# Mathematical Masterpieces: Further Chronicles by the Explorers

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 Autorská práva

### Oblíbené pasáže

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.