This book was first published in 1970 and reprinted four years later, but since then it has sold out. The helpful comments of my colleagues have strengthened my con viction that some changes and corrections were to be done in the third edition. These are summarized below, supplementing a nearly unaltered part of the piace to the original edition. Any work on the history of science spanning a considerably long period of time has to satisfy a great number of criteria as the discipline under scrutiny has to be examined as it evolved, embedded in the intricate network of relations in the national and universal history of culture. Ta compound the problem, the rise of mathematics out of backwardness in Hungary was fraught with relapses instead of leading in a straight line to today's heights. To begin with, the author of a book on science history encounters the problem of what material to include and how to treat it. To stretch the point a little, one might say that as many authors and reviewers, as many opinions. It is almost impossible to coordinate all the divergent points of view and expectations.

Springer-Verlag has invited me to bring out my Selected Works. Being aware that Springer-Verlag enjoys high esteem in the scientific world as a reputed publisher, I have willingly accepted the offer. Immediately, I was faced with two problems. The first was that of acquaint ing the reader with the important stages in my scientific aetivities. For this purpose, I have included in the Selected Works eertain of my early works that have greatly influeneed my later studies. For the same reason, I have also in cluded in the book those works that contain the first, erude versions of the proofs for many of my basic theorems. The second problem was that of giving the reader the best possible opportunity to familiarize himself with the most important results and to learn to use my method. For this reason I have included the later improved versions of the proofs for my basic results, as weil as the monographs The Method of Trigo nometric Sums in Number Theory (Seeond Edition) and Special Variants of the Method of Trigonometric Sums."

The official book behind the Academy Award-winning film The Imitation Game, starring Benedict Cumberbatch and Keira Knightley Alan Turing was the mathematician whose cipher-cracking transformed the Second World War. Taken on by British Intelligence in 1938, as a shy young Cambridge don, he combined brilliant logic with a flair for engineering. In 1940 his machines were breaking the Enigma-enciphered messages of Nazi Germany's air force. He then headed the penetration of the super-secure U-boat communications. But his vision went far beyond this achievement. Before the war he had invented the concept of the universal machine, and in 1945 he turned this into the first design for a digital computer. Turing's far-sighted plans for the digital era forged ahead into a vision for Artificial Intelligence. However, in 1952 his homosexuality rendered him a criminal and he was subjected to humiliating treatment. In 1954, aged 41, Alan Turing took his own life.

The scientific personalities of Luigi Cremona, Eugenio Beltrami, Salvatore Pincherle, Federigo Enriques, Beppo Levi, Giuseppe Vitali, Beniamino Segre and of several other mathematicians who worked in Bologna in the century 1861-1960 are examined by different authors, in some cases providing different view points. Most contributions in the volume are historical; they are reproductions of original documents or studies on an original work and its impact on later research. The achievements of other mathematicians are investigated for their present-day importance.

In this textbook the authors present first-year geometry roughly in the order in which it was discovered. The first five chapters show how the ancient Greeks established geometry, together with its numerous practical applications, while more recent findings on Euclidian geometry are discussed as well. The following three chapters explain the revolution in geometry due to the progress made in the field of algebra by Descartes, Euler and Gauss. Spatial geometry, vector algebra and matrices are treated in chapters 9 and 10. The last chapteroffers an introduction to projective geometry, which emerged in the19thcentury.

Complemented by numerous examples, exercises, figures and pictures, the book offers both motivation and insightful explanations, and provides stimulating and enjoyable reading for students and teachers alike.
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This book contains new translations and a new analysis of the procedure texts of Babylonian mathematical astronomy, the earliest known form of mathematical astronomy of the ancient world. The translations are based on a modern approach incorporating recent insights from Assyriology and translation science.

The work contains updated and expanded interpretations of the astronomical algorithms and investigations of previously ignored linguistic, mathematical and other aspects of the procedure texts.

Special attention is paid to issues of mathematical representation and over 100 photos of cuneiform tablets dating from 350-50 BCE are presented.

In 2-3 years, theauthor intends to continue his study of Babylonian mathematicalastronomy with a new publication which will contain new editions and reconstructions of approx. 250 tabular texts and a new philological, astronomical and mathematical analysis of these texts. Tabular texts are end products of Babylonian math astronomy, computed with algorithms that are formulated in the present volume, Procedure Texts."

This book is a collection of essays on the reception of Leibniz's thinking in the sciences and in the philosophy of science in the 19th and 20th centuries. Authors studied include C.F. Gauss, Georg Cantor, Kurd Lasswitz, Bertrand Russell, Ernst Cassirer, Louis Couturat, Hans Reichenbach, Hermann Weyl, Kurt Goedel and Gregory Chaitin. In addition, we consider concepts and problems central to Leibniz's thought and that of the later authors: the continuum, space, identity, number, the infinite and the infinitely small, the projects of a universal language, a calculus of logic, a mathesis universalis etc. The book brings together two fields of research in the history of philosophy and of science (research on Leibniz, and the research concerned with some major developments in the 19th and 20th centuries); it describes how Leibniz's thought appears in the works of these authors, in order to better understand Leibniz's influence on contemporary science and philosophy; but it also assesses that reception critically, confronting it in particular with the current state of Leibniz research and with the various editions of his work.

Ernst Zermelo (1871-1953) is regarded as the founder of axiomatic set theory and is best-known for the first formulation of the axiom of choice. However, his papers also include pioneering work in applied mathematics and mathematical physics. This edition of his collected papers consists of two volumes. The present Volume II covers Ernst Zermelo's work on the calculus of variations, applied mathematics, and physics. The papers are each presented in their original language together with an English translation, the versions facing each other on opposite pages. Each paper or coherent group of papers is preceded by an introductory note provided by an acknowledged expert in the field who comments on the historical background, motivation, accomplishments, and influence.

This book, for a broad readership, examines the young Einstein from a variety of perspectives - personal, scientific, historical, and philosophical.

The focus of this book is the fundamental influence of the cyphering tradition on mathematics education in North American colleges, schools, and apprenticeship training classes between 1607 and 1861. It is the first book on the history of North American mathematics education to be written from that perspective. The principal data source is a set of 207 handwritten cyphering books that have never previously been subjected to careful historical analysis.

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 third volume comprises selected papers written between 1965 and 1979.

The 16th-Century intellectual Robert Recorde is chiefly remembered for introducing the equals sign into algebra, yet the greater significance and broader scope of his work is often overlooked.

"Robert Recorde: Tudor Polymath, Expositor and Practitioner of Computation" presents an authoritative and in-depth analysis of the man, his achievements and his historical importance. This scholarly yet accessible work examines the latest evidence on all aspects of Recorde s life, throwing new light on a character deserving of greater recognition.

Topics and features: presents a concise chronology of Recorde s life; examines his published works; "The Grounde of Artes," "The Pathway to Knowledge," "The Castle of Knowledge," and "The Whetstone of Witte"; describes Recorde s professional activities in the minting of money and the mining of silver, as well as his dispute with William Herbert, Earl of Pembroke; investigates Recorde s work as a physician, his linguistic and antiquarian interests, and his religious beliefs; discusses the influence of Recorde s publisher, Reyner Wolfe, in his life; reviews his legacy to 17th-Century science, and to modern computer science and mathematics.

This fascinating insight into a much under-appreciated figure is a must-read for researchers interested in the history of computer science and mathematics, and for scholars of renaissance studies, as well as for the general reader."

Among the group of physics honors students huddled in 1957 on a Colorado mountain watching Sputnik bisect the heavens, one young scientist was destined, three short years later, to become a key player in America s own top-secret spy satellite program. One of our era s most prolific mathematicians, Karl Gustafson was given just two weeks to write the first US spy satellite s software. The project would fundamentally alter America s Cold War strategy, and this autobiographical account of a remarkable academic life spent in the top flight tells this fascinating inside story for the first time.

Gustafson takes you from his early pioneering work in computing, through fascinating encounters with Nobel laureates and Fields medalists, to his current observations on mathematics, science and life. He tells of brushes with death, being struck by lightning, and the beautiful women who have been a part of his journey."

Asymptotic methods belong to the, perhaps, most romantic area of modern mathematics. They are widely known and have been used in me chanics, physics and other exact sciences for many, many decades. But more than this, asymptotic ideas are found in all branches of human knowledge, indeed in all areas of life. In this broader context they have not and perhaps cannot be fully formalized. However, they are mar velous, they leave room for fantasy, guesses and intuition; they bring us very near to the border of the realm of art. Many books have been written and published about asymptotic meth ods. Most of them presume a mathematically sophisticated reader. The authors here attempt to describe asymptotic methods on a more accessi ble level, hoping to address a wider range of readers. They have avoided the extreme of banishing formulae entirely, as done in some popular science books that attempt to describe mathematical methods with no mathematics. This is impossible (and not wise). Rather, the authors have tried to keep the mathematics at a moderate level. At the same time, using simple examples, they think they have been able to illustrate all the key ideas of asymptotic methods and approaches, to depict in de tail the results of their application to various branches of knowledg- from astronomy, mechanics, and physics to biology, psychology and art. The book is supplemented by several appendices, one of which con tains the profound ideas of R. G.

L.E.J. Brouwer (1881-1966) is best known for his revolutionary ideas on topology and foundations of mathematics (intuitionism). The present collection contains a mixture of letters; university and faculty correspondence has been included, some of which shed light on the student years, and in particular on the exchange of letters with his PhD adviser, Korteweg. Acting as the natural sequel to the publication of Brouwer's biography, this book provides instrumental reading for those wishing to gain a deeper understanding of Brouwer and his role in the twentieth century. Striking a good balance of biographical and scientific information, the latter deals with innovations in topology (Cantor-Schoenflies style and the new topology) and foundations. The topological period in his research is well represented in correspondence with Hilbert, Schoenflies, Poincare, Blumenthal, Lebesgue, Baire, Koebe, and foundational topics are discussed in letters exchanged with Weyl, Fraenkel, Heyting, van Dantzig and others. There is also a large part of correspondence on matters related to the interbellum scientific politics. This book will appeal to both graduate students and researchers with an interest in topology, the history of mathematics, the foundations of mathematics, philosophy and general science.

Mathematics is as much a science of the real world as biology is. It is the science of the world's quantitative aspects (such as ratio) and structural or patterned aspects (such as symmetry). The book develops a complete philosophy of mathematics that contrasts with the usual Platonist and nominalist options.

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."

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.

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 is a collection of Harald Cramer's extensive work on number theory, probability, mathematical statistics and insurance mathematics. Many of these are not easily found in their original sources nowadays, for instance his pioneering work on risk theory published in the Skandia Insurance Company's jubilee volumes in 1930 and 1955. Despite their age, these eminent examples of Cramer's expository style remain highly readable. Cramer was one of the "fathers" of modern mathematical statistics. His famous book on the subject is still an important reference. His statistical papers included here were seminal to the subsequent development of the subject. The collection includes a complete bibliography of Cramer's work.

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.

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.

This two volume set presents over 50 of the most groundbreaking contributions of Menahem M Schiffer. All of the reprints of Schiffer's works herein have extensive annotation and invited commentaries, giving new clarity and insight into the impact and legacy of Schiffer's work. A complete bibliography and brief biography make this a rounded and invaluable reference.