Book:George F. Simmons/Calculus Gems

George F. Simmons: Calculus Gems: Brief Lives and Memorable Mathematics

Published $\text {1992}$, McGraw-Hill

ISBN 0-07-057566-5.

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 Simmons's $1972$ work Differential Equations.

Contents

Preface

Part A: Brief Lives

The Ancients

A.1 Thales (ca. $\text {625}$ – $\text {547}$ B.C.)

A.2 Pythagoras (ca. $\text {580}$ – $\text {500}$ B.C.)

A.3 Democritus (ca. $\text {460}$ – $\text {370}$ B.C.)

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

A.5 Archimedes (ca. $\text {287}$ – $\text {212}$ B.C.)

Appendix: The Text of Archimedes

A.6 Apollonius (ca. $\text {262}$ – $\text {190}$ B.C.)

Appendix: Apollonius' General Preface to His Treatise

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

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

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

A.9 Hypatia ($\text {370?}$ – $\text {425}$)

A Proof of Diophantus' Theorem on Pythagorean Triples

The Forerunners

A.10 Kepler ($\text {1571}$ – $\text {1630}$)

A.11 Descartes ($\text {1596}$ – $\text {1650}$)

A.12 Mersenne ($\text {1588}$ – $\text {1648}$)

A.13 Fermat ($\text {1601}$ – $\text {1665}$)

A.14 Cavalieri ($\text {1598}$ – $\text {1647}$)

A.15 Torricelli ($\text {1608}$ – $\text {1647}$)

A.16 Pascal ($\text {1623}$ – $\text {1662}$)

A.17 Huygens ($\text {1629}$ – $\text {1695}$)

The Early Moderns

A.18 Newton ($\text {1642}$ – $\text {1727}$)

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

A.19 Leibniz ($\text {1646}$ – $\text {1716}$)

A.20 The Bernoulli Brothers (James $\text {1654}$ – $\text {1705}$, John $\text {1667}$ – $\text {1748}$)

A.21 Euler ($\text {1707}$ – $\text {1783}$)

A.22 Lagrange ($\text {1736}$ – $\text {1813}$)

A.23 Laplace ($\text {1749}$ – $\text {1827}$)

A.24 Fourier ($\text {1768}$ – $\text {1830}$)

The Mature Moderns

A.25 Gauss ($\text {1777}$ – $\text {1855}$)

A.26 Cauchy ($\text {1789}$ – $\text {1857}$)

A.27 Abel ($\text {1802}$ – $\text {1829}$)

A.28 Dirichlet ($\text {1805}$ – $\text {1859}$)

A.29 Liouville ($\text {1809}$ – $\text {1882}$)

A.30 Hermite ($\text {1822}$ – $\text {1901}$)

A.31 Chebyshev ($\text {1821}$ – $\text {1894}$)

A.32 Riemann ($\text {1826}$ – $\text {1866}$)

A.33 Weierstrass ($\text {1815}$ – $\text {1897}$)

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 $n$

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 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

Critical View

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.

Errata

Power Series Expansion for Tangent Function

Chapter $\text {B}.20$: The Bernoulli Numbers and some Wonderful Discoveries of Euler: The Power Series for the Tangent:

Based on our knowledge of the Bernoulli numbers, the first few terms of the [ Power Series Expansion for Tangent Function ] are easy to calculate explicitly,

$\tan x = x + \dfrac 1 3 x^3 + \dfrac 2 {15} x^5 + \dfrac {17} {315} x^7 + \dfrac {67} {2835} x^9 + \cdots$