Mathematical Masterpieces Further Chronicles by the Explorers

ISBN-10: 0387330615
ISBN-13: 9780387330617
Edition: 2007
List price: $59.99 Buy it from $25.49
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Description: This book traces the historical development of four different mathematical concepts by presenting readers with the original sources. Although primary sources can be more demanding, the investment yields the rewards of a deeper understanding of the  More...

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Book details

List price: $59.99
Copyright year: 2007
Publisher: Springer
Publication date: 8/14/2007
Binding: Paperback
Pages: 340
Size: 6.25" wide x 9.25" long x 0.75" tall
Weight: 1.100
Language: English

This book traces the historical development of four different mathematical concepts by presenting readers with the original sources. Although primary sources can be more demanding, the investment yields the rewards of a deeper understanding of the subject, an appreciation of the details, and a glimpse into the direction research has taken. Each chapter contains a different story, each anchored around a sequence of selected primary sources showcasing a masterpiece of mathematical achievement. The authors begin by studying the interplay between the discrete and continuous, with a focus on sums of powers. They proceed to the development of algorithms for finding numerical solutions of equations as developed by Newton, Simpson and Smale. Next they explore our modern understanding of curvature, with its roots in the emerging calculus of the 17 th century, while the final chapter ends with an exploration of the elusive properties of prime numbers, and the patterns found therein. This book emerged from a course taught at New Mexico State University to juniors and seniors majoring in mathematics. The intended audience is juniors and seniors majoring in mathematics, as well as anyone pursuing independent study. The authors have included exercises, numerous historical photographs, and an annotated bibliography.

Preface
The Bridge Between Continuous and Discrete
Introduction
Archimedes Sums Squares to Find the Area Inside a Spiral
Fermat and Pascal Use Figurate Numbers, Binomials, and the Arithmetical Triangle to Calculate Sums of Powers
Jakob Bernoulli Finds a Pattern
Euler's Summation Formula and the Solution for Sums of Powers
Euler Solves the Basel Problem
Solving Equations Numerically: Finding Our Roots
Introduction
Qin Solves a Fourth-Degree Equation by Completing Powers
Newton's Proportional Method
Simpson's Fluxional Method
Smale Solves Simpson
Curvature and the Notion of Space
Introduction
Huygens Discovers the Isochrone
Newton Derives the Radius of Curvature
Euler Studies the Curvature of Surfaces
Gauss Defines an Independent Notion of Curvature
Riemann Explores Higher-Dimensional Space
Patterns in Prime Numbers: The Quadratic Reciprocity Law
Introduction
Euler Discovers Patterns for Prime Divisors of Quadratic Forms
Lagrange Develops a Theory of Quadratic Forms and Divisors
Legendre Asserts the Quadratic Reciprocity Law
Gauss Proves the "Fundamental Theorem"
Eisenstein's Geometric Proof
Gauss Composes Quadratic Forms: The Class Group
Appendix on Congruence Arithmetic
References
Credits
Name Index
Subject Index

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