This is a collection of gems from the literature of mathematics that shine as brightly today as when they first appeared in print. They deserve to be seen and admired.
The selections include two opposing views on the purpose of mathematics, The Strong Law of Small Numbers, the treatment of calculus in the 1771 Encyclopaedia Britannica, several proofs that the number of legs on a horse is infinite, a deserved refutation of the ridiculous Euler-Diderot anecdote, the real story of π and the Indiana Legislature, the reason why Theodorus stopped proving that square roots were irrational when he got to the square root of 17, an excerpt from Mathematics Made Difficult, a glimpse into the mind of a calculating prodigy….There will be something of interest here for almost anyone interested in mathematics.
Table of Contents
1. Dieudonné on Mathematics
2. Why is Mathematics?
3. Is Mathematics Inevitable?
4. A Defense of Quadratic Equations
5. Obtuse Triangles
6. A Small Paradox
7. Applied Mathematics
8. The Law of Small Numbers
9. The Parallel Postulate
10. Arithmetic in the United States
11. The Moore Method
12. Early Calculus
14. A Tangled Tale
15. A Brief Life
17. Boole and finite Differences
18. Calculating Prodigies
19. James Smith, Circle-Squarer
20. Legislating π
21. Mathematics and Music
22. Mathematics Books
23. Irrational Square Roots
24. The Euler-Diderot Anecdote
25. Mathematics Made Difficult
26. Mathematical Humor
About the Editor
Excerpt: Chapter 9 The Parallel Postulate (p. 103)
Euler was amazing. This can be said even though we know nothing about him other than that he lived in Alexandria some time during the reign of the Ptolemy who ruled between 325 and 285 BC and was the author of the Elements. There is an anecdote or two told about him, but, given the propensity of the human race to make up stories about extraordinary people, there is no reason to think that they ever occurred as related. One, though, is worth repeating. Euclid was asked by a student, the anecdote goes, that perennial question, "What is it all good for?" Euclid instructed his slave to give the student the equivalent of a dollar, so that he could say that he had gained something from mathematics. Modern teachers of mathematics have neither slaves nor enough dollars to follow Euclid's example, even if they wanted to.
Euclid was amazing because the Elements was so well done that it lived on, as the first (and until the Renaissance, the last) word on geometry for two thousand years, a longer life than has been had by any other non-religious book. Euclid did not create all the mathematics in the Elements, and it is even possible that he did nothing new, but he was a superb compiler and organizer. (There is an anecdote about Blaise Pascal—this shows how far anecdotes can be depended on—that when he was a child he reinvented geometry for himself, independently proving the theorems in the Elements, and doing then in the same order as Euclid. The Elements were so close to holy writ that the deviser of the anecdote must have thought that any departure from Euclid's system would be heresy.)
On the basis of intuition alone, I think the Euclid in fact did not originally prove any of the theorems in the Elements. In our day, many mathematicians, if not most of them, are a bit impatient with proof. The fun in doing mathematics is getting the new result and seeing where it leads. Writing it up in a proper form is, if not drudgery, then considerably less fun. Euclid, I think, was a magnificent editor, taking the results of his predecessors, some of them seeming to him muddled, and by using his wonderfully clear mind, putting them in the proper form, shape, and order. This is the merest speculation, but one of the purposes of history is to give opportunities for such things.
The basis of Euclidean geometry is Euclid's five postulates. Here they are, paraphrased:
A straight line can be drawn joining any two points.
Finite straight lines can be extended indefinitely.
A circle can be drawn with any center and radius.
All right angles are equal.
If a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, then the two straight lines, if extended, will meet on the side where the angles are less than two right angles.
About the Author
Underwood Dudley earned his B.S. and M.S. degrees from the Carnegie Institute of Technology and his doctorate (in number theory) from the University of Michigan. He taught briefly at the Ohio State University and then at DePauw University from 1967-2004. Woody has written six books and many papers, reviews, and commentaries. He has served in many editing positions, including editor of The Pi Mu Epsilon Journal, 1993-96 and The College Mathematics Journal, 1999-2003. He is widely known and admired for his speaking ability―especially his ability to find humor in mathematics. He was the PME J. Sutherland Frame lecturer in 1992 and the MAA Pólya lecturer in 1995-96. Woody’s contributions to mathematics have earned him many awards, including the Trevor Evans award, from the MAA in 1996, the Distinguished Service Award, from the Indiana Section of the MAA in 2000, and the Meritorious Service Award , from the MAA in 2004.