
George Gabriel Stokes was one of the most significant mathematicians and natural philosophers of the nineteenth century. Serving as Lucasian professor at Cambridge he made wideranging contributions to optics, fluid dynamics and mathematical analysis. As Secretary of the Royal Society he played a major role in the direction of British science acting as both a sounding board and a gatekeeper. Outside his own area he was a distinguished public servant and MP for Cambridge University. He was keenly interested in the relation between science and religion and wrote extensively on the matter. This edited collection of essays brings together experts in mathematics, physics and the history of science to cover the many facets of Stokes’s life in a scholarly but accessible way.

This book brings together important thinkers in mathematics, history, and philosophy to explore the relationship between mathematics and narrative. “Circles disturbed” reflect the last words of Archimedes before he was slain by a Roman soldier—“Don't disturb my circles”—words that seem to refer to two radically different concerns: that of the practical person living in the concrete world of reality, and that of the theoretician lost in a world of abstraction. Stories and theorems are, in a sense, the natural languages of these two worlds—stories representing the way we act and interact, and theorems giving us pure thought, distilled from the hustle and bustle of reality. Yet, though the voices of stories and theorems seem totally different, they share profound connections and similarities. This book delves into topics such as the way in which historical and biographical narratives shape our understanding of mathematics and mathematicians, the development of “myths of origins” in mathematics, the structure and importance of mathematical dreams, the role of storytelling in the formation of mathematical intuitions, the ways mathematics helps us organize the way we think about narrative structure, and much more.

In line with the emerging field of philosophy of mathematical practice, this book pushes the philosophy of mathematics away from questions about the reality and truth of mathematical entities and statements and toward a focus on what mathematicians actually do—and how that evolves and changes over time. How do new mathematical entities come to be? What internal, natural, cognitive, and social constraints shape mathematical cultures? How do mathematical signs form and reform their meanings? How can we model the cognitive processes at play in mathematical evolution? And how does mathematics tie together ideas, reality, and applications? This book combines philosophical, historical, and cognitive studies to paint a fully rounded image of mathematics not as an absolute ideal but as a human endeavor that takes shape in specific social and institutional contexts. The book builds on case studies to confront philosophical reconstructions and cuttingedge cognitive theories. It focuses on the contingent semiotic and interpretive dimensions of mathematical practice, rather than on mathematics' claim to universal or fundamental truths, in order to explore not only what mathematics is, but also what it could be. Along the way, the book challenges conventional views that mathematical signs represent fixed, ideal entities; that mathematical cognition is a rigid transfer of inferences between formal domains; and that mathematics' exceptional consensus is due to the subject's underlying reality. The result is a revisionist account of mathematical philosophy that will interest mathematicians, philosophers, and historians of science alike.

Boolean algebra, also called Boolean logic, is at the heart of the electronic circuitry in everything we use—from our computers and cars, to home appliances. How did a system of mathematics established in the Victorian era become the basis for such incredible technological achievements a century later? This book combines engaging problems and a colorful historical narrative to tell the remarkable story of how two men in different eras—mathematician and philosopher George Boole and electrical engineer and pioneering information theorist Claude Shannon—advanced Boolean logic and became founding fathers of the electronic communications age. The book takes readers from fundamental concepts to a deeper and more sophisticated understanding of modern digital machines, in order to explore computing and its possible limitations in the twentyfirst century and beyond.

The history of mathematics is filled with major breakthroughs resulting from solutions to recreational problems. Problems of interest to gamblers led to the modern theory of probability, for example, and surreal numbers were inspired by the game of Go. Yet even with such groundbreaking findings and a wealth of popularlevel books, research in recreational mathematics has often been neglected. This book returns with a brandnew compilation of fascinating problems and solutions in recreational mathematics. It gathers together the top experts in recreational math and presents a compelling look at board games, card games, dice, toys, computer games, and much more. The book is divided into five parts: puzzles and brainteasers, geometry and topology, graph theory, games of chance, and computational complexity. Readers will discover what origami, roulette wheels, and even the game of Trouble can teach about math. Chapters contain new results, and include short expositions on the topic's background, providing a framework for understanding the relationship between serious mathematics and recreational games. Mathematical areas explored include combinatorics, logic, graph theory, linear algebra, geometry, topology, computer science, operations research, probability, game theory, and music theory.

The history of mathematics is filled with major breakthroughs resulting from solutions to recreational problems. Problems of interest to gamblers led to the modern theory of probability, for example, and surreal numbers were inspired by the game of Go. Yet even with such groundbreaking findings and a wealth of popularlevel books exploring puzzles and brainteasers, research in recreational mathematics has often been neglected. This book brings together authors from a variety of specialties to present fascinating problems and solutions in recreational mathematics. The chapters show how sophisticated mathematics can help construct mazes that look like famous people, how the analysis of crossword puzzles has much in common with understanding epidemics, and how the theory of electrical circuits is useful in understanding the classic Towers of Hanoi puzzle. The card game SET® is related to the theory of errorcorrecting codes, and simple tictactoe takes on a new life when played on an affine plane. Inspirations for the book's wealth of problems include board games, card tricks, fake coins, flexagons, pencil puzzles, poker, and so much more. Looking at a plethora of eclectic games and puzzles, this book is sure to entertain, challenge, and inspire academic mathematicians and avid math enthusiasts alike.

This book contains complete transcriptions, with notes, of the 133 surviving letters of Charles Hutton (1737–1823). The letters span the period 1770–1823 and are drawn from nearly thirty different archives. Most have not been published before. Hutton was one of the most prominent British mathematicians of his generation. He played roles at the Royal Society, the Royal Military Academy, the Board of Longitude, the ‘philomath’ network, and elsewhere. He worked on the explosive force of gunpowder and the mean density of the earth, winning the Royal Society’s Copley Medal in 1778; he was also at the focus of a celebrated row at the Royal Society in 1784 over the place of mathematics there. He is of particular historical interest because of the variety of roles he played in British mathematics, the dexterity with which he navigated, exploited, and shaped personal and professional networks in mathematics and science, and the length and public profile of his career. Hutton corresponded nationally and internationally, and his correspondence illustrates the overlapping, intersection, and interaction of the different networks in which Hutton moved. It therefore provides new information about how Georgian mathematics was structured socially and how mathematical careers worked in that period. It provides a rare and valuable view of a mathematical culture that would substantially cease to exist when British mathematics embraced continental methods from the early nineteenth century onwards.

This book casts new light on the work of Thomas Harriot (c.15601621), an innovative thinker and practitioner in several branches of the mathematical sciences, including navigation, astronomy, optics, geometry, and algebra. On his death Harriot left behind over 4,000 manuscript sheets, but most of his work still remains unpublished. This book focuses on 140 of those sheets, those concerned with the structure and solution of equations. The original material has been carefully ordered, translated, and annotated to provide the first complete edition of Harriot's treatise, and an extended introduction provides the reader with a lucid background to the work. Illustrations from the manuscripts provide additional interest. The appendices discuss correlations between Harriot's manuscripts and those of this contemporaries, Viète, Warner, and Torporley.

Reverse mathematics is a new field that seeks to find the axioms needed to prove given theorems. Reverse mathematics began as a technical field of mathematical logic, but its main ideas have precedents in the ancient field of geometry and the early twentiethcentury field of set theory. This book offers a historical and representative view, emphasizing basic analysis and giving a novel approach to logic. It concludes that mathematics is an arena where theorems cannot always be proved outright, but in which all of their logical equivalents can be found. This creates the possibility of reverse mathematics, where one seeks equivalents that are suitable as axioms. By using a minimum of mathematical logic in a wellmotivated way, the book will engage advanced undergraduates and all mathematicians interested in the foundations of mathematics.

This book traces the development of Egyptian mathematics, from the end of the fourth millennium BC—and the earliest hints of writing and number notation—to the end of the pharaonic period in GrecoRoman times. The book surveys three thousand years of Egyptian history to present an integrated picture of theoretical mathematics in relation to the daily practices of Egyptian life and social structures. It shows that from the earliest beginnings, pharaonic civilization used numerical techniques to efficiently control and use their material resources and labor. Even during the Old Kingdom, a variety of metrological systems had already been devised. By the Middle Kingdom, procedures had been established to teach mathematical techniques to scribes in order to make them proficient administrators for their king. The book looks at counterparts to the notation of zero, suggests an explanation for the evolution of unit fractions, and analyzes concepts of arithmetic techniques. It draws connections and comparisons to Mesopotamian mathematics, examines which individuals in Egyptian society held mathematical knowledge, and considers which scribes were trained in mathematical ideas and why. Of interest to historians of mathematics, mathematicians, Egyptologists, and all those curious about Egyptian culture, the book sheds new light on a civilization's unique mathematical evolution.

The history of mathematics starts in earnest with one of Pythagoras’ most important proofs, the Pythagorean theorem. This proof was the first link in a chain of groundbreaking ideas, all interconnected with each other, that turned mathematics into an “art of the mind.” The chain continues to be extended today. There would be no computers, science, engineering, or philosophy without Pythagoras’ legacy. This book sketches an outline of that legacy by presenting and discussing ten of the greatest ideas in the mathematical chain. Its aim is to illustrate why mathematics can be designated an intellectual art, a creative enterprise that mirrors any art, from music to painting. Pythagoras actually connected music and mathematics into a theory of the world called the Harmony of the Spheres. The book is intended for a general audience, and especially those who may think that mathematics is uninteresting or boring. Each of its ten chapter ends with five exploratory puzzles that will allow readers to become engaged in some of the ideas treated in the chapter without any technical knowledge. This will allow readers to use this book as well as a collection of fairly easy math problems.

This book is a history of the development of mathematical astronomy in China, from the late third century BCE, to the early third century CE—a period often referred to as ‘early imperial China’. It narrates the changes in ways of understanding the movements of the heavens and the heavenly bodies that took place during those four and a half centuries, and tells the stories of the institutions and individuals involved in those changes. It gives clear explanations of technical practice in observation, instrumentation and calculation, and the steady accumulation of data over many years—but it centres on the activity of the individual human beings who observed the heavens, recorded what they saw, and made calculations to analyse and eventually make predictions about the motions of the celestial bodies. It is these individuals, their observations, their calculations and the words they left to us that provide the narrative thread that runs through this work. Throughout the book, the author gives clear translations of original material that allow the reader direct access to what the people in this book said about themselves and what they tried to do. This book is designed to be accessible to a broad readership interested in the history of science, the history of China and the comparative history of ancient cultures, while still being useful to specialists in the history of astronomy.

Martin Folkes (1690–1754): Newtonian, Antiquary, Connoisseur is a cultural and intellectual biography of the only President of both the Royal Society and the Society of Antiquaries. Sir Isaac Newton’s protégé, astronomer, mathematician, freemason, art connoisseur, Voltaire’s friend and Hogarth’s patron, his was an intellectually vibrant world. Folkes was possibly the bestconnected natural philosopher and antiquary of his age, an epitome of Enlightenment sociability, and yet he was a surprisingly neglected figure, the long shadow of Newton eclipsing his brilliant disciple. A complex figure, Folkes edited Newton’s posthumous works in biblical chronology, yet was a religious skeptic and one of the first members of the gentry to marry an actress. His interests were multidisciplinary, from his authorship of the first complete history of the English coinage, to works concerning ancient architecture, statistical probability, and astronomy. Rich archival material, including Folkes’s travel diary, correspondence, and his library and art collections permit reconstruction through Folkes’s eyes of what it was like to be a collector and patron, a Masonic freethinker, and antiquarian and virtuoso in the days before ‘science’ became subspecialised. Folkes’s virtuosic sensibility and possible role in the unification of the Society of Antiquaries and the Royal Society tells against the historiographical assumption that this was the age in which the ‘two cultures’ of the humanities and sciences split apart, never to be reunited. In Georgian England, antiquarianism and ‘science’ were considered largely part of the same endeavour.

Algorithms are the hidden methods that computers apply to process information and make decisions. The book tells the story of algorithms from their ancient origins to the present day and beyond. The book introduces readers to the inventors and events behind the genesis of the world’s most important algorithms. Along the way, it explains, with the aid of examples and illustrations, how the most influential algorithms work. The first algorithms were invented in Mesopotamia 4,000 years ago. The ancient Greeks refined the concept, creating algorithms for finding prime numbers and enumerating Pi. AlKhawrzmi’s 9th century books on algorithms ultimately became their conduit to the West. The invention of the electronic computer during World War II transformed the importance of the algorithm. The first computer algorithms were for military applications. In peacetime, researchers turned to grander challenges  forecasting the weather, route navigation, choosing marriage partners, and creating artificial intelligences. The success of the Internet in the 70s depended on algorithms for transporting data and correcting errors. A clever algorithm for ranking websites was the spark that ignited Google. Recommender algorithms boosted sales at Amazon and Netflix, while the EdgeRank algorithm drove Facebook’s NewsFeed. In the 21st century, an algorithm that mimics the operation of the human brain was revisited with the latest computer technology. Suddenly, algorithms attained humanlevel accuracy in object and speech recognition. An algloirthm defeated the world champion at Go  the most complex of board games. Today, algorithms for cryptocurrencies and quantum computing look set to change the world.

Charles Lutwidge Dodgson is best known for his ‘Alice’ books, Alice’s Adventures in Wonderland and Through the LookingGlass, written under his penname of Lewis Carroll. He is also remembered as a pioneer of Victorian photography. But his everyday job was a lecturer in Mathematics at Christ Church, Oxford University. What mathematics did he do? How good a mathematician was he? And how influential was his work, both at the time and since? This book investigates these questions by outlining his mathematical life, describing in an accessible way his writings in geometry, algebra, logic, the theory of voting, and recreational mathematics, and discussing his mathematical legacy. There is also a full mathematical bibliography of Dodgson’s mathematical publications. This is the first academic work that collects the research on Dodgson’s wideranging mathematical achievements into a single accessible volume, and is written by acknowledged world experts on these activities. Much material is collected here for the first time, including the results of recent research. It has been carefully edited and is presented in an introductory and accessible form with many illustrations, both explanatory and historical. Expected to become the standard work on the subject, it should be of great interest to anyone interested in Lewis Carroll, Oxford, Victorian Britain, or mathematics.

While all of us regularly use basic mathematical symbols such as those for plus, minus, and equals, few of us know that many of these symbols weren't available before the sixteenth century. What did mathematicians rely on for their work before then? And how did mathematical notations evolve into what we know today? This book explains the fascinating history behind the development of our mathematical notation system. It shows how symbols were used initially, how one symbol replaced another over time, and how written math was conveyed before and after symbols became widely adopted. Traversing mathematical history and the foundations of numerals in different cultures, the book looks at how historians have disagreed over the origins of the number system for the past two centuries. It follows the transfigurations of algebra from a rhetorical style to a symbolic one, demonstrating that most algebra before the sixteenth century was written in prose or in verse employing the written names of numerals. It also investigates the subconscious and psychological effects that mathematical symbols have had on mathematical thought, moods, meaning, communication, and comprehension. It considers how these symbols influence us (through similarity, association, identity, resemblance, and repeated imagery), how they lead to new ideas by subconscious associations, how they make connections between experience and the unknown, and how they contribute to the communication of basic mathematics. From words to abbreviations to symbols, this book shows how math evolved to the familiar forms we use today.

This book is the first of a six volume edition of the complete correspondence of John Wallis (16161703). It begins with his earliest known letters written shortly before the outbreak of the first Civil War while he was serving as a private chaplain, and ends on the eve of the restoration of the monarchy in 1660, by which time he was already an established figure within the Republic of Letters. The period covered is thus a momentous one in Wallis's life. It witnesses his election to Savilian professor of geometry at the University of Oxford in 1649 and his subsequent rise to become one of the leading mathematicians of his day, particularly through his introduction of new arithmetical approaches to Cavalieri's method of quadratures. The correspondence reflects the full breadth of his professional activities in theology and mathematics, and provides insights not only into religious debates taking place during the revolutionary years but also into the various questions with which the mathematicallyorientated scientific community was concerned. Many of the previously unpublished letters also throw light on University affairs. After his controversial election to the post of Keeper of the Archives in 1658, Wallis fought vigorously to uphold the rights of the University of Oxford whenever necessary, and to prevent as far as possible outside interference from political and religious quarters.

This book traces the rich history of spherical trigonometry, revealing how the cultures of classical Greece, medieval Islam, and the modern West used this forgotten art to chart the heavens and the Earth. Once at the heart of astronomy and oceangoing navigation for two millennia, the discipline was also a mainstay of mathematics education for centuries and taught widely until the 1950s. The book explores this exquisite branch of mathematics and its role in ancient astronomy, geography, and cartography; Islamic religious rituals; celestial navigation; polyhedra; stereographic projection; and more. The book conveys the sheer beauty of spherical trigonometry, providing readers with a new appreciation of its elegant proofs and often surprising conclusions. It is illustrated throughout with stunning historical images and informative drawings and diagrams. It also features easytouse appendices as well as exercises that originally appeared in textbooks from the eighteenth to the early twentieth centuries.

What is algebra? For some, it is an abstract language of x's and y's. For mathematics majors and professional mathematicians, it is a world of axiomatically defined constructs like groups, rings, and fields. This book considers how these two seemingly different types of algebra evolved and how they relate. The book explores the history of algebra, from its roots in the ancient civilizations of Egypt, Mesopotamia, Greece, China, and India, through its development in the medieval Islamic world and medieval and early modern Europe, to its modern form in the early twentieth century. Defining algebra originally as a collection of techniques for determining unknowns, the book traces the development of these techniques from geometric beginnings in ancient Egypt and Mesopotamia and classical Greece. It shows how similar problems were tackled in Alexandrian Greece, in China, and in India, then looks at how medieval Islamic scholars shifted to an algorithmic stage, which was further developed by medieval and early modern European mathematicians. With the introduction of a flexible and operative symbolism in the sixteenth and seventeenth centuries, algebra entered into a dynamic period characterized by the analytic geometry that could evaluate curves represented by equations in two variables, thereby solving problems in the physics of motion. This new symbolism freed mathematicians to study equations of degrees higher than two and three, ultimately leading to the present abstract era. The book follows algebra's remarkable growth through different epochs around the globe.

Medieval Europe was a meeting place for the Christian, Jewish, and Islamic civilizations, and the fertile intellectual exchange of these cultures can be seen in the mathematical developments of the time. This book presents original Latin, Hebrew, and Arabic sources of medieval mathematics, and shows their crosscultural influences. Most of the Hebrew and Arabic sources appear here in translation for the first time. Readers will discover key mathematical revelations, foundational texts, and sophisticated writings by Latin, Hebrew, and Arabicspeaking mathematicians, including Abner of Burgos's elegant arguments proving results on the conchoid—a curve previously unknown in medieval Europe; Levi ben Gershon's use of mathematical induction in combinatorial proofs; AlMuʾtaman Ibn Hūd's extensive survey of mathematics, which included proofs of Heron's Theorem and Ceva's Theorem; and Muhyī alDīn alMaghribī's interesting proof of Euclid's parallel postulate. The book includes a general introduction, section introductions, footnotes, and references.