## Mathematical Masterpieces: Further Chronicles by the ExplorersIn 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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### Contents

The Bridge Between Continuous and Discrete | 1 |

Finding Our Roots | 83 |

Curvature and the Notion of Space 159 | 158 |

311 | |

Credits | 323 |

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### Common terms and phrases

algebraic algorithm approximate Archimedes arithmetic progressions arithmetical triangle Basel problem Bernoulli numbers binomial calculate Cardano circle coefficients compute convergence cube cubic equations curved surface derivatives determine differential digit Disquisitiones divisor problem divisors of numbers Eisenstein’s equation Euler Exercise expression Fermat figurate numbers form x2 function fundamental theorem Gauss Gaussian curvature geometry given Huygens infinite integers iteration Lagrange Lagrange’s Legendre Legendre symbol Leonhard Euler linear manifold mathematicians mathematics measure of curvature modern modulo multiplied Newton nontrivial prime divisors notation number theory obtain odd prime osculating circle Pascal’s pattern pendulum plane polynomial positive prime numbers proof prove Qin’s quadratic forms quadratic reciprocity law quadratic residue quantity reader relatively prime Riemann root sequence Simpson’s fluxional method solution solve sphere summation formula sums of powers synthetic division Tableau theorema egregium values