# Book:George F. Simmons/Calculus Gems

## Contents

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

Published $1992$, **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. $625$ – $547$ 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 ($1$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 ($370?$ – $425$)
- A Proof of Diophantus' Theorem on Pythagorean Triples

**The Forerunners**

**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 ($1768$ – $1830$)

- A.18 Newton ($1642$ – $1727$)

**The Mature Moderns**

#### 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,*

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