The last one hundred years have seen many important achievements in the classical part of number theory. After the proof of the Prime Number Theorem in 1896, a quick development of analytical tools led to the invention of various new methods, like Brun's sieve method and the circle method of Hardy, Littlewood and Ramanujan; developments in topics such as prime and additive number theory, and the solution of Fermat s problem. Rational Number Theory in the 20th Century: From PNT to FLT offers a short survey of 20th century developments in classical number theory, documenting between the proof of the Prime Number Theorem and the proof of Fermat's Last Theorem. The focus lays upon the part of number theory that deals with properties of integers and rational numbers. Chapters are divided into five time periods, which are then further divided into subject areas. With the introduction of each new topic, developments are followed through to the present day. This book will appeal to graduate researchers and student in number theory, however the presentation of main results without technicalities will make this accessible to anyone with an interest in the area."

This publication was made possible through a bequest from my beloved late wife. United together in this present collection are those works by the author which have not previously appeared in book form. The following are excepted: Vorlesungen uber Differential und Integralrechnung (Lectures on Differential and Integral Calculus) Vols 1-3, Birkhauser Verlag, Basel (1965-1968); Aufgabensammlung zur Infinitesimalrechnung (Exercises in Infinitesimal Calculus) Vols 1, 2a, 2b, and 3, Birkhauser Verlag, Basel (1967-1977); two issues from Memorial des Sciences on Conformal Mapping (written together with C. Gattegno), Gauthier-Villars, Paris (1949); Solution of Equations in Euc1idean and Banach Spaces, Academic Press, New York (1973); and Stu- dien uber den Schottkyschen Satz (Studies on Schottky's Theorem), Wepf & Co., Basel (1931). Where corrections have had to be implemented in the text of certain papers, references to these are made at the conc1usion of each paper. In the few instances where this system does not, for technical reasons, seem appropriate, an asterisk in the page margin indicates wherever a correction is necessary and this is then given at the end of the paper. (There is one exception: the correc- tions to the paper on page 561 are presented on page 722. The works are published in 6 volumes and are arranged under 16 topic headings. Within each heading, the papers are ordered chronologically according to the date of original publication.

In recognition of professor Shiing-Shen Chern s long and distinguished service to mathematics and to the University of California, the geometers at Berkeley held an International Symposium in Global Analysis and Global Geometry in his honor in June 1979. The output of this Symposium was published in a series of three separate volumes, comprising approximately a third of Professor Chern s total publications up to 1979. Later, a fourth volume was published, focusing on papers written during the Eighties. This first volume comprises selected papers written between 1932 and 1975. In making the selections, Professor Chern gave preference to shorter and lesser-known papers."

The Origins of Husserl's Totalizing Act At noon on Monday, October 24th, 1887, Dr. Edmund G. Husserl defended the dissertation that would qualify him as a university lecturer at Halle. Entitled "On the Concept of Number," it was written under Carl Stumpf who, like Husserl, had been a student of Franz Brentano. In this, his first published philosophical work, Husserl sought to secure the foundations of mathematics by deriving its most fundamental concepts from psychical acts. In the same year, Heinrich Hertz published an article entitled, "Con cerning an Influence of Ultraviolet Light on the Electrical Discharge." The article detailed his discovery of a new "relation between two entirely different forces," those of light and electricity. Hermann von Helmholtz, whose theory guided Hertz's initial research, called it the "most important physical discovery of the century," and Hertz became an immediate sensation. He lectured on his discovery in 1889 before a general session of the German Association meeting in Heidelberg. In this lecture that, as he wrote beforehand to Emil Cohn, he was deter mined should not be "entirely unintelligible to the laity," Hertz explained that light ether and electro-magnetic forces were interdependent. He went on to tell his audience that they need not expect their senses to grant them access to these phenomena. Indeed, he said, the latter are not only insusceptible of sense perception, but are false from the standpoint of the senses."

A unique series of fascinating research papers on subjects related to the work of Niels Henrik Abel, written by some of the foremost specialists in their fields. Some of the authors have been specifically invited to present papers, discussing the influence of Abel in a mathematical-historical context. Others have submitted papers presented at the Abel Bicentennial Conference, Oslo June 3-8, 2002. The idea behind the book has been to produce a text covering a substantial part of the legacy of Abel, as perceived at the beginning of the 21st century.

In recognition of professor Shiing-Shen Chern's long and distinguished service to mathematics and to the University of California, the geometers at Berkeley held an International Symposium in Global Analysis and Global Geometry in his honor in June 1979. The output of this Symposium was published in a series of three separate volumes, comprising approximately a third of Professor Chern's total publications up to 1979. Later, this fourth volume was published, focusing on papers written during the Eighties.

Apollonius's Conics was one of the greatest works of advanced mathematics in antiquity. The work comprised eight books, of which four have come down to us in their original Greek and three in Arabic. By the time the Arabic translations were produced, the eighth book had already been lost. In 1710, Edmond Halley, then Savilian Professor of Geometry at Oxford, produced an edition of the Greek text of the Conics of Books I-IV, a translation into Latin from the Arabic versions of Books V-VII, and a reconstruction of Book VIII. The present work provides the first complete English translation of Halley's reconstruction of Book VIII with supplementary notes on the text. It also contains 1) an introduction discussing aspects of Apollonius's Conics 2) an investigation of Edmond Halley's understanding of the nature of his venture into ancient mathematics, and 3) an appendices giving a brief account of Apollonius's approach to conic sections and his mathematical techniques. This book will be of interest to students and researchers interested in the history of ancient Greek mathematics and mathematics in the early modern period.

Aus Paulis letztem Lebensjahrzehnt sind uber 2000 Briefe erhalten und in diesem grundlegenden Werk zur Physikgeschichte der Nachkriegszeit zusammengefasst. Neben der Physik wird hier auch der allgemeinere geistesgeschichtliche Hintergrund unserer Naturwissenschaft beleuchtet. Dieser Teilband enthalt wissenschaftliche Korrespondez uber grundlegende und andere allgemeine Fragen der Physik der Jahre 1955-1956. In diese Zeit fallen der Beginn der axiomatischen Feldtheorie, die Anfange von CERN, der Berner Relativitatskongress zur 50-Jahr-Feier von Einsteins Entdeckung sowie N. Bohrs 70. Geburtstag und naturlich die fruhe Geschichte des Neutrinos. Pauli und seine Briefpartner nehmen aktiv an diesen Ereignissen teil und beleuchten sie in eindrucksvoller Weise in ihrer Korrespondenz. Die reich annotierten und kommentierten Briefe sind chronologisch angeordnet und durch Verzeichnisse und Register erschlossen. From the final decade of Pauli's life, nearly 2000 letters survive. These are collected in this fundamental work on the history of physics in the post-war era. Going beyond physics, these letters also shed light on the cultural and philosophical background of our natural sciences. This part of the collection contains scientific correspondence about fundamental and other general questions concerning physics in the years 1955-1956. This period saw the beginnings of axiomatic field theories, the birth of CERN, the Berner Relativity Congress marking the 50th Anniversary of Einstein's discovery, and N. Bohr's 70th birthday. They were also the formative years of neutrino physics. Pauli and his correspondence partners played an active role in these events and illuminate them in an impressive manner in their letters. The extensively annotated and commented letters are organized chronologically and complemented by indexes and references.

How our understanding of calculus has evolved over more than three centuries, how this has shaped the way it is taught in the classroom, and why calculus pedagogy needs to change Calculus Reordered takes readers on a remarkable journey through hundreds of years to tell the story of how calculus evolved into the subject we know today. David Bressoud explains why calculus is credited to seventeenth-century figures Isaac Newton and Gottfried Leibniz, and how its current structure is based on developments that arose in the nineteenth century. Bressoud argues that a pedagogy informed by the historical development of calculus represents a sounder way for students to learn this fascinating area of mathematics. Delving into calculus's birth in the Hellenistic Eastern Mediterranean-particularly in Syracuse, Sicily and Alexandria, Egypt-as well as India and the Islamic Middle East, Bressoud considers how calculus developed in response to essential questions emerging from engineering and astronomy. He looks at how Newton and Leibniz built their work on a flurry of activity that occurred throughout Europe, and how Italian philosophers such as Galileo Galilei played a particularly important role. In describing calculus's evolution, Bressoud reveals problems with the standard ordering of its curriculum: limits, differentiation, integration, and series. He contends that the historical order-integration as accumulation, then differentiation as ratios of change, series as sequences of partial sums, and finally limits as they arise from the algebra of inequalities-makes more sense in the classroom environment. Exploring the motivations behind calculus's discovery, Calculus Reordered highlights how this essential tool of mathematics came to be.

The astonishing variety and beauty of mathematical elements in stamp design is brought to life in this collection of more than 350 stamps, illustrated with mathematical figures, people, and content, each reproduced in enlarged format, in full color. It's a perfect gift book for anyone interested in stamps, or in the surprising use of mathematics in the real world. The author is widely known in the math community for his regular column on stamps in the magazine The Mathematical Intelligencer.

Emmy Noether (1882-1935) was one of the most influential mathematicians of the 20th century. The development of abstract algebra, which is one of the most distinctive innovations of 20th century mathematics, can largely be traced back to her - in her published papers, lectures and her personal influence on her contemporaries. By now her contributions have become so thoroughly absorbed into our mathematical culture that only rarely are they specifically attributed to her. This book presents an extensive collection of her work. Albert Einstein wrote in a letter to the New York Times of May 1st, 1935: "In the judgment of the most competent living mathematicians, Fraulein Noether was the most significant creative mathematical genius thus far produced since the higher education of women began. In the realm of algebra, in which the most gifted mathematicians have been busy for centuries, she discovered methods which have proved of enormous importance in the development of the present-day younger generation of mathematicians." Emmy Noether leistete grundlegende Arbeiten zur Abstrakten Algebra. Ihre Auffassung von Mathematik war sehr nutzlich fur die damalige Physik, aber wurde auch kontrovers diskutiert. Die Debatte ging darum, ob Mathematik eher konzeptuell und abstract (intuitionistisch) oder mehr physikalisch basiert und angewandt (konstruktionistisch) sein sollte. Noethers konzeptuelle Auffassung der Algebra fuhrte zu neuen Grundlagen, die Algebra, Geometrie, Lineare Algebra, Topologie und Logik vereinheitlichten."

This monograph is concerned with the fitting of linear relationships in the context of the linear statistical model. As alternatives to the familiar least squared residuals procedure, it investigates the relationships between the least absolute residuals, the minimax absolute residual and the least median of squared residuals procedures. It is intended for graduate students and research workers in statistics with some command of matrix analysis and linear programming techniques.

During his lifetime, Henri Poincare published three major philosophical books which achieved great success: "La science et l'hypothese" (1902), "La valeur de la science" (1905) and "Science et methode" (1908). After his death in 1913, a fourth volume of his philosophical works was published by his heirs as "Dernieres pensees" (1913). The four books constitute the core of Poincare's philosophic works and were given an ovation by scientific and general public. Around 1919, Gustave Le Bon wrote to Poincare's widow. As the director of the "Bibliotheque de Philosophie Scientifique at Flammarion," he asked her permission to publish a second posthumous volume. "L'Opportunisme scientifique" was intended to be the fifth and final volume of Poincare's philosophical writings. Louis Rougier had elaborated the project, with the collaboration of Gustave Le Bon, and the approval of the philosopher Emile Boutroux and his son Pierre. Because of the reservations of the mathematician's heirs, this book was never published and Dernieres pensees remained his last philosophical book. Nevertheless Poincare's correspondence - which is kept in the Poincare Archives at University Nancy 2 - contains a large amount of documents concerning the project, its justification and the discussions between Louis Rougier and the mathematician's heirs. The aim of this book is to restore this episode, which gives some crucial informations about editorial practices of Poincare and about the posterity of his philosophic thinking."

The ancient Greeks played a fundamental role in the history of mathematics and their ideas were reused and developed in subsequent periods all the way down to the scientific revolution and beyond. In this, the first complete history for a century. Reviel Netz offers a panoramic view of the rise and influence of Greek mathematics and its significance in world history. He explores the Near Eastern antecedents and the social and intellectual developments underlying the subject's beginnings in Greece in the fifth century BCE. He leads the reader through the proofs and arguments of key figures like Archytas, Euclid and Archimedes, and considers the totality of the Greek mathematical achievement which also includes, in addition to pure mathematics, such applied fields as optics, music, mechanics and, above all, astronomy. This is the story not only of a major historical development, but of some of the finest mathematics ever created.

A glorious period of Hungarian mathematics started in 1900 when Lipot Fejer discovered the summability of Fourier series.This was followed by the discoveries of his disciples in Fourier analysis and in the theory of analytic functions. At the same time Frederic (Frigyes) Riesz created functional analysis and Alfred Haar gave the first example of wavelets. Later the topics investigated by Hungarian mathematicians broadened considerably, and included topology, operator theory, differential equations, probability, etc. The present volume, the first of two, presents some of the most remarkable results achieved in the twentieth century by Hungarians in analysis, geometry and stochastics.

The book is accessible to anyone with a minimum knowledge of mathematics. It is supplemented with an essay on the history of Hungary in the twentieth century and biographies of those mathematicians who are no longer active. A list of all persons referred to in the chapters concludes the volume. "

To mark the centenary of the 1910 to 1913 publication of the monumental Principia Mathematica by Alfred N. Whitehead and Bertrand Russell, this collection of fifteen new essays by distinguished scholars considers the influence and history of PM over the last hundred years.

In 1915 and 1916 Emmy Noether was asked by Felix Klein and David Hilbert to assist them in understanding issues involved in any attempt to formulate a general theory of relativity, in particular the new ideas of Einstein. She was consulted particularly over the difficult issue of the form a law of conservation of energy could take in the new theory, and she succeeded brilliantly, finding two deep theorems.

But between 1916 and 1950, the theorem was poorly understood and Noether's name disappeared almost entirely. People like Klein and Einstein did little more then mention her name in the various popular or historical accounts they wrote. Worse, earlier attempts which had been eclipsed by Noether's achievements were remembered, and sometimes figure in quick historical accounts of the time.

This book carries a translation of Noether's original paper into English, and then describes the strange history of its reception and the responses to her work. Ultimately the theorems became decisive in a shift from basing fundamental physics on conservations laws to basing it on symmetries, or at the very least, in thoroughly explaining the connection between these two families of ideas. The real significance of this book is that it shows very clearly how long it took before mathematicians and physicists began to recognize the seminal importance of Noether's results. This book is thoroughly researched and provides careful documentation of the textbook literature. Kosmann-Schwarzbach has thus thrown considerable light on this slow dance in which the mathematical tools necessary to study symmetry properties and conservation laws were apparently provided long before the orchestra arrives and the party begins."

The work of Professor Eduard Cech had a si~ificant influence on the development of algebraic and general topology and differential geometry. This book, which appears on the occasion of the centenary of Cech's birth, contains some of his most important papers and traces the subsequent trends emerging from his ideas. The body of the book consists of four chapters devoted to algebraic topology, Cech-Stone compactification, dimension theory and differential geometry. Each of these includes a selection of Cech's papers, a brief summary of some results which followed from his work or constituted solutions to the problems he posed, and several selected papers by various authors concerning the areas of study he initiated. The book also contains a concise biography borrowed with minor changes from the book Topological papers of E. tech, a list of Cech's publications and a very brief note on his activity in the didactics of mathematics. The editors wish to express their sincere gratitude to all who contributed to the completion and publication of this book.

This study discusses the history of the central limit theorem and related probabilistic limit theorems from about 1810 through 1950. In this context the book also describes the historical development of analytical probability theory and its tools, such as characteristic functions or moments. The central limit theorem was originally deduced by Laplace as a statement about approximations for the distributions of sums of independent random variables within the framework of classical probability, which focused upon specific problems and applications.

Making this theorem an autonomous mathematical object was very important for the development of modern probability theory.

This systematic and historical treatment of Russell's contributions to analytic philosophy, from his embrace of analysis in 1898 to his landmark theory of descriptions in 1905, draws important connections between his philosophically motivated conception of analysis and the technical apparatus he devised to facilitate analyses in mathematics

Viewed as a flashpoint of the Scientific Revolution, early modern astronomy witnessed a virtual explosion of ideas about the nature and structure of the world. This study explores these theories in a variety of intellectual settings, challenging our view of modern science as a straightforward successor to Aristotelian natural philosophy. It shows how astronomers dealt with celestial novelties by deploying old ideas in new ways and identifying more subtle notions of cosmic rationality. Beginning with the celestial spheres of Peurbach and ending with the evolutionary implications of the new star Mira Ceti, it surveys a pivotal phase in our understanding of the universe as a place of constant change that confirmed deeper patterns of cosmic order and stability.

The editors of the present series had originally intended to publish an integrated work on the history of mathematics in the nineteenth century, passing systemati cally from one discipline to another in some natural order. Circumstances beyond their control, mainly difficulties in choosing authors, led to the abandonment of this plan by the time the second volume appeared. Instead of a unified mono graph we now present to the reader a series of books intended to encompass all the mathematics of the nineteenth century, but not in the order of the accepted classification of the component disciplines. In contrast to the first two books of The Mathematics of the Nineteenth Century, which were divided into chapters, this third volume consists of four parts, more in keeping with the nature of the publication. 1 We recall that the first book contained essays on the history of mathemati 2 cal logic, algebra, number theory, and probability, while the second covered the history of geometry and analytic function theory. In the present third volume the reader will find: 1. An essay on the development of Chebyshev's theory of approximation of functions, later called "constructive function theory" by S. N. Bernshtein. This highly original essay is due to the late N. I. Akhiezer (1901-1980), the author of fundamental discoveries in this area. Akhiezer's text will no doubt attract attention not only from historians of mathematics, but also from many specialists in constructive function theory."

Moritz Pasch (1843-1930) is justly celebrated as a key figure in the history of axiomatic geometry. Less well known are his contributions to other areas of foundational research. This volume features English translations of 14 papers Pasch published in the decade 1917-1926. In them, Pasch argues that geometry and, more surprisingly, number theory are branches of empirical science; he provides axioms for the combinatorial reasoning essential to Hilbert's program of consistency proofs; he explores "implicit definition" (a generalization of definition by abstraction) and indicates how this technique yields an "empiricist" reconstruction of set theory; he argues that we cannot fully understand the logical structure of mathematics without clearly distinguishing between decidable and undecidable properties; he offers a rare glimpse into the mind of a master of axiomatics, surveying in detail the thought experiments he employed as he struggled to identify fundamental mathematical principles; and much more. This volume will: Give English speakers access to an important body of work from a turbulent and pivotal period in the history of mathematics, help us look beyond the familiar triad of formalism, intuitionism, and logicism, show how deeply we can see with the help of a guide determined to present fundamental mathematical ideas in ways that match our human capacities, will be of interest to graduate students and researchers in logic and the foundations of mathematics.