Springer 2008

Metascience (2008) 17:475–479 DOI 10.1007/s11016-008-9215-0

REVIEW

THE BOOK THAT FRANKLIN NEVER WROTE

Paul C. Pasles, Benjamin Franklin’s Numbers: An Unsung Mathematical Odyssey. Princeton and Oxford: Princeton University Press, 2008. Pp. 254. £15.95 US$26.95 HB

By Carla Mulford Few books on Benjamin Franklin offer significantly new information. Paul C. Pasles’s Benjamin Franklin’s Numbers is that rare book among contemporary studies of Franklin’s science that develops new information and offers important new insights about his mathematical genius. Scholars of Franklin have traditionally considered him deficient in the science of mathematics. Perhaps they were misled by Franklin’s own demurrals about the qualities of his magic squares, mere ‘‘Arithmetical Curiosities’’, as he called them in his letter to Peter Collinson later published in Collinson’s fourth edition of Franklin’s Experiments and Observations on Electricity, Made in Philadelphia in America in 1769 (p. 124). Franklin wrote: ‘‘I still think I might have employed more usefully’’ the leisure time spent, when a youth, making magic squares. He pointed to ‘‘the good sense of our English mathematicians, that they would not spend their time in things that were mere difficiles nugæ, incapable of any useful application’’. Yet, Franklin averred: ‘‘[p]erhaps the considering and answering such questions … may not be altogether useless, if it produces by practice an habitual readiness and exactness in mathematical disquisitions, which readiness may, on many occasions, be of real use’’ (pp. 125–126). Here we see Franklin’s rhetorical skill: acknowledging that spending time making magic squares might be perceived as wasting time, Franklin persisted in justifying his activity as potentially useful. We thus observe his characteristic deference to pragmatism as opposed to speculative science. Undeterred by Franklin’s allusion to his contemporaries’ negative attitudes about mathematical gamesmanship and unde-

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terred too by scholars’ traditional views that Franklin lacked mathematical genius, Pasles has created the first complete account of Franklin’s mathematical mind and the means by which he applied mathematics to contemporary problems. Pasles examines Franklin’s preoccupations with mathematical games ranging from magic squares to what we today call modern utility theory (including marginal utility theory and consumer utility analysis), population theory (more commonly called demography), cost–benefit analysis, laws of transitivity, and, in the realm of electrical science, the conservation of charge and the inverse square law (eventually proven by his friend, Joseph Priestley, based on Franklin’s theoretical hypotheses). Even Franklin’s definition of electricity as occurring in positive and negative charges, Pasles points out, derived from his effective use of mathematics. ‘‘Franklin seems to have been utterly convinced that the most subjective aspects of life could be measured, quantified, and thus made mathematical’’ (p. 97), Pasles contends. His book ably demonstrates that Franklin ‘‘brought quantitative reasoning to the realm of decision making at a time when this was still a radical notion’’ (p. 98). After a first chapter setting up the book’s project, Pasles treats ‘‘A Brief History of Magic’’ as Chapter 2. In an unconventional and sweeping gesture, Pasles locates the possible origination of magic squares in the same principles as those of scapulimancy, or ‘‘the art of reading patterns on animal bones or shells for the purpose of divination (that is, prophecy and other insight by magical means)’’ (p. 21). According to Pasles, the 2200 B.C.E. Ôlo shu’ matrix is the probable antecedent for the magic squares adopted in the West in the centuries before Franklin started playing with numbers. Pasles tells a fascinating (and controversial) story, explaining that the Ômeme’ or basic, original code for magic squares and cubes, crossed the East and Middle East, reaching Europe by the thirteenth century C.E. (p. 22). Although Franklin himself attributed his mathematical training to George Brownell’s school, Hodder’s Arithmetic, Edward Cocker’s books, John Locke, the Port Royal logicians, and Samuel Sturmy, Pasles identifies various other influences on Franklin’s mathematical thinking, including the work of Cornelius Agrippa and Albrecht Du¨rer, along with that of Girolamo Cardano, Blaise Pascal, Bernard Fre´nicle de Bessy, and their important, unacknowledged antecedents from China, Persia, Arabia, and India. Using Edwin Wolf’s work on Franklin’s library,

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Pasles identifies immediate sources for Franklin’s magic squares in the work of Jacques Ozanam, John Tipper, and Tycho Wing and he speculates on the uses of magic squares in masonry. Succeeding chapters are less historically expansive and more detailed in their materials on Franklin. Chapter 3 discusses Franklin’s almanacs and his work with the Pennsylvania Assembly. Pasles examines several mathematical games and magic squares that Franklin developed at the time he started the almanacs, pointing out that he was fascinated with combinations, ‘‘that branch of mathematics that counts patterns and arrangements of objects’’, and perfect numbers. Using evidence from Poor Richard (Franklin’s almanac) and the Pennsylvania Gazette (Franklin’s newspaper), Pasles shows how Franklin attempted to show readers the importance of learning numbers. Franklin’s remarkable calculations of population in the Observations Concerning the Increase of Mankind (written 1751; published 1755) have always been used as ample evidence of his skill in mathematics; Pasles clarifies the fact that Franklin made demographic analyses and offered them in Poor Richard much earlier than most scholars have noted (pp. 70–71). Chapters 4 and 5 treat Franklin’s mathematical and scientific achievements of the 1740s and 1750s in the light of his public projects to enhance colonial Britons’ lives. Linking Franklin’s mathematical genius to his insistence that human lives could be improved if individuals used their skills appropriately, Pasles clarifies how Franklin’s projects – from developing a lottery and funding a militia to proposing the Academy, charity school, and daylight saving time – all derived from his steady impulse to quantify and improve on the quotidian. Franklin’s Ômoral algebra’ (also called his Ôprudential algebra’), a kind of utility theory, had a clear impact on Jeremy Bentham’s formulations of utilitarianism. Indeed, Pasles posits that Bentham likely drew his ideas from Franklin through their mutual friends, Richard Price and Joseph Priestley (pp. 103–107). In discussing Franklin’s work with electricity, Pasles shows how Franklin’s innovations in the field revolutionised a relatively stagnant area of inquiry – stagnant, that is, until Franklin came along and charted new electrical territory for what Pasles, following J. L. Heilbron, calls ‘‘The Age of Franklin’’ (p. 105). Chapter 5 acknowledges the importance of Franklin’s friendship and collaboration with James Logan and, through Logan, with Peter Collinson

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and Edmund Halley and John Flamsteed, First and Second Astronomers Royal. At the height of his scientific experiments with lightning, Franklin was developing new magic squares, creating innovations such as Ôbent row’ sequences and hidden riddles that Pasles uncovers for readers for the first time. Franklin significantly outperformed his French and German precursors. Chapters 6, 7, and 8 on Franklin’s continued innovations in magic squares, trace their evolution through the 1760s and 1770s. The 1767 publication of Franklin’s Ômagic 16-square’ triggered a transformation of the early meme into a new and extravagant form, the Ômagic circle’, perfected by Franklin (discussed in Chapter 7). Published in James Ferguson’s Tables and Tracts, Relative to Several Arts and Sciences (London, 1767), Franklin’s Ômagic 16-square’ became a matter of inquiry in meetings of the American Philosophical Society in Philadelphia and then in England and Europe, as well. Franklin wrote (from London) to his friend John Winthrop, a professor at Harvard, 2 July, 1768: ‘‘The magic square and circle, I am told, have occasioned a good deal of puzzling among the mathematicians here; but no one has desired me to show him my method of disposing the numbers. It seems they wish rather to investigate it themselves’’ (p. 152). Pasles shows that Franklin’s work was particularly important to Charles Hutton, Fellow of the Royal Society and professor of mathematics at the Royal Military Academy, and his colleague, Isaac Dalby. These scholars, like many others, were intrigued by Franklin’s arrangements of numbers. So when his ÔMagic Circle of Circles’ was published in the 1769 edition of his Experiments and Observations, Franklin caught and held the attention of Britain and Europe. Franklin’s contemporaries stood in awe of Franklin’s mathematical skill. Pasles establishes the long tradition of the meme from which magic circles derived (pp. 161–166). The work of Franklin’s predecessors from China and Japan was improved on in Franklin’s circles: Franklin combined two kinds of magic circles into one, multi-chromatic one. Pasles writes, ‘‘Magic semicircles, excentric [Franklin’s term] circles, and excentric semicircles are all innovations that began with Franklin’’ (p. 167). In addition, Pasles reveals – for the first time – that underlying Franklin’s magic circles is a calculation based on pi, as a constant. Pasles explains how Franklin (by manipulating the paper on which magic squares appeared) folded his magic squares into magic circles: ‘‘rows become radii,

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columns become circles,’’ and ‘‘[t]en excentric circles are derived from bent rows’’ (p. 175). Pasles also shows how a ‘‘little-heralded property built into most of Franklin’s magic squares, the magic zigzag’’ thus comes to light (p. 175). Replete with biographical insight, scholarship in mathematics and cultural and intellectual history and series of mathematical puzzles, Pasles’ book is, as he contends, ‘‘the book that Franklin never wrote’’ (p. [1]). Pasles’s style – mixing intellectual history and scholarly materials with intriguing puzzles and explanations of mathematical truths and problems – mimics Franklin’s own method in his almanacs and letters, where Franklin would offer explanation and background, then puzzles, then more background and explanation, sometimes in unrelated sequences. Little mathematical games and puzzles, along with the brief ÔInterlude’ called ÔPhilomath Math’ (bound into the book between Chapters 3 and 4), and the Appendix – all are intriguingly reminiscent of the little mathematical games offered in Poor Richard, in Franklin’s letters to friends, and in his writings on population and trade. This is a delightful and interesting book on an often misunderstood aspect of Franklin’s life and career. Department of English Pennsylvania State University University Park USA