## Benjamin Franklin's Mathematics

## © Copyright 2002-2012 Paul C. Pasles.

## THE BOOK!

Read about Benjamin Franklin's Numbers! Check your local bookstore or order from Princeton University Press or Amazon. You can read the reviews here and the first chapter here. Interviews can be found at Radio Times and Technica, capsule reviews at NPR.org and New Scientist , and the latest review at Convergence. Eminent Franklin biographer James Srodes weighs in for the Washington Times. Find out more at Science News or at my Franklin page.

## Fast Forward

What is this page all about? Articles by me ## For several years now I have been devoted to educating the world about Benjamin Franklin's little-known role as a mathematician. He has been portrayed as a mathematical dilettante, but I can assure you, that is far from the truth! This page outlines some basics, but you can find much, much more in Benjamin Franklin's Numbers.

What is this page all about?

## What is a magic square?

We will sooner see agreement on most other pressing issues of the day before the world sees eye-to-eye on a standard definition of a "magic square".You draw a square array -- a square

matrix, in the mathematical lingo-- and fill in the spaces with numbers. On that much, we can all agree. Oh, occasionally one sees letters used instead of numbers, especially in mystical contexts; some "New Age" books include this old example:

SATORAREPOTENETOPERAROTASwhich is supposed to ward off the evil eye. (I suppose it works... noone has successfully given me the evil eye since I started typing today.) But that superstitious nonsense is not so interesting mathematically, so let's stick with numbers.

Not only should the entries be numeric, they are usually presumed to be

wholenumbers. Franklin (and others) chose to use 1,2,3,... in some order. Thus the highest entry isn^{2}, withnbeing the number of cells on a side. Here is one example withn= 3:

294753618and you can see more examples here. Every row and column adds up to the same number (in the figure above, the common sum is 15), and moreover the two diagonal sums also have that same "magic" sum.

## What does any of this have to do with Ben Franklin?

Articles by Paul PaslesArticles on other subjects can be found here.

- "Benjamin Franklin, Magician?"
Franklin Gazette, Fall 2000. While conducting preliminary research in 1999-2000, I communicated frequently with the dedicated folks at the American Philosophical Society. Several of them are members of the Friends of Franklin, and they encouraged me to write a preliminary piece for their newsletter. (This was the 2nd piece I wrote, but the first to be published.)- "The Lost Squares of Dr. Franklin." lead article
American Mathematical Monthly, June-July 2001. I won't be posting this one on the web, since there is some overlap with a chapter in my forthcoming book on the subject. However, the abstract/summary is online, as are some addenda:

- Appendix 1: Franklin Squares. Here's one possible method BF might have used to build his most famous squares.
- Appendix 2: More Magic Squares. See some magic squares that figure prominently in Ben's story.
- Appendix 3: The Magic Circle. This is one of Ben's finest creations.
- "Benjamin Franklin." MacTutor entry, June 2001.
- "Digging For Squares."
Math Horizons, April 2002. solutions to the exercises.- "Franklin's Other 8-Square."
Journal of Recreational Mathematics, 31:3, 2003.- "A Bent for Magic."
Mathematics Magazine, 79:1, 2006.

## News Articles About Paul's Research + Book Mentions, Etc.

The "lost squares" research was reportedin the journalScience, in "Random Samples" (4 May 2001). The same writeup was previewed at Postech's Science Now and Academic Press's Daily inScight. An undated review appears in Mathematical Digest. In July 2001 there was full-page piece in a European edition of Scientific American,Pour la Science(Edition française de Scientific American). An excerpt is here. Around the same time, my paper was mentioned in a review of Robin Wilson's new book.

In July and August 2001, the "lost squares"research was Math in the Media top story! The "Poor Richard's Magic Squares" teaser even decorated the AMS home page. That summer, the world's most famous computer scientist wrote me a nice letter about "The Lost Squares". In response to my paper he is making a change in the new edition of his famous Selected Papers on Computer Science. The abstract section of November'sHistoria Mathematicalisted "The Lost Squares". In December, Mathematics Magazine reviewed theScienceandMonthlyarticles. Also, I was quoted in an article that appeared in theFranklin & Marshall Gazette.

The August 2002 issueofPour la Scienceprinted another of the 8-squares, plus a few statements by me on the subject.Historiaabstracts the "Digging for Squares" piece. Cliff Pickover's book (2002) mentions my work on the "lost squares". Says he: "It is amazing that these new squares ... did not come to light until the twenty-first century" (pp. 382-3). TheMonthlypaper was reviewed in the usual places. Here's the Zentralblatt review. You can also read the MR review if your institution has a MathSciNet subscription.

More recently, my papers were cited in Polyhedral Cones of Magic Cubes and Squares by Ahmed, De Loera & Hemmecke; in The number of "magic" squares and hypercubes by Beck, et al.; and in How Many Squares Are There, Mr. Franklin?: Constructing and Enumerating Franklin Squares by Ahmed, which appeared in the May 2004American Mathematical Monthly.My Franklin Squares page is recommended in the wonderful Everything Kids' Math Puzzles Book: Brain Teasers, Games, and Activities for Hours of Fun.The Toronto

Globe and Mail, Canada's "newspaper of record", ran a front-page story about Peter Loly's new result on Franklin squares (March 2006). I had the honor of being quoted. Read Peter's paper and the news article.In January, Ed Rendell and I (together with 298 other people) lit the candles on a very big cake at the National Constitution Center. We were the " 300 Modern-Day Franklins"!

## Talks

The most dangerous place to be is between a mathematician and a microphone.In the summer of 1999 I spoke on "A Reluctant Colonial Mathematician" at the Institute for the History of Mathematics and Its Use in Teaching. A few months later, an updated version of this talk was delivered at the 2000 Joint Meetings of the AMS & MAA. I created a sequel for the 2001 Joint Meetings, and yet another for the Spring 2003 AMS Eastern Section Meeting. Here are the abstracts as they were printed in the

Abstracts of Papers Presented to the American Mathematical Society:Washington, DC meeting (January 2000). A Reluctant Colonial Mathematician.

Two centuries after their creation, Franklin semi-magic squares are unsurpassed in ingenuity. Two examples are quite well-known, two more are far less well-known, and a final pair seem to have been lost. I will discuss the circumstances surrounding the creation and subsequent fate of these half-dozen gems from a reluctant colonial mathematician.

AMS contributed paper session on Mathematics Education and History. Meeting 2074,abstract 950-01-98.Ben had a hand in founding two great institutions of higher learning: the University of Pennsylvania and Franklin & Marshall College. In 2001 I was invited to preach the gospel at both schools. (Here's a press release.)New Orleans, LA meeting (January 2001).The Lost Squares of Dr. Franklin.

B. Franklin's mathematical side extends beyond the two magic squares frequently seen in recreational mathematics books. This talk will cover his unpublished squares, almanac mathematics, and personal library. It will also describe his procedure for constructing "magic circles" similar to those seen in Smith & Mikami's "A History of Japanese Mathematics".

AMS special session on the History of Mathematics. Meeting 2025, abstract 962-01-113.

New York, NY meeting (April 2003).The Followers of Franklin.

For two hundred thirty-five years, recreational math enthusiasts have devoted themselves to decoding the methods used by a minor mathematician of major renown. Like the Founder himself, most of the contributors to this program were untrained amateurs working in isolation. This talk surveys the work and lives of those who bothered to leave a written record.

AMS special session on the History of Mathematics. Meeting 2094,abstract 986-01-44.Other places where I've delivered different versions of this spiel are Bryn Mawr College, Villanova University, Adelphi University (at the Pohle Colloquium), Temple University, Cabrini College, Manhattan College, Rowan University and Haverford College.

In April 2004 I spoke at the NCTM (National Council of Teachers of Mathematics) meeting in Philadelphia, and in May I was keynote speaker at the CMC3 (California Mathematics Council) conference in Lake Tahoe. I led the Franklin-themed workshop at Mathfest 2005 in Albuquerque. In July 2006 I was a resident scholar at the NEH workshop for K-12 teachers, Benjamin Franklin and the Invention of America .

In 2009, I presented at the annual meeting of the History of Science Society in Phoenix and at Science & Math Night at Labyrinth Books in Princeton. In 2010, I spoke at Rutgers University.

More items of interest

(section under development) email questions, comments, suggestions to paul.pasles@villanova.edu.

this page © copyright 2002-2012 Paul C. PaslesLast updated August 30, 2012.