Nonplussed! by Julian Havil Read Free Book Online Page A

rules and no objectives.
    Ian Ellis
    The Society for the Diffusion of Useful Knowledge , founded (mainly by Lord Brougham) in 1828, had the object of publishing information for people who were unable to obtain formal teaching, or who preferred self-education. The celebrated English mathematician and logician Augustus De Morgan was a gifted educator who contributed no less than 712 articles to one of the society’s publications, the Penny Cyclopaedia : one of them (published in 1838 and titled Induction ) detailed (possibly for the first time) a rigorous basis for mathematical induction.
    It would appear that De Morgan was contacted by more than his fair share of people whom we might now call mathematical cranks or, to use his own word, paradoxers , defined by him in the following way:
    A great many individuals, ever since the rise of the mathematical method, have, each for himself, attacked its direct and indirect consequences. I shall call each of these persons a paradoxer, and his system a paradox. I use the word in theold sense: a paradox is something which is apart from general opinion, either in subject matter, method, or conclusion.
    His unwelcome exposure to
    squarers of the circle, trisectors of the angle, duplicators of the cube, constructors of perpetual motion, subverters of gravitation, stagnators of the earth, builders of the universe…
    inspired the (posthumously published) Budget of Paradoxes , a revised and extended collection of letters to another significant publication, the Athenæum journal.
    The Budget is an eclectic collection of comments, opinions and reviews of ‘paradoxical’ books and articles which De Morgan had accumulated in his own considerable library, partly by purchase at bookstands, partly from books sent to him for review or by the authors themselves. It seems that one James Smith, a successful Liverpool merchant working at the Mersey Dock Board, was the most persistent cause of such aggravation:
    Mr. Smith continues to write me long letters, to which he hints that I am to answer. In his last of 31 closely written sides of note paper…
    Mr Smith’s conviction was that(which he seemed to ‘prove’ by assuming the result and showing that all other possible values then led to a contradiction). The reader may enjoy delving a little deeper into the world of mathematical cranks by reading Woody Dudley’s delightful book of that name.
    It is small wonder then that De Morgan picked on probability as a rich seam for the paradoxers to mine, but he recognized it as a seam which contained more than fool’s gold. Again, from the Budget we read
    The paradoxes of what is called chance, or hazard, might themselves make a small volume. All the world understands that there is a long run, a general average; but a great part of the world is surprised that this general average should be computed and predicted. There are many remarkable cases of verification; and one of them relates to the quadrature ofthe circle…. I now come to the way in which such considerations have led to a mode in which mere pitch-and-toss has given a more accurate approach to the quadrature of the circle than has been reached by some of my paradoxers. The method is as follows: Suppose a planked floor of the usual kind, with thin visible seams between the planks. Let there be a thin straight rod, or wire, not so long as the breadth of the plank. This rod, being tossed up at hazard, will either fall quite clear of the seams, or will lay across one seam. Now Buffon, and after him Laplace, proved the following: That in the long run the fraction of the whole number of trials in which a seam is intersected will be the fraction which twice the length of the rod is of the circumference of the circle having the breadth of a plank for its diameter. In 1855 Mr. Ambrose Smith, of Aberdeen, made 3,204 trials with a rod three-fifths of the distance between the planks: there were 1,213 clear intersections, and 11 contacts on which it was