# Mathematics

**An exploration of mathematical style through 99 different proofs of the same theorem**

This book offers a multifaceted perspective on mathematics by demonstrating 99 different proofs of the same theorem. Each chapter solves an otherwise unremarkable equation in distinct historical, formal, and imaginative styles that range from Medieval, Topological, and Doggerel to Chromatic, Electrostatic, and Psychedelic. With a rare blend of humor and scholarly aplomb, Philip Ording weaves these variations into an accessible and wide-ranging narrative on the nature and practice of mathematics.

Inspired by the experiments of the Paris-based writing group known as the Oulipo--whose members included Raymond Queneau, Italo Calvino, and Marcel Duchamp--Ording explores new ways to examine the aesthetic possibilities of mathematical activity. *99 Variations on a Proof* is a mathematical take on Queneau's *Exercises in Style*, a collection of 99 retellings of the same story, and it draws unexpected connections to everything from mysticism and technology to architecture and sign language. Through diagrams, found material, and other imagery, Ording illustrates the flexibility and creative potential of mathematics despite its reputation for precision and rigor.

Readers will gain not only a bird's-eye view of the discipline and its major branches but also new insights into its historical, philosophical, and cultural nuances. Readers, no matter their level of expertise, will discover in these proofs and accompanying commentary surprising new aspects of the mathematical landscape.

**The intellectual and human story of a mathematical proof that transformed our ideas about mathematics.**

In 1824 a young Norwegian named Niels Henrik Abel proved conclusively that algebraic equations of the fifth order are not solvable in radicals. In this book Peter Pesic shows what an important event this was in the history of thought. He also presents it as a remarkable human story. Abel was twenty-one when he self-published his proof, and he died five years later, poor and depressed, just before the proof started to receive wide acclaim. Abel's attempts to reach out to the mathematical elite of the day had been spurned, and he was unable to find a position that would allow him to work in peace and marry his fiancé.

But Pesic's story begins long before Abel and continues to the present day, for Abel's proof changed how we think about mathematics and its relation to the real world. Starting with the Greeks, who invented the idea of mathematical proof, Pesic shows how mathematics found its sources in the real world (the shapes of things, the accounting needs of merchants) and then reached beyond those sources toward something more universal. The Pythagoreans' attempts to deal with irrational numbers foreshadowed the slow emergence of abstract mathematics. Pesic focuses on the contested development of algebra--which even Newton resisted--and the gradual acceptance of the usefulness and perhaps even beauty of abstractions that seem to invoke realities with dimensions outside human experience. Pesic tells this story as a history of ideas, with mathematical details incorporated in boxes. The book also includes a new annotated translation of Abel's original proof.

**The true story that inspired the 2020 film.**

The autobiography of mathematician Stanislaw Ulam, one of the great scientific minds of the twentieth century, tells a story rich with amazingly prophetic speculations and peppered with lively anecdotes. As a member of the Los Alamos National Laboratory from 1944 on, Ulam helped to precipitate some of the most dramatic changes of the postwar world. He was among the first to use and advocate computers for scientific research, originated ideas for the nuclear propulsion of space vehicles, and made fundamental contributions to many of today's most challenging mathematical projects.

With his wide-ranging interests, Ulam never emphasized the importance of his contributions to the research that resulted in the hydrogen bomb. Now Daniel Hirsch and William Mathews reveal the true story of Ulam's pivotal role in the making of the "Super," in their historical introduction to this behind-the-scenes look at the minds and ideas that ushered in the nuclear age. An epilogue by Françoise Ulam and Jan Mycielski sheds new light on Ulam's character and mathematical originality.

This volume emulates the quality and diverse interests of Knorr's innovative, exact, and far-reaching research. Topics inspired by Knorr include a study of geometric analysis and synthesis in ancient Greece and medieval Islam; examination of Eudoxus as originator for the ideas of proportionality underlying Book V of Euclid's Elements; and the extent that Renaissance theorists of linear perspective had access to ancient sources. This book considers the status of Eudoxus's theory of homocentric spheres in Greek astronomy and the examination of the status of in Greek mathematics. A detailed discussion of the geometrical chemistry of Plato's Timaeus and its interpretation in antiquity stems from Knorr's work, and a study of Plato's concept of numbers and its relation to the Theory of Forms. Knorr's varied interests motivate investigation into the representation of numbers in the Latin middle ages, or why we read Arabic numbers backwards, and the history of science in a chronology of the three dynasties in ancient China.

**The first history of postwar mathematics, offering a new interpretation of the rise of abstraction and axiomatics in the twentieth century.**

Why did abstraction dominate American art, social science, and natural science in the mid-twentieth century? Why, despite opposition, did abstraction and theoretical knowledge flourish across a diverse set of intellectual pursuits during the Cold War? In recovering the centrality of abstraction across a range of modernist projects in the United States, Alma Steingart brings mathematics back into the conversation about midcentury American intellectual thought. The expansion of mathematics in the aftermath of World War II, she demonstrates, was characterized by two opposing tendencies: research in pure mathematics became increasingly abstract and rarified, while research in applied mathematics and mathematical applications grew in prominence as new fields like operations research and game theory brought mathematical knowledge to bear on more domains of knowledge. Both were predicated on the same abstractionist conception of mathematics and were rooted in the same approach: modern axiomatics.

For American mathematicians, the humanities and the sciences did not compete with one another, but instead were two complementary sides of the same epistemological commitment. Steingart further reveals how this mathematical epistemology influenced the sciences and humanities, particularly the postwar social sciences. As mathematics changed, so did the meaning of mathematization.

*Axiomatics* focuses on American mathematicians during a transformative time, following a series of controversies among mathematicians about the nature of mathematics as a field of study and as a body of knowledge. The ensuing debates offer a window onto the postwar development of mathematics band Cold War epistemology writ large. As Steingart's history ably demonstrates, mathematics is the social activity in which styles of truth--here, abstraction--become synonymous with ways of knowing.

**From a zebra's stripes to a spider's web: an engaging examination of patterns in nature and the mathematics that underlie them.**

From a zebra's stripes to a spider's web, from sand dunes to snowflakes, nature is full of patterns underlaid by mathematical principles. In *The Beauty of Numbers in Nature*, Ian Stewart shows how life forms from the principles of mathematics. Each chapter in *The Beauty of Numbers in Nature* explores a different kind of patterning system and its mathematical underpinnings. In doing do, the book also uncovers some universal patterns--both in nature and made by humans--from the basic geometry of ancient Greece to the complexities of fractals.

Stewart draws on a wide range of sources to examine the mathematics of patterns: the Pythagoreans' obsession with numbers as the philosophical basis of the universe; a great mathematician who wondered about how a violin makes music; a clerk in a patent office who realized that space and time can get mixed together; a maverick mathematician who questioned why nature spurns such regular geometric shapes as spheres and cylinders in favor of jagged lightning bolts, asymmetrically branching trees, and the uneven terrain of mountainsides.

The book begins with a simple and often-asked question about the shape and individual uniqueness of snowflakes. How can such a strange mixture of regularity and irregularity exist in a tiny bit of frozen water? By the end of the book, readers will have learned that mathematical patterns can come in many guises, some of which don't resemble patterns at all.

Optimists believe this is the best of all possible worlds. And pessimists fear that might really be the case. But what *is* the best of all possible worlds? How do we define it? Is it the world that operates the most efficiently? Or the one in which most people are comfortable and content? Questions such as these have preoccupied philosophers and theologians for ages, but there was a time, during the seventeenth and eighteenth centuries, when scientists and mathematicians felt they could provide the answer.

This book is their story. Ivar Ekeland here takes the reader on a journey through scientific attempts to envision the best of all possible worlds. He begins with the French physicist Maupertuis, whose least action principle asserted that everything in nature occurs in the way that requires the least possible action. This idea, Ekeland shows, was a pivotal breakthrough in mathematics, because it was the first expression of the concept of *optimization*, or the creation of systems that are the most efficient or functional. Although the least action principle was later elaborated on and overshadowed by the theories of Leonhard Euler and Gottfried Leibniz, the concept of optimization that emerged from it is an important one that touches virtually every scientific discipline today.

Tracing the profound impact of optimization and the unexpected ways in which it has influenced the study of mathematics, biology, economics, and even politics, Ekeland reveals throughout how the idea of optimization has driven some of our greatest intellectual breakthroughs. The result is a dazzling display of erudition--one that will be essential reading for popular-science buffs and historians of science alike.

**The year's finest writing on mathematics from around the world, with a foreword by Nobel Prize**-**winning physicist Roger Penrose**

This annual anthology brings together the year's finest mathematics writing from around the world. Featuring promising new voices alongside some of the foremost names in the field, *The Best Writing on Mathematics 2013* makes available to a wide audience many articles not easily found anywhere else--and you don't need to be a mathematician to enjoy them. These writings offer surprising insights into the nature, meaning, and practice of mathematics today. They delve into the history, philosophy, teaching, and everyday occurrences of math, and take readers behind the scenes of today's hottest mathematical debates. Here Philip Davis offers a panoramic view of mathematics in contemporary society; Terence Tao discusses aspects of universal mathematical laws in complex systems; Ian Stewart explains how in mathematics everything arises out of nothing; Erin Maloney and Sian Beilock consider the mathematical anxiety experienced by many students and suggest effective remedies; Elie Ayache argues that exchange prices reached in open market transactions transcend the common notion of probability; and much, much more.

In addition to presenting the year's most memorable writings on mathematics, this must-have anthology includes a foreword by esteemed mathematical physicist Roger Penrose and an introduction by the editor, Mircea Pitici. This book belongs on the shelf of anyone interested in where math has taken us--and where it is headed.

**The year's finest writing on mathematics from around the world**

This annual anthology brings together the year's finest mathematics writing from around the world. Featuring promising new voices alongside some of the foremost names in the field, *The Best Writing on Mathematics 2014* makes available to a wide audience many articles not easily found anywhere else--and you don't need to be a mathematician to enjoy them. These writings offer surprising insights into the nature, meaning, and practice of mathematics today. They delve into the history, philosophy, teaching, and everyday occurrences of math, and take readers behind the scenes of today's hottest mathematical debates. Here John Conway presents examples of arithmetical statements that are almost certainly true but likely unprovable; Carlo Séquin explores, compares, and illustrates distinct types of one-sided surfaces known as Klein bottles; Keith Devlin asks what makes a video game good for learning mathematics and shows why many games fall short of that goal; Jordan Ellenberg reports on a recent breakthrough in the study of prime numbers; Stephen Pollard argues that mathematical practice, thinking, and experience transcend the utilitarian value of mathematics; and much, much more.

In addition to presenting the year's most memorable writings on mathematics, this must-have anthology includes an introduction by editor Mircea Pitici. This book belongs on the shelf of anyone interested in where math has taken us--and where it is headed.

**The year's finest writing on mathematics from around the world**

This annual anthology brings together the year's finest mathematics writing from around the world. Featuring promising new voices alongside some of the foremost names in the field, *The Best Writing on Mathematics 2015* makes available to a wide audience many articles not easily found anywhere else--and you don't need to be a mathematician to enjoy them. These writings offer surprising insights into the nature, meaning, and practice of mathematics today. They delve into the history, philosophy, teaching, and everyday occurrences of math, and take readers behind the scenes of today's hottest mathematical debates.

Here David Hand explains why we should actually expect unlikely coincidences to happen; Arthur Benjamin and Ethan Brown unveil techniques for improvising custom-made magic number squares; Dana Mackenzie describes how mathematicians are making essential contributions to the development of synthetic biology; Steven Strogatz tells us why it's worth writing about math for people who are alienated from it; Lisa Rougetet traces the earliest written descriptions of Nim, a popular game of mathematical strategy; Scott Aaronson looks at the unexpected implications of testing numbers for randomness; and much, much more.

In addition to presenting the year's most memorable writings on mathematics, this must-have anthology includes a bibliography of other notable writings and an introduction by the editor, Mircea Pitici. This book belongs on the shelf of anyone interested in where math has taken us--and where it is headed.

-- "Publishers Weekly"**The year's finest mathematics writing from around the world**

This annual anthology brings together the year's finest mathematics writing from around the world. Featuring promising new voices alongside some of the foremost names in the field, *The Best Writing on Mathematics 2016* makes available to a wide audience many articles not easily found anywhere else--and you don't need to be a mathematician to enjoy them. These writings offer surprising insights into the nature, meaning, and practice of mathematics today. They delve into the history, philosophy, teaching, and everyday occurrences of math, and take readers behind the scenes of today's hottest mathematical debates.

Here Burkard Polster shows how to invent your own variants of the Spot It! card game, Steven Strogatz presents young Albert Einstein's proof of the Pythagorean Theorem, Joseph Dauben and Marjorie Senechal find a treasure trove of math in New York's Metropolitan Museum of Art, and Andrew Gelman explains why much scientific research based on statistical testing is spurious. In other essays, Brian Greene discusses the evolving assumptions of the physicists who developed the mathematical underpinnings of string theory, Jorge Almeida examines the misperceptions of people who attempt to predict lottery results, and Ian Stewart offers advice to authors who aspire to write successful math books for general readers. And there's much, much more.

In addition to presenting the year's most memorable writings on mathematics, this must-have anthology includes a bibliography of other notable writings and an introduction by the editor, Mircea Pitici. This book belongs on the shelf of anyone interested in where math has taken us--and where it is headed.

-- "Math Tango"**The year's finest mathematics writing from around the world**

This annual anthology brings together the year's finest mathematics writing from around the world. Featuring promising new voices alongside some of the foremost names in the field, *The Best Writing on Mathematics 2017* makes available to a wide audience many articles not easily found anywhere else--and you don't need to be a mathematician to enjoy them. These writings offer surprising insights into the nature, meaning, and practice of mathematics today. They delve into the history, philosophy, teaching, and everyday occurrences of math, and take readers behind the scenes of today's hottest mathematical debates.

Here Evelyn Lamb describes the excitement of searching for incomprehensibly large prime numbers, Jeremy Gray speculates about who would have won math's highest prize--the Fields Medal--in the nineteenth century, and Philip Davis looks at mathematical results and artifacts from a business and marketing viewpoint. In other essays, Noson Yanofsky explores the inherent limits of knowledge in mathematical thinking, Jo Boaler and Lang Chen reveal why finger-counting enhances children's receptivity to mathematical ideas, and Carlo Séquin and Raymond Shiau attempt to discover how the Renaissance painter Fra Luca Pacioli managed to convincingly depict his famous rhombicuboctahedron, a twenty-six-sided Archimedean solid. And there's much, much more.

In addition to presenting the year's most memorable writings on mathematics, this must-have anthology includes a bibliography of other notable writings and an introduction by the editor, Mircea Pitici. This book belongs on the shelf of anyone interested in where math has taken us--and where it is headed.

**The year's finest mathematical writing from around the world**

This annual anthology brings together the year's finest mathematics writing from around the world. Featuring promising new voices alongside some of the foremost names in the field, *The Best Writing on Mathematics 2018* makes available to a wide audience many pieces not easily found anywhere else--and you don't need to be a mathematician to enjoy them. These essays delve into the history, philosophy, teaching, and everyday aspects of math, offering surprising insights into its nature, meaning, and practice--and taking readers behind the scenes of today's hottest mathematical debates.

James Grime shows how to build subtly mischievous dice for playing slightly unfair games and Michael Barany traces how our appreciation of the societal importance of mathematics has developed since World War II. In other essays, Francis Su extolls the inherent values of learning, doing, and sharing mathematics, and Margaret Wertheim takes us on a mathematical exploration of the mind and the world--with glimpses at science, philosophy, music, art, and even crocheting. And there's much, much more.

In addition to presenting the year's most memorable math writing, this must-have anthology includes an introduction by the editor and a bibliography of other notable pieces on mathematics.

This is a must-read for anyone interested in where math has taken us--and where it is headed.

**The year's finest mathematical writing from around the world**

This annual anthology brings together the year's finest mathematics writing from around the world. Featuring promising new voices alongside some of the foremost names in the field, *The Best Writing on Mathematics 2019* makes available to a wide audience many articles not easily found anywhere else--and you don't need to be a mathematician to enjoy them. These essays delve into the history, philosophy, teaching, and everyday aspects of math, offering surprising insights into its nature, meaning, and practice--and taking readers behind the scenes of today's hottest mathematical debates.

In this volume, Moon Duchin explains how geometric-statistical methods can be used to combat gerrymandering, Jeremy Avigad illustrates the growing use of computation in making and verifying mathematical hypotheses, and Kokichi Sugihara describes how to construct geometrical objects with unusual visual properties. In other essays, Neil Sloane presents some recent additions to the vast database of integer sequences he has catalogued, and Alessandro Di Bucchianico and his colleagues highlight how mathematical methods have been successfully applied to big-data problems. And there's much, much more.

In addition to presenting the year's most memorable math writing, this must-have anthology includes an introduction by the editor and a bibliography of other notable writings on mathematics.

This is a must-read for anyone interested in where math has taken us--and where it is headed.

Contents include On the Hypotheses which Lie at the Foundations of Geometry by Georg Friedrich Riemann; On the Facts which Lie at the Foundations of Geometry and On the Origin and Significance of Geometrical Axioms by Hermann von Helmholtz; A Comparative Review of Recent Researches in Geometry by Felix Klein; On the Space Theory of Matter by William Kingdon Clifford; On the Foundations of Geometry by Henri Poincaré; Euclidean Geometry and Riemannian Geometry by Elie Cartan; and The Problem of Space, Ether, and the Field in Physics by Albert Einstein.

These remarkably accessible papers will appeal to students of modern physics and mathematics, as well as anyone interested in the origins and sources of Einstein's most profound work. Peter Pesic of St. John's College in Santa Fe, New Mexico, provides an introduction, as well as notes that offer insights into each paper.

**BACK IN PRINT with a new preface and a new chapter**

In this unusual book, first published by The MIT Press in 1980 and now updated with a new chapter, Steven Brams applies the mathematical theory of games to the Hebrew Bible. Brams's thesis is that God and the human biblical characters acted rationally--that is, given their preferences and their knowledge of other players' preferences, they made strategy choices that led to the best attainable outcomes. Beginning with the Creation and focusing on those stories richest in conflict and intrigue, Brams uses elementary game-theoretic tools to elucidate the rational calculations of biblical players and to show precisely the manner in which they sought to achieve their goals. He relies almost exclusively on noncooperative theory, making use of both game tree and matrix forms of games. Brams uses his strategic analyses to build a detailed assessment of God's character and motivations, including the reasons for His frequently wrathful behavior. Brams's insights have application to biblical studies, the philosophy of religion, political theory, and game theory and methodology.In the new chapter, Brams surveys the literature of the past twenty years on political-strategic interpretations of the Hebrew Bible. He also extends the game-theoretic analysis, using the theory of moves, to study a counterfactual situation--what if Abraham had refused God's command to sacrifice Isaac?--and to examine the rationality of believing in a superior being.

More than three centuries after its creation, calculus remains a dazzling intellectual achievement and the gateway into higher mathematics. This book charts its growth and development by sampling from the work of some of its foremost practitioners, beginning with Isaac Newton and Gottfried Wilhelm Leibniz in the late seventeenth century and continuing to Henri Lebesgue at the dawn of the twentieth--mathematicians whose achievements are comparable to those of Bach in music or Shakespeare in literature. William Dunham lucidly presents the definitions, theorems, and proofs. "Students of literature read Shakespeare; students of music listen to Bach," he writes. But this tradition of studying the major works of the "masters" is, if not wholly absent, certainly uncommon in mathematics. This book seeks to redress that situation.

Like a great museum, *The Calculus Gallery* is filled with masterpieces, among which are Bernoulli's early attack upon the harmonic series (1689), Euler's brilliant approximation of pi (1779), Cauchy's classic proof of the fundamental theorem of calculus (1823), Weierstrass's mind-boggling counterexample (1872), and Baire's original "category theorem" (1899). Collectively, these selections document the evolution of calculus from a powerful but logically chaotic subject into one whose foundations are thorough, rigorous, and unflinching--a story of genius triumphing over some of the toughest, most subtle problems imaginable.

Anyone who has studied and enjoyed calculus will discover in these pages the sheer excitement each mathematician must have felt when pushing into the unknown. In touring *The Calculus Gallery*, we can see how it all came to be.

*The Calculus of Friendship* is the story of an extraordinary connection between a teacher and a student, as chronicled through more than thirty years of letters between them. What makes their relationship unique is that it is based almost entirely on a shared love of calculus. For them, calculus is more than a branch of mathematics; it is a game they love playing together, a constant when all else is in flux. The teacher goes from the prime of his career to retirement, competes in whitewater kayaking at the international level, and loses a son. The student matures from high school math whiz to Ivy League professor, suffers the sudden death of a parent, and blunders into a marriage destined to fail. Yet through it all they take refuge in the haven of calculus--until a day comes when calculus is no longer enough.

Like calculus itself, *The Calculus of Friendship* is an exploration of change. It's about the transformation that takes place in a student's heart, as he and his teacher reverse roles, as they age, as they are buffeted by life itself. Written by a renowned teacher and communicator of mathematics, *The Calculus of Friendship* is warm, intimate, and deeply moving. The most inspiring ideas of calculus, differential equations, and chaos theory are explained through metaphors, images, and anecdotes in a way that all readers will find beautiful, and even poignant. Math enthusiasts, from high school students to professionals, will delight in the offbeat problems and lucid explanations in the letters.

For anyone whose life has been changed by a mentor, *The Calculus of Friendship* will be an unforgettable journey.

When first published posthumously in 1963, this bookpresented a radically different approach to the teaching of calculus. In sharp contrast to the methods of his time, Otto Toeplitz did not teach calculus as a static system of techniques and facts to be memorized. Instead, he drew on his knowledge of the history of mathematics and presented calculus as an organic evolution of ideas beginning with the discoveries of Greek scholars, such as Archimedes, Pythagoras, and Euclid, and developing through the centuries in the work of Kepler, Galileo, Fermat, Newton, and Leibniz. Through this unique approach, Toeplitz summarized and elucidated the major mathematical advances that contributed to modern calculus.

Reissued for the first time since 1981 and updated with a new foreword, this classic text in the field of mathematics is experiencing a resurgence of interest among students and educators of calculus today.

Such ubiquitous mathematical entities as pi and the Fibonacci numbers are found throughout the natural world and are also the foundation of our technological civilization. By exploring the intriguing games presented here, you'll come away with a greater appreciation for the beauty and importance of these and many more math concepts.

This is the perfect book for people who were turned off by math in school but now as adults wonder what they may have missed.

**The internationally bestselling authors of The Cartoon Introduction to Economics return to make calculus fun**

The award-winning illustrator Grady Klein has teamed up once again with the world's only stand-up economist, Yoram Bauman, Ph.D., to take on the daunting subject of calculus. A supplement to traditional textbooks, *The Cartoon Introduction to Calculus *focuses on the big ideas rather than all the formulas you have to memorize.

With Klein and Bauman as our guides, we scale the dual peaks of Mount Derivative and Mount Integral, and from their summits, we see how calculus relates to the rest of mathematics. Beginning with the problems of speed and area, Klein and Bauman show how the discipline is unified by a fundamental theorem. We meet geniuses like Archimedes, Liu Hui, and Bonaventura Cavalieri, who survived the slopes on intuition but prepared us for the avalanche-like dangers posed by mathematical rigor. Then we trek onward and scramble through limits and extreme values, optimization and integration, and learn how calculus can be applied to economics, physics, and so much more. We discover that calculus isn't the pinnacle of mathematics after all, but its tools are foundational to everything that follows. Klein and Bauman round out the book with a handy glossary of symbols and terms, so you don't have to worry about mixing up constants and constraints. With a witty and engaging narrative full of jokes and insights, *The Cartoon Introduction to Calculus *is an essential primer for students or for anyone who is curious about math.

We all played tag when we were kids. What most of us don't realize is that this simple chase game is in fact an application of pursuit theory, and that the same principles of games like tag, dodgeball, and hide-and-seek are also at play in military strategy, high-seas chases by the Coast Guard, and even romantic pursuits. In *Chases and Escapes*, Paul Nahin gives us the first complete history of this fascinating area of mathematics, from its classical analytical beginnings to the present day.

Drawing on game theory, geometry, linear algebra, target-tracking algorithms, and much more, Nahin also offers an array of challenging puzzles with their historical background and broader applications. *Chases and Escapes* includes solutions to all problems and provides computer programs that readers can use for their own cutting-edge analysis.

Now with a gripping new preface on how the Enola Gay escaped the shock wave from the atomic bomb dropped on Hiroshima, this book will appeal to anyone interested in the mathematics that underlie pursuit and evasion.

*Shu-shu chiu-chang.*Subsequent studies examine the entire range of mathematical techniques and problems found within Ch'in's book.

The core of this book consists of an in-depth study of what modern mathematicians still refer to as the Chinese remainder theorem for the solution of indeterminate equations of the first degree. This was Ch'in's most original contribution to mathematics--so original that no one could correctly explain Ch'in's procedure until the early nineteenth century. This volume's concluding study unites information on artisanal, economic, administrative, and military affairs dispersed throughout Ch'in's writings, providing rare insights into thirteenth-century China.

*Achilles paradox*to

*zero matrix*, it covers all commonly encountered terms and concepts from pure and applied mathematics and statistics, for example, linear algebra, optimisation, nonlinear equations, and differential equations. In addition, there are entries on major mathematicians and on topics of more general interest, such as fractals, game theory, and chaos.

Using graphs, diagrams, and charts to render definitions as comprehensible as possible, entries are clear and accessible. Almost 200 new entries have been added to this edition, including terms such as **arrow paradox**, **nested set**, and **symbolic logic**. Useful appendices follow the A-Z dictionary and include lists of Nobel Prize winners and Fields' medallists, Greek letters, formulae, and tables of inequalities, moments of inertia, Roman numerals, a geometry summary, additional trigonometric values of special angles, and much more. This edition contains recommended web links, which are accessible and kept up to date via the *Dictionary of Mathematics* companion website.

An expert in the fields of analysis and number theory, author G. H. Hardy taught for decades at both Cambridge and Oxford.

*A Course of Pure Mathematics*is suitable for college and high school students and teachers of calculus as well as fans of pure math. Each chapter includes demanding problem sets that allow students to apply the principles directly, and four helpful Appendixes supplement the text.

From the Ishango Bone of central Africa and the Inca *quipu* of South America to the dawn of modern mathematics, *The Crest of the Peacock* makes it clear that human beings everywhere have been capable of advanced and innovative mathematical thinking. George Gheverghese Joseph takes us on a breathtaking multicultural tour of the roots and shoots of non-European mathematics. He shows us the deep influence that the Egyptians and Babylonians had on the Greeks, the Arabs' major creative contributions, and the astounding range of successes of the great civilizations of India and China.

The third edition emphasizes the dialogue between civilizations, and further explores how mathematical ideas were transmitted from East to West. The book's scope is now even wider, incorporating recent findings on the history of mathematics in China, India, and early Islamic civilizations as well as Egypt and Mesopotamia. With more detailed coverage of proto-mathematics and the origins of trigonometry and infinity in the East, *The Crest of the Peacock* further illuminates the global history of mathematics.