A quick search on Amazon.com turns up literally hundreds of biographies of Ben Franklin, ranging from his own Autobiography of Benjamin Franklin to the children's book How Ben Franklin Stole The Lightning and many many others. It seems that this would make it hard to find room for another book on aspects of Franklin's life. However, Paul C. Pasles' new book Benjamin Franklin's Numbers: An Unsung Mathematical Odyssey fills a niche that had not yet been explored: his life as an amateur mathematician.
As many mathematicians know, either from their own readings or from the many mugs and T-shirts one can find it emblazoned on at places like the Joint Meetings, Benjamin Franklin once asked "What science can there be more noble, more excellent, more useful for men, more admirably high and demonstrative, than this of mathematics?" A cynic might point out that Franklin also wrote of Thomas Godfrey that "like most great mathematicians I have met with, he expected universal precision in everything said, or was forever denying or distinguishing upon trifles, to the disturbance of all conversation." So it is probably an understatement to say that Franklin had a conflicted relationship with mathematics and mathematicians. But there is no denying that he loved puzzles and mathematical tidbits, and as the editor of magazines such as The Pennsylvania Gazette and The General Magazine, as well as the almanacs he is famous for, he indulged this interest often. One example was published in the 1738 edition of Poor Richard:
A Frugal Thought
In an acre of land there are 43560 square feet. In 100 acres are 4356000 square feet; Twenty pounds will buy 100 acres of the proprietor, In £ 20 are 4800 pence; by which divide the Number of feet in 100 acres; and you will find That one penny will buy 907 square feet; or A lot of 30 feet square — Save your pence.
Another type of mathematical puzzle which Benjamin Franklin had quite a bit of interest in was the study of Magic Squares, and it is these objects which the bulk of Pasles' book is devoted to. An n×n magic square is a square arrangement of the numbers from 1 to n2 so that the entries in each row, column, and the two main diagonals add to the same sum. These objects have been an interest of study dating back to at least 2200 BC, and Pasles' book contains an interesting history of their evolution as well as some methods to count the number of magic squares of a given size and to create your own.
Franklin's interest in magic squares has been documented before, but never in as much detail as in Benjamin Franklin's Numbers . Pasles has collected a number of letters and notes written by Franklin about the squares, including methods to build magic squares with various properties. One particularly interesting example is an 8 × 8 square that has a large number of symmetries in addition to the ones necessary to be a magic square. Franklin also developed a number of 'magic circles' which are, as the name suggests, a variation on magic squares which involve placing numbers in a concentric circular pattern so that the sum of the numbers along certain radii and circles obey prescribed properties. These circles are not as well-studied as their square counterparts, so they may be brand new to many readers (they certainly were to this reviewer) and they have a number of interesting patterns worth exploring.
Benjamin Franklin's Numbers touches on quite a few interesting topics, but I have to admit that the book left me unsatisfied in several ways. First and foremost, the topic is so specific that I found myself wanting more historical perspective, both about the mathematics at the time and about other parts of Franklin's life. I understand why the author chose to stick with a small, well-defined niche. As someone who has not studied that period of American history since high school, however, this reader would have benefited from a wider perspective. And, barring that larger context, I am not sure the narrow topic in the book warranted 250 pages.
I also think many mathematicians will balk at some of the types of results that Pasles classifies as 'mathematics'. While there is certainly some nontrivial mathematics that underlies the study of magic squares, many of the things that Franklin did are better classified as number games or diversions. Referring to him as a mathematician — even an amateur one — doesn't feel right. It probably betrays a certain snobbishness on behalf of this reviewer, but I have to admit that it worries me that a reader who didn't know better could walk away with the impression that playing with magic squares is what mathematicians do for a living.
Finally, the book often uses a tone that I found overly informal and chatty to the point of being distracting, referring to the internet as "the misinformation superhighway" and to magic squares as a meme that is not particularly useful and "just catchy, like a cold."
Despite these misgivings, there are a number of things to like about Pasles' book. There are quite a few interesting anecdotes about Franklin's life and about the history of recreational mathematics, and some of the quotations from primary sources and the artwork really help to make the stories come to life. It is also one of the only biographies I have ever read which is littered with sample problems (and in some cases worked-out solutions) for the reader to do.
The book is extremely well-annotated, and one can imagine it being a good gateway for mathy people who want to learn early American history as well as for history buffs to start learning about recreational mathematics (and hopefully move on to more serious mathematical pursuits). In the preface to Benjamin Franklin's Numbers, the author writes: "As with all great lives, Franklin's life appears to us as a magnificent jigsaw puzzle with a few pieces missing. With this book, I hope to fit one more piece firmly into place." Pasles certainly succeeds in this goal — I just would have liked to see a bigger segment of the puzzle.
Darren Glass (firstname.lastname@example.org) is an Assistant Professor at Gettysburg College.