Author: C.H.Jr. Edwards

Publisher: Springer Science & Business Media

ISBN: 1461262305

Category: Mathematics

Page: 368

View: 6500

The calculus has served for three centuries as the principal quantitative language of Western science. In the course of its genesis and evolution some of the most fundamental problems of mathematics were first con fronted and, through the persistent labors of successive generations, finally resolved. Therefore, the historical development of the calculus holds a special interest for anyone who appreciates the value of a historical perspective in teaching, learning, and enjoying mathematics and its ap plications. My goal in writing this book was to present an account of this development that is accessible, not solely to students of the history of mathematics, but to the wider mathematical community for which my exposition is more specifically intended, including those who study, teach, and use calculus. The scope of this account can be delineated partly by comparison with previous works in the same general area. M. E. Baron's The Origins of the Infinitesimal Calculus (1969) provides an informative and reliable treat ment of the precalculus period up to, but not including (in any detail), the time of Newton and Leibniz, just when the interest and pace of the story begin to quicken and intensify. C. B. Boyer's well-known book (1949, 1959 reprint) met well the goals its author set for it, but it was more ap propriately titled in its original edition-The Concepts of the Calculus than in its reprinting.
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Author: David Perkins

Publisher: MAA

ISBN: 0883855755

Category: Mathematics

Page: 165

View: 8444

" ... Is primarily a collection of results that show how calculus came to be, beginning in ancient Greece and climaxing with the discovery of calculus. The book requires only a basic knowledge of high school geometry and algebra. Exercises introduce further historical figures and their results." -- Cover, p.[4].
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Author: Ernst Hairer,Gerhard Wanner

Publisher: Springer Science & Business Media

ISBN: 0387770364

Category: Mathematics

Page: 379

View: 5761

This book presents first-year calculus roughly in the order in which it was first discovered. The first two chapters show how the ancient calculations of practical problems led to infinite series, differential and integral calculus and to differential equations. The establishment of mathematical rigour for these subjects in the 19th century for one and several variables is treated in chapters III and IV. Many quotations are included to give the flavor of the history. The text is complemented by a large number of examples, calculations and mathematical pictures and will provide stimulating and enjoyable reading for students, teachers, as well as researchers.
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Author: Margaret E. Baron

Publisher: Elsevier

ISBN: 1483280926

Category: Mathematics

Page: 312

View: 5882

The Origins of Infinitesimal Calculus focuses on the evolution, development, and applications of infinitesimal calculus. The publication first ponders on Greek mathematics, transition to Western Europe, and some center of gravity determinations in the later 16th century. Discussions focus on the growth of kinematics in the West, latitude of forms, influence of Aristotle, axiomatization of Greek mathematics, theory of proportion and means, method of exhaustion, discovery method of Archimedes, and curves, normals, tangents, and curvature. The manuscript then examines infinitesimals and indivisibles in the early 17th century and further advances in France and Italy. Topics include the link between differential and integral processes, concept of tangent, first investigations of the cycloid, and arithmetization of integration methods. The book reviews the infinitesimal methods in England and Low Countries and rectification of arcs. The publication is a vital source of information for historians, mathematicians, and researchers interested in infinitesimal calculus.
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Author: H. H. Goldstine

Publisher: Springer Science & Business Media

ISBN: 1461381061

Category: Mathematics

Page: 410

View: 1671

The calculus of variations is a subject whose beginning can be precisely dated. It might be said to begin at the moment that Euler coined the name calculus of variations but this is, of course, not the true moment of inception of the subject. It would not have been unreasonable if I had gone back to the set of isoperimetric problems considered by Greek mathemati cians such as Zenodorus (c. 200 B. C. ) and preserved by Pappus (c. 300 A. D. ). I have not done this since these problems were solved by geometric means. Instead I have arbitrarily chosen to begin with Fermat's elegant principle of least time. He used this principle in 1662 to show how a light ray was refracted at the interface between two optical media of different densities. This analysis of Fermat seems to me especially appropriate as a starting point: He used the methods of the calculus to minimize the time of passage cif a light ray through the two media, and his method was adapted by John Bernoulli to solve the brachystochrone problem. There have been several other histories of the subject, but they are now hopelessly archaic. One by Robert Woodhouse appeared in 1810 and another by Isaac Todhunter in 1861.
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Author: John Stillwell

Publisher: Springer Science & Business Media

ISBN: 144196052X

Category: Mathematics

Page: 662

View: 6527

From a review of the second edition: "This book covers many interesting topics not usually covered in a present day undergraduate course, as well as certain basic topics such as the development of the calculus and the solution of polynomial equations. The fact that the topics are introduced in their historical contexts will enable students to better appreciate and understand the mathematical ideas involved...If one constructs a list of topics central to a history course, then they would closely resemble those chosen here." (David Parrott, Australian Mathematical Society) This book offers a collection of historical essays detailing a large variety of mathematical disciplines and issues; it’s accessible to a broad audience. This third edition includes new chapters on simple groups and new sections on alternating groups and the Poincare conjecture. Many more exercises have been added as well as commentary that helps place the exercises in context.
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Author: N. Bourbaki

Publisher: Springer Science & Business Media

ISBN: 3642616933

Category: Mathematics

Page: 301

View: 990

Each volume of Nicolas Bourbakis well-known work, The Elements of Mathematics, contains a section or chapter devoted to the history of the subject. This book collects together those historical segments with an emphasis on the emergence, development, and interaction of the leading ideas of the mathematical theories presented in the Elements. In particular, the book provides a highly readable account of the evolution of algebra, geometry, infinitesimal calculus, and of the concepts of number and structure, from the Babylonian era through to the 20th century.
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Author: Benno Artmann

Publisher: Springer Science & Business Media

ISBN: 1461214122

Category: Mathematics

Page: 349

View: 5489

Euclid presents the essential of mathematics in a manner which has set a high standard for more than 2000 years. This book, an explanation of the nature of mathematics from its most important early source, is for all lovers of mathematics with a solid background in high school geometry, whether they be students or university professors.
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Author: Joseph W. Dauben,Christoph J. Scriba

Publisher: Springer Science & Business Media

ISBN: 9783764361679

Category: Mathematics

Page: 689

View: 7043

As an historiographic monograph, this book offers a detailed survey of the professional evolution and significance of an entire discipline devoted to the history of science. It provides both an intellectual and a social history of the development of the subject from the first such effort written by the ancient Greek author Eudemus in the Fourth Century BC, to the founding of the international journal, Historia Mathematica, by Kenneth O. May in the early 1970s.
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The Calculus and the Development of Theoretical Physics in Europe, 1750–1914

Author: Elizabeth Garber

Publisher: Springer Science & Business Media

ISBN: 1461217660

Category: Science

Page: 399

View: 3858

This work is the first explicit examination of the key role that mathematics has played in the development of theoretical physics and will undoubtedly challenge the more conventional accounts of its historical development. Although mathematics has long been regarded as the "language" of physics, the connections between these independent disciplines have been far more complex and intimate than previous narratives have shown. The author convincingly demonstrates that practices, methods, and language shaped the development of the field, and are a key to understanding the mergence of the modern academic discipline. Mathematicians and physicists, as well as historians of both disciplines, will find this provocative work of great interest.
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Author: Jeremy Gray

Publisher: Springer

ISBN: 3319237152

Category: Mathematics

Page: 350

View: 5119

This book contains a history of real and complex analysis in the nineteenth century, from the work of Lagrange and Fourier to the origins of set theory and the modern foundations of analysis. It studies the works of many contributors including Gauss, Cauchy, Riemann, and Weierstrass. This book is unique owing to the treatment of real and complex analysis as overlapping, inter-related subjects, in keeping with how they were seen at the time. It is suitable as a course in the history of mathematics for students who have studied an introductory course in analysis, and will enrich any course in undergraduate real or complex analysis.
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Author: John Wallis

Publisher: Springer Science & Business Media

ISBN: 1475743122

Category: Mathematics

Page: 192

View: 7904

John Wallis (1616-1703) was the most influential English mathematician prior to Newton. He published his most famous work, Arithmetica Infinitorum, in Latin in 1656. This book studied the quadrature of curves and systematised the analysis of Descartes and Cavelieri. Upon publication, this text immediately became the standard book on the subject and was frequently referred to by subsequent writers. This will be the first English translation of this text ever to be published.
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Author: W.M. Priestley

Publisher: Springer Science & Business Media

ISBN: 1468493493

Category: Mathematics

Page: 441

View: 8004

This book is for students being introduced to calculus, and it covers the usual topics, but its spirit is different from wh at might be expected. Though the approach is basically historical in nature, emphasis is put upon ideas and their place-not upon events and their dates. Its purpose is to have students to learn calculus first, and to learn incidentally something about the nature of mathematics. Somewhat to the surprise of its author, the book soon became animated by a spirit of opposition to the darkness that separates the sciences from the humanities. To fight the speil of that darkness anything at hand is used, even a few low tricks or bad jokes that seemed to offer a slight promise of success. To lighten the darkness, to illuminate some of the common ground shared by the two cultures, is a goal that justifies almost any means. It is possible that this approach may make calculus more fun as weil. Whereas the close ties of mathematics to the sciences are weil known, the ties binding mathematics to the humanities are rarely noticed. The result is a distorted view of mathematics, placing it outside the mainstream of liberal arts studies. This book tries to suggest gently, from time to time, where a kinship between mathematics and the humanities may be found.
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The Theory of Calculus

Author: Kenneth A. Ross

Publisher: Springer Science & Business Media

ISBN: 1461462711

Category: Mathematics

Page: 412

View: 9724

For over three decades, this best-selling classic has been used by thousands of students in the United States and abroad as a must-have textbook for a transitional course from calculus to analysis. It has proven to be very useful for mathematics majors who have no previous experience with rigorous proofs. Its friendly style unlocks the mystery of writing proofs, while carefully examining the theoretical basis for calculus. Proofs are given in full, and the large number of well-chosen examples and exercises range from routine to challenging. The second edition preserves the book’s clear and concise style, illuminating discussions, and simple, well-motivated proofs. New topics include material on the irrationality of pi, the Baire category theorem, Newton's method and the secant method, and continuous nowhere-differentiable functions.
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A Course in the History of Geometry in the 19th Century

Author: Jeremy Gray

Publisher: Springer Science & Business Media

ISBN: 9780857290601

Category: Mathematics

Page: 384

View: 3745

Based on the latest historical research, Worlds Out of Nothing is the first book to provide a course on the history of geometry in the 19th century. Topics covered in the first part of the book are projective geometry, especially the concept of duality, and non-Euclidean geometry. The book then moves on to the study of the singular points of algebraic curves (Plücker’s equations) and their role in resolving a paradox in the theory of duality; to Riemann’s work on differential geometry; and to Beltrami’s role in successfully establishing non-Euclidean geometry as a rigorous mathematical subject. The final part of the book considers how projective geometry rose to prominence, and looks at Poincaré’s ideas about non-Euclidean geometry and their physical and philosophical significance. Three chapters are devoted to writing and assessing work in the history of mathematics, with examples of sample questions in the subject, advice on how to write essays, and comments on what instructors should be looking for.
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Author: J. Lützen

Publisher: Springer Science & Business Media

ISBN: 1461394724

Category: Mathematics

Page: 232

View: 5639

I first learned the theory of distributions from Professor Ebbe Thue Poulsen in an undergraduate course at Aarhus University. Both his lectures and the textbook, Topological Vector Spaces, Distributions and Kernels by F. Treves, used in the course, opened my eyes to the beauty and abstract simplicity of the theory. However my incomplete study of many branches of classical analysis left me with the question: Why is the theory of distributions important? In my continued studies this question was gradually answered, but my growing interest in the history of mathematics caused me to alter my question to other questions such as: For what purpose, if any, was the theory of distributions originally created? Who invented distributions and when? I quickly found answers to the last two questions: distributions were invented by S. Sobolev and L. Schwartz around 1936 and 1950, respectively. Knowing this answer, however, only created a new question: Did Sobolev and Schwartz construct distributions from scratch or were there earlier trends and, if so, what were they? It is this question, concerning the pre history of the theory of distributions, which I attempt to answer in this book. Most of my research took place at the History of Science Department of Aarhus University. I wish to thank this department for its financial and intellectual support. I am especially grateful to Lektors Kirsti Andersen from the History of Science Department and Lars Mejlbo from the Mathematics Department, for their kindness, constructive criticism, and encouragement.
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An Annotated Translation

Author: Robert E. Bradley,C. Edward Sandifer

Publisher: Springer Science & Business Media

ISBN: 1441905499

Category: Mathematics

Page: 412

View: 464

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.
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Author: R. Rashed

Publisher: Springer Science & Business Media

ISBN: 9401732744

Category: History

Page: 382

View: 5858

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.
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Author: W.M. Priestley

Publisher: Springer Science & Business Media

ISBN: 1461216583

Category: Mathematics

Page: 404

View: 6373

Presenting mathematics as forming a natural bridge between the humanities and the sciences, this book makes calculus accessible to those in the liberal arts. Much of the necessary geometry and algebra are exposed through historical development, and a section on the development of calculus offers insights into the place of mathematics in the history of thought.
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A Concrete Introduction to Algebraic Curves

Author: Robert Bix

Publisher: Springer Science & Business Media

ISBN: 1475729758

Category: Mathematics

Page: 292

View: 6537

Algebraic curves are the graphs of polynomial equations in two vari 3 ables, such as y3 + 5xy2 = x + 2xy. By focusing on curves of degree at most 3-lines, conics, and cubics-this book aims to fill the gap between the familiar subject of analytic geometry and the general study of alge braic curves. This text is designed for a one-semester class that serves both as a a geometry course for mathematics majors in general and as a sequel to college geometry for teachers of secondary school mathe matics. The only prerequisite is first-year calculus. On the one hand, this book can serve as a text for an undergraduate geometry course for all mathematics majors. Algebraic geometry unites algebra, geometry, topology, and analysis, and it is one of the most exciting areas of modem mathematics. Unfortunately, the subject is not easily accessible, and most introductory courses require a prohibitive amount of mathematical machinery. We avoid this problem by focusing on curves of degree at most 3. This keeps the results tangible and the proofs natural. It lets us emphasize the power of two fundamental ideas, homogeneous coordinates and intersection multiplicities.
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