Mathematical masterpieces : further chronicles by the explorers

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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In a manner similar to Mathematical Expeditions (1999), Knoebel and colleagues help students learn mathematics by getting them to both experience and explore original works in the history of mathematics. The book provides four adventures, exploring how the Bernoulli numbers formed a bridge between the continuous and the discrete; how algorithms were developed to solve equations; the developing concept of curvature relative to geometric space; and the constant quest to find patterns in prime numbers. As part of these adventures, students are introduced to the works and discoveries of Archimedes, Fermat, Pascal, Bernoulli, Euler, Qin, Newton, Simpson, Smale, Huygens, Gauss, Riemann, Lagrange, Legendre, and Eisenstein. This book is written at a more difficult level mathematically than Mathematical Expeditions, requiring multivariate calculus as a minimal background. It is well organized with a helpful "guide" through each exploration and a broad collection of interesting problems integrated into the explorations, all complemented by an extensive reference list and index. Readers or journeyers--especially students, mathematics teachers, and mathematicians--will find these explorations refreshing and revealing. Summing Up: Highly recommended. Lower-division undergraduates through faculty. J. Johnson Western Washington University

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

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