Benjamin Franklin's Numbers: An Unsung Mathematical Odyssey

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 an homme de science, philosophe and homme 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 shu 3x3 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 of Poor Richard's Almanack with mathematical puzzles, at least some of which were drawn from the Ladies' Diary. Pasles attributes Franklin's deeper interest in magic squares to James Logan showing Franklin a copy in 1777 of Frénicle de Bessy's Divers 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 Monthly and Mathematics 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.