Benjamin Franklin's Numbers

  • Paul C. Pasles
Princeton University Press: 2007. 266 pp. $26.95, £15.95 9780691129563 | ISBN: 978-0-6911-2956-3

Benjamin Franklin was a brilliant amateur scientist in an era when amateur science mattered. His experiments on electricity won him the Royal Society's Copley Medal in 1753 — it is to Franklin that we owe the notion of positive and negative charges. He charted (and named) the Gulf Stream. His prodigious inventions include the lightning rod, the glass harmonica, bifocals and the Franklin stove.

In Franklin's Numbers, a book mixing intellectual history and mathematical puzzles (with solutions appended), Paul Pasles brings out a less-celebrated sphere of Franklin's intellect. He makes the case for the founding father as a mathematician. Franklin's main contribution to the subject was his work on magic squares, and these are the focus of the book.

Even bent rows (1, 8, 13, 12, for instance) of Franklin's notes sum to 34.

A magic square is, in its traditional formulation, an n × n grid containing the numbers 1 to n2, such that all rows, all columns, and both diagonals sum to the same number. Franklin, characteristically, improved on the usual form, producing squares that could be summed in more intriguing ways, along 'bent rows', for example. He also concocted several magic circles as a further novelty.

Franklin was diffident on the subject of his work on magic squares, sheepishly admitting to having spent time on them out of proportion to the subject's utility. Pasles takes up the defence of Franklin's squares, correctly pointing out that utility is not a suitable measure for a piece of mathematics and that future applications are notoriously hard to predict.

It is here that the case starts to become shaky. The number theorist G. H. Hardy wrote in his Mathematician's Apology in 1940 that “the best mathematics is serious as well as beautiful”, going on to assert that “the 'seriousness' of a mathematical theorem lies not in its practical consequences ... but in the significance of the mathematical ideas which it connects”. By this measure, magic squares, entertaining though they are, rank mathematically just a little higher than chess problems (Hardy's example of real but unimportant mathematics).

Perhaps Franklin just came too late to pure mathematics, already a mature field in his era, but early to electricity, where the work of a gentleman researcher could still be ground-breaking. It was Franklin's electrical work, viewed in the light of Maxwell's equations, that gave us genuine mathematical magic.