understanding of why you lose." * Cosmos *"Ten Great Ideas about Chance isn’t just about 18th century philosophical arguments, World War II events or tests of expensive, hard-to-pronounce drugs. The book’s ideas are as down to earth and as current as your busted bracket for NCAA Men’s Basketball." * Herald Business Journal *"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, written in a conversational style with many examples and asides and an emphasis throughout on the people who have built the theory."---David M. Bressoud, UMAP Journal"A terrific book. The authors explain 10 great ideas in probability, starting from their history and pursuing their philosophical implications."---Eric S. Rosenthal, Mathematics Magazine

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Book SynopsisAmong the many constants that appear in mathematics, ?, e, and i are the most familiar. Following closely behind is ?,, or gamma, a constant that arises in many mathematical areas yet maintains a profound sense of mystery. In a tantalizing blend of history and mathematics, Julian Havil takes the reader on a journey through logarithms and the harmonTrade Review"[A] wonderful book... Havil's emphasis on historical context and his conversational style make this a pleasure to read... Gamma is a gold mine of irresistible mathematical nuggets. Anyone with a serious interest in maths will find it richly rewarding."--Ben Longstaff, New Scientist "This book is a joy from start to finish."--Gerry Leversha, Mathematical Gazette "Wonderful... Havil's emphasis on historical context and his conversational style make this a pleasure to read...Gammais a gold mine of irresistible mathematical nuggets. Anyone with a serious interest in math will find it richly rewarding."--New Scientist"A joy from start to finish."--Mathematical Gazette"[Gamma] is not a book about mathematics, but a book of mathematics... [It] is something like a picaresque novel; the hero, Euler's constantg, serves as the unifying motif through a wide range of mathematical adventures."--Notices of the American Mathematical Society "[Gamma] is enjoyable for many reasons. Here are just two. First, the explanations are not only complete, but they have the right amount of generality... Second, the pleasure Havil has in contemplating this material is infectious."--MAA Online "It is only fitting that someone should write a book about gamma, or Euler's constant. Havil takes on this task and does an excellent job."--Choice "Mathematics is presented throughout as something connected to reality... Many readers will find in [Gamma] exactly what they have been missing."--Mohammad Akbar, Plus Magazine, Millennium Mathematics Project, University of Cambridge "This book is written in an informal, engaging, and often amusing style. The author takes pains to make the mathematics clear. He writes about the mathematical geniuses of the past with reverence and awe. It is especially nice that the mathematical topics are discussed within a historical context."--Ward R. Stewart, Mathematics TeacherTable of ContentsForeword xv Acknowledgements xvii Introduction xix Chapter One The Logarithmic Cradle 1 1.1 A Mathematical Nightmare- and an Awakening 1 1.2 The Baron's Wonderful Canon 4 1.3 A Touch of Kepler 11 1.4 A Touch of Euler 13 1.5 Napier's Other Ideas 16 Chapter Two The Harmonic Series 21 2.1 The Principle 21 2.2 Generating Function for Hn 21 2.3 Three Surprising Results 22 Chapter Three Sub-Harmonic Series 27 3.1 A Gentle Start 27 3.2 Harmonic Series of Primes 28 3.3 The Kempner Series 31 3.4 Madelung's Constants 33 Chapter Four Zeta Functions 37 4.1 Where n Is a Positive Integer 37 4.2 Where x Is a Real Number 42 4.3 Two Results to End With 44 Chapter Five Gamma's Birthplace 47 5.1 Advent 47 5.2 Birth 49 Chapter Six The Gamma Function 53 6.1 Exotic Definitions 53 6.2 Yet Reasonable Definitions 56 6.3 Gamma Meets Gamma 57 6.4 Complement and Beauty 58 Chapter Seven Euler's Wonderful Identity 61 7.1 The All-Important Formula 61 7.2 And a Hint of Its Usefulness 62 Chapter Eight A Promise Fulfilled 65 Chapter Nine What Is Gamma Exactly? 69 9.1 Gamma Exists 69 9.2 Gamma Is What Number? 73 9.3 A Surprisingly Good Improvement 75 9.4 The Germ of a Great Idea 78 Chapter Ten Gamma as a Decimal 81 10.1 Bernoulli Numbers 81 10.2 Euler -Maclaurin Summation 85 10.3 Two Examples 86 10.4 The Implications for Gamma 88 Chapter Eleven Gamma as a Fraction 91 11.1 A Mystery 91 11.2 A Challenge 91 11.3 An Answer 93 11.4 Three Results 95 11.5 Irrationals 95 11.6 Pell's Equation Solved 97 11.7 Filling the Gaps 98 11.8 The Harmonic Alternative 98 Chapter Twelve Where Is Gamma? 101 12.1 The Alternating Harmonic Series Revisited 101 12.2 In Analysis 105 12.3 In Number Theory 112 12.4 In Conjecture 116 12.5 In Generalization 116 Chapter Thirteen It's a Harmonic World 119 13.1 Ways of Means 119 13.2 Geometric Harmony 121 13.3 Musical Harmony 123 13.4 Setting Records 125 13.5 Testing to Destruction 126 13.6 Crossing the Desert 127 13.7 Shuffiing Cards 127 13.8 Quicksort 128 13.9 Collecting a Complete Set 130 13.10 A Putnam Prize Question 131 13.11 Maximum Possible Overhang 132 13.12 Worm on a Band 133 13.13 Optimal Choice 134 Chapter Fourteen It's a Logarithmic World 139 14.1 A Measure of Uncertainty 139 14.2 Benford's Law 145 14.3 Continued-Fraction Behaviour 155 Chapter Fifteen Problems with Primes 163 15.1 Some Hard Questions about Primes 163 15.2 A Modest Start 164 15.3 A Sort of Answer 167 15.4 Picture the Problem 169 15.5 The Sieve of Eratosthenes 171 15.6 Heuristics 172 15.7 A Letter 174 15.8 The Harmonic Approximation 179 15.9 Different-and Yet the Same 180 15.10 There are Really Two Questions, Not Three 182 15.11 Enter Chebychev with Some Good Ideas 183 15.12 Enter Riemann, Followed by Proof(s)186 Chapter Sixteen The Riemann Initiative 189 16.1 Counting Primes the Riemann Way 189 16.2 A New Mathematical Tool 191 16.3 Analytic Continuation 191 16.4 Riemann's Extension of the Zeta Function 193 16.5 Zeta's Functional Equation 193 16.6 The Zeros of Zeta 193 16.7 The Evaluation of (x) and p(x)196 16.8 Misleading Evidence 197 16.9 The Von Mangoldt Explicit Formula-and How It Is Used to Prove the Prime Number Theorem 200 16.10 The Riemann Hypothesis 202 16.11 Why Is the Riemann Hypothesis Important? 204 16.12 Real Alternatives 206 16.13 A Back Route to Immortality-Partly Closed 207 16.14 Incentives, Old and New 210 16.15 Progress 213 Appendix A The Greek Alphabet 217 Appendix B Big Oh Notation 219 Appendix C Taylor Expansions 221 C.1 Degree 1 221 C.2 Degree 2 221 C.3 Examples 223 C.4 Convergence 223 Appendix D Complex Function Theory 225 D.1 Complex Differentiation 225 D.2 Weierstrass Function 230 D.3 Complex Logarithms 231 D.4 Complex Integration 232 D.5 A Useful Inequality 235 D.6 The Indefinite Integral 235 D.7 The Seminal Result 237 D.8 An Astonishing Consequence 238 D.9 Taylor Expansions-and an Important Consequence 239 D.10 Laurent Expansions-and Another Important Consequence 242 D.11 The Calculus of Residues 245 D.12 Analytic Continuation 247 Appendix E Application to the Zeta Function 249 E.1 Zeta Analytically Continued 249 E.2 Zeta's Functional Relationship 253 References 255 Name Index 259 Subject Index 263

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Book SynopsisIn the title, "[the square root of minus one]" appears as a radical over "-1."Trade ReviewOne of Choice's Outstanding Academic Titles for 1999 Honorable Mention for the 1998 Award for Best Professional/Scholarly Book in Mathematics, Association of American Publishers "A book-length hymn of praise to the square root of minus one."--Brian Rotman, Times Literary Supplement "An Imaginary Tale is marvelous reading and hard to put down. Readers will find that Nahin has cleared up many of the mysteries surrounding the use of complex numbers."--Victor J. Katz, Science "[An Imaginary Tale] can be read for fun and profit by anyone who has taken courses in introductory calculus, plane geometry and trigonometry."--William Thompson, American Scientist "Someone has finally delivered a definitive history of this 'imaginary' number... A must read for anyone interested in mathematics and its history."--D. S. Larson, Choice "Attempting to explain imaginary numbers to a non-mathematician can be a frustrating experience... On such occasions, it would be most useful to have a copy of Paul Nahin's excellent book at hand."--A. Rice, Mathematical Gazette "Imaginary numbers! Threeve! Ninety-fifteen! No, not those kind of imaginary numbers. If you have any interest in where the concept of imaginary numbers comes from, you will be drawn into the wonderful stories of how i was discovered."--Rebecca Russ, Math Horizons "There will be something of reward in this book for everyone."--R.G. Keesing, Contemporary Physics "Nahin has given us a fine addition to the family of books about particular numbers. It is interesting to speculate what the next member of the family will be about. Zero? The Euler constant? The square root of two? While we are waiting, we can enjoy An Imaginary Tale."--Ed Sandifer, MAA Online "Paul Nahin's book is a delightful romp through the development of imaginary numbers."--Robin J. Wilson, London Mathematical Society Newsletter "You will definitely enjoy it. In fact it clearly reflects the the joy and delight that the author experienced when he was confronted with complex analysis during his engineering studies."--Adhemar Bultheel, European Mathematical SocietyTable of Contents*FrontMatter, pg. i*A Note to the Reader, pg. vii*Contents, pg. ix*Illustrations, pg. xi*Preface to the Paperback Edition, pg. xiii*Preface, pg. xxi*Introduction, pg. 1*CHAPTER ONE The Puzzles of Imaginary Numbers, pg. 8*CHAPTER TWO. A First Try at Understanding the Geometry of -1, pg. 31*CHAPTER THREE. The Puzzles Start to Clear, pg. 48*CHAPTER FOUR. Using Complex Numbers, pg. 84*CHAPTER FIVE. More Uses of Complex Numbers, pg. 105*CHAPTER SIX. Wizard Mathematics, pg. 142*CHAPTER SEVEN. The Nineteenth Century, Cauchy, and the Beginning of Complex Function Theory, pg. 187*APPENDIX A. The Fundamental Theorem of Algebra, pg. 227*APPENDIX B. The Complex Roots of a Transcendental Equation, pg. 230*APPENDIX C. ( -1)( -1) to 135 Decimal Places, and How It Was Computed, pg. 235*APPENDIX D. Solving Clausen's Puzzle, pg. 238*APPENDIX E. Deriving the Differential Equation for the Phase-Shift Oscillator, pg. 240*APPENDIX F. The Value of the Gamma Function on the Critical Line, pg. 244*Notes, pg. 247*Name Index, pg. 261*Subject Index, pg. 265*Acknowledgments, pg. 269

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Book SynopsisDescribes the developments in number theory by looking at one of the most exciting unsolved problems in contemporary mathematics - the Birch and Swinnerton-Dyer Conjecture. The Clay Mathematics Institute is offering a prize of $1 million to anyone who can discover a general solution to the problem.Trade Review"The authors present their discussion in an informal, sometimes playful manner and with detail that will appeal to an audience with a basic understanding of calculus. This book will captivate math enthusiasts as well as readers curious about an intriguing and still unanswered question."--Margaret Dominy, Library Journal "Minimal prerequisites and its clear writing make this book (which even has a few exercises) a great choice for a seminar for mathematics majors, who at some point should have such an excursion to one of the frontiers of mathematics."--Mathematics Magazine "The authors of Elliptic Tales do a superb job in demonstrating the approach that mathematicians take when they confront unsolved problems involving elliptic curves."--Sungkon Chang, Times Higher Education "One cannot help being impressed, in reading the book and pursuing a few of the references, by the magnitude of the enterprise it chronicles."--James Case, SIAM News "Ash and Gross thoroughly explain the statement and significance of the linchpin Birch and Swinnerton-Dyer conjection... [A]sh and Gross deliver ample and current intellectual and technical substance."--Choice "I would envision this book as an excellent text for an undergraduate 'capstone' course in mathematics; the book lends itself to independent reading, but topics may be explored in much greater depth and rigor in the classroom. Additionally, the book indeed brings together ideas from calculus, complex variables and algebra, showing how a single mathematical research question may require an integrated understanding of the various branches of mathematics. Thus, it encourages students to reinforce their understanding of these various fields, while simultaneously introducing them to an open question in mathematics and a vibrant field of study."--Lisa A. Berger, Mathematical Reviews Clippings "The book is very pleasantly written, and in my opinion, the authors have done an admirable job in giving an idea to non-experts what the Birch-Swinnerton Dyer conjecture is about."--Jan-Hendrik Evertse, Zentralblatt MATH "The book's most important contributions ... are the sense of discovery, invention, and insight into the habits of mind used by mathematicians on this journey. I would recommend this book to anyone who wants to be challenged mathematically or who wants to experience mathematics as creative and exciting."--Jacqueline Coomes, Mathematics Teacher "[T]his book is a wonderful introduction to what is arguably one of the most important mathematical problems of our time and for that reason alone it deserves to be widely read. Another reason to recommend this book is the opportunity to share in the readily apparent joy the authors have for their subject and the beauty they see in it, not least because ... joy and beauty are the most important reasons for doing mathematics, irrespective of its dollar value."--Rob Ashmore, Mathematics Today "This book has many nice aspects. Ash and Gross give a truly stimulating introduction to elliptic curves and the BSD conjecture for undergraduate students. The main achievement is to make a relative easy exposition of these so technical topics."--Jonathan Sanchez-Hernandez, Mathematical SocietyTable of ContentsPreface xiii Acknowledgments xix Prologue 1 PART I. DEGREE Chapter 1. Degree of a Curve 13 1.Greek Mathematics 13 2.Degree 14 3.Parametric Equations 20 4.Our Two Definitions of Degree Clash 23 Chapter 2. Algebraic Closures 26 1.Square Roots of Minus One 26 2.Complex Arithmetic 28 3.Rings and Fields 30 4.Complex Numbers and Solving Equations 32 5.Congruences 34 6.Arithmetic Modulo a Prime 38 7.Algebraic Closure 38 Chapter 3. The Projective Plane 42 1.Points at Infinity 42 2.Projective Coordinates on a Line 46 3.Projective Coordinates on a Plane 50 4.Algebraic Curves and Points at Infinity 54 5.Homogenization of Projective Curves 56 6.Coordinate Patches 61 Chapter 4. Multiplicities and Degree 67 1.Curves as Varieties 67 2.Multiplicities 69 3.Intersection Multiplicities 72 4.Calculus for Dummies 76 Chapter 5. B'ezout's Theorem 82 1.A Sketch of the Proof 82 2.An Illuminating Example 88 PART II. ELLIPTIC CURVES AND ALGEBRA Chapter 6. Transition to Elliptic Curves 95 Chapter 7. Abelian Groups 100 1.How Big Is Infinity? 100 2.What Is an Abelian Group? 101 3.Generations 103 4.Torsion 106 5.Pulling Rank 108 Appendix: An Interesting Example of Rank and Torsion 110 Chapter 8. Nonsingular Cubic Equations 116 1.The Group Law 116 2.Transformations 119 3.The Discriminant 121 4.Algebraic Details of the Group Law 122 5.Numerical Examples 125 6.Topology 127 7.Other Important Facts about Elliptic Curves 131 5.Two Numerical Examples 133 Chapter 9. Singular Cubics 135 1.The Singular Point and the Group Law 135 2.The Coordinates of the Singular Point 136 3.Additive Reduction 137 4.Split Multiplicative Reduction 139 5.Nonsplit Multiplicative Reduction 141 6.Counting Points 145 7.Conclusion 146 Appendix A: Changing the Coordinates of the Singular Point 146 Appendix B: Additive Reduction in Detail 147 Appendix C: Split Multiplicative Reduction in Detail 149 Appendix D: Nonsplit Multiplicative Reduction in Detail 150 Chapter 10. Elliptic Curves over Q 152 1.The Basic Structure of the Group 152 2.Torsion Points 153 3.Points of Infinite Order 155 4.Examples 156 PART III. ELLIPTIC CURVES AND ANALYSIS Chapter 11. Building Functions 161 1.Generating Functions 161 2.Dirichlet Series 167 3.The Riemann Zeta-Function 169 4.Functional Equations 171 5.Euler Products 174 6.Build Your Own Zeta-Function 176 Chapter 12. Analytic Continuation 181 1.A Difference that Makes a Difference 181 2.Taylor Made 185 3.Analytic Functions 187 4.Analytic Continuation 192 5.Zeroes, Poles, and the Leading Coefficient 196 Chapter 13. L-functions 199 1.A Fertile Idea 199 2.The Hasse-Weil Zeta-Function 200 3.The L-Function of a Curve 205 4.The L-Function of an Elliptic Curve 207 5.Other L-Functions 212 Chapter 14. Surprising Properties of L-functions 215 1.Compare and Contrast 215 2.Analytic Continuation 220 3.Functional Equation 221 Chapter 15. The Conjecture of Birch and Swinnerton-Dyer 225 1.How Big Is Big? 225 2.Influences of the Rank on the Np's 228 3.How Small Is Zero? 232 4.The BSD Conjecture 236 5.Computational Evidence for BSD 238 6.The Congruent Number Problem 240 Epilogue 245 Retrospect 245 Where DoWe Go from Here? 247 Bibliography 249 Index 251

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Book SynopsisThis volume is a collection of articles on themes related to the book Laws of Form by George Spencer-Brown. Laws of Form was first published in 1969 and brings forth a new articulation of the foundations of thought. In Laws of Form we have a mathematical formalism based on one symbol and an approach to the question how the world would appear if a distinction could be drawn. Laws of Form does not answer the question how, given nothing as a beginning, a distinction can, indeed must, inevitably take place. This second question must, in its own structure, be left to each individual thinker. Nevertheless, Laws of Form, beautifully written and content free (form is emptiness, emptiness is form) is the most powerful mathematical text on the edge of nothing that has been produced since Euclid''s Elements. These papers are a tribute to Spencer-Brown and his singular achievement.

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Book SynopsisTrade Review"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"A popular account of important ideas and their development."—Peter M. Neumann, Times Higher Education Supplement"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."—Amy Shell-Gellasch, MAA Reviews"At last, a popular book that isn't afraid to print a mathematical formula in all its symbolic glory! Thanks to Eli Maor for proving—in his delightful, playful way—the eternal importance of a three-sided idea as old as humankind."—Dava Sobel, author of Longitude

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Book SynopsisTrade Review"[Stillwell] writes clearly and engagingly... [Elements of Mathematics] can appeal to various constituencies at different levels of mathematical sophistication."--Mark Hunacek, MAA Reviews "A great exploration of elementary mathematics, its limitations, how infinity complicates things, and how various branches of mathematics fit together."--Antonio Cangiano, Math-Blog "Stillwell is ... One of the better current mathematical authors: he writes clearly and engagingly, and makes more of an effort than most to provide historical detail and a sense of how various mathematical ideas tie in with one another... The features we have learned to expect from Stillwell (including, but not limited to, excellent writing) are present in [Elements of Mathematics] as well."--MAA Reviews "An accessible read... Stillwell breaks down the basics, providing both historical and practical perspectives from arithmetic to infinity."--Gemma Tarlach, Discover "[A] sophisticated treatment of topics usually described as elementary."--John Allen Paulos "[Elements of Mathematics] is quite a tour de force, organized by areas of mathematics--arithmetic, computation, algebra, geometry, calculus, and so on--and in each area Stillwell manages to distill down the big ideas and the connections with other areas. He is a master expositor, and the text manages to be engaging and accessible without watering down the mathematics. I definitely learned new things from the book!"--Brent Yorgey, Math Less Traveled blog "From a lifetime of teaching, Stillwell has distilled some nice examples from the entire gamut of elementary mathematics."--Mathematical Reviews Clippings "[A] wonderful book... I think that [Elements of Mathematics] will itself become a modern classic and a reference work for anyone trying to learn basic topics in any of the major fields of mathematics."--Victor Katz, Bulletin of the American Mathematical Society "Elements of Mathematicsis a fine ... overview of the field of mathematics... The writing is clear, succinct, organized, and the diagrams [and] illustrations excellent... While some of the discussion is introductory or elementary, it always leads to deeper, more challenging ideas... [T]his will make a fine basic addition to most mathematicians' bookshelves."--Math Tango "Stillwell uses his broad and impressive command of mathematics to transport a reader through each topic and to a higher level of understanding and questioning."--Convergence "[A] wonderful book ... I think that [Elements of Mathematics] will itself become a modern classic and a reference work for anyone trying to learn basic topics in any of the major fields of mathematics."--Victor Katz, Bulletin of the American Mathematical Society "[Elements of Mathematics] is a book that everybody should read. You will be the better for it."--Reuben Hersh, American Mathematical MonthlyTable of Contents*Frontmatter, pg. i*Contents, pg. vii*Preface, pg. xi*1. Elementary Topics, pg. 1*2. Arithmetic, pg. 35*3. Computation, pg. 73*4. Algebra, pg. 106*5. Geometry, pg. 148*6. Calculus, pg. 193*7. Combinatorics, pg. 243*8. Probability, pg. 279*9. Logic, pg. 298*10. Some Advanced Mathematics, pg. 336*Bibliography, pg. 395*Index, pg. 405

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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 SynopsisTrade Review“[A] jaunty book.”—James Ryerson, New York Times Book Review“Devlin leads a cheerful pursuit to rediscover the hero of 13th-century European mathematics, taking readers across centuries and through the back streets of medieval and modern Italy in this entertaining and surprising history.”—Publishers Weekly“Finding Fibonacci showcases Devlin’s writerly flair.”—Davide Castelvecchi, Nature“[Devlin] talks his way into Italian research libraries in search of early manuscripts, photographs all 11 street signs on Via Leonardo Fibonacci in Florence and strives to cultivate a love for numbers in his readers.”—Andrea Marks, Scientific American“Engaging and entertaining.”—Library Journal“Personal and lively.”—Adhemar Bultheel, European Mathematical Society“Devlin’s enthusiasm for his subject is infectious.”—Tony Mann, Times Higher Education

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Book SynopsisTrade Review"One of Choice's Outstanding Academic Titles for 2013""Masterly. . . . Gray encapsulates Poincaré's multiple dimensions; his intellectual biography is both a tour de force and a triumph of readability."---George Szpiro, Nature"Gray shows us the full dazzling sweep of what Poincaré accomplished, including the work on dynamical systems and chaos that only came into its own in recent years. A tour de force, Gray's masterful treatment will long remain an invaluable resource for all who want to understand Poincaré, so embedded within his times and yet so far ahead of them."---Peter Pesic, Science"[A] comprehensive but uncluttered guide to Poincaré's extensive oeuvres."---Madeline Muntersbjorn, Times Higher Education"Full of the mathematical, physical and metaphysical ideas of a man who was not only a dispassionate observer of the world around us, but of our way of understanding it."---Mark Ronan, Standpoint Magazine"[A] comprehensive assessment of Poincare's work and its importance, essential for anyone interested in Poincare's scholarship or the history of mathematics."---Laura Tarwater Scharp, Sacramento Book Review"Comprehensive." * Science News *"A fundamental study of the scientific work of one of the greatest mathematicians and mathematical physicists of the three decades straddling the 19th and 20th centuries. . . . Chapters are organized topically, not chronologically. Each illuminates in depth one or other of Poincaré's works but all are set in context both historical and temathic such that each can serve as an introduction into the many subjects to which Poincaré made a contribution."---Alexander Bogomolny, CTK Insights"Poincare's work is fully alive in science today. This biography is one of the first thorough introductions to his work, it should get the attention of mathematicians, natural scientists and philosophers."---Ferdinand Verhulst, European Legacy"Gray, a mathematics historian and scholar on the life and work of Henry Poincaré, has, with the support of a Leverhulme Research Fellowship, produced this comprehensive and definitive 'scientific biography.' Gray offers abundant rich information on Poincaré's ideas and scientific process, the evolution and maturity of his mathematics including missteps, the dexterity of his reasoning, and the influences that shaped his thought." * Choice *"I recommend [this] book highly."---Robert E. O'Malley, Jr., SIAM Review"Jeremy Gray's book on Poincaré's mathematics, physics, and philosophy is an important contribution to the literature and a huge step towards a full biography of this pioneer ofmodern science."---Reinhard Siegmund-Schultze, Zentralblatt MATH"Gray's book is a comprehensive scientific biography of Poincare. It embraces the broad scope of Poincare's work, from his philosophical speculations to his popular writing, and gives a thorough overview of his extensive mathematical researches."---Peter Lynch, Irish Mathematical Society Bulletin"[T]he author does not simply give platitudes when writing about Poincare's ideas: mathematicians will enjoy reading about his discoveries concerning the three-body problem, the theory of functions, topology, number theory, Lie theory, algebraic geometry, and probability. This scientific biography is the first to comprehensively cover all of Poincare's main contributions to mathematics, philosophy, and physics."---Alan S. McRae, Mathemematical Reviews Clippings"Jeremy Gray has done a marvelous job of exposition and of binding together the many different cognitive, social and biographical strands into the coherent whole of a challenging, but highly rewarding, 'scientific biography'."---Klaus Hentschel, British Journal for the History of Science"A good intellectual biography of an artist should help the reader see how a particular worldview shapes the pursuit of art. Gray's book does that most admirably."---Daniel S. Alexander, H-France Review"Henry Poincaré is likely to remain the standard by which scientific biographies, at least those that concern physicists and mathematicians, are judged for some time."---Christopher Cumo, Canadian Journal of History"I warmly recommend the book to anyone with an interest in the development of modern mathematics. It will surely be the definitive scientific biography of Poincare for the foreseeable future."---John Stillwell, Notices of the AMS"Gray describes Poincaré's scientific epoch in a beautiful way. Due attention is paid to the mathematical and further scientific aspects of his life, and the intellectual complexity of his achievements, both in their range and their depth, are amply discussed. Gray displays a mastery of his material that is rare even among historians of mathematics and science, and his biography is richly rewarding, engrossing, and informative. He deserves our congratulations."---H. W. Broer, Journal of the British Society for the History of Mathematics"Gray succeeds admirably in presenting both the conceptual and the historical context necessary to appreciate Poincaré's contributions. Gray's masterful biography may well serve as a standard example for future endeavors of this kind."---Tilman Sauer, Isis"The obvious virtue of this book is its comprehensiveness. The deeper virtue is to connect Poincaré's views of all the parts of his work and to encourage more of that. Gray gives us Poincaré's view of Science as a whole."---Colin McLarty, Mathematical Intelligencer"The book is an endless source of interesting insights by Poincaré. . . . I would recommend the book for mathematicians, mathematics educators, and philosophers in higher education who want a rich understanding of Poincaré, his work, and his times."---Mary L. Garner, Mathematics Teacher

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Book SynopsisFirst published in 1919, Introduction to Mathematical Philosophy shows Russell drawing on his formidable knowledge of philosophy and mathematics to write a brilliant introduction to the subject. This Routledge Classics edition includes a new Foreword by Michael Potter.Table of ContentsForeword to the Routledge Classics Edition Michael Potter Preface 1. The Series of Natural Numbers 2. Definition of Number 3. Finitude and Mathematical Induction 4. The Definition of Order 5. Kinds of Relations 6. Similarity of Relations 7. Rational, Real, and Complex Numbers 8. Infinite Cardinal Numbers 9. Infinite Series and Ordinals 10. Limits and Continuity 11. Limits and Continuity of Functions 12. Selections and the Multiplicative Axiom 13. The Axiom of Infinity and Logical Types 14. Incompatibility and the Theory of Deduction 15. Propositional Functions 16. Descriptions 17. Classes 18. Mathematics and Logic. Index

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Book SynopsisTrade Review"This book could well serve as a history of mathematics. … [Stillwell] has done an amazing job of collecting and categorizing many of the most important ideas in this area."---Jim Stein, New Books in Mathematics"Stillwell’s [The Story of Proof] joins his two other Princeton University Press books in having my highest recommendation. I just wish they had been around when I was a student."---George Hacken, Computing Reviews"I hugely enjoyed this book."---Jonathan Shock, Mathemafrica"This book would be perfect for any keen undergraduate, keen amateur, or indeed a teacher of mathematics, who wants a book to dip into to use for the classroom."---Jonathan Shock, Mathemafrica"A well-crafted, thought-provoking meditation on the concept of proof in mathematics. . . .It is a substantive book that deserves to be read and reflected upon."---Tommy Murphy, Irish Mathematical Society Bulletin"This is a work that mathematicians, historians, and philosophers will find especially engaging, as will anyone with a serious interest in mathematics and the limits of certainty that it is constantly probing."---J.W. Dauben, Choice

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Book SynopsisSpherical Geometry and Its Applications introduces spherical geometry and its practical applications in a mathematically rigorous form. The text can serve as a course in spherical geometry for mathematics majors. Readers from various academic backgrounds can comprehend various approaches to the subject. The book introduces an axiomatic system for spherical geometry and uses it to prove the main theorems of the subject. It also provides an alternate approach using quaternions. The author illustrates how a traditional axiomatic system for plane geometry can be modified to produce a different geometric world but a geometric world that is no less real than the geometric world of the plane. Features: A well-rounded introduction to spherical geometry Provides several proofs of some theorems to appTable of ContentsReview of three-dimensional geometry Geometry in a plane Geometry in space Plane trigonometry Coordinates and vectors The sphere in space Great circles Distance and angles Area Spherical coordinates Axiomatic spherical geometry Basic axioms Angles Triangles Congruence Inequalities Area Trigonometry Spherical Pythagorean theorem and law of sines Spherical law of cosines and analogue formula Right triangles The four-parts and half angle formulas Dualization Solution of triangles Astronomy The celestial sphere Changing coordinates Rise and set of objects in the sky The measurement of time Rise and set times in standard time Polyhedra Regular solids Crystals Spherical mappings Rotations and reflections Spherical projections Quaternions Review of complex numbers Quaternions: Definitions and basic properties Application to the sphere Triangles Rotations and Reflections Selected solutions to exercises

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Book SynopsisThink of a number between one and ten.No, hang on, let's make this interesting. Between zero and infinity. Even if you stick to the whole numbers, there are a lot to choose from - an infinite number in fact. Throw in decimal fractions and infinity suddenly gets an awful lot bigger (is that even possible?) And then there are the negative numbers, the imaginary numbers, the irrational numbers like pi which never end. It literally never ends.The world of numbers is indeed strange and beautiful. Among its inhabitants are some really notable characters - pi, e, the "imaginary" number i and the famous golden ratio to name just a few. Prime numbers occupy a special status. Zero is very odd indeed: is it a number, or isn't it?How Numbers Work takes a tour of this mind-blowing but beautiful realm of numbers and the mathematical rules that connect them. Not only that, but take a crash course on the biggest unsolved problems that keep mathematicians up at night, find out about the strange and unexpected ways mathematics influences our everyday lives, and discover the incredible connection between numbers and reality itself.ABOUT THE SERIESNew Scientist Instant Expert books are definitive and accessible entry points to the most important subjects in science; subjects that challenge, attract debate, invite controversy and engage the most enquiring minds. Designed for curious readers who want to know how things work and why, the Instant Expert series explores the topics that really matter and their impact on individuals, society, and the planet, translating the scientific complexities around us into language that's open to everyone, and putting new ideas and discoveries into perspective and context.

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Book SynopsisWhy is 7 such a lucky number and 13 so unlucky? Why does a jury traditionally have `12 good men and true', and why are there 24 hours in the day and 60 seconds in a minute? This fascinating new book explores the world of numbers from pin numbers to book titles, and from the sixfold shape of snowflakes to the way our roads, houses and telephone numbers are designated in fact and fiction. Using the numbers themselves as its starting point it investigates everything from the origins and meaning of counting in early civilizations to numbers in proverbs, myths and nursery rhymes and the ancient `science' of numerology. It also focuses on the quirks of odds and evens, primes, on numbers in popular sports - and much, much more. So whether you've ever wondered why Heinz has 57 varieties, why 999 is the UK's emergency phone number but 911 is used in America, why Coco Chanel chose No. 5 for her iconic perfume, or how the title Catch 22 was chosen, then this is the book for you. Dip in anywhere and you'll find that numbers are not just for adding and measuring but can be hugely entertaining and informative whether you're buying a diamond or choosing dinner from the menu.Table of Contents1 Introduction 8 2 Numbers of many sorts 10 3 Numbers and counting 12 4 1 - The number of unity 17 5 2 - The duality 19 6 3 - The trinity of perfection 21 7 4 - Truth and justice 23 8 5 - The number of nature 26 9 6 - Without a fault Six and the snowflake 29 10 7 - The symbol of fortune 32 11 The seven wonders of the world 35 12 Puzzles to solve 37 13 8 - Harmony and balance 38 14 9 - Unbounded 40 15 The nine muses 43 16 10 - On our fingers and toes 44 17 11 - The final hour 46 18 12 - A number in time 47 19 The twelve labours of Hercules 49 20 13 - And other teens 53 21 0 - The story of zero 54 22 Lucky - and unlucky - numbers 56 23 Odds and evens 59 24 Many favourite numbers 60 25 The secrets of numerology 62 26 Big numbers 66 27 Small numbers 68 28 The appeal of primes 69 29 Making shapes with numbers 72 30 Numbers - more different types 74 31 - the most famous number 75 32 Fibonacci - the brilliant number sequence 77 33 The golden ratio 79 34 Numbers in use 81 35 The world we live in 84 36 Our planet earth 86 37 Lines on the map 89 38 Measuring the world 94 39 A matter of weight 98 40 By volume 101 41 More about money 104 42 Inventing the calendar 106 43 Time and the circle 109 44 The living world 110 45 The human body 114 46 A number for your home 120 47 Addresses in fiction 121 48 A dark history 123 49 The streets of power 124 50 Numbers for the post 126 51 Roads to take - navigating by numbers 127 52 The number to call 130 53 PIN - what's your number? 132 54 Edible connections 134 55 Sizing up our drinks 137 56 Of yarns, fabrics and clothes 139 57 All that glitters 140 58 Beauty by numbers 142 59 For our leisure and entertainment 144 60 Proverbs and sayings 146 61 Bingo lingo 147 62 Books with numbers 149 63 In the film title 156 64 Counting in song 161 65 Jazz numbers 163 66 A musical miscellany 165 67 Poetry's secrets revealed 167 68 The beautiful game 170 69 The oval ball game - rugby football 174 70 The game of golf 176 71 Throwing darts 178 72 Cricket - bat on ball 179 73 On court - the game of tennis 182 74 Snooker and other cue games 185 75 The game of baseball 187 76 Chancing your luck 189 77 Throwing dice 192 78 Playing dominoes 194 79 Solving the square 195 80 Index 200 81 About the author 208

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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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