Hailed as one of the greatest mathematical results of the twentieth century, the recent proof of Fermat's Last Theorem by Andrew Wiles brought to public attention the enigmatic problem-solver Pierre de Fermat, who centuries ago stated his famous conjecture in a margin of a book, writing that he did not have enough room to show his "truly marvelous demonstration." Along with formulating this proposition--xn+yn=zn has no rational solution for "n" > 2--Fermat, an inventor of analytic geometry, also laid the foundations of differential and integral calculus, established, together with Pascal, the conceptual guidelines of the theory of probability, and created modern number theory. In one of the first full-length investigations of Fermat's life and work, Michael Sean Mahoney provides rare insight into the mathematical genius of a hobbyist who never sought to publish his work, yet who ranked with his contemporaries Pascal and Descartes in shaping the course of modern mathematics.

Prime numbers are beautiful, mysterious, and beguiling mathematical objects. The mathematician Bernhard Riemann made a celebrated conjecture about primes in 1859, the so-called Riemann hypothesis, which remains one of the most important unsolved problems in mathematics. Through the deep insights of the authors, this book introduces primes and explains the Riemann hypothesis. Students with a minimal mathematical background and scholars alike will enjoy this comprehensive discussion of primes. The first part of the book will inspire the curiosity of a general reader with an accessible explanation of the key ideas. The exposition of these ideas is generously illuminated by computational graphics that exhibit the key concepts and phenomena in enticing detail. Readers with more mathematical experience will then go deeper into the structure of primes and see how the Riemann hypothesis relates to Fourier analysis using the vocabulary of spectra. Readers with a strong mathematical background will be able to connect these ideas to historical formulations of the Riemann hypothesis.

Die Gesammelten Abhandlungen von Ferdinand Georg Frobenius erscheinen in drei Banden. Band I enthalt in chronologischer Abfolge seine Veroeffentlichungen von 1870 bis 1880, Band II jene von 1880 bis 1896, und Band III die Artikel von 1896 bis 1917. Band III umfasst die Veroeffentlichungen Nr. 53 bis 107. R. Brauer: ...if the reader wants to get an idea about the importance of Frobenius work today, all he has to do is to look at books and papers on groups...

Die Gesammelten Abhandlungen von Ferdinand Georg Frobenius erscheinen in drei Banden. Band I enthalt in chronologischer Abfolge seine Veroeffentlichungen von 1870 bis 1880, Band II jene von 1880 bis 1896, und Band III die Artikel von 1896 bis 1917. Band II umfasst die Artikel Nr. 22 bis 52. R. Brauer: ...if the reader wants to get an idea about the importance of Frobenius work today, all he has to do is to look at books and papers on groups...

From the Preface (K. Chandrasekharan, 1966): "The publication of this collection of papers is intended as a service to the mathematical community, as well as a tribute to the genius of CARL LUDWIG SIEGEL, who is rising seventy.In the wide range of his interests, in his capacity to uncover, to attack, and to subdue problems of great significance and difficulty, in his invention of new concepts and ideas, in his technical prowess, and in the consummate artistry of his presentation, SIEGEL resembles the classical figures of mathematics. In his combination of arithmetical, analytical, algebraical, and geometrical methods of investigation, and in his unerring instinct for the conceptual and structural, as distinct from the merely technical, aspects of any concrete problem, he represents the best type of modern mathematical thought. At once classical and modern, his work has profoundly influenced the mathematical culture of our time."Volume I includes Siegel's papers written between 1921 and 1937.

From the Preface: "The name of Hermann Weyl is enshrined in the history of mathematics. A thinker of exceptional depth, and a creator of ideas, Weyl possessed an intellect which ranged far and wide over the realm of mathematics, and beyond. His mind was sharp and quick, his vision clear and penetrating. Whatever he touched he adorned. His personality was suffused with humanity and compassion, and a keen aesthetic sensibility. Its fullness radiated charm. He was young at heart to the end. By precept and example, he inspired many mathematicians, and influenced their lives. The force of his ideas has affected the course of science. He ranks among the few universalists of our time. This collection of papers is a tribute to his genius. It is intended as a service to the mathematical community....These papers will no doubt be a source of inspirations to scholars through the ages." Volume II comprises 38 articles written between 1918 and 1926.

On the road toward a history of turbulence, this book focuses on what the actors in this research field have identified as the "turbulence problem". Turbulent flow rose to prominence as one of the most persistent challenges in science. At different times and in different social and disciplinary settings, the nature of this problem has changed in response to changing research agendas. This book does not seek to provide a comprehensive account, but instead an exemplary exposition on the environments in which problems become the subjects of research agendas, with particular emphasis on the first half of the 20th century.

This book contains a history of real and complex analysis in the nineteenth century, from the work of Lagrange and Fourier to the origins of set theory and the modern foundations of analysis. It studies the works of many contributors including Gauss, Cauchy, Riemann, and Weierstrass. This book is unique owing to the treatment of real and complex analysis as overlapping, inter-related subjects, in keeping with how they were seen at the time. It is suitable as a course in the history of mathematics for students who have studied an introductory course in analysis, and will enrich any course in undergraduate real or complex analysis.

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.

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.

Who first presented Pascal's triangle? (It was not Pascal.) Who first presented Hamiltonian graphs? (It was not Hamilton.) Who first presented Steiner triple systems? (It was not Steiner.) 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: 1) it constitutes the first book-length survey of the history of combinatorics; and 2) 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 by Donald E. Knuth 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 by the distinguished combinatorialist, Peter J. Cameron. This book is not expected to be read from cover to cover, although it can be. Rather, it aims to serve as a valuable resource to a variety of audiences. Combinatorialists with little or no knowledge about the development of their subject will find the historical treatment stimulating. A historian of mathematics will view its assorted surveys as an encouragement for further research in combinatorics. The more general reader will discover an introduction to a fascinating and too little known subject that continues to stimulate and inspire the work of scholars today.

Dieser Buchtitel ist Teil des Digitalisierungsprojekts Springer Book Archives mit Publikationen, die seit den Anfangen des Verlags von 1842 erschienen sind. Der Verlag stellt mit diesem Archiv Quellen fur die historische wie auch die disziplingeschichtliche Forschung zur Verfugung, die jeweils im historischen Kontext betrachtet werden mussen. Dieser Titel erschien in der Zeit vor 1945 und wird daher in seiner zeittypischen politisch-ideologischen Ausrichtung vom Verlag nicht beworben.

This is the first volume of a collection of papers in honor of the fiftieth birthday of Jean-Yves Beziau. These 25 papers have been written by internationally distinguished logicians, mathematicians, computer scientists, linguists and philosophers, including Arnon Avron, John Corcoran, Wilfrid Hodges, Laurence Horn, Lloyd Humbertsone, Dale Jacquette, David Makinson, Stephen Read, and Jan Wolenski. It is a state-of-the-art source of cutting-edge studies in the new interdisciplinary field of universal logic. The papers touch upon a wide range of topics including combination of logic, non-classical logic, square and other geometrical figures of opposition, categorical logic, set theory, foundation of logic, philosophy and history of logic (Aristotle, Avicenna, Buridan, Schroeder, MacColl). This book offers new perspectives and challenges in the study of logic and will be of interest to all students and researchers interested the nature and future of logic.

From the Preface: "The name of Hermann Weyl is enshrined in the history of mathematics. A thinker of exceptional depth, and a creator of ideas, Weyl possessed an intellect which ranged far and wide over the realm of mathematics, and beyond. His mind was sharp and quick, his vision clear and penetrating. Whatever he touched he adorned. His personality was suffused with humanity and compassion, and a keen aesthetic sensibility. Its fullness radiated charm. He was young at heart to the end. By precept and example, he inspired many mathematicians, and influenced their lives. The force of his ideas has affected the course of science. He ranks among the few universalists of our time. This collection of papers is a tribute to his genius. It is intended as a service to the mathematical community....These papers will no doubt be a source of inspirations to scholars through the ages." Volume III comprises 52 articles written between 1926 and 1940.

From the Preface: "The name of Hermann Weyl is enshrined in the history of mathematics. A thinker of exceptional depth, and a creator of ideas, Weyl possessed an intellect which ranged far and wide over the realm of mathematics, and beyond. His mind was sharp and quick, his vision clear and penetrating. Whatever he touched he adorned. His personality was suffused with humanity and compassion, and a keen aesthetic sensibility. Its fullness radiated charm. He was young at heart to the end. By precept and example, he inspired many mathematicians, and influenced their lives. The force of his ideas has affected the course of science. He ranks among the few universalists of our time. This collection of papers is a tribute to his genius. It is intended as a service to the mathematical community....These papers will no doubt be a source of inspirations to scholars through the ages." Volume I comprises 29 articles written between 1908 and 1917.

This classic study by the eminent Dutch historian of science E. J. Dijksterhuis (1892-1965) presents the work of the Greek mathematician and mechanical engineer to the modern reader. With meticulous scholarship, Dijksterhuis surveys the whole range of evidence on Archimedes' life and the 2000-year history of the manuscripts and editions of the text, and then undertakes a comprehensive examination of all the extant writings.

Originally published in 1987.

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 paperback editions preserve the original texts of these important books while presenting them in durable paperback 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.

Felix Hausdorff gehort zu den herausragenden Mathematikern der ersten Halfte des 20. Jahrhunderts. Er hinterliess einen ungewohnlich reichhaltigen Korpus wissenschaftlicher Manuskripe. Sein Gesamtwerk soll nun in 9 Banden, jeweils mit detaillierten Kommentaren, herausgegeben werden. Der vorliegende Band II enthalt Hausdorffs wohl wichtigstes Werk, die "Grundzuge der Mengenlehre" Dieses Buch gehort zu den Klassikern der mathematischen Literatur und hat auf die Entwicklung der Mathematik im 20. Jahrhundert einen bedeutenden Einfluss ausgeubt. Daher erschien es geboten, ausfuhrliche Kommentare beizufugen. In diesen Kommentaren werden vor allem die bedeutenden originellen Beitrage, die Hausdorff in den "Grundzugen" zur Topologie, allgemeinen und deskriptiven Mengenlehre geleistet hat, eingehend behandelt. Insbesondere wird versucht, Hausdorffs Leistungen in die historische Entwicklung einzuordnen und ihre jeweilige Wirkungsgeschichte zu skizzieren."

As any student of art will tell you, one of the chief accomplishments of the Renaissance was the development of perspective in painting--the introduction of spatial perception that led to the legendary beauty and majesty of works by Giotto, Botticelli, and da Vinci. In The Invention ofInfinity, Dr. J. V. Field, a noted historian on math and the arts, tells the remarkable story of how the "practical" mathematics of Renaissance artists actually influenced the development of "proper" mathematics--a true story of life imitating art.
Here is the fascinating history of the emergence of modern mathematics during the Renaissance, and its intimate relationship with the artisan and artistic traditions of the time. The book covers the period from 1300 to 1650, when craftsmen were educated in "practical mathematics," and when the field of mathematics was gradually taking up a more significant place on the intellectual landscape. Field traces the influence of the mathematics of perspective in the arts, and shows how this led to the invention of a new kind of geometry in the 17th century--the new projective geometry of Desargues--which proved to be a highly significant contribution to the development of modern mathematics. Additionally, the author explores the 14th and 15th-century "abacus" schools popular among merchants and craftsmen, and the contrast between these practical, widely used tools and the abstract arithmetic and geometry taught in the universities of the time, and their application in the theory of music and elementary astronomy.
Extensively illustrated with superb color and black and white plates, and including selected extracts from the original mathematical texts, this clear and entertaining account will delight anyone interested in the history of mathematics and art, as well as in the multi-layered social history of the Renaissance.

Gerhard Gentzen (1909-1945) ist der Begrunder der modernen mathematischen Beweistheorie. Die nachhaltige Bedeutung der von ihm entwickelten Methoden, Regeln und Strukturen zeigt sich heute in wichtigen Teilgebieten der Informatik, in der Verifikation von Programmen. Die Arbeiten Gentzens uber das naturliche Schliessen, der Sequenzenkalkul und die Ordinal-Beweistheorie beeindrucken noch heute durch ihre Einsicht und Eleganz. Der Autor dokumentiert in dieser ersten umfassenden Biografie Leben und Werk Gerhard Gentzens, seinen tragischen Lebensweg, Festnahme 1945 in Prag, Gefangenschaft und Tod. Die Bedingungen wissenschaftlicher Forschung, in diesem Fall der mathematischen Logik, im nationalsozialistischen Deutschland, den ideologischen Kampf um eine "Deutsche Logik" und deren Protagonisten ist ein weiterer Schwerpunkt des Buches. Zahlreiche, bislang unveroffentlichte Quellen, Fotos und Dokumente aus Korrespondenzen und Nachlass sowie der Abdruck dreier Vortrage von Gerhard Gentzen machen dieses Buch zu einer erstrangigen Informationsquelle uber diesen bedeutenden Mathematiker und seine Zeit. Der Band wird erganzt durch ein Essay von Jan von Plato uber Gentzens Beweistheorie und deren Entwicklung bis zur Gegenwart."

eine Assistentenstelle bei GERHARD HARIG am bereits 1906 gegrundeten Karl-Sudhoff-Institut fur Geschichte der Medizin und Naturwissenschaften in Leipzig, die er anderen Angeboten (z. B. beim Flugzeugbau) vorzog. Nach dem Tode von Professor HARIG bekam HANS WUSSING 1967 (als einziger habilitierter Wissenschaftshistoriker in der DDR) eine Dozentur fur Geschichte der Mathematik und der Naturwissenschaften und wurde zum kommissarischen Direktor des Sudhoff-Instituts eingesetzt. Ein Jahr spater wurde er zum a. o. Professor fur Geschichte der Mathematik und der Naturwissenschaften berufen, 1970 erfolgte die Ernennung zum ordent lichen Professor. Von 1977 bis 1982 war er Direktor des Sudhoff-Instituts und ist seit 1982 Leiter der Abteilung fur Geschichte der Mathematik und der Naturwissenschaften. Die Reihe von WUSSINGs Publikationen ist lang. Eine Liste seiner Veroffentlichungen bis 1985 findet sich in der Zeitschrift NTM, Bd. 24 (1987), S. 1-5. Es ist hier nicht der Ort, all seine Arbeiten im einzelnen zu wurdigen. Erwahnt seien nur die wichtigsten Buchpublikationen: 1962 erschien bei B. G. Teubner Leipzig die Mathematik in der Antike. WUSSING verfasste Biographien von COPERNICUS, GAUSS, NEWTON und ADAM RIES. Auch seine neueste Publikation hat mit dem bekannten deutschen Rechenmeister zu tun: Die Goss von ADAM RIES konnte er trotz schwie rigster Umstande zusammen mit WOLFGANG KAUNZNER noch rechtzeitig im Jubilaumsjahr 1992 herausgeben. WUSSING ist auch ein erfolgreicher Hochschullehrer."

Wieviele Schritte muss ein Mensch gehen und wie gefahrlich sind seine Wege, wieviel Leid ertragt ein Mensch und woher nimmt er die Kraft, um die einzige Bahn zu beschreiten, die fur ihn wichtig ist: die zur Freiheit ohne Verlust seiner Rechtschaffenheit? Egon Balas erzahlt in dieser ungemein fesselnd geschriebenen Autobiographie von den Wegen, die ihn aus Transsilvanien nach Pennsylvania fuhrten, auf denen der 1922 in Klausenburg (Kolozsvar - ung., Cluj - rum.) geborene Sohn einer ungarisch-judischen Familie zum beruhmten Mathematiker wurde, der seit 1966 in den USA lebt. Die Leser dieses Buches werden auf eine erstaunliche Lebensreise mitgenommen, sie erfahren von grossem Mut und grenzenlosem Optimismus. Egon Balas - geboren als Egon Blatt - erlebte als Heranwachsender den Zusammenbruch der alten, vermeintlich sicheren Ordnung, schloss sich 1942 der Kommunistischen Partei Ungarns an, kampfte im Untergrund gegen den Faschismus, wurde eingesperrt, gefoltert und konnte schliesslich in den Kriegswirren fliehen. Mit seiner Geschichte gibt der Autor zugleich einen Einblick in die Tragoedie der Siebenburger Juden, von denen die meisten in den Jahren 1942 bis 1944 ermordet wurden. Von den dreissig Mitgliedern der Balas-Familie uberlebten nur sieben das Inferno. Egon Balas' spatere Frau Edith gehoerte zu den wenigen, die aus Auschwitz zuruckkehrten. Nach dem zweiten Weltkrieg hatte Balas infolge seines Widerstandes wahrend des Krieges wichtige Funktionen im kommunistischen Rumanien inne, unter anderem als Sekretar der Rumanischen Gesandtschaft in London und danach als Direktor fur Wirtschaftsangelegenheiten im Aussenministerium. Er geriet zunehmend in Widerspruch zum stalinistischen Regime, sass als politischer Haftling mehr als zwei Jahre in Einzelhaft bei der rumanischen Staatssicherheit Securitate und kam erst nach Stalins Tod wieder frei. Der Leser erfahrt in diesem spannungsgeladenen Buch von den vielen uberraschenden und unerwarteten Wendungen, die das standige Auf und Ab dieses aufregenden Lebens begleiten, das reich ist an physischen und psychischen Schicksalsprufungen. Das Buch ist ein Tatsachenbericht, der sich wie ein Roman liest. Dabei wird der Leser auch in skurrile, in groteske Situationen einbezogen, die - nach beklemmenden Schilderungen - befreiend wirken. Egon Balas beschreibt in diesem Buch seinen Weg vom idealistischen jungen Kommunisten zum desillusionierten Dissidenten, der 1966 aus Rumanien auswanderte. Er schildert seine berufliche Laufbahn, die ihn - trotz aller politischen Schwierigkeiten und widrigen Umstande - von der OEkonomie zur Mathematik fuhrte.

At the time of David Hilbert's death in 1943, his leading disciple, Her- mann Weyl, wrote that " . . . the era of mathematics upon which he impressed the seal of his spirit and which is now sinking below the horizon achieved a more perfect balance than prevailed before and after, between the mastering of concrete problems and the formation of general abstract concepts. "l Weyl attributed this "happy equilibrium" in no small part to Hilbert 's work and its influence, adding that "no mathematician of equal stature has risen from our generation., 2 Surely, it would be difficult to exaggerate the importance of Hilbert's contributions to twentieth-century mathematics or even to conceive of what mathematics today would be like without them. He overturned the concep- tual framework of older fields ranging from invariant theory and algebraic number theory to the foundations of geometry. He rehabilitated the Dirich- let Principle, propelled integral equation theory to the forefront of active research, derived the field equations governing Einstein's general theory of relativity, created modern proof theory and metamathematics, and through- out his career he championed the power and efficacy of the axiomatic method not only for mathematics but for all of the exact sciences. Every educated mathematician knows something about Hilbert space, the Hilbert problems, and Hilbert 's formalist program.

One of the great algebraists of the nineteenth century, Marie Ennemond Camille Jordan (1838 1922) became known for his work on matrices, Galois theory and group theory. However, his most profound effect on how we see mathematics came through his Cours d'analyse, which appeared in three editions. Reissued here is the first edition, which was published in three volumes between 1882 and 1887. While highly influential in its time, it now appears to us a transitional work between the partially rigorous 'epsilon delta' calculus of Cauchy and his successors, and the new 'real number' analysis of Weierstrass and Cantor. The first two volumes follow the old tradition while the third volume incorporates a substantial amount of the new analysis. Ten years later, the even more influential second edition followed the new point of view from its start. Volume 1 (1882) covers differential calculus."

Die allgemeine Relativitastheorie lasst sich nur mit Hilfe des Tensorkalkuls formulieren. Diesen lernte Einstein 1912 in Form des absoluten Differentialkalkuls kennen. Dessen Schopfer war Gregorio Ricci, dem zusammen mit Sophus Lie und anderen der Ausbau der Theorie der Differentialinvarianten gelang. Der absolute Differentialkalkul passte zur allgemeinen Relativitatstheorie wie ein Schlussel zum Schloss: der in den Jahren 1884-92 von Ricci entwickelte Kalkul erfullte in der Tat genau das physikalische Konzept der allgemeinen Relativitatstheorie, das Einstein 1907-15 ausarbeitete. Ein derartiges Zusammenpassen war nur dadurch moglich, weil sowohl Ricci innerhalb der Mathematik als auch Einstein innerhalb der Physik vergleichbare Fragen stellten, namlich Fragen nach Invarianten bei speziellen Transformationen. Es wird versucht, den historischen Weg so genau wie moglich anhand der Quellen nachzuzeichnen. Neu ist die Herausarbeitung des invariantentheoretischen Aspekts, dem gegenuber die Bedeutung der Differentialgeometrie fur die Entwicklung des Tensorkalkuls in den Hintergrund treten muss."

Discrete mathematics has been rising in prominence in the past fifty years, both as a tool with practical applications and as a source of new and interesting mathematics. The topics in discrete mathematics have become so well developed that it is easy to forget that common threads connect the different areas, and it is through discovering and using these connections that progress is often made. For over fifty years, Ron Graham has been able to illuminate some of these connections and has helped to bring the field of discrete mathematics to where it is today. To celebrate his contribution, this volume brings together many of the best researchers working in discrete mathematics, including Fan Chung, Erik D. Demaine, Persi Diaconis, Peter Frankl, Alfred W. Hales, Jeffrey C. Lagarias, Allen Knutson, Janos Pach, Carl Pomerance, N. J. A. Sloane, and of course, Ron Graham himself.