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Journey Through Mathematics

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Table of Contents

Preface.- 1 TRIGONOMETRY.- 1.1 The Hellenic Period.- 1.2 Ptolemy's Table of Chords.- 1.3 The Indian Contribution.- 1.4 Trigonometry in the IslamicWorld.- 1.5 Trigonometry in Europe.- 1.6 From Viete to Pitiscus.- 2 LOGARITHMS.- 2.1 Napier's First Three Tables.- 2.2 Napier's Logarithms.- 2.3 Briggs' Logarithms.- 2.4 Hyperbolic Logarithms.- 2.5 Newton's Binomial Series.- 2.6 The Logarithm According to Euler.- 3 COMPLEX NUMBERS.- 3.1 The Depressed Cubic.- 3.2 Cardano's Contribution.- 3.3 The Birth of Complex Numbers.- 3.4 Higher-Order Roots of Complex Numbers.- 3.5 The Logarithms of Complex Numbers.- 3.6 CasparWessel's Breakthrough.- 3.7 Gauss and Hamilton Have the FinalWord.- 4 INFINITE SERIES.- 4.1 The Origins.- 4.2 The Summation of Series.- 4.3 The Expansion of Functions.- 4.4 The Taylor and Maclaurin Series.- 5 THE CALCULUS.- 5.1 The Origins.- 5.2 Fermat's Method of Maxima and Minima.- 5.3 Fermat's Treatise on Quadratures.- 5.4 Gregory's Contributions.- 5.5 Barrow's Geometric Calculus.- 5.6 From Tangents to Quadratures.- 5.7 Newton's Method of Infinite Series.- 5.8 Newton's Method of Fluxions.- 5.9 Was Newton's Tangent Method Original?.- 5.10 Newton's First and Last Ratios.- 5.11 Newton's Last Version of the Calculus.- 5.12 Leibniz' Calculus: 1673-1675.- 5.13 Leibniz' Calculus: 1676-1680.- 5.14 The Arithmetical Quadrature.- 5.15 Leibniz' Publications.- 5.16 The Aftermath.- 6 CONVERGENCE.- 6.1 To the Limit.- 6.2 The Vibrating String MakesWaves.- 6.3 Fourier Puts on the Heat.- 6.4 The Convergence of Series.- 6.5 The Difference Quotient.- 6.6 The Derivative.- 6.7 Cauchy's Integral Calculus.- 6.8 Uniform Convergence.- BIBLIOGRAPHY.- Index

About the Author

Enrique A. Gonzalez-Velasco is a professor of mathematics at the University of Massachusetts at Lowell. His specialty is the history of math. He has also published a textbook on Fourier analysis and boundary value problems with a historical focus.


From the reviews: "This book is certainly to be recommended to any student of the history of mathematics who has a particular interest in any of the topics covered, particularly those whose history is less usually taught in detail at undergraduate level ... . the friendly writing style, interesting tangents on etymology and biography, and the chance to discover the work of some less celebrated mathematicians still make Journey through mathematics a very agreeable read." (Katherine Steiner, BSHM Bulletin, Vol. 28 (1), 2013) Choice - Outstanding Academic Title in 2012 "This detailed, carefully written volume is a selective history of mathematics, up to what might be considered the `early modern period' ... with emphasis on 17th- and 18th-century advances. ... the book particularly suitable for undergraduate mathematics students, including those planning to teach in secondary schools. The volume contains an extensive bibliography with many original sources ... . A worthwhile addition to the literature. Summing Up: Highly recommended. Upper-division undergraduates through researchers/faculty." (S. J. Colley, Choice, Vol. 49 (7), March, 2012) "Describes the history of mathematics that gave rise to our modern concepts in calculus ... . Gonzalez-Velasco has done a marvelous job by sketching this very readable historical tale. ... not only compulsory reading for a course on the history of mathematics, but everyone teaching a calculus course should be aware of the roots and the wonderful achievements of the mathematical giants of the past centuries. They boldly went where nobody had gone before and paved the road for what we take for granted today." (A. Bultheel, The European Mathematical Society, March, 2012) "Gonzalez-Velasco has put together an immense amount of material that can be used in teaching some central topics in mathematics using the words of the originators. He is to be commended for all of his research and his use of sources that are not easily available. ... this book will have readers who will benefit from it, most of those will be mature readers very experienced in mathematics and its history." (Victor J. Katz, Mathematical Reviews, Issue 2012 g) "The book is affectionately written and one feels on nearly every page that the author eventually fell in love with his subjects. ... Many illustrations from that sources as well as modern drawings and copper engravings of the mathematicians involved contribute to the readability of the text and invite the reader rummaging in the book. This is an honest book. ... Students of mathematics, mathematicians with an inclination towards the history of their subject, and teachers of mathematics will profit much from this book." (Thomas Sonar, Zentralblatt MATH, Vol. 1243, 2012) "The book basically consists of the history of six well-known branches of mathematics ... . Throughout the book there is cultural, political and biographical background as a context for the historical enquiry. There are many illustrations and excerpts from primary sources, and the author provides a feel for the way in which mathematics was created, why it was created, and how it was expressed. ... this book is a highly recommended addition to the existing literature." (P. N. Ruane, MAA Reviews, December, 2011)

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