This book, first published in 1977, discusses the Muslim contribution to mathematics during the golden age of Muslim learning from the seventh to the thirteenth century. It was during this period that Muslim culture exerted powerful economic, political and religious influence over a large part of the civilised world. The work of the Muslim scholars was by no means limited to religion, business and government. They researched and extended the theoretical and applied science of the Greeks and Romans of an earlier era in ways that preserved and strengthened man's knowledge in these important fields. Although the main object of this book is to trace the history of the Muslim contribution to mathematics during the European Dark Ages, some effort is made to explain the progress of mathematical thought and its effects upon present day culture. Certain Muslim mathematicians are mentioned because of the important nature of their ideas in the evolution of mathematical thinking during this earlier era. Muslim mathematicians invented the present arithmetical decimal system and the fundamental operations connected with it - addition, subtraction, multiplication, division, raising to a power, and extracting the square root and the cubic root. They also introduced the 'zero' symbol to Western culture which simplified considerably the entire arithmetical system and its fundamental operations; it is no exaggeration if it is said that this specific invention marks the turning point in the development of mathematics into a science.

This extensive selection of William Feller's scientific papers shows the breadth of his oeuvre as well as the historical development of his scientific interests. Six seminal papers - originally written in German - on the central limit theorem, the law of large numbers, the foundations of probability theory, stochastic processes and mathematical biology are now, for the first time, available in English. The material is accompanied by detailed scholarly comments on Feller's work and its impact, a complete bibliography, a list of his PhD students as well as a biographic sketch of his life with a sample of pictures from Feller's family album. William Feller was one of the leading mathematicians in the development of probability theory in the 20th century. His work continues to be highly influential, in particular in the theory of stochastic processes, limit theorems and applications of mathematics to biology. These volumes will be of value to all those interested in probability t heory, analysis, mathematical biology and the history of mathematics.

There exist literary histories of probability and scientific histories of probability, but it has generally been thought that the two did not meet. Campe begs to differ. Mathematical probability, he argues, took over the role of the old probability of poets, orators, and logicians, albeit in scientific terms. Indeed, mathematical probability would not even have been possible without the other probability, whose roots lay in classical antiquity.
"The Game of Probability" revisits the seventeenth and eighteenth-century "probabilistic revolution," providing a history of the relations between mathematical and rhetorical techniques, between the scientific and the aesthetic. This was a revolution that overthrew the "order of things," notably the way that science and art positioned themselves with respect to reality, and its participants included a wide variety of people from as many walks of life. Campe devotes chapters to them in turn. Focusing on the interpretation of games of chance as the model for probability and on the reinterpretation of aesthetic form as verisimilitude (a critical question for theoreticians of that new literary genre, the novel), the scope alone of Campe's book argues for probability's crucial role in the constitution of modernity.

There exist literary histories of probability and scientific histories of probability, but it has generally been thought that the two did not meet. Campe begs to differ. Mathematical probability, he argues, took over the role of the old probability of poets, orators, and logicians, albeit in scientific terms. Indeed, mathematical probability would not even have been possible without the other probability, whose roots lay in classical antiquity.
"The Game of Probability" revisits the seventeenth and eighteenth-century probabilistic revolution, providing a history of the relations between mathematical and rhetorical techniques, between the scientific and the aesthetic. This was a revolution that overthrew the order of things, notably the way that science and art positioned themselves with respect to reality, and its participants included a wide variety of people from as many walks of life. Campe devotes chapters to them in turn. Focusing on the interpretation of games of chance as the model for probability and on the reinterpretation of aesthetic form as verisimilitude (a critical question for theoreticians of that new literary genre, the novel), the scope alone of Campe's book argues for probability's crucial role in the constitution of modernity.

Essentials of Mathematical Thinking addresses the growing need to better comprehend mathematics today. Increasingly, our world is driven by mathematics in all aspects of life. The book is an excellent introduction to the world of mathematics for students not majoring in mathematical studies. The author has written this book in an enticing, rich manner that will engage students and introduce new paradigms of thought. Careful readers will develop critical thinking skills which will help them compete in today's world. The book explains: What goes behind a Google search algorithm How to calculate the odds in a lottery The value of Big Data How the nefarious Ponzi scheme operates Instructors will treasure the book for its ability to make the field of mathematics more accessible and alluring with relevant topics and helpful graphics. The author also encourages readers to see the beauty of mathematics and how it relates to their lives in meaningful ways.

Originally published in 1950, this book was based on a short series of lectures given by the author at the University of Illinois in 1948. Aimed at the non-specialist, the chief aim of the text was to provide a general introduction to contemporary developments in the field of calculating instruments and machines. But there is some treatment of the historical side of the subject, with appreciation shown for the vision and foresight of key pioneers Charles Babbage and Lord Kelvin. This is a concise and informative volume that will be of value to anyone with an interest in the development and history of computation.

Der Band 1A beginnt mit einem Vorwort zur Gesamtedition. Den Hauptteil des Bandes bilden Hausdorffs Arbeiten uber geordnete Mengen aus den Jahren 1901-1909. Diese haben der Entwicklung der Mengenlehre nachhaltige Impulse verliehen. Sie enthalten zahlreiche fur die Untersuchung geordneter Mengen grundlegende neue Begriffe sowie tiefliegendere Resultate. Alle diese Arbeiten sind sorgfaltig kommentiert. Die Kommentare zeigen, dass einige von Hausdorff's Ideen und Resultaten fur die moderne Grundlagenforschung hochaktuell sind.

Ferner enthalt der Band Hausdorff's kritische Besprechung von Russells "The Principles of Mathematics," aus dem Nachlass seine Vorlesung "Mengenlehre" von 1901 (eine der ersten Vorlesungen uber dieses Gebiet uberhaupt) sowie einen Essay "Hausdorff als akademischer Lehrer."

Theory of Conics, Geometrical Constructions and Practical Geometry: A History of Arabic Sciences and Mathematics Volume 3, provides a unique primary source on the history and philosophy of mathematics and science from the mediaeval Arab world. The present text is complemented by two preceding volumes of A History of Arabic Sciences and Mathematics, which focused on founding figures and commentators in the ninth and tenth centuries, and the historical and epistemological development of 'infinitesimal mathematics' as it became clearly articulated in the oeuvre of Ibn al-Haytham. This volume examines the increasing tendency, after the ninth century, to explain mathematical problems inherited from Greek times using the theory of conics. Roshdi Rashed argues that Ibn al-Haytham completes the transformation of this 'area of activity,' into a part of geometry concerned with geometrical constructions, dealing not only with the metrical properties of conic sections but with ways of drawing them and properties of their position and shape. Including extensive commentary from one of world's foremost authorities on the subject, this book contributes a more informed and balanced understanding of the internal currents of the history of mathematics and the exact sciences in Islam, and of its adaptive interpretation and assimilation in the European context. This fundamental text will appeal to historians of ideas, epistemologists and mathematicians at the most advanced levels of research.

This vividly illustrated history of the International Congress of Mathematicians - a meeting of mathematicians from around the world held roughly every four years - acts as a visual history of the 25 congresses held between 1897 and 2006, as well as a story of changes in the culture of mathematics over the past century. Because the congress is an international meeting, looking at its history allows us a glimpse into the effect of wars and strained relations between nations on the scientific community.

Statistics-driven thinking is ubiquitous in modern society. In this ambitious and sophisticated study of the history of statistics, which begins with probability theory in the seventeenth century, Alain Desrosieres shows how the evolution of modern statistics has been inextricably bound up with the knowledge and power of governments. He traces the complex reciprocity between modern governments and the mathematical artifacts that both dictate the duties of the state and measure its successes.

No other work, in any language, covers such a broad spectrum--probability, mathematical statistics, psychology, economics, sociology, surveys, public health, medical statistics--in accurately synthesizing the history of statistics, with an emphasis on the conceptual development of social statistics, culminating in twentieth-century applied econometrics.

Al-Khwarizmi was a mathematician, astronomer and geographer. He worked most of his life as a scholar in the House of Wisdom in Baghdad during the first half of the 9th century and is considered by many to be the father of algebra. His Algebra (Kitab al-Jabr wa-al-muqabala), written around 820, was the first scientific text in history to systematically present algebra as a mathematical discipline that is independent of geometry and arithmetic. This groundbreaking work is divided into two main sections: one dealing with algebraic theory, and the other focusing on the calculation of inheritances and legacies. Al-Khwarizmi's book laid down the groundwork for a scientific field where mathematics and juridical learning meet, which was furthermore developed through the efforts of successive generations of mathematicians and jurists. This text also highlighted for the first time the deep-rooted possibilities in algebra to extend the use of mathematical disciplines from one to another, such as the application of arithmetic to algebra, or of geometry into algebra, and vice-versa for these three disciplines into one another; hence opening up novel areas of mathematical research. Latin translations of al-Khwarizmi's book began in the 12th century, and these texts held a continuous influence over algebra and mathematics until the 16th century.

Heavenly Mathematics 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. Glen Van Brummelen 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. He conveys the sheer beauty of spherical trigonometry, providing readers with a new appreciation of its elegant proofs and often surprising conclusions. Heavenly Mathematics is illustrated throughout with stunning historical images and informative drawings and diagrams. This unique compendium also features easy-to-use appendixes as well as exercises that originally appeared in textbooks from the eighteenth to the early twentieth centuries.

"Plato's Ghost" is the first book to examine the development of mathematics from 1880 to 1920 as a modernist transformation similar to those in art, literature, and music. Jeremy Gray traces the growth of mathematical modernism from its roots in problem solving and theory to its interactions with physics, philosophy, theology, psychology, and ideas about real and artificial languages. He shows how mathematics was popularized, and explains how mathematical modernism not only gave expression to the work of mathematicians and the professional image they sought to create for themselves, but how modernism also introduced deeper and ultimately unanswerable questions.

"Plato's Ghost" evokes Yeats's lament that any claim to worldly perfection inevitably is proven wrong by the philosopher's ghost; Gray demonstrates how modernist mathematicians believed they had advanced further than anyone before them, only to make more profound mistakes. He tells for the first time the story of these ambitious and brilliant mathematicians, including Richard Dedekind, Henri Lebesgue, Henri Poincare, and many others. He describes the lively debates surrounding novel objects, definitions, and proofs in mathematics arising from the use of naive set theory and the revived axiomatic method--debates that spilled over into contemporary arguments in philosophy and the sciences and drove an upsurge of popular writing on mathematics. And he looks at mathematics after World War I, including the foundational crisis and mathematical Platonism.

"Plato's Ghost" is essential reading for mathematicians and historians, and will appeal to anyone interested in the development of modern mathematics."

More than three centuries after its creation, calculus remains a dazzling intellectual achievement and the gateway to 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. Now with a new preface by the author, this book documents 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, subtlest problems imaginable. In touring The Calculus Gallery, we can see how it all came to be.

While Eugenio Calabi is best known for his contributions to the theory of Calabi-Yau manifolds, this Steele-Prize-winning geometer's fundamental contributions to mathematics have been far broader and more diverse than might be guessed from this one aspect of his work. His works have deep influence and lasting impact in global differential geometry, mathematical physics and beyond. By bringing together 47 of Calabi's important articles in a single volume, this book provides a comprehensive overview of his mathematical oeuvre, and includes papers on complex manifolds, algebraic geometry, Kahler metrics, affine geometry, partial differential equations, several complex variables, group actions and topology. The volume also includes essays on Calabi's mathematics by several of his mathematical admirers, including S.K. Donaldson, B. Lawson and S.-T. Yau, Marcel Berger; and Jean Pierre Bourguignon. This book is intended for mathematicians and graduate students around the world. Calabi's visionary contributions will certainly continue to shape the course of this subject far into the future.

Did you know there are 17 possible types of symmetric wallpaper pattern? Do you know what 'casting out the nines' is? Or why 88 is the fourth 'untouchable' number? Or how 7 is used to test for the onset of dementia. Number fanatic Derrick Niederman has a mission to bring numbers to life. He explores the unique properties of the most exciting numbers from 1 to 200, wherever they may crop up: from mathematics to sport, from history to the natural world, from language to pop culture. Packed with illustrations, amusing facts, puzzles, brainteasers and anecdotes, this is an enthralling and thought-provoking numerical voyage through the history of mathematics, investigating problems of logic, geometry and arithmetic along the way. ***PRAISE FOR THE REMARKABLE LIVES OF NUMBERS*** 'A hugely entertaining pick-and-mix of history, culture and mathematical puzzles.' BBC Focus 'This book is a complete joy. It made me smile. A lot.' Carol Vorderman 'Entertaining and engaging... Once you start reading it's just like the number system itself - impossible to stop.' Ian Stewart 'A fun book... definitely challenging.' Vanity Fair 'All sorts of fascinating mathematical minutiae.' Time Out

Leonhard 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. "Euler's Gem" tells the illuminating story of this indispensable mathematical idea.

From ancient Greek geometry to today's cutting-edge research, Euler's Gem celebrates the discovery of Euler's beloved polyhedron formula and its far-reaching impact on topology, the study of shapes. In 1750, Euler observed that any polyhedron composed of "V" vertices, "E" edges, and "F" faces satisfies the equation "V"-"E"+"F"=2. David Richeson tells how the Greeks missed the formula entirely; how Descartes almost discovered it but fell short; how nineteenth-century mathematicians widened the formula's scope in ways that Euler never envisioned by adapting it for use with doughnut shapes, smooth surfaces, and higher dimensional shapes; and how twentieth-century mathematicians discovered that every shape has its own Euler's formula. Using wonderful examples and numerous illustrations, Richeson presents the formula's many elegant and unexpected applications, such as showing why there is always some windless spot on earth, how to measure the acreage of a tree farm by counting trees, and how many crayons are needed to color any map.

Filled with a who's who of brilliant mathematicians who questioned, refined, and contributed to a remarkable theorem's development, "Euler's Gem" will fascinate every mathematics enthusiast.

This eminently readable book focuses on the people of mathematics and draws the reader into their fascinating world. In a monumental address, given to the International Congress of Mathematicians in Paris in 1900, David Hilbert, perhaps the most respected mathematician of his time, developed a blueprint for mathematical research in the new century. Jokingly called a natural introduction to thesis writing with examples, this collection of problems has indeed become a guiding inspiration to many mathematicians, and those who succeeded in solving or advancing their solutions form an Honors Class among research mathematicians of this century. In a remarkable labor of love and with the support of many of the major players in the field, Ben Yandell has written a fascinating account of the achievements of this Honors Class, covering mathematical substance and biographical aspects.

In 1913, Russian imperial marines stormed an Orthodox monastery at Mt. Athos, Greece, to haul off monks engaged in a dangerously heretical practice known as Name Worshipping. Exiled to remote Russian outposts, the monks and their mystical movement went underground. Ultimately, they came across Russian intellectuals who embraced Name Worshipping and who would achieve one of the biggest mathematical breakthroughs of the twentieth century, going beyond recent French achievements.

Loren Graham and Jean-Michel Kantor take us on an exciting mathematical mystery tour as they unravel a bizarre tale of political struggles, psychological crises, sexual complexities, and ethical dilemmas. At the core of this book is the contest between French and Russian mathematicians who sought new answers to one of the oldest puzzles in math: the nature of infinity. The French school chased rationalist solutions. The Russian mathematicians, notably Dmitri Egorov and Nikolai Luzin who founded the famous Moscow School of Mathematics were inspired by mystical insights attained during Name Worshipping. Their religious practice appears to have opened to them visions into the infinite and led to the founding of descriptive set theory.

The men and women of the leading French and Russian mathematical schools are central characters in this absorbing tale that could not be told until now. "Naming Infinity" is a poignant human interest story that raises provocative questions about science and religion, intuition and creativity.

This Element aims to present an outline of mathematics and its history, with particular emphasis on events that shook up its philosophy. It ranges from the discovery of irrational numbers in ancient Greece to the nineteenth- and twentieth-century discoveries on the nature of infinity and proof. Recurring themes are intuition and logic, meaning and existence, and the discrete and the continuous. These themes have evolved under the influence of new mathematical discoveries and the story of their evolution is, to a large extent, the story of philosophy of mathematics.

One of the most important mathematical achievements of the past several decades has been A. Grothendieck's work on algebraic geometry. In the early 1960s, he and M. Artin introduced etale cohomology in order to extend the methods of sheaf-theoretic cohomology from complex varieties to more general schemes. This work found many applications, not only in algebraic geometry, but also in several different branches of number theory and in the representation theory of finite and p-adic groups. Yet until now, the work has been available only in the original massive and difficult papers. In order to provide an accessible introduction to etale cohomology, J. S. Milne offers this more elementary account covering the essential features of the theory. The author begins with a review of the basic properties of flat and etale morphisms and of the algebraic fundamental group. The next two chapters concern the basic theory of etale sheaves and elementary etale cohomology, and are followed by an application of the cohomology to the study of the Brauer group. After a detailed analysis of the cohomology of curves and surfaces, Professor Milne proves the fundamental theorems in etale cohomology -- those of base change, purity, Poincare duality, and the Lefschetz trace formula. He then applies these theorems to show the rationality of some very general L-series. Originally published in 1980. The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.

This book is an attempt to describe the gradual development of the major schools of research on number theory in South India, Punjab, Mumbai, Bengal, and Bihar-including the establishment of Tata Institute of Fundamental Research (TIFR), Mumbai, a landmark event in the history of research of number theory in India. Research on number theory in India during modern times started with the advent of the iconic genius Srinivasa Ramanujan, inspiring mathematicians around the world. This book discusses the national and international impact of the research made by Indian number theorists. It also includes a carefully compiled, comprehensive bibliography of major 20th century Indian number theorists making this book important from the standpoint of historic documentation and a valuable resource for researchers of the field for their literature survey. This book also briefly discusses the importance of number theory in the modern world of mathematics, including applications of the results developed by indigenous number theorists in practical fields. Since the book is written from the viewpoint of the history of science, technical jargon and mathematical expressions have been avoided as much as possible.

Welcher Mathematiker berechnete das Datum des Weltuntergangs? Wer war die Frau, die ihre Liebe zur Mathematik durch Tapeten entdeckte? Welcher Pionier des Computerzeitalters trug seinen Pyjama unter dem Sakko? Und welche Mathematikerin musste sich als Mann ausgeben, um studieren zu koennen? Das erfahren Sie in diesem Buch. Sie lernen nicht, was eine abelsche Varietat ist oder wann genau Emmy Noether Abitur machte. Das koennen Sie in Lehrbuchern lesen oder auf Wikipedia nachschlagen. Hier geht es um die Menschen hinter der Mathematik: Wie haben sie gelebt, was hat sie bewegt und wie sahen sie aus? Der oft leider geschichtslos vermittelten Mathematik wird durch 111 gezeichnete Portrats und Geschichten ein Gesicht gegeben. Es geht also um Mathematik, jedoch (keine Angst!) nicht um die "richtige" Mathematik mit Formeln und Herleitungen, sondern um ihren Platz in der Kultur und in der Geschichte - und um das, was sie mit den Menschen macht, die sie machen. Im Buch geht es locker zu und Sie koennen es nach Lust und Laune irgendwo aufschlagen und dort einfach mit dem Lesen anfangen. Damit Sie sich - im wahrsten Sinne des Wortes - ein Bild machen koennen.

What do pure mathematicians do, and why do they do it? Looking beyond the conventional answers--for the sake of truth, beauty, and practical applications--this book offers an eclectic panorama of the lives and values and hopes and fears of mathematicians in the twenty-first century, assembling material from a startlingly diverse assortment of scholarly, journalistic, and pop culture sources.

Drawing on his personal experiences and obsessions as well as the thoughts and opinions of mathematicians from Archimedes and Omar Khayyam to such contemporary giants as Alexander Grothendieck and Robert Langlands, Michael Harris reveals the charisma and romance of mathematics as well as its darker side. In this portrait of mathematics as a community united around a set of common intellectual, ethical, and existential challenges, he touches on a wide variety of questions, such as: Are mathematicians to blame for the 2008 financial crisis? How can we talk about the ideas we were born too soon to understand? And how should you react if you are asked to explain number theory at a dinner party?

Disarmingly candid, relentlessly intelligent, and richly entertaining, "Mathematics without Apologies" takes readers on an unapologetic guided tour of the mathematical life, from the philosophy and sociology of mathematics to its reflections in film and popular music, with detours through the mathematical and mystical traditions of Russia, India, medieval Islam, the Bronx, and beyond."