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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- Following on from the 2000 edition of Jan De Witt's Elementa Curvarum Linearum, Liber Primus, this book provides the accompanying translation of the second volume of Elementa Curvarum Linearum (Foundations of Curved Lines). One of the first books to be published on Analytic Geometry, it was originally written in Latin by the Dutch statesman and mathematician Jan de Witt, soon after Descartes' invention of the subject.

- Born in 1625, Jan de Witt served with distinction as Grand Pensionary of Holland for much of his adult life. In mathematics, he is best known for his work in actuarial mathematics as well as extensive contributions to analytic geometry.

- Elementa Curvarum Linearum, Liber Secondus moves forward from the construction of the familiar conic sections covered in the Liber Primus, with a discussion of problems connected with their classification; given an equation, it covers how one can recover the standard form, and additionally how one can find the equation's geometric properties.

- This volume, begun by Albert Grootendorst (1924-2004) and completed after his death by Jan Aarts, Reinie Erne and Miente Bakker, is supplemented by:

- annotation explaining finer points of the translation;

- extensive commentary on the mathematics These features make the work of Jan de Witt broadly accessible to today's mathematicians."

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

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.

The aim of this book is to present and analyze philosophical conceptions concerning mathematics and logic as formulated by Polish logicians, mathematicians and philosophers in the 1920s and 1930s. It was a remarkable period in the history of Polish science, in particular in the history of Polish logic and mathematics. Therefore, it is justified to ask whether and to what extent the development of logic and mathematics was accompanied by a philosophical reflection. We try to answer those questions by analyzing both works of Polish logicians and mathematicians who have a philosophical temperament as well as their research practice. Works and philosophical views of the following Polish scientists will be analyzed: Waclaw Sierpinski, Zygmunt Janiszewski, Stefan Mazurkiewicz, Stefan Banach Hugo Steinhaus, Eustachy Zylinsk and Leon Chwistek, Jan Lukasiewicz, Zygmunt Zawirski, Stanislaw Lesniewski, Tadeusz Kotarbinski, Kazimierz Ajdukiewicz, Alfred Tarski, Andrzej Mostowski and Henryk Mehlberg, Jan Sleszynski, Stanislaw Zaremba and Witold Wilkosz. To indicate the background of scientists being active in the 1920s and 1930s we consider in Chapter 1 some predecessors, in particular: Jan Sniadecki, Jozef Maria Hoene-Wronski, Samuel Dickstein and Edward Stamm.

This work counters historiographies that search for the origins of modern science within the experimental practices of Europe 's first scientific institutions, such as the Cimento. It proposes that we should look beyond the experimental rhetoric found in published works, to find that the Cimento academicians were participants in a culture of natural philosophical theorising that existed throughout Europe.

Diese Arbeit enthiilt zwei grof3ere Fallstudien zur Beziehung zwischen theo- retischer Mathematik und Anwendungen im 19. Jahrhundert. Sie ist das Ergebnis eines mathematikhistorischen Forschungsprojekts am Mathemati- schen Fachbereich der Universitiit-Gesamthochschule Wuppertal und wurde dort als Habilitationsschrift vorgelegt. Ohne das wohlwollende Interesse von Herrn H. Scheid und den Kollegen der Abteilung fUr Didaktik der Mathema- tik ware das nicht moglich gewesen: Inhaltlich verdankt sie - direkt oder indirekt - vielen Beteiligten et- was. So wurde mein Interesse an den kristallographischen Symmetriekon- zepten, dem Thema der ersten Fallstudie, durch Anregungen und Hinweise von Herrn E. Brieskorn geweckt. Sowohl von seiner Seite als auch von Herrn J. J. Burckhardt stammen uberdies viele wert volle Hinweise zum Manuskript von Kapitel I. Herrn C. J. Scriba mochte ich fur seine die gesamte Arbeit betreffenden priizisen Anmerkungen danken und Herrn W. Borho ebenso fUr seine ubergreifenden Kommentare und Vorschlage. Beziiglich der in Kapitel II behandelten projektiven Methoden in der Baustatik des 19. Jahrhunderts gilt mein besonderer Dank den Herren K. -E. Kurrer und T. Hiinseroth fUr ihre zum Teil sehr detaillierten Anmerkungen aus dem Blickwinkel der Geschichte der Bauwissenschaften. Schliefilich geht mein Dank an alle nicht namentlich Erwiihnten, die in Gesprachen, technisch oder auch anderweitig zur Fertig- stellung dieser Arbeit beigetragen haben. Fur die vorliegende Publikation habe ich einen Anhang mit einer Skizze von in unserem Zusammenhang besonders wichtig erscheinenden Aspekten der Theorie der kristallographischen Raumgruppen hinzugefUgt. Ich hoffe, daB er zum Verstiindnis des mathematischen Hintergrunds der historischen Arbeiten des ersten Kapitels beitragt.

This book deals with Newton's understanding of the original divine design hidden in the mathematical laws of nature and delivered to humanity by messengers, such as Noah, Moses and Christ. It is written to an audience of laymen and professionals alike. It is the first scholarly work to point out that for Newton the three laws of motion the Principia and the two central Biblical Commandments (worshipping and loving God alone and loving thy neighbour) touch upon the practical applications of God's original design. The book interprets Newton's mathematical method of fluxions (the calculus) as a divine method through which human beings can purify and guard themselves against material bondage (idolatry), whilst becoming more in tune with the simplicity of the spiritual commandments of the true ancient religion. A comparison with Leibniz's calculus and theological beliefs is given in order to emphasize the uniqueness of Newton's science and spirituality.

Intellectual History and the Identity of John Dee In April 1995, at Birkbeck College, University of London, an interdisciplinary colloquium was held so that scholars from diverse fields and areas of expertise could 1 exchange views on the life and work of John Dee. Working in a variety of fields - intellectual history, history of navigation, history of medicine, history of science, history of mathematics, bibliography and manuscript studies - we had all been drawn to Dee by particular aspects of his work, and participating in the colloquium was to c- front other narratives about Dee's career: an experience which was both bewildering and instructive. Perhaps more than any other intellectual figure of the English Renaissance Dee has been fragmented and dispersed across numerous disciplines, and the various attempts to re-integrate his multiplied image by reference to a particular world-view or philosophical outlook have failed to bring him into focus. This volume records the diversity of scholarly approaches to John Dee which have emerged since the synthetic accounts of I. R. F. Calder, Frances Yates and Peter French. If these approaches have not succeeded in resolving the problematic multiplicity of Dee's activities, they will at least deepen our understanding of specific and local areas of his intellectual life, and render them more historiographically legible.

An understanding of developments in Arabic mathematics between the IXth and XVth century is vital to a full appreciation of the history of classical mathematics. This book draws together more than ten studies to highlight one of the major developments in Arabic mathematical thinking, provoked by the double fecondation between arithmetic and the algebra of al-Khwarizmi, which led to the foundation of diverse chapters of mathematics: polynomial algebra, combinatorial analysis, algebraic geometry, algebraic theory of numbers, diophantine analysis and numerical calculus. Thanks to epistemological analysis, and the discovery of hitherto unknown material, the author has brought these chapters into the light, proposes another periodization for classical mathematics, and questions current ideology in writing its history. Since the publication of the French version of these studies and of this book, its main results have been admitted by historians of Arabic mathematics, and integrated into their recent publications. This book is already a vital reference for anyone seeking to understand history of Arabic mathematics, and its contribution to Latin as well as to later mathematics. The English translation will be of particular value to historians and philosophers of mathematics and of science.

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.

In this volume specialists in mathematics, physics, and linguistics present the first comprehensive analysis of the ideas and influence of Hermann G. Grassmann (1809-1877), the remarkable universalist whose work recast the foundations of these disciplines and shaped the course of their modern development.

This clearly written and enlightening textbook provides a concise, introductory guide to the key mathematical concepts and techniques used by computer scientists. Topics and features: ideal for self-study, offering many pedagogical features such as chapter-opening key topics, chapter introductions and summaries, review questions, and a glossary; places our current state of knowledge within the context of the contributions made by early civilizations, such as the ancient Babylonians, Egyptians and Greeks; examines the building blocks of mathematics, including sets, relations and functions; presents an introduction to logic, formal methods and software engineering; explains the fundamentals of number theory, and its application in cryptography; describes the basics of coding theory, language theory, and graph theory; discusses the concept of computability and decideability; includes concise coverage of calculus, probability and statistics, matrices, complex numbers and quaternions.

This book, in three parts, describes three phases in the development of the modern theory and calculation of the Moon's motion. Part I explains the crisis in lunar theory in the 1870s that led G.W. Hill to lay a new foundation for an analytic solution, a preliminary orbit he called the "variational curve." Part II is devoted to E.W. Brown's completion of the new theory as a series of successive perturbations of Hill's variational curve. Part III describes the revolutionary developments in time-measurement and the determination of Earth-Moon and Earth-planet distances that led to the replacement of the Hill Brown theory in 1984.

This volume presents a selection of 434 letters from and to the Dutch physicist and Nobel Prize winner Hendrik Antoon Lorentz (1853-1928), covering the period from 1883 until a few months before his death in February 1928. The sheer size of the available correspondence (approximately 6000 letters from and to Lorentz) preclude a full publication. The letters included in this volume have been selected according to various criteria, the most important of which is scientific importance. A second criterion has been the availability of letters both from and to Lorentz, so that the reader can follow the exchange between Lorentz and his correspondent. Within such correspondences a few unimportant items, dealing with routine administrative or organizational matters, have been omitted. An exception to the scientific criterion is the exchange of letters between Lorentz and Albert Einstein, Max Planck, Woldemar Voigt, and Wilhelm Wien during World War I: these letters have been included because they shed important light on the disruption of the scientific relations during the war and on the political views of these correspondents as well as of Lorentz. similar reasons the letters exchanged with Einstein and Planck on post-war political issues have been included. Biographical sketch Hendrik Antoon Lorentz was born on July 18, 1853 in the Dutch town of Arnhem. He was the son of a relatively well-to-do owner of a nursery.

Covering both the history of mathematics and of philosophy, Descartes's Mathematical Thought reconstructs the intellectual career of Descartes most comprehensively and originally in a global perspective including the history of early modern China and Japan. Especially, it shows what the concept of "mathesis universalis" meant before and during the period of Descartes and how it influenced the young Descartes. In fact, it was the most fundamental mathematical discipline during the seventeenth century, and for Descartes a key notion which may have led to his novel mathematics of algebraic analysis.

The theories of V. V. Wagner (1908-1981) on abstractions of systems of binary relations are presented here within their historical and mathematical contexts. This book contains the first translation from Russian into English of a selection of Wagner's papers, the ideas of which are connected to present-day mathematical research. Along with a translation of Wagner's main work in this area, his 1953 paper 'Theory of generalised heaps and generalised groups,' the book also includes translations of three short precursor articles that provide additional context for his major work. Researchers and students interested in both algebra (in particular, heaps, semiheaps, generalised heaps, semigroups, and groups) and differential geometry will benefit from the techniques offered by these translations, owing to the natural connections between generalised heaps and generalised groups, and the role played by these concepts in differential geometry. This book gives examples from present-day mathematics where ideas related to Wagner's have found fruitful applications.

This work offers a re-edition of twelve mathematical tablets from the site of Tell Harmal, in the borders of present-day Baghdad. In ancient times, Tell Harmal was Saduppum, a city representative of the region of the Diyala river and of the kingdom of Esnunna, to which it belonged for a time. These twelve tablets were originally published in separate articles in the beginning of the 1950s and mostly contain solved problem texts. Some of the problems deal with abstract matters such as triangles and rectangles with no reference to daily life, while others are stated in explicitly empirical contexts, such as the transportation of a load of bricks, the size of a vessel, the number of men needed to build a wall and the acquisition of oil and lard. This new edition of the texts is the first to group them, and takes into account all the recent developments of the research in the history of Mesopotamian mathematics. Its introductory chapters are directed to readers interested in an overview of the mathematical contents of these tablets and the language issues involved in their interpretation, while a chapter of synthesis discusses the ways history of mathematics has typically dealt with the mathematical evidence and inquires how and to what degree mathematical tablets can be made part of a picture of the larger social context. Furthermore, the volume contributes to a geography of the Old Babylonian mathematical practices, by evidencing that scribes at Saduppum made use of cultural material that was locally available. The edited texts are accompanied by translations, philological, and mathematical commentaries.

The Theory of the Top was originally presented by Felix Klein as an 1895 lecture at Goettingen University that was broadened in scope and clarified as a result of collaboration with Arnold Sommerfeld. The Theory of the Top: Volume IV. Technical Applications of the Theory of the Top is the fourth and final installment in a series of self-contained English translations that provide insights into kinetic theory and kinematics.

This contributed volume investigates the active role of the different contexts of mathematics teaching on the evolution of the practices of mathematical concepts, with particular focus on their foundations. The book aims to deconstruct the strong and generally wide-held conviction that research in mathematics constitutes the only driving force for any progress in the development of mathematics as a field. In compelling and convincing contrast, these chapters aim to show the productive function of teaching, showcasing investigations from countries and regions throughout various eras, from Old Babylonia through the 20th Century. In so doing, they provide a critical reflection on the foundations of mathematics, as well as instigate new research questions, and explore the interfaces between teaching and research.

We see numbers on automobile license plates, addresses, weather reports, and, of course, on our smartphones. Yet we look at these numbers for their role as descriptors, not as an entity in and unto themselves. Each number has its own history of meaning, usage, and connotation in the larger world. The Secret Lives of Numbers takes readers on a journey through integers, considering their numerological assignments as well as their significance beyond mathematics and in the realm of popular culture. Of course we all know that the number 13 carries a certain value of unluckiness with it. The phobia of the number is called Triskaidekaphobia; Franklin Delano Roosevelt was known to invite and disinvite guests to parties to avoid having 13 people in attendance; high-rise buildings often skip the 13th floor out of superstition. There are many explanations as to how the number 13 received this negative honor, but from a mathematical point of view, the number 13 is also the smallest prime number that when its digits are reversed is also a prime number. It is honored with a place among the Fibonacci numbers and integral Pythagorean triples, as well as many other interesting and lesser-known occurrences. In The Secret Lives of Numbers, popular mathematician Alfred S. Posamentier provides short and engaging mini-biographies of more than 100 numbers, starting with 1 and featuring some especially interesting numbers -like 6,174, a number with most unusual properties -to provide readers with a more comprehensive picture of the lives of numbers both mathematically and socially.

The present book is a translation into English of Elernenta CU'f'Varurn Linearurn-Liber Prirnus, written in Latin, by the Dutch statesman and mathematician Jan de Witt (1625-1672). Together with its sequel, Ele- rnenta CU'f'Varurn Linearurn-Liber Secundus, it constitutes the first text- book on Analytic Geometry, based on the ideas of Descartes, as laid down in his Geornetrie of 1637. The first edition of de Witt's work appeared in 1659 and this translation is its first translation into English. For more details the reader is referred to the Introduction. Apart from this translation and this introduction, the present work con- tains an extensive summary, annotations to the translation, and two ap- pendices on the role of the conics in Greek mathematics. The translation has been made from the second edition, printed by the Blaeu Company in Amsterdam in 1684. In 1997 the translator published a translation into Dutch of the same work, likewise supplied with an introduction, a summary, notes, and two appendices. This edition appeared as a publication of the Stichting Mathe- matisch Centrum Amsterdam. The present translation, however, is a direct translation of the Latin text. The rest of this work is an English version of the introduction, the summary, the notes, and the appendices, based on the Dutch original.

The smartphones in our pockets and computers like brains. The vagaries of game theory and evolutionary biology. Nuclear weapons and self-replicating spacecrafts. All bear the fingerprints of one remarkable, yet largely overlooked, man: John von Neumann. Born in Budapest at the turn of the century, von Neumann is one of the most influential scientists to have ever lived. A child prodigy, he mastered calculus by the age of eight, and in high school made lasting contributions to mathematics. In Germany, where he helped lay the foundations of quantum mechanics, and later at Princeton, von Neumann's colleagues believed he had the fastest brain on the planet-bar none. He was instrumental in the Manhattan Project and the design of the atom bomb; he helped formulate the bedrock of Cold War geopolitics and modern economic theory; he created the first ever programmable digital computer; he prophesized the potential of nanotechnology; and, from his deathbed, he expounded on the limits of brains and computers-and how they might be overcome. Taking us on an astonishing journey, Ananyo Bhattacharya explores how a combination of genius and unique historical circumstance allowed a single man to sweep through a stunningly diverse array of fields, sparking revolutions wherever he went. The Man from the Future is an insightful and thrilling intellectual biography of the visionary thinker who shaped our century.