Book:George F. Simmons/Calculus Gems

Subject Matter

 * History of Mathematics

It is worth pointing out that in the section Brief Lives, it is not the lives themselves that were necessarily brief, merely the accounts of those lives.


 * Some readers will recognise that this book has been reconstructed out of two massive appendices in my 1985 calculus book, with many additions, rearrangements and minor adjustments.
 * --- from Preface

It is also worth pointing out that a significant quantity of the material has also appeared in exactly the same form in 's $1972$ work.

Contents

 * Preface

Part A: Brief Lives

 * The Ancients
 * A.1 (ca. 625 – 547 B.C.)
 * A.2 (ca. 580 – 500 B.C.)
 * A.3 (ca. 460 – 370 B.C.)
 * A.4 (ca. 300 B.C.)
 * A.5 (ca. 287 – 212 B.C.)
 * Appendix: The Text of
 * A.6 (ca. 262 – 190 B.C.)
 * Appendix: ' General Preface to His Treatise
 * A.7 (1st century A.D.)
 * A.8 (4th century A.D.)
 * Appendix: The Focus-Directrix-Eccentricity Definitions of the Conic Sections
 * A.9 (370? – 425)
 * A Proof of Diophantus' Theorem on Pythagorean Triples


 * The Forerunners
 * A.10 (1571 – 1630)
 * A.11 (1596 – 1650)
 * A.12 (1588 – 1648)
 * A.13 (1601 – 1665)
 * A.14 (1598 – 1647)
 * A.15 (1608 – 1647)
 * A.16 (1623 – 1662)
 * A.17 (1629 – 1695)


 * The Early Moderns
 * A.18 (1642 – 1727)
 * Appendix: 's 1714(?) Memorandum of the Two Plague Years of 1665 and 1666
 * A.19 (1646 – 1716)
 * A.20 The Bernoulli Brothers ( 1654 – 1705, 1667 – 1748)
 * A.21 (1707 – 1783)
 * A.22 (1736 – 1813)
 * A.23 (1749 – 1827)
 * A.24 (1768 – 1830)


 * The Mature Moderns
 * A.25 (1777 – 1855)
 * A.26 (1789 – 1857)
 * A.27 (1802 – 1829)
 * A.28 (1805 – 1859)
 * A.29 (1809 – 1882)
 * A.30 (1822 – 1901)
 * A.31 (1821 – 1894)
 * A.32 (1826 – 1866)
 * A.33 (1815 – 1897)

Part B: Memorable Mathematics

 * B.1 The Pythagorean Theorem
 * Appendix: The Formulas of Heron and Brahmagupta
 * B.2 More about Numbers: Irrational, Perfect Numbers, and Mersenne Primes
 * B.3 's Quadrature of the Parabola
 * B.4 The Lunes of Hippocrates
 * B.5 's Calculation of $\int_0^b x^n \mathrm d x$ for Positive Rational $n$
 * B.6 How Archimedes Discovered Integration
 * B.7 A Simple Approach to $E = M c^2$
 * B.8 Rocket Propulsion in Outer Space
 * B.9 A Proof of Vieta's Formula
 * B.10 An Elementary Proof of Leibniz's Formula $\frac \pi 4 = 1 - \frac 1 3 + \frac 1 5 - \frac 1 7 + \cdots$
 * B.11 The Catenary, or Curve of a Hanging Chain
 * B.12 Wallis's Product
 * B.13 How Discovered His Formula $\frac \pi 4 = 1 - \frac 1 3 + \frac 1 5 - \frac 1 7 + \cdots$
 * B.14 's Discovery of the Formula $\sum_i^\infty \frac 1 {n^2} = \frac {\pi^2} 6$
 * B.15 A Rigorous Proof of Euler's Formula $\sum_i^\infty \frac 1 {n^2} = \frac {\pi^2} 6$
 * B.16 The Sequence of Primes
 * B.17 More About Irrational Numbers. $\pi$ Is Irrational
 * Appendix: A Proof that $e$ Is Irrational
 * B.18 Algebraic and Transcendental Numbers. $e$ Is Transcendental.
 * B.19 The Series $\sum \frac 1 {p_n}$ of the Reciprocals of the Primes
 * B.20 The Bernoulli Numbers and Some Wonderful Discoveries of
 * B.21 The Cycloid
 * B.22 's Solution of the Brachistochrone Problem
 * B.23 Evolutes and Involutes. The Evolute of a Cycloid
 * B.24 Euler's Formula $\sum_i^\infty \frac 1 {n^2} = \frac {\pi^2} 6$ by Double Integration
 * B.25 Kepler's Laws and Newton's Law of Gravitation
 * B.26 Extensions of the Complex Number System. Algebras, Quaternions, and Lagrange's Four Squares Theorem


 * Answers to Problems


 * Index

Overview
Beware that some of the dates given are incorrect, as are some of the attributions. But these are minor flaws, of interest only to a historian.