Mathematical Masterpieces: Further Chronicles by the Explorers

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Springer Science & Business Media, 16. 10. 2007 - Počet stran: 340
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? Hasn’t the Origin of Species been read untold millions of times? Hasn’t every paragraph been subjected to overt scholarly scrutiny and exegesis? Letmeshareasecretrootedingeneralhumanfoibles. . . . Veryfew people, including authors willing to commit to paper, ever really read primary sources—certainly 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 them—in 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 Gould’s 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.
 

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Obsah

The Bridge Between Continuous and Discrete 11 Introduction
1
12 Archimedes Sums Squares to Find the Area Inside a Spiral
18
13 Fermat and Pascal Use Figurate Numbers Binomials and the Arithmetical Triangle to Calculate Sums of Powers
26
14 Jakob Bernoulli Finds a Pattern
41
15 Eulers Summation Formula and the Solution for Sums of Powers
50
16 Euler Solves the Basel Problem
70
Solving Equations Numerically Finding Our Roots
83
22 Qin Solves a FourthDegree Equation by Completing Powers
110
35 Gauss Defines an Independent Notion of Curvature
196
36 Riemann Explores HigherDimensional Space
214
Patterns in Prime Numbers The Quadratic Reciprocity Law
229
42 Euler Discovers Patterns for Prime Divisors of Quadratic Forms
251
43 Lagrange Develops a Theory of Quadratic Forms and Divisors
261
44 Legendre Asserts the Quadratic Reciprocity Law
279
45 Gauss Proves the Fundamental Theorem
286
46 Eisensteins Geometric Proof
292

23 Newtons Proportional Method
125
24 Simpsons Fluxional Method
132
25 Smale Solves Simpson
140
Curvature and the Notion of Space
158
32 Huygens Discovers the Isochrone
167
33 Newton Derives the Radius of Curvature
181
34 Euler Studies the Curvature of Surfaces
187
The Class Group
301
48 Appendix on Congruence Arithmetic
306
References
311
Credits
323
Name Index
325
Subject Index
328
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Strana 5 - If a straight line one extremity of which remains fixed be made to revolve at a uniform rate in a plane until it returns to the position from which it started, and if, at the same time as the straight line is revolving, a point move at a uniform rate along the straight line, starting from the fixed extremity, the point will describe a spiral in the plane.

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