This book presents the contributions of the 20th century to economic theory in a mathematical language and in historical sequence. General equilibrium is the focal point of the book; but also a number of macroeconomic models, especially with respect to the first half of the century, are considered. Dynamic models are extensively studied per se, and not merely as extensions of their static counterparts. The book with its extensive bibliography gives a broad view over the developments in mathematical economics and is therefore an invaluable source of information for researchers and students working in this field.

The focus of this book is on the birth and historical development of permutation statistical methods from the early 1920s to the near present. Beginning with the seminal contributions of R.A. Fisher, E.J.G. Pitman, and others in the 1920s and 1930s, permutation statistical methods were initially introduced to validate the assumptions of classical statistical methods. Permutation methods have advantages over classical methods in that they are optimal for small data sets and non-random samples, are data-dependent, and are free of distributional assumptions. Permutation probability values may be exact, or estimated via moment- or resampling-approximation procedures. Because permutation methods are inherently computationally-intensive, the evolution of computers and computing technology that made modern permutation methods possible accompanies the historical narrative. Permutation analogs of many well-known statistical tests are presented in a historical context, including multiple correlation and regression, analysis of variance, contingency table analysis, and measures of association and agreement. A non-mathematical approach makes the text accessible to readers of all levels.

This volume presents a selection of papers by Henry P. McKean, which illustrate the various areas in mathematics in which he has made seminal contributions. Topics covered include probability theory, integrable systems, geometry and financial mathematics. Each paper represents a contribution by Prof. McKean, either alone or together with other researchers, that has had a profound influence in the respective area.

Hermann Gunther Gramann was one of the 19th century's most remarkable scientists, but many aspects of his life have remained in the dark. This book assembles essential, first-hand information on the Gramann family. It sheds light on the family's struggle for scientific knowledge, progress and education. It puts a face on the protagonists of an exciting development in the history of science. And it highlights the peculiar set of influences which led Hermann Gramann to brilliant insights in mathematics, philology and physics.

This book of sources is meant to complement the biography of Gramann and the proceedings of the 2009 Gramann Bicentennial Conference (Birkhauser 2010). "Roots and Traces" will interest all scholars working on Hermann Gramann and related topics. It offers newly discovered pictures of family members, historical texts documenting life in this exceptional family and an English translation of these previously unpublished papers.

Text in German and English.

* Examines the history and philosophy of the mathematical sciences in a cultural context, tracing their evolution from ancient times up to the twentieth century * 176 articles contributed by authors of 18 nationalities * Chronological table of main events in the development of mathematics * Fully integrated index of people, events and topics * Annotated bibliographies of both classic and contemporary sources * Unique coverage of Ancient and non-Western traditions of mathematics

This book presents new insights into Leibniz's research on planetary theory and his system of pre-established harmony. Although some aspects of this theory have been explored in the literature, others are less well known. In particular, the book offers new contributions on the connection between the planetary theory and the theory of gravitation. It also provides an in-depth discussion of Kepler's influence on Leibniz's planetary theory and more generally, on Leibniz's concept of pre-established harmony. Three initial chapters presenting the mathematical and physical details of Leibniz's works provide a frame of reference. The book then goes on to discuss research on Leibniz's conception of gravity and the connection between Leibniz and Kepler.

Leon Ehrenpreis has been one of the leading mathematicians in the twentieth century. His contributions to the theory of partial differential equations were part of the golden era of PDEs, and led him to what is maybe his most important contribution, the Fundamental Principle, which he announced in 1960, and fully demonstrated in 1970. His most recent work, on the other hand, focused on a novel and far reaching understanding of the Radon transform, and offered new insights in integral geometry. Leon Ehrenpreis died in 2010, and this volume collects writings in his honor by a cadre of distinguished mathematicians, many of which were his collaborators.

Every age and every culture has relied on the incorporation of mathematics in their works of architecture to imbue the built environment with meaning and order. Mathematics is also central to the production of architecture, to its methods of measurement, fabrication and analysis. This two-volume edited collection presents a detailed portrait of the ways in which two seemingly different disciplines are interconnected. Over almost 100 chapters it illustrates and examines the relationship between architecture and mathematics. Contributors of these chapters come from a wide range of disciplines and backgrounds: architects, mathematicians, historians, theoreticians, scientists and educators. Through this work, architecture may be seen and understood in a new light, by professionals as well as non-professionals.Volume I covers architecture from antiquity through Egyptian, Mayan, Greek, Roman, Medieval, Inkan, Gothic and early Renaissance eras and styles. The themes that are covered range from symbolism and proportion to measurement and structural stability. From Europe to Africa, Asia and South America, the chapters span different countries, cultures and practices.

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 completeEnglish translation of Halley's reconstruction of Book VIII withsupplementary notes on the text. It also contains 1)an introduction discussing aspects of Apollonius's Conics 2) an investigation of Edmond Halley's understanding ofthe nature of his venture into ancient mathematics, and 3) anappendices 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 ofancientGreekmathematics and mathematics in the early modern period."

The book contains reproductions of the most important papers that gave birth to the first developments in nonlinear programming. Of particular interest is W. Karush's often quoted Master Thesis, which is published for the first time. The anthology includes an extensive preliminary chapter, where the editors trace out the history of mathematical programming, with special reference to linear and nonlinear programming. "

This monograph investigates the development of hydrostatics as a science. In the process, it sheds new light on the nature of science and its origins in the Scientific Revolution. Readers will come to see that the history of hydrostatics reveals subtle ways in which the science of the seventeenth century differed from previous periods. The key, the author argues, is the new insights into the concept of pressure that emerged during the Scientific Revolution. This came about due to contributions from such figures as Simon Stevin, Pascal, Boyle and Newton. The author compares their work with Galileo and Descartes, neither of whom grasped the need for a new conception of pressure. As a result, their contributions to hydrostatics were unproductive. The story ends with Newton insofar as his version of hydrostatics set the subject on its modern course. He articulated a technical notion of pressure that was up to the task. Newton compared the mathematical way in hydrostatics and the experimental way, and sided with the former. The subtleties that lie behind Newton's position throws light on the way in which developments in seventeenth-century science simultaneously involved mathematization and experimentation. This book serves as an example of the degree of conceptual change that new sciences often require. It will be of interest to those involved in the study of history and philosophy of science. It will also appeal to physicists as well as interested general readers.

Philanthropies funded by the Rockefeller family have been prominent in the social history of the twentieth century for their involvement in medicine and applied science. This book provides the first detailed study of their relatively brief but nonetheless influential foray into the field of mathematics. The careers of a generation of pathbreakers in modern mathematics, such as S.Banach, B.L.van der Waerden and Andre Weil, were decisively affected by their becoming fellows of the Rockefeller-funded International Education Board in the 1920s. To help promote cooperation between physics and mathematics Rockefeller funds supported the erection of the new Mathematical Institute in Gottingen between 1926 and 1929, while the rise of probability and mathematical statistics owes much to the creation of the Institut Henri Poincare in Paris by American philanthropy at about the same time. This account draws upon the documented evaluation processes behind these personal and institutional involvements of philanthropies. It not only sheds light on important events in the history of mathematics and physics of the 20th century but also analyzes the comparative developments of mathematics in Europe and the United States. Several of the documents are given in their entirety as significant witnesses to the gradual shift of the centre of world mathematics to the USA. This shift was strengthened by the Nazi purge of German and European mathematics after 1933 to which the Rockefeller Foundation reacted with emergency programs that subsequently contributed to the American war effort. The general historical and political background of the events discussed in this book is the mixture of competition and cooperation between the various European countries and the USA after World War I, and the consequences of the Nazi dictatorship after 1933. Ideological positions of both the philanthropists and mathematicians mattered heavily in that process. Cultural bias in the selection of fellows and of disciplines supported, and the economic predominance of American philanthropy, led among other things to a restriction of the programs to Europe and America, to an uneven consideration of European candidates, and to preferences for Americans. Political self-isolation of the Soviet Union contributed to an increasing alienation of that important mathematical culture from Western mathematics. By focussing on a number of national cultures the investigation aims to represent a step toward a true inter-cultural comparison in mathematics."

In the early modern period, a crucial transformation occurred in the classical conception of number and magnitude. Traditionally, numbers were merely collections of discrete units that measured some multiple. Magnitude, on the other hand, was usually described as being continuous, or being divisible into parts that are infinitely divisible. This traditional idea of discrete number versus continuous magnitude was challenged in the early modern period in several ways.

This detailed study explores how the development of algebraic symbolism, logarithms, and the growing practical demands for an expanded number concept all contributed to a broadening of the number concept in early modern England. An interest in solving practical problems was not, in itself, enough to cause a generalisation of the number concept. It was the combined impact of novel practical applications together with the concomitant development of such mathematical advances as algebraic notation and logarithms that produced a broadened number concept.

Karl Menger, one of the founders of dimension theory, belongs to the most original mathematicians and thinkers of the twentieth century. He was a member of the Vienna Circle and the founder of its mathematical equivalent, the Viennese Mathematical Colloquium. Both during his early years in Vienna and, after his emigration, in the United States, Karl Menger made significant contributions to a wide variety of mathematical fields, and greatly influenced some of his colleagues. The Selecta Mathematica contain Menger's major mathematical papers, based on his own selection from his extensive writings. They deal with topics as diverse as topology, geometry, analysis and algebra, as well as writings on economics, sociology, logic, philosophy and mathematical results. The two volumes are a monument to the diversity and originality of Menger's ideas.

This book traces the life of Cholesky (1875-1918), and gives his family history. After an introduction to topography, an English translation of an unpublished paper by him where he explained his method for linear systems is given, studied and replaced in its historical context. His other works, including two books, are also described as well as his involvement in teaching at a superior school by correspondence. The story of this school and its founder, Leon Eyrolles, are addressed. Then, an important unpublished book of Cholesky on graphical calculation is analyzed in detail and compared to similar contemporary publications. The biography of Ernest Benoit, who wrote the first paper where Choleskys method is explained, is provided. Various documents, highlighting the life and the personality of Cholesky, end the book."

In 1821, Augustin-Louis Cauchy (1789-1857) published a textbook, the Cours d analyse, to accompany his course in analysis at the Ecole Polytechnique. It is one of the most influential mathematics books ever written. Not only did Cauchy provide a workable definition of limits and a means to make them the basis of a rigorous theory of calculus, but he also revitalized the idea that all mathematics could be set on such rigorous foundations. Today, the quality of a work of mathematics is judged in part on the quality of its rigor, and this standard is largely due to the transformation brought about by Cauchy and the Cours d analyse.

For this translation, the authors have also added commentary, notes, references, and an index.

Metamathematics and the Philosophical Tradition is the first work to explore in such historical depth the relationship between fundamental philosophical quandaries regarding self-reference and meta-mathematical notions of consistency and incompleteness. Using the insights of twentieth-century logicians from Goedel through Hilbert and their successors, this volume revisits the writings of Aristotle, the ancient skeptics, Anselm, and enlightenment and seventeenth and eighteenth century philosophers Leibniz, Berkeley, Hume, Pascal, Descartes, and Kant to identify ways in which these both encode and evade problems of a priori definition and self-reference. The final chapters critique and extend more recent insights of late 20th-century logicians and quantum physicists, and offer new applications of the completeness theorem as a means of exploring "metatheoretical ascent" and the limitations of scientific certainty. Broadly syncretic in range, Metamathematics and the Philosophical Tradition addresses central and recurring problems within epistemology. The volume's elegant, condensed writing style renders accessible its wealth of citations and allusions from varied traditions and in several languages. Its arguments will be of special interest to historians and philosophers of science and mathematics, particularly scholars of classical skepticism, the Enlightenment, Kant, ethics, and mathematical logic.

The book presents the main features of the Wasan tradition, which is the indigenous mathematics that developed in Japan during the Edo period. (1600-1868). It begins with a description of the first mathematical textbooks published in the 17th century, then shifts to the work of the two leading mathematicians of this tradition, Seki Takakazu and Takebe Katahiro. The book provides substantial information on the historical and intellectual context, the role played by the Chinese mathematical treatises introduced at the late 16th century, and an analysis of Seki 's and Takebe 's contribution to the development of algebra and calculus in Japan.

This book honors the career of historian of mathematics J.L. Berggren, his scholarship, and service to the broader community. The first part, of value to scholars, graduate students, and interested readers, is a survey of scholarship in the mathematical sciences in ancient Greece and medieval Islam. It consists of six articles (three by Berggren himself) covering research from the middle of the 20th century to the present. The remainder of the book contains studies by eminent scholars of the ancient and medieval mathematical sciences. They serve both as examples of the breadth of current approaches and topics, and as tributes to Berggren's interests by his friends and colleagues.

In this book the author presents a comprehensive study of Diophantos' monumental work known as Arithmetika, a highly acclaimed and unique set of books within the known Greek mathematical corpus. Its author, Diophantos, is an enigmatic figure of whom we know virtually nothing. Starting with Egyptian, Babylonian and early Greek mathematics the author paints a picture of the sources the Arithmetika may have had. Life in Alexandria, where Diophantos lived, is described and, on the basis of the limited available evidence, his biography is outlined. Of Arithmetika's 13 books only 6 survive in Greek. It was not until 1971 that these were complemented by the discovery of 4 other books in an Arab translation. This allows the author to describe the structure, the contents and the mathematics of the Arithmetika in detail. Furthermore it is shown that Diophantos had a remarkable skill to solve higher degree equations. In the second part, the author draws our attention to the survival of Diophantos' work in both Arab and European mathematical cultures. Once Xylander's critical 1575 edition reached its European public, the fame of the Arithmetika grew. It was studied, translated and modified by such authors as Bombelli, Stevin and Viete. It reached its pinnacle of fame in 1621 with the publication of Bachet's translation into Latin. The marginal notes by Fermat in his copy of Diophantos, including his famous "Last Theorem," were the starting point of a whole new research subject: the theory of numbers.

In this book, several world experts present (one part of) the mathematical heritage of Kolmogorov. Each chapter treats one of his research themes or a subject invented as a consequence of his discoveries. The authors present his contributions, his methods, the perspectives he opened to us, and the way in which this research has evolved up to now. Coverage also includes examples of recent applications and a presentation of the modern prospects.

The tremendous success of indivisibles methods in geometry in the seventeenth century, responds to a vast project: installation of infinity in mathematics. The pathways by the authors are very diverse, as are the characterizations of indivisibles, but there are significant factors of unity between the various doctrines of indivisible; the permanence of the language used by all authors is the strongest sign. These efforts do not lead to the stabilization of a mathematical theory (with principles or axioms, theorems respecting these first statements, followed by applications to a set of geometric situations), one must nevertheless admire the magnitude of the results obtained by these methods and highlights the rich relationships between them and integral calculus. The present book aims to be exhaustive since it analyzes the works of all major inventors of methods of indivisibles during the seventeenth century, from Kepler to Leibniz. It takes into account the rich existing literature usually devoted to a single author. This book results from the joint work of a team of specialists able to browse through this entire important episode in the history of mathematics and to comment it. The list of authors involved in indivisibles field is probably sufficient to realize the richness of this attempt; one meets Kepler, Cavalieri, Galileo, Torricelli, Gregoire de Saint Vincent, Descartes, Roberval, Pascal, Tacquet, Lalouvere, Guldin, Barrow, Mengoli, Wallis, Leibniz, Newton.

This undergraduate textbook is intended primarily for a transition course into higher mathematics, although it is written with a broader audience in mind. The heart and soul of this book is problem solving, where each problem is carefully chosen to clarify a concept, demonstrate a technique, or to enthuse. The exercises require relatively extensive arguments, creative approaches, or both, thus providing motivation for the reader. With a unified approach to a diverse collection of topics, this text points out connections, similarities, and differences among subjects whenever possible. This book shows students that mathematics is a vibrant and dynamic human enterprise by including historical perspectives and notes on the giants of mathematics, by mentioning current activity in the mathematical community, and by discussing many famous and less well-known questions that remain open for future mathematicians.

Ideally, this text should be used for a two semester course, where the first course has no prerequisites and the second is a more challenging course for math majors; yet, the flexible structure of the book allows it to be used in a variety of settings, including as a source of various independent-study and research projects.
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The year 2007 marks the 300th anniversary of the birth of one of the Enlightenment's most important mathematicians and scientists, Leonhard Euler. This volume is a collection of 24 essays by some of the world's best Eulerian scholars from seven different countries about Euler, his life and his work.
Some of the essays are historical, including much previously unknown information about Euler's life, his activities in the St. Petersburg Academy, the influence of the Russian Princess Dashkova, and Euler's philosophy. Others describe his influence on the subsequent growth of European mathematics and physics in the 19th century. Still others give technical details of Euler's innovations in probability, number theory, geometry, analysis, astronomy, mechanics and other fields of mathematics and science.
- Over 20 essays by some of the best historians of mathematics and science, including Ronald Calinger, Peter Hoffmann, Curtis Wilson, Kim Plofker, Victor Katz, Ruediger Thiele, David Richeson, Robin Wilson, Ivor Grattan-Guinness and Karin Reich
- New details of Euler's life in two essays, one by Ronald Calinger and one he co-authored with Elena Polyakhova
- New information on Euler's work in differential geometry, series, mechanics, and other important topics including his influence in the early 19th century

In1713, PierreRem onddeMontmortwrotetothemathematicianNicolasBernoulli: It would bedesirable if someone wanted totake thetrouble toinstruct how and inwhat order the discoveries in mathematics have come about . . . The histories of painting, of music, of medicine have been written. A good history of mathematics, especially of geometry, would bea much more interesting and useful work . . . Such a work, ifdone well, could be regarded to some extent as a history of the human mind, since it is in this science, more than in anything else, that man makes known that gift of intelligence that God has given him to rise 1 above all other creatures. Ahalf-centurylater, Jean-EtienneMontuclaprovidedsuchanaccountinhisHistoire des mathem atiques ( rst printed in 1758, and reissued in a greatly expanded form 2 in 1799). Montucla's great work is generally acknowledged as the rst genuine history of mathematics. According to modern historians, previous attempts at such a history had amounted to little more than collections of anecdotes, biographies or exhaustive bibliographies: "jumbles of names, dates and titles," as one writer in the 3 Dictionary of Scienti c Biography characterizes them. Montucla, in contrast, was thoroughly animated by the Enlightenment project expressed in de Montmort's l- ter. In his Histoire he set out to provide a philosophicalhistory of the "development 4 of the human mind," as he himself described it."