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REVIEWS Contents
Leonhard Euler and the Bernoullis
Leonhard Euler and the Bernoullis by M.B.W. Tent, 2009, A.K. Peters, 296 pp, £22.00, ISBN 978-1-56881-464-3. Although some adults will enjoy reading Leonhard Euler and the Bernoullis, it’s a book written for young readers. It is probably best suited to those between 14 and 16 years of age, although one can imagine a younger reader being captivated by this story of mathematics and mathematicians. It has 39 short chapters, and is written in the form of a novel. It is rich in more or less realistic-sounding dialogue, as imagined by an author who has already written similar books about Carl Friedrich Gauss (1777–1855) and Emmy Noether (1882–1935). Tent had a career teaching middle school mathematics in the United States and is excellent at writing history of science for adolescents. Leonhard Euler (1707–83) was arguably the most influential and productive mathematician of all time. His collected works fill more than 80 volumes. He single-handedly made number theory a part of mainstream mathematics and did seminal work in fields like combinatorics that would not even be named until long after his death. He was born in Basel, Switzerland, but spent most of his career in St Petersburg and Berlin. Basel also produced a family of mathematicians named Bernoulli. Brothers Jacob (1654–1705) and Johann (1667–1748) were pioneers of Leibniz’ differential and integral calculus. Collectively, their achievements and influence in mathematics were as great as Euler’s. Johann’s son Daniel (1700–82) was one of Euler’s greatest contemporaries. At least half a dozen minor mathematicians are to be found among Daniel’s brothers, cousins and nephews. The Bernoulli and Euler families were closely intertwined: Euler’s father attended Jacob’s lectures when in university, Euler was mentored by Johann during his own student days, and he subsequently worked with Daniel at the St. Petersburg Academy. Essentially, Tent’s book is a saga of this Basel clan, including the extended Bernoulli family and Leonhard Euler. Not surprisingly, Tent’s novel also teaches some European history. For example, the action opens with the persecution of the Huguenots, which resulted in a Bernoulli patriarch fleeing from Antwerp to Frankfurt. Later on, readers learn about St. Petersburg and Berlin during their ’Great’ times: the reigns of Peter, Frederick and Catherine. Even more importantly, young readers are exposed to some real mathematics. Interspersed within the story are polar coordinates, the St. Petersburg Paradox in probability theory, Euler’s polyhedral formula, magic squares, and some number theory. Much of this is done through narrative prose, but Tent communicates some mathematical content in dialogue, for example when Jacob explains his polar coordinates to his younger brother. The Bernoullis were a quarrelsome lot, but Tent keeps her story positive. There is very little about the falling out between Jacob and Johann at the end of the 17th century and nothing at all about Daniel Bernoulli’s cold relations with Euler in the middle of the next. The story ends not with the familiar legend of Euler’s last day on earth, but with an uplifting, if entirely fictional, conversation between the elderly Euler and his ten-year-old grandson. Tent’s attention to detail, both mathematically and historically, is excellent. Whatever factual quibbles I might have are so minor as not to be worth mentioning. A bibliography would have been a welcome addition for those students who might be inspired to further reading. Leonhard Euler and the Bernoullis is not for the mathphobic. On the other hand, it assumes only a modest mathematical background, one quite appropriate to the intended audience. It is a welcome addition to the mathematics education literature and would make an excellent gift for a young person with mathematical interests. Rob Bradley
The Monty Hall Problem: The Remarkable Story of Math’s Most Contentious Brain Teaser by Jason Rosenhouse, 2009, OUP, 208 pp, £15.99, ISBN 978-0-19-536789-8. The classic Monty Hall problem is as follows: A game-show host offers a contestant the choice of one of three doors: behind one door lies a new car, and behind each of the other two lies a goat. After the contestant chooses a door, the host opens one of the other doors which reveals a goat, picking a door at random if he has two goats to choose from. He then offers the contestant the opportunity to switch doors – whichever door the contestant chooses is opened to reveal a goat or a car. What is the best strategy? The answer? The contestant should always switch – this gives them a 2/3 chance of winning the car. Many people find this unintuitive, feeling that the remaining two options are equally good. I should admit that I don’t generally use the Monty Hall problem with students, as I am not convinced anyone is enlightened by having it explained. But I have had fun teasing people with it, from my girlfriend 35 years ago, to a senior QC at a dinner. However, my first thought was: how can anyone write a whole book on this? Well, Jason Rosenhouse has, and it’s surprisingly good. I like his opening claim that probability is unintuitive: in fact the whole book is about counter-intuitive results. Chapter 1, Ancestral Monty, gives a history going back to the 3-prisoners problem posed by Martin Gardner in 1959, the translation to the game show format in 1975, and the well-told story of the Marilyn vos Savant column in Parade, which led to a series of articles in which she countered arguments, some apparently from mathematicians, that there is no advantage in switching. Rosenhouse proceeds to Classical Monty, in which he uses the problem to introduce solution methods using (a rather confusing) enumeration of sample spaces and Monte Carlo simulations. In Bayesian Monty a variation is introduced in which the host himself does not know where the car is, which provides a justification for ideas of independence and Bayes theorem. New versions come thick and fast, and are used to widen the discussion to different interpretations of probability. Things get more technical in Progressive Monty in which there are n doors and a strategy in which the contestant sticks to the same door until there are two left, and then switches, is shown to be optimal. Miscellaneous Monty deals with more variations than I thought possible, with multiple players, two hosts, and so on, while Cognitive Monty explores people’s responses to the problem. Philosophical Monty considers how versions of the problem have been used by philosophers to contrast beliefs about the probabilities for single cases of the game with beliefs about long-run statistical properties. This gets difficult. The book finishes with a fine, and presumably fairly exhaustive, set of references. The book is chatty and welcoming, and the author’s enthusiasm is infectious. There is, however, a rather uneven use of mathematics, with binomial coefficients introduced without definition on page 11, and fairly basic ideas of probability coming much later. I am not sure of the intended audience: serious enthusiasts may find it too basic, while beginners will grind to an exhausted halt well before the end. To be honest, after 194 pages of Monty Hall I still am not inclined to use the problem in my efforts to inspire people about the joys of probability, statistics and risk. But I am impressed at how much material can be hung onto a single problem, and the author has my sincere admiration for being such a dedicated, if not obsessive, exponent. David Spiegelhalter
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