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E-grāmata: Mathematical Masterpieces: Further Chronicles by the Explorers

  • Formāts: PDF+DRM
  • Sērija : Readings in Mathematics
  • Izdošanas datums: 16-Oct-2007
  • Izdevniecība: Springer-Verlag New York Inc.
  • Valoda: eng
  • ISBN-13: 9780387330624
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  • Formāts: PDF+DRM
  • Sērija : Readings in Mathematics
  • Izdošanas datums: 16-Oct-2007
  • Izdevniecība: Springer-Verlag New York Inc.
  • Valoda: eng
  • ISBN-13: 9780387330624
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In introducing his essays on the study and understanding of nature and e- lution, biologist Stephen J. Gould writes: [ W]e acquire a surprising source of rich and apparently limitless novelty from the primary documents of great thinkers throughout our history. But why should any nuggets, or even ?akes, be left for int- lectual miners in such terrain? Hasnt the Origin of Species been read untold millions of times? Hasnt every paragraph been subjected to overt scholarly scrutiny and exegesis? Letmeshareasecretrootedingeneralhumanfoibles. . . . Veryfew people, including authors willing to commit to paper, ever really read primary sourcescertainly not in necessary depth and completion, and often not at all. . . . I can attest that all major documents of science remain cho- full of distinctive and illuminating novelty, if only people will study themin full and in the original editions. Why would anyone not yearn to read these works; not hunger for the opportunity? [ 99, p. 6f] It is in the spirit of Goulds insights on an approach to science based on p- mary texts that we o er the present book of annotated mathematical sources, from which our undergraduate students have been learning for more than a decade. Although teaching and learning with primary historical sources require a commitment of study, the investment yields the rewards of a deeper understanding of the subject, an appreciation of its details, and a glimpse into the direction research has taken. Our students read sequences of primary sources.

Recenzijas

From the reviews:









"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)

Preface v
The Bridge Between Continuous and Discrete
1(82)
Introduction
1(17)
Archimedes Sums Squares to Find the Area Inside a Spiral
18(8)
Fermat and Pascal Use Figurate Numbers, Binomials, and the Arithmetical Triangle to Calculate Sums of Powers
26(15)
Jakob Bernoulli Finds a Pattern
41(9)
Euler's Summation Formula and the Solution for Sums of Powers
50(20)
Euler Solves the Basel Problem
70(13)
Solving Equations Numerically: Finding Our Roots
83(76)
Introduction
83(27)
Qin Solves a Fourth-Degree Equation by Completing Powers
110(15)
Newton's Proportional Method
125(7)
Simpson's Fluxional Method
132(8)
Smale Solves Simpson
140(19)
Curvature and the Notion of Space
159(70)
Introduction
159(8)
Huygens Discovers the Isochrone
167(14)
Newton Derives the Radius of Curvature
181(6)
Euler Studies the Curvature of Surfaces
187(9)
Gauss Defines an Independent Notion of Curvature
196(18)
Riemann Explores Higher-Dimensional Space
214(15)
Patterns in Prime Numbers: The Quadratic Reciprocity Law
229(82)
Introduction
229(22)
Euler Discovers Patterns for Prime Divisors of Quadratic Forms
251(10)
Lagrange Develops a Theory of Quadratic Forms and Divisors
261(18)
Legendre Asserts the Quadratic Reciprocity Law
279(7)
Gauss Proves the ``Fundamental Theorem''
286(6)
Eisenstein's Geometric Proof
292(9)
Gauss Composes Quadratic Forms: The Class Group
301(5)
Appendix on Congruence Arithmetic
306(5)
References 311(12)
Credits 323(2)
Name Index 325(4)
Subject Index 329