This is the second part of a two volume anthology comprising a selection of 49 articles that illustrate the depth, breadth and scope of Nigel Kalton's research. Each article is accompanied by comments from an expert on the respective topic, which serves to situate the article in its proper context, to successfully link past, present and hopefully future developments of the theory and to help readers grasp the extent of Kalton's accomplishments. Kalton's work represents a bridge to the mathematics of tomorrow, and this book will help readers to cross it. Nigel Kalton (1946-2010) was an extraordinary mathematician who made major contributions to an amazingly diverse range of fields over the course of his career.

This is the first comprehensive International Handbook on the History of Mathematics Education, covering a wide spectrum of epochs and civilizations, countries and cultures. Until now, much of the research into the rich and varied history of mathematics education has remained inaccessible to the vast majority of scholars, not least because it has been written in the language, and for readers, of an individual country. And yet a historical overview, however brief, has become an indispensable element of nearly every dissertation and scholarly article. This handbook provides, for the first time, a comprehensive and systematic aid for researchers around the world in finding the information they need about historical developments in mathematics education, not only in their own countries, but globally as well. Although written primarily for mathematics educators, this handbook will also be of interest to researchers of the history of education in general, as well as specialists in cultural and even social history.

This is the first extensive biography of the influential German mathematician, Peter Gustav Lejeune Dirichlet (1805 - 1859). Dirichlet made major contributions to number theory in addition to clarifying concepts such as the representation of functions as series, the theory of convergence, and potential theory. His mathematical methodology was explicitly based on a thorough knowledge of the work of his predecessors and his belief in the underlying unity of the branches of mathematics. This unified approach is exemplified in a paper that effectively launched the field of analytic number theory. The same orientation pervaded his teaching, which had a profound influence on the work of many mathematicians of subsequent generations. Chapters dealing with his mathematical work alternate with biographical chapters that place Dirichlet's life and those of some of his notable associates in the context of the political, social, and artistic culture of the period. This book will appeal not only to mathematicians but also to historians of mathematics and sciences, and readers interested in the cultural and intellectual history of the nineteenth century.

This book provides the only critical edition and English translation of Mahmud al-Jaghmini's al-Mulakhkhas fi al-hay'a al-basita, the most widely circulated Arabic treatise on Ptolemaic astronomy ever written. Composed in the early 13th century, this introductory textbook played a crucial role in the teaching, dissemination, and institutional instruction of Islamic astronomy well into the 19th century (and beyond). Establishing the base text is a fundamental prerequisite for gaining insights into what was considered an elementary astronomical textbook in Islam and also for understanding the extensive commentary tradition that built upon it. Within this volume, the Mulakhkhas is situated within the broader context of the genre of literature termed 'ilm al-hay'a, which has become the subject of intensive research over the past 25 years. In so doing, it provides a survey of summary accounts of theoretical astronomy of Jaghmini's predecessors, both Ancient and Islamic, which could have served as potential sources for the Mulakhkhas. Jaghmini's dates (which until now remained unsettled) are established, and it is definitively shown that he composed not only the Mulakhkhas but also other scientific treatises, including the popular medical treatise al-Qanunca, during a period that has been deemed one of scientific decline and stagnation in Islamic lands. The book will be of particular interest to scholars engaged in the study of Islamic theoretical astronomy, but is accessible to a general readership interested in learning what constituted an introduction to Ptolemaic astronomy in Islamic lands.

This biography traces the life and work of Mary Fairfax Somerville, whose extraordinary mathematical talent only came to light through fortuitous circumstances. Barely taught to read and write as a child, all the science she learned and mastered was self taught. In this delightful narrative the author takes up the challenge of discovering how Somerville came to be one of the most outstanding British women scientists and, furthermore, a popular writer. Particular attention is paid to the gender aspects of Somerville's success in what was, to put it mildly, a predominantly male domain.

This unique volume presents an ecocultural and embodied perspective on understanding numbers and their history in indigenous communities. The book focuses on research carried out in Papua New Guinea and Oceania, and will help educators understand humanity's use of numbers, and their development and change. The authors focus on indigenous mathematics education in the early years and shine light on the unique processes and number systems of non-European styled cultural classrooms. This new perspective for mathematics education challenges educators who have not heard about the history of number outside of Western traditions, and can help them develop a rich cultural competence in their own practice and a new vision of foundational number concepts such as large numbers, groups, and systems. Featured in this invaluable resource are some data and analyses that chief researcher Glendon Angove Lean collected while living in Papua New Guinea before his death in 1995. Among the topics covered: The diversity of counting system cycles, where they were established, and how they may have developed. A detailed exploration of number systems other than base 10 systems including: 2-cycle, 5-cycle, 4- and 6-cycle systems, and body-part tally systems. Research collected from major studies such as Geoff Smith's and Sue Holzknecht's studies of Morobe Province's multiple counting systems, Charly Muke's study of counting in the Wahgi Valley in the Jiwaka Province, and Patricia Paraide's documentation of the number and measurement knowledge of her Tolai community. The implications of viewing early numeracy in the light of this book's research, and ways of catering to diversity in mathematics education. In this volume Kay Owens draws on recent research from diverse fields such as linguistics and archaeology to present their exegesis on the history of number reaching back ten thousand years ago. Researchers and educators interested in the history of mathematical sciences will find History of Number: Evidence from Papua New Guinea and Oceania to be an invaluable resource.

Einfuhrung.- Qualitat und Quantitat in der antiken Philosophie: Zur Genese einer Fragestellung.- Quantity and Quality in Scholastic Aristotelian Natural Philosophy: John Dumbleton's Summa Logicae et Philosophiae Naturalis.- Der Arzt Paracelsus zwischen Physica und Virtus.- (Bearbeitung: Hans Bieri; Nachwort: Gunter Scharf) Qualitat und Quantitat in Keplers Weltharmonik.- Farbenlehre bei Newton und Goethe.- Laboratoriumspraxis, Quantitaten und die Produktion von Erkenntnis: Transformationen der Chemie zwischen Lavoisier und Liebig.- Die Einheit von Mikro- und Makrokosmos: Quantitat und Qualitat in der Physik.- Qualitas und Quantitas und die beiden Hirnhalften.- Rechnen und Unterscheiden: Qualitat und Quantitat zwischen Philosophie und Wissenschaft.- Gibt es eine Wiedergeburt der Qualitat in der Mathematik?.- Literaturauswahl zur Einfuhrung.- UEber die Autoren.

This book is the first part of a two volume anthology comprising a selection of 49 articles that illustrate the depth, breadth and scope of Nigel Kalton's research. Each article is accompanied by comments from an expert on the respective topic, which serves to situate the article in its proper context, to successfully link past, present and hopefully future developments of the theory, and to help readers grasp the extent of Kalton's accomplishments. Kalton's work represents a bridge to the mathematics of tomorrow, and this book will help readers to cross it. Nigel Kalton (1946-2010) was an extraordinary mathematician who made major contributions to an amazingly diverse range of fields over the course of his career.

This book documents the process of transformation from natural philosophy, which was considered the most important of the sciences until the early modern era, into modern disciplines such as mathematics, physics, natural history, chemistry, medicine and engineering. It focuses on the 18th century, which has often been considered uninteresting for the history of science, representing the transition from the age of genius and the birth of modern science (the 17th century) to the age of prodigious development in the 19th century. Yet the 18th century, the century of Enlightenment, as will be demonstrated here, was in fact characterized by substantial ferment and novelty. To make the text more accessible, little emphasis has been placed on the precise genesis of the various concepts and methods developed in scientific enterprises, except when doing so was necessary to make them clear. For the sake of simplicity, in several situations reference is made to the authors who are famous today, such as Newton, the Bernoullis, Euler, d'Alembert, Lagrange, Lambert, Volta et al. - not necessarily because they were the most creative and original minds, but mainly because their writings represent a synthesis of contemporary and past studies. The above names should, therefore, be considered more labels of a period than references to real historical characters.

This book provides the first critical edition of Ibn al-Haytham's On the Shape of the Eclipse with English translation and commentary, which records the first scientific analysis of the camera obscura. On the Shape of the Eclipse includes pioneering research on the conditions of formation of the image, in a time deemed to be committed to aniconism. It also provides an early attempt to merge the two branches of Ancient optics-the theory of light and theory of vision. What perhaps most strongly characterizes this treatise is the close interaction of a geometric analysis of light and experimental reasoning. Ibn al-Haytham conducted his experiments in a systematic way by varying all that could be changed: the shape and size of the aperture, the focal length of the camera obscura, the distance and shape of the celestial bodies. This way, he achieved a thorough understanding. This work represents a decisive step in both the history of optics and the application of the experimental method that was just as efficient in medieval Islam as today.

This book deals with a topic that has been largely neglected by philosophers of science to date: the ability to refer and analyze in tandem. On the basis of a set of philosophical case studies involving both problems in number theory and issues concerning time and cosmology from the era of Galileo, Newton and Leibniz up through the present day, the author argues that scientific knowledge is a combination of accurate reference and analytical interpretation. In order to think well, we must be able to refer successfully, so that we can show publicly and clearly what we are talking about. And we must be able to analyze well, that is, to discover productive and explanatory conditions of intelligibility for the things we are thinking about. The book's central claim is that the kinds of representations that make successful reference possible and those that make successful analysis possible are not the same, so that significant scientific and mathematical work typically proceeds by means of a heterogeneous discourse that juxtaposes and often superimposes a variety of kinds of representation, including formal and natural languages as well as more iconic modes. It demonstrates the virtues and necessity of heterogeneity in historically central reasoning, thus filling an important gap in the literature and fostering a new, timely discussion on the epistemology of science and mathematics.

A sweeping exploration of the development and far-reaching applications of harmonic analysis such as signal processing, digital music, Fourier optics, radio astronomy, crystallography, medical imaging, spectroscopy, and more. Featuring a wealth of illustrations, examples, and material not found in other harmonic analysis books, this unique monograph skillfully blends together historical narrative with scientific exposition to create a comprehensive yet accessible work. While only an understanding of calculus is required to appreciate it, there are more technical sections that will charm even specialists in harmonic analysis. From undergraduates to professional scientists, engineers, and mathematicians, there is something for everyone here. The second edition of The Evolution of Applied Harmonic Analysis contains a new chapter on atmospheric physics and climate change, making it more relevant for today's audience. Praise for the first edition: "...can be thoroughly recommended to any reader who is curious about the physical world and the intellectual underpinnings that have lead to our expanding understanding of our physical environment and to our halting steps to control it. Everyone who uses instruments that are based on harmonic analysis will benefit from the clear verbal descriptions that are supplied." - R.N. Bracewell, Stanford University "The book under review is a unique and splendid telling of the triumphs of the fast Fourier transform. I can recommend it unconditionally... Elena Prestini... has taken one major mathematical idea, that of Fourier analysis, and chased down and described a half dozen varied areas in which Fourier analysis and the FFT are now in place. Her book is much to be applauded." - Society for Industrial and Applied Mathematics "This is not simply a book about mathematics, or even the history of mathematics; it is a story about how the discipline has been applied (to borrow Fourier's expression) to 'the public good and the explanation of natural phenomena.' ... This book constitutes a significant addition to the library of popular mathematical works, and a valuable resource for students of mathematics." - Mathematical Association of America Reviews

The second edition of this book updates and expands upon a historically important collection of mathematical problems first published in the United States by Birkhauser in 1981. These problems serve as a record of the informal discussions held by a group of mathematicians at the Scottish Cafe in Lwow, Poland, between the two world wars. Many of them were leaders in the development of such areas as functional and real analysis, group theory, measure and set theory, probability, and topology. Finding solutions to the problems they proposed has been ongoing since World War II, with prizes offered in many cases to those who are successful. In the 35 years since the first edition published, several more problems have been fully or partially solved, but even today many still remain unsolved and several prizes remain unclaimed. In view of this, the editor has gathered new and updated commentaries on the original 193 problems. Some problems are solved for the first time in this edition. Included again in full are transcripts of lectures given by Stanislaw Ulam, Mark Kac, Antoni Zygmund, Paul Erdoes, and Andrzej Granas that provide amazing insights into the mathematical environment of Lwow before World War II and the development of The Scottish Book. Also new in this edition are a brief history of the University of Wroclaw's New Scottish Book, created to revive the tradition of the original, and some selected problems from it. The Scottish Book offers a unique opportunity to communicate with the people and ideas of a time and place that had an enormous influence on the development of mathematics and try their hand on the unsolved problems. Anyone in the general mathematical community with an interest in the history of modern mathematics will find this to be an insightful and fascinating read.

This book contains around 80 articles on major writings in mathematics published between 1640 and 1940. All aspects of mathematics are covered: pure and applied, probability and statistics, foundations and philosophy. Sometimes two writings from the same period and the same subject are taken together. The biography of the author(s) is recorded, and the circumstances of the preparation of the writing are given. When the writing is of some lengths an analytical table of its contents is supplied. The contents of the writing is reviewed, and its impact described, at least for the immediate decades. Each article ends with a bibliography of primary and secondary items.
.First book of its kind
.Covers the period 1640-1940 of massive development in mathematics
.Describes many of the main writings of mathematics
.Articles written by specialists in their field

This is a volume of chapters on the historical study of information, computing, and society written by seven of the most senior, distinguished members of the History of Computing field. These are edited, expanded versions of papers presented in a distinguished lecture series in 2018 at the University of Colorado Boulder - in the shadow of the Flatirons, the front range of the Rocky Mountains. Topics range widely across the history of computing. They include the digitalization of computer and communication technologies, gender history of computing, the history of data science, incentives for innovation in the computing field, labor history of computing, and the process of standardization. Authors were given wide latitude to write on a topic of their own choice, so long as the result is an exemplary article that represents the highest level of scholarship in the field, producing articles that scholars in the field will still look to read twenty years from now. The intention is to publish articles of general interest, well situated in the research literature, well grounded in source material, and well-polished pieces of writing. The volume is primarily of interest to historians of computing, but individual articles will be of interest to scholars in media studies, communication, computer science, cognitive science, general and technology history, and business.

Introducing the Collins Modern Classics, a series featuring some of the most significant books of recent times, books that shed light on the human experience - classics which will endure for generations to come. 'Maths is one of the purest forms of thought, and to outsiders mathematicians may seem almost otherworldly' In 1963, schoolboy Andrew Wiles stumbled across the world's greatest mathematical problem: Fermat's Last Theorem. Unsolved for over 300 years, he dreamed of cracking it. Combining thrilling storytelling with a fascinating history of scientific discovery, Simon Singh uncovers how an Englishman, after years of secret toil, finally solved mathematics' most challenging problem. Fermat's Last Theorem is remarkable story of human endeavour, obsession and intellectual brilliance, sealing its reputation as a classic of popular science writing. 'To read it is to realise that there is a world of beauty and intellectual challenge that is denied to 99.9 per cent of us who are not high-level mathematicians' The Times

The present volume provides a fascinating overview of geometrical ideas and perceptions from the earliest cultures to the mathematical and artistic concepts of the 20th century. It is the English translation of the 3rd edition of the well-received German book "5000 Jahre Geometrie," in which geometry is presented as a chain of developments in cultural history and their interaction with architecture, the visual arts, philosophy, science and engineering. Geometry originated in the ancient cultures along the Indus and Nile Rivers and in Mesopotamia, experiencing its first "Golden Age" in Ancient Greece. Inspired by the Greek mathematics, a new germ of geometry blossomed in the Islamic civilizations. Through the Oriental influence on Spain, this knowledge later spread to Western Europe. Here, as part of the medieval Quadrivium, the understanding of geometry was deepened, leading to a revival during the Renaissance. Together with parallel achievements in India, China, Japan and the ancient American cultures, the European approaches formed the ideas and branches of geometry we know in the modern age: coordinate methods, analytical geometry, descriptive and projective geometry in the 17th an 18th centuries, axiom systems, geometry as a theory with multiple structures and geometry in computer sciences in the 19th and 20th centuries. Each chapter of the book starts with a table of key historical and cultural dates and ends with a summary of essential contents of geometr y in the respective era. Compelling examples invite the reader to further explore the problems of geometry in ancient and modern times. The book will appeal to mathematicians interested in Geometry and to all readers with an interest in cultural history. From letters to the authors for the German language edition I hope it gets a translation, as there is no comparable work. Prof. J. Grattan-Guinness (Middlesex University London) "Five Thousand Years of Geometry" - I think it is the most handsome book I have ever seen from Springer and the inclusion of so many color plates really improves its appearance dramatically! Prof. J.W. Dauben (City University of New York) An excellent book in every respect. The authors have successfully combined the history of geometry with the general development of culture and history. ... The graphic design is also excellent. Prof. Z. Nadenik (Czech Technical University in Prague)

Chapters "Turing and Free Will: A New Take on an Old Debate" and "Turing and the History of Computer Music" are available open access under a Creative Commons Attribution 4.0 International License via link.springer.com.

For a century, the neuron doctrine has been the basis for our concepts of nervous organization and brain function. Formulated in 1891 by Wilhelm Waldeyer, it stated that the cell theory applies to the nervous system. Santiago Ramon y Cajal, Spain's greatest scientist, was its main architect; his main tool was a capricious nerve cell stain discovered by Camillo Golgi. This book reviews the original papers on which the neuron doctrine was based, showing that the evidence came from a much wider base of contributions than is generally realized, including such diverse and brilliant personalities as Albrecht Kolliker, Sigmund Freud, Wilhelm His, August Forel, Fritdjof Nansen and Gustav Retzius. Furthermore, many questions about terminology of the parts of the neuron and about the organization of neurons into reflex pathways and networks were raised and debated, questions that remain relevant to this day. Electron microscopical studies in the 1950s appeared to confirm the classical doctrine, but subsequent studies have revealed complexities that were not anticipated. This book reviews these new studies against the background of the classical work, and suggests some new directions for revising our concept of the neuron as a basis for the functional organization of the nervous system.

This book studies the brilliant Danish 19th Century astronomer, T.N. Thiele who made important contributions to statistics, actuarial science, astronomy and mathematics. The most important of these contributions in statistics are translated into English for the first time, and the text includes comments that set his achievements in a modern and historical perspective.

The book deals with expounding the nature of Reality as it is understood in contemporary times in Quantum Physics. It also explains the classical Indian theory of Sunya in its diverse facets. Thereafter it undertakes comparison between the two which is an area of great topical interest. It is a cross-disciplinary study by erudite Indian and western scholars between traditional Indian knowledge system and contemporary researches in Physical sciences. It points out how the theory of 'Sunyata has many seminal ideas and theories in common with contemporary Quantum Physics. The learned authors have tried to dissolve the "mysteries" of Quantum Physics and resolved its "weird paradoxes" with the help of theory of Sunyata. The issue of non-separability or entanglement has been approached with the help of the Buddhist theory of Pratityasamutpada. The paradoxical situation of "wave-particle duality" has been explained with the help of Upanisadic theory of complementarity of the two opposites. The measurement problem represented by "Schrodinger's cat" has been dealt with by resorting to two forms of the calculation of probabilities. Some writers have argued for Sunyata-like non-essentialist position to understand quantum reality. To make sense of quantum theory some papers provide a happy symbiosis of technical understanding and personal meditative experience by drawing multifarious parallels. This book will be of interest to philosophically inclined physicists and philosophers with interest in quantum mechanics.

The contributions in this proceedings volume offer a new perspective on the mathematical ties between France and Italy, and reveal how mathematical developments in these two countries affected one another. The focus is above all on the Peninsula's influence on French mathematicians, counterbalancing the historically predominant perception that French mathematics was a model for Italian mathematicians. In the process, the book details a subtle network of relations between the two countries, where mathematical exchanges fit into the changing and evolving framework of Italian political and academic structures. It reconsiders the issue of nationalities in all of its complexity, an aspect often neglected in research on the history of mathematics. The works in this volume are selected contributions from a conference held in Lille and Lens (France) in November 2013 on Images of Italian Mathematics in France from Risorgimento to Fascism. The authors include respected historians of mathematics, philosophers of science, historians, and specialists for Italy and intellectual relations, ensuring the book will be of great interest to their peers.

This book addresses the mathematical rationality contained in the making of string figures. It does so by using interdisciplinary methods borrowed from anthropology, mathematics, history and philosophy of mathematics. The practice of string figure-making has long been carried out in many societies, and particularly in those of oral tradition. It consists in applying a succession of operations to a string (knotted into a loop), mostly using the fingers and sometimes the feet, the wrists or the mouth. This succession of operations is intended to generate a final figure. The book explores different modes of conceptualization of the practice of string figure-making and analyses various source material through these conceptual tools: it looks at research by mathematicians, as well as ethnographical publications, and personal fieldwork findings in the Chaco, Paraguay, and in the Trobriand Islands, Papua New Guinea, which all give evidence of the rationality that underlies this activity. It concludes that the creation of string figures may be seen as the result of intellectual processes, involving the elaboration of algorithms, and concepts such as operation, sub-procedure, iteration, and transformation.

In 1842 the Belgian mathematician Eugene Charles Catalan asked whether 8 and 9 are the only consecutive pure powers of non-zero integers. 160 years after, the question was answered affirmatively by the Swiss mathematician of Romanian origin Preda Mihailescu. In other words, 32 - 23 = 1 is the only solution of the equation xp - yq = 1 in integers x, y, p, q with xy 0 and p, q 2. In this book we give a complete and (almost) self-contained exposition of Mihailescu's work, which must be understandable by a curious university student, not necessarily specializing in Number Theory. We assume a very modest background:a standard university course of algebra, including basic Galois theory, and working knowledge of basic algebraic number theory.

Charles Dupin was a multifaceted figure in the history of France, where his life spanned several regimes. He produced an enormous number of publications in mathematics, engineering, economics, and education. Long neglected by historians, he is at last beginning to receive attention. In his youth, he championed many causes, including the education of women, perhaps because of the influence of his dynamic and learned mother. He was already very ambitious as a youth and left behind the usual youthful desires in pursuit of his goals. Dupin began as a brilliant mathematician as a student at the Ecole polytechnique in Paris and proceeded to become a fine naval engineer, that is until visits to Britain inspired him to change his course of direction. As the French industry was undergoing expansion, Dupin saw in Britain that workers were more efficient and healthier if they were educated. He greatly admired the freedom he witnessed in Britain, and this did not endear him to the French government of the Restoration. Indeed, the high honours to which he so much aspired eluded him for a considerable time. He saw the British savings banks and regular saving by workers within industry as a system to be introduced in France and one that should be propagated. As an economist, he considered the welfare of French workers as vital to an efficient industry. He was particularly concerned with the protection of children in work and the education of workers. In fact, he might be considered the father of workers' education in France. This was a subject very close to his heart and, from his early years, he devoted himself to making public lectures available to all, including women. However, Dupin's popularity declined as the importance of a thriving economy began to take precedence over the workers' needs, with the workers focusing mainly on having a living wage. This is the first published study of Charles Dupin and his entire life's work. It illuminates his work and contribution in so many spheres, as well as his contacts with other scientists and educators. His mathematics have long interested scholars in the field, and he would have been an outstanding naval engineer. He was a linguist and highly cultured; with his aesthetic sense he might well have rivaled San, but because of his driving ambition he was a great man manqu . Against a background of tremendous changes in France, he made important contributions in many areas, as evidenced by the bibliography in this book. This work will be of interest to mathematicians, historians of science, sociologists, economists, engineers, and educators.