# The Harmony of the World: 75 Years of Mathematics Magazine

Share this page *Edited by *
*Gerald L. Alexanderson; Peter Ross*

MAA Press: An Imprint of the American Mathematical Society

Who would expect to find in the pages of Mathematics Magazine the first full treatment of one of the more important and oft-cited twentieth century theorems in analysis, the Stone-Weierstrass Theorem, in an article by Marshall Stone himself? Where else would one look for proofs of trigonometric identities using commutative ring theory? Or one of the earliest and best expository articles on the then new Jones knot polynomials, an article that won the prestigious Chauvenet Prize? Or an amusing article purporting to show that the value of has been time dependent over the years? These and much more are in this collection of the best from Mathematics Magazine. Readers are inundated with new material in the many mathematical journals. Gems from past issues of Mathematics Magazine or the Monthly or the College Mathematics Journal are read with pleasure when they appear, but get pushed into the background when the next issues arrive. So from time to time it is rewarding to go back and see just what marvelous material has been published over many years, articles now to some extent forgotten. There is history of mathematics (algebraic numbers, inequalities, probability and the Lebesgue integral, quaternions, Pólya's enumeration theorem, and group theory) and stories of mathematicians (Hypatia, Gauss, E. T. Bell, Hamilton, and Euler). The list of authors is star-studded: E. T. Bell, Otto Neugebauer, D. H. Lehmer, Morris Kline, Einar Hille, Richard Bellman, Judith Grabiner, Paul Erdos, B. L. van der Waerden, Paul R. Halmos, Doris Schattschneider, J. J. Burckhardt, Branko Grunbaum, and many more.

#### Reviews & Endorsements

A wonderful resource of information about the recent history of mathematics as well as some gems about how important advances in earlier mathematical history were made.

-- Mary Coupland, Australian Math Teacher

In all, this is an attractive, urbane, and inviting selection of mathematics, culture, and history, seen through one important journal's lens. Editor Alexanderson is well qualified, having edited the Magazine himself in the lates 1980s.

-- Paul Zorn, Mathematical Reviews

# Table of Contents

## The Harmony of the World: 75 Years of Mathematics Magazine

- Cover cover11
- Title page iii4
- Introduction vii8
- Contents ix10
- A Brief History of Mathematics Magazine xi12
- Part I The First Fifteen Years 116
- Perfect Numbers by Zena Garrett 318
- Rejected Papers of Three Famous Mathematicians by Arnold Emch 520
- Review of Men of Mathematics by E. T. Bell by G. Waldo Dunnington 924
- Oslo under the Integral Sign by G. Waldo Dunnington 1126
- Vigeland’s Monument to Abel in Oslo by G. Waldo Dunnington 1934
- The History of Mathematics by Otto Neugebauer 2338
- Numerical Notations and Their Influence on Mathematics by D. H. Lehmer 2944

- Part II The 1940s 3348
- Part III The 1950s 6984
- Part IV The 1960s 107122
- Generalizations of Theorems about Triangles by Carl B. Allendoerfer 109124
- A Radical Suggestion by Roy J. Dowling 115130
- Topology and Analysis by R. C. Buck 117132
- The Sequence {sin n} by Stanley Ogilvy 121136
- Probability Theory and the Lebesgue Integral by Truman Botts 123138
- On Round Pegs in Square Holes and Square Pegs in Round Holes by David Singmaster 129144
- pi_t : 1832–1879 by Underwood Dudley 133148

- Part V The 1970s 135150
- Trigonometric Identities by Andy R. Magid 137152
- A Property of 70 by Paul Erdös 139154
- Hamilton’s Discovery of Quaternions by B. L. van der Waerden 143158
- Geometric Extremum Problems by G. D. Chakerian and L. H. Lange 151166
- Pólya’s Enumeration Theorem by Example by Alan Tucker 161176
- Logic from A to G by Paul R. Halmos 169184
- Tiling the Plane with Congruent Pentagons by Doris Schattschneider 175190
- Unstable Polyhedral Structures by Michael Goldberg 191206

- Part VI The 1980s 197212
- Leonhard Euler,1707–1783 by J. J. Burckhardt 199214
- Love Affairs and Differential Equations by Steven H. Strogatz 211226
- The Evolution of Group Theory by Israel Kleiner 213228
- Design of an Oscillating Sprinkler by Bart Braden 229244
- The Centrality of Mathematics in the History of WesternThought by Judith V. Grabiner 237252
- Geometry Strikes Again by Branko Grünbaum 247262
- Why Your Classes Are Larger Than ‘‘Average’’ bt David Hemenway 255270
- The New Polynomial Invariants of Knots and Links by W. B. R. Lickorish and Kenneth C. Millett 257272

- Briefly Noted 273288
- The Problem Section 279294
- Index 281296
- About the Editors 287302