Mathematical Masterpieces: Further Chronicles by the Explorers / Edition 1

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Overview

Experience the discovery of mathematics by reading the original work of some of the greatest minds throughout history. Here are the stories of four mathematical adventures, including the Bernoulli numbers as the passage between discrete and continuous phenomena, the search for numerical solutions to equations throughout time, the discovery of curvature and geometric space, and the quest for patterns in prime numbers. Each story is told through the words of the pioneers of mathematical thought. Particular advantages of the historical approach include providing context to mathematical inquiry, perspective to proposed conceptual solutions, and a glimpse into the direction research has taken. The text is ideal for an undergraduate seminar, independent reading, or a capstone course, and offers a wealth of student exercises with a prerequisite of at most multivariable calculus.
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Editorial Reviews

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"This book is closely related to courses of mathematics held for students at New Mexico State University … . An important aspect of the book is the numerous exercises, which should help students to gain a deeper insight into the presented material. Numerous references and well-organized indices make the book easy to use. It can be recommended for university libraries and students with an interest in the history of mathematics presented from a modern point of view." (EMS Newsletter, September, 2008)

"This book consists of four chapters, each of which presents a ‘sequence of selected primary sources’ leading up to a ‘masterpiece of mathematical achievement’. … Each chapter contains … lots of historical comments sketching the further development of the topic. There are also a lot of exercises. … This is a well written and entertaining book that can (and should) be used in seminars or reading courses." (Franz Lemmermeyer, Zentralblatt MATH, Vol. 1140, 2008)

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Product Details

Table of Contents


Preface     V
The Bridge Between Continuous and Discrete     1
Introduction     1
Archimedes Sums Squares to Find the Area Inside a Spiral     18
Fermat and Pascal Use Figurate Numbers, Binomials, and the Arithmetical Triangle to Calculate Sums of Powers     26
Jakob Bernoulli Finds a Pattern     41
Euler's Summation Formula and the Solution for Sums of Powers     50
Euler Solves the Basel Problem     70
Solving Equations Numerically: Finding Our Roots     83
Introduction     83
Qin Solves a Fourth-Degree Equation by Completing Powers     110
Newton's Proportional Method     125
Simpson's Fluxional Method     132
Smale Solves Simpson     140
Curvature and the Notion of Space     159
Introduction     159
Huygens Discovers the Isochrone     167
Newton Derives the Radius of Curvature     181
Euler Studies the Curvature of Surfaces     187
Gauss Defines an Independent Notion of Curvature     196
Riemann Explores Higher-Dimensional Space     214
Patterns in Prime Numbers: The Quadratic Reciprocity Law     229
Introduction     229
Euler Discovers Patterns forPrime Divisors of Quadratic Forms     251
Lagrange Develops a Theory of Quadratic Forms and Divisors     261
Legendre Asserts the Quadratic Reciprocity Law     279
Gauss Proves the "Fundamental Theorem"     286
Eisenstein's Geometric Proof     292
Gauss Composes Quadratic Forms: The Class Group     301
Appendix on Congruence Arithmetic     306
References     311
Credits     323
Name Index     325
Subject Index     329
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