ProofWiki:Books/George F. Simmons/Calculus Gems

George F. Simmons: Calculus Gems

Published 1992, McGraw-Hill.

ISBN 0-07-057566-5

• Subtitled: Brief Lives and Memorable Mathematics

Subject Matter

• History of Mathematics

It's 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.

Contents

Preface

Part A: Brief Lives

The Ancients

A.1 Thales (ca. 625 – 547 B.C.)

A.2 Pythagoras (ca. 580 – 500 B.C.)

A.3 Democritus (ca. 460 – 370 B.C.)

A.4 Euclid (ca. 300 B.C.)

A.5 Archimedes (ca. 287 – 212 B.C.)

Appendix: The Text of Archimedes

A.6 Apollonius (ca. 262 – 190 B.C.)

Appendix: Apollonius' General Preface to His Treatise

A.7 Heron (1st century A.D.)

A.8 Pappus (4th century A.D.)

Appendix: The Focus-Directrix-Eccentricity Definitions of the Conic Sections

A.9 Hypatia (370? – 425)

A Proof of Diophantus' Theorem on Pythagorean Triples

The Forerunners

A.10 Kepler (1571 – 1630)

A.11 Descartes (1596 – 1650)

A.12 Mersenne (1588 – 1648)

A.13 Fermat (1601 – 1665)

A.14 Cavalieri (1598 – 1647)

A.15 Torricelli (1608 – 1647)

A.16 Pascal (1623 – 1662)

A.17 Huygens (1629 – 1695)

The Early Moderns

A.18 Newton (1642 – 1727)

Appendix: Newton's 1714(?) Memorandum of the Two Plague Years of 1665 and 1666

A.19 Leibniz (1646 – 1716)

A.20 The Bernoulli Brothers (James 1654 – 1705, John 1667 – 1748)

A.21 Euler (1707 – 1783)

A.22 Lagrange (1736 – 1813)

A.23 Laplace (1749 – 1827)

A.24 Fourier (1768 – 1830)

The Mature Moderns

A.25 Gauss (1777 – 1855)

A.26 Cauchy (1789 – 1857)

A.27 Abel (1802 – 1829)

A.28 Dirichlet (1805 – 1859)

A.29 Liouville (1809 – 1882)

A.30 Hermite (1822 – 1901)

A.31 Chebyshev (1821 – 1894)

A.32 Riemann (1826 – 1866)

A.33 Weierstrass (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 Archimedes's Quadrature of the Parabola

B.4 The Lunes of Hippocrates

B.5 Fermat's Calculation of $\int_0^b x^n \mathrm d x$ for Positive Rational $x$

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 Leibniz Discovered His Formula $\frac \pi 4 = 1 - \frac 1 3 + \frac 1 5 - \frac 1 7 + \cdots$

B.14 Euler'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 Euler

B.21 The Cycloid

B.22 Bernoulli'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