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 17th 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. TOC:Preface.- The Bridge Between the Continuous and the Discrete.- Solving Equations Numerically: Finding our Roots.- Curvature and the Notion of Space.- Patterns in Prime Numbers: The Quadratic Reciprocity Law.- References.- Credits. 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. Preface V 1 The Bridge Between Continuous and Discrete 1 1.1 Introduction 1 1.2 Archimedes Sums quares to Find the Area Inside a Spiral 18 1.3 Fermat and Pascal Use Figurate Numbers, Binomials, and the Arithmetical Triangle to Calculate Sums of Powers 26 1.4 Jakob Bernoulli Finds a Pattern 41 1.5 Euler's Summation Formula and the Solution for Sums of Powers 50 1.6 Euler Solves the Basel Problem 70 2 Solving Equations Numerically: Finding Our Roots 83 2.1 Introduction 83 2.2 Qin Solves a Fourth-Degree Equation by Completing Powers 110 2.3 Newton's Proportional Method 125 2.4 Simpson's Fluxional Method 132 2.5 Smale Solves Simpson 140 3 Curvature and the Notion of Space 159 3.1 Introduction 159 3.2 Huygens Discovers the Isochrone 167 3.3 Newton Derives the Radius of Curvature 181 3.4 Euler Studies the Curvature of Surfaces 187 3.5 Gauss Defines an Independent Notion of Curvature 196 3.6 Riemann Explores Higher-Dimensional Space 214 4 Patterns in Prime Numbers: The Quadratic Reciprocity Law 229 4.1 Introduction 229 4.2 Euler Discovers Patterns for Prime Divisors of Quadratic Forms 251 4.3 Lagrange Develops a Theory of Quadratic Forms and Divisors 261 4.4 Legendre Asserts the Quadratic Reciprocity Law 279 4.5 Gauss Proves the "Fundamental Theorem" 286 4.6 Eisenstein's Geometric Proof 292 4.7 Gauss Composes Quadratic Forms: The Class Group 301 4.8 Appendix on Congruence Arithmetic 306 References 311 Credits 323 Name Index 325 Subject Index 329