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 n^{2} 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 (dglass@gettysburg.edu) is an Assistant Professor at Gettysburg College.

## Comments

## Marvin Schaefer

This little book is a well-researched biography that is oriented towards Benjamin Franklin's mathematical and scientific accomplishments. While the "numbers' in the title are primarily those that show up in Franklin's magic squares and magic circles, Paul C. Pasles shows that the polymath Franklin was engaged in far more than simple numerical manipulation. In this book's nine illustrated chapters, Pasles describes many of Franklin's excursions into economics, population growth, physics, astronomy, as well as Franklin's publication of recreational mathematics puzzles in

Poor Richard's Almanack(1732-1758).Many of us grew up with a fragmented view of Franklin's life, much of which came from popular folklore. Referred to familiarly as Ben Franklin, we learned of his popular homilies ("Early to bed, early to rise, makes a man healthy, wealthy and wise") and some of his inventions (bifocals, printer's towel, glass harmonica, Franklin stove), and his experimentation with electricity and the kite-flying episode that nearly killed him but that led to the invention of the lightning rod. Every school child was also taught about Franklin's strong contributions to writing the

Declaration of Independence,Treaty of Paris, and the United States Constitution. As an adult living in France, I was surprised to find that Dr Franklin was highly regarded as anhomme de science, philosopheandhomme d'état, with "scientist' frequently getting top billing on various plaques and monuments commemorating this great American polymath.Pasles' book adds significantly to our knowledge of Franklin's work as a scientist well-versed in mathematics. By the time he was 42, Franklin was able to retire from printing and engage in other pursuits including astronomy, electricity, optics, music, and recreational mathematics. Many of these activities were influenced by his business acumen. For example, we learn that in 1751 Franklin published a paper on population growth (some eleven years before Malthus' birth, and at least 47 years before Malthus'

Essay on the Principle of Population!), and then that the population of the New World colonies would double every 20-25 years, soon to outstrip that of England as it did in the early 19th century. In 1784 Franklin wrote a satirical financial analysis of the cost of lighting homes in Paris, wherein which he proposed a form of summer daylight saving time (q.v.at http://webexhibits.org/daylightsaving/franklin3.html).As the title suggests, most of this book looks at Franklin's magic squares and circles. It contains a brief history of magic squares, beginning with the Chinese

lo shu3x3 square (2800-2200 BCE) and culminating in Franklin's first dabbling with magic squares in the 1730s. Pasles documents much of Franklin's communications with contemporary scientists and his familiarity with scientific and mathematical literature. Franklin ornamented issues ofPoor Richard's Almanackwith mathematical puzzles, at least some of which were drawn from theLadies' Diary. Pasles attributes Franklin's deeper interest in magic squares to James Logan showing Franklin a copy in 1777 of Frénicle de Bessy'sDivers ouvrages de mathématique et de physique(1693), a book filled with magic squares.But, where Frénicle showed a variety of fully-magic squares and methods for their construction, Franklin became increasingly interested in constructing squares where "bent" rows and columns and various blocks of squares also totaled to the magic sum. Franklin claimed that he could produce his panmagic squares literally as fast as he could write the numbers down.Pasles presents a fresh and entertaining view of Franklin's magic squares and circles, detailing their properties and hypothesizing the methods Franklin may have derived to construct them. He uses color plates to advantage in showing their "most magically magical" properties, a tool that is not so much necessary with the squares as it is with Franklin's magic circles. Indeed, Pasles shows that most texts simplified the magic circles, not exposing all of the ways in which the magic sum could be computed -- for not only do the numbers along the radii and concentric circles total to the same number, but also the numbers along four families of eccentrically-centered circles! The depth of Pasles' years of research is impressive, and in his explorations he has also uncovered a previously unknown fully-magical square of side 16, where for once the diagonals also total to the magic sum 2056.

Pasles has maintained a website that gives supplementary information on magic squares and on Franklin's numerical recreations. Alternative methods of construction are given at these sites. See http://pasles.org/Franklin/index.html and http://www.pasles.org/magic.html.

Despite its many strengths, this book does have some blemishes. The book does not have an explicit bibliography. End notes are copiously provided following the text of each chapter. Some of these are bibliographic citations, while others provide considerable elaboration over what is in the text or cross references to other parts of the book. I found the frequency of these end notes to be distracting and the need to constantly page to them was often frustrating. Several references are to the author's papers published in past issues of the

MonthlyandMathematics Magazine.While these papers are easily accessible to anyone with access to a university library or JSTORE, the text would have been improved by the inclusion of either the articles or a summary in an appendix. The level of the text is generally accessible to the advanced middle school or high school student. The author occasionally presents a box containing a problem for the reader to solve. Most of these problems are quite trivial and the solution is overly detailed; in other cases, the solution in the appendix does not give much insight on how Franklin might have solved the problem. In particular, a pursuit curve problem is given where the "solution' points to a calculus text for a contemporary method of finding its solution. There are also short appendices giving a basis from which every 4x4 Franklin square can be obtained and a Franklin-Strachey method for constructing even-order Franklin magic squares. The method uses the Kronecker product which, for some reason, the author chooses not to define directly. Were a short definition presented, I think this appendix would have been accessible to most of the book's readers.Overall, I enjoyed reading this book and believe that you will also. It would certainly make a splendid gift to a young reader or anyone interested in recreational mathematics.

Marvin Schaefer (bwapast@erols.com ) is a computer security expert and was chief scientist at the National Computer Security Center at the NSA, and at Arca Systems. He has been a member of the MAA since 1961 and now operates an antiquarian book store called

Books With a Past.