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The mathematical genius Alan Turing (1912-1954) was one of the greatest scientists and thinkers of the 20th century. Now well known for his crucial wartime role in breaking the ENIGMA code, he was the first to conceive of the fundamental principle of the modern computer — the idea of controlling a computing machine's operations by means of coded instructions, stored in the machine's ‘memory’. In 1945, Turing drew up his revolutionary design for an electronic computing machine — his Automatic Computing Engine (‘ACE’). A pilot model of the ACE ran its first programme in 1950 and the production version, the ‘DEUCE’, went on to become a cornerstone of the fledgling British computer industry. The first ‘personal’ computer was based on Turing's ACE. This book describes Turing's struggle to build the modern computer. It contains first-hand accounts by Turing and by the pioneers of computing who worked with him. The book describes the hardware and software of the ACE and contains chapters describing Turing's path-breaking research in the fields of Artificial Intelligence (AI) and Artificial Life (A-Life).

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.

The history of mathematics is a well-studied and vibrant area of research, with books and scholarly articles published on various aspects of the subject. Yet, the history of combinatorics seems to have been largely overlooked. This book goes some way to redress this and serves two main purposes: it constitutes the first book-length survey of the history of combinatorics, and it assembles, for the first time in a single source, researches on the history of combinatorics that would otherwise be inaccessible to the general reader. Individual chapters have been contributed by sixteen experts. The book opens with an introduction to two thousand years of combinatorics. This is followed by seven chapters on early combinatorics, leading from Indian and Chinese writings on permutations to late-Renaissance publications on the arithmetical triangle. The next seven chapters trace the subsequent story, from Euler’s contributions to such wide-ranging topics as partitions, polyhedra, and latin squares to the 20th-century advances in combinatorial set theory, enumeration, and graph theory. The book concludes with some combinatorial reflections.

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 is the first of a six volume edition of the complete correspondence of John Wallis (1616-1703). 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 mathematically-orientated 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 is the second book of a six volume edition of the complete correspondence of one of the leading figures in the scientific revolution of the 17th century, the Oxford mathematician and theologian John Wallis (1616–1703). It covers the period 1660 to September 1668 and thus some of the most decisive years of political and scientific reorganization in England during that century. The volume begins shortly before the restoration of the monarchy in 1660 and witnesses the emergence of the Royal Society from scientific circles, which had existed earlier in London and Oxford. Wallis's involvement in the Royal Society stretches back to its beginnings. After its official establishment, he became one of its most active members, corresponding regularly with its secretary Henry Oldenburg and attending meetings whenever he was in London. Wallis contributed extensively to contemporary scientific debate both in England and on the continent, and many of his letters to Oldenburg on mathematical and physical topics were edited and published in the journal Philosophical Transactions to this purpose. The correspondence contained in the volume, much of which is previously unpublished, throws new light on the background to the scientific revolution and on university politics during this time. As Keeper of the Archives, Wallis was often called upon to prepare papers aimed at defending the University of Oxford's ancient rights and privileges, and was also required to spend a considerable amount of his time in London. To this extent, at least his university commitments and scientific interests were able to go hand-in-hand.

This book provides an accessible account of the rise of algebra in England from the medieval period to the later years of the 17th century. The book includes new research and is the most detailed study to date of early modern English algebra. In its structure and content this book builds on a much earlier history of algebra, A treatise of algebra, published in 1685 by John Wallis (Savilian Professor of Geometry at Oxford). This book both analyses Wallis' text and moves beyond it. Thus, it explores the reception and dissemination of important ideas from continental Europe up to the end of the 16th century, and the subsequent revolution in English mathematics in the 17th century. In particular, the book includes chapters on the work of Thomas Harriot, William Oughtred, John Pell, and William Brouncker, as well as of Wallis himself.

This book explores how the mathematics the Jesuits brought to China was reconstructed as a branch of imperial learning so that the emperor Kangxi (r. 1662–1722) could consolidate his power over the most populous empire in the world. Kangxi forced a return to the use of what became known as ‘Western’ methods in official astronomy. In his middle life he studied astronomy, musical theory, and mathematics in person, with Jesuits as his teachers. In his last years he sponsored a book that was intended to compile these three disciplines, and he set several of his sons to work on this project. All this activity formed a vital part of his plan for establishing Manchu authority over the Chinese. This book sets out to explain how and why Kangxi made the sciences a tool for laying the foundations of empire, and to show how, as part of this process, mathematics was reconstructed as a branch of imperial learning.

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.

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 wide-ranging 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 casts new light on the work of Thomas Harriot (c.1560-1621), 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.

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 ocean-going 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 easy-to-use appendices as well as exercises that originally appeared in textbooks from the eighteenth to the early twentieth centuries.

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.

The oldest known mathematical table was found in the ancient Sumerian city of Shuruppag in southern Iraq. Since then, tables have been an important feature of mathematical activity; table making and printed tabular matter are important precursors to modern computing and information processing. This book contains a series of chapters summarizing the technical, institutional, and intellectual history of mathematical tables from earliest times until the late 20th century. It covers mathematical tables (the most important computing aid for several hundred years until the 1960s), data tables (e.g., Census tables), professional tables (e.g., insurance tables), and spreadsheets — the most recent tabular innovation. This book captures the history of tables through eleven chapters. The contributors describe the various information processing techniques and artefacts whose unifying concept is ‘the mathematical table’.

The mathematician John Pell was a member of the Royal Society and one of the generation of scientists that included Boyle, Wren, and Hooke. Although he left a huge body of manuscript materials, he has remained a neglected figure, whose papers have never been properly explored. This book is a full-length study of Pell and presents an in-depth account of his life and mathematical thinking based on a detailed study of his manuscripts. It also brings to life a strange, appealing, but awkward character, whose failure to publish his discoveries was caused by powerful scruples. In addition, this book shows that the range of Pell's interests extended far beyond mathematics. He was a key member of the circle of the ‘intelligencer’ Samuel Hartlib; he prepared translations of works by Descartes and Comenius; in the 1650s he served as Cromwell's envoy to Switzerland; and in the last part of his life he was an active member of the Royal Society, interested in the whole range of its activities. The study of Pell's life and thought thus illuminates many different aspects of 17th-century intellectual life. The book is in three parts. The first is a detailed biography of Pell; the second is an extended essay on his mathematical work; the third is a richly annotated edition of his correspondence with Sir Charles Cavendish. This correspondence, which has often been cited by scholars but has never been published in full, is concerned not only with mathematics but also with optics, philosophy, and many other subjects. Conducted mainly while Pell was in the Netherlands and Cavendish was also on the Continent, it is a fascinating example of the correspondence that flourished in the 17th-century ‘Republic of Letters’.

Lord Kelvin was one of the greatest physicists of the Victorian era. Widely known for the development of the Kelvin scale of temperature measurement, Kelvin's interests ranged across thermodynamics, the age of the Earth, the laying of the first transatlantic telegraph cable, not to mention inventions such as an improved maritime compass and a sounding device, which allowed depths to be taken both quickly and while the ship was moving. He was an academic engaged in fundamental research, while also working with industry and technological advances. He corresponded and collaborated with other eminent men of science such as Stokes, Joule, Maxwell, and Helmholtz; was raised to the peerage as a result of his contributions to science, and finally buried in Westminster Abbey next to Newton. This book contains a collection of chapters covering the life and wide-ranging scientific contributions made by William Thomson, Lord Kelvin (1824-1907).

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 twenty-first century and beyond.

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 cutting-edge 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.

Charles Lutwidge Dodgson is best known for his ‘Alice’ books, Alice’s Adventures in Wonderland and Through the Looking-Glass, written under his pen-name 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 wide-ranging 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.

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 Greco-Roman 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.