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.  B.C.)
 * A.2 (ca.  B.C.)
 * A.3 (ca.  B.C.)
 * A.4 (ca. $300$ B.C.)
 * A.5 (ca.  B.C.)
 * Appendix: The Text of
 * A.6 (ca.  B.C.)
 * Appendix: ' General Preface to His Treatise
 * A.7 ($1$st century A.D.)
 * A.8 ($4$th century A.D.)
 * Appendix: The Focus-Directrix-Eccentricity Definitions of the Conic Sections
 * A.9
 * A Proof of Diophantus' Theorem on Pythagorean Triples


 * The Forerunners
 * A.10
 * A.11
 * A.12
 * A.13
 * A.14
 * A.15
 * A.16
 * A.17


 * The Early Moderns
 * A.18
 * Appendix: 's $1714$(?) Memorandum of the Two Plague Years of $1665$ and $1666$
 * A.19
 * A.20 The Bernoulli Brothers
 * A.21
 * A.22
 * A.23
 * A.24


 * The Mature Moderns
 * A.25
 * A.26
 * A.27
 * A.28
 * A.29
 * A.30
 * A.31
 * A.32
 * A.33

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

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.

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: