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REVIEWS Contents
A History of the Study of Mathematics at Cambridge
A History of the Study of Mathematics at Cambridge by Walter William Rouse Ball, 1st edn. 1889, facsimile paperback reprint 2009, Cambridge University Press, £15.99, US$23.99, ISBN 978-1-108-00207-3. When Rouse Ball, fellow and lecturer at Trinity College Cambridge, wrote on the history of mathematics, there were few competitors in English. His A Short Account of the History of Mathematics (1888) long remained in print and influenced British views of the development of mathematics. The book presently reviewed has a much narrower focus, although it covers a long time-span, from Cambridge University’s origins in the late twelfth century until 1858, when new Victorian statutes were introduced. Its re-issue is part of CUP’s Cambridge Library Collection devoted to the history and influence of the University. These are all facsimiles, available on a print-on-demand basis. It is easy to criticise the shortcomings of Rouse Ball’s book: it is outdated and inaccurate in various ways, and he adopts a parochial view that fails to situate mathematics at Cambridge within European, or indeed British, mathematics. But his broader Short Account… provides some defence for the restricted focus of the present work. Also, Rouse Ball wrote at a time when few research aids were available. The (old) Dictionary of National Biography then covered only the first few letters of the alphabet, and his work precedes most of the now numerous college and university histories. Those interested in the organisation of teaching and examining in the University and its colleges, and the lives of the students, should begin with chapter XI, ’Outlines of the history of the university’; then proceed to chapters VIII, IX and X on ’The organisation and subjects of education’, ’The exercises in the schools’ and ’The mathematical tripos’. In the mediaeval period, arrangements for teaching and examination were cursory. The Elizabethan statutes of 1570 sought to improve matters, and they remained in force until 1858, by which time many were ignored or accorded lip-service. Rouse Ball quotes, without translation, some late eighteenth-century Latin oral disputations: these later became farcical travesties before abandonment. The senate-house examination, or mathematical tripos, began about 1725, with order-of-merit lists printed from 1747 onwards. Questions were dictated, rather than written or printed, until after 1786; and Rouse Ball gives those from 1785 and 1786. The problem papers from 1802 are also reproduced. With the growth in prestige of the tripos, candidates increasingly had recourse to private tutors. The outstanding results of William Hopkins and Edward J. Routh are briefly mentioned, but fuller modern accounts exist. The first seven chapters address the teachers and researchers under the headings ’Mediaeval mathematics’, ’The mathematics of the renaissance’, ’The commencement of modern mathematics’, ’The life and works of Newton’, ’The rise of the Newtonian school’, ’The later Newtonian school’, and ’The analytical school’. These have least stood the test of time, for most individuals mentioned have been the subjects of more modern scholarship. A historian friend recently described this as ’a nice old book’ and so it is. Though no longer of much value as a historical source, it gives a concise overview of mathematics at Cambridge, and some of Rouse Ball’s personal comments are trenchant. For instance, we learn that Isaac Barrow was slovenly in dress and an inveterate smoker; that Richard Bentley’s features were "indicative of cruelty and selfishness"; and that sympathy for William Frend, banished in 1793, "will probably be dissipated by reading his own account of the proceedings". Alternative versions of this out-of-copyright title are available free of charge from www.archive.org/details/texts in electronic format. Alex D.D. Craik
Those Fascinating Numbers by Jean-Marie de Koninck, American Mathematical Society, 2009, 426 pp, £36.95 US$49, €41.00, ISBN 978-0-8218-4807-4. Biscuits of Number Theory edited by A.T. Benjamin and E. Brown, Mathematical Association of America, 2009, 336 pp, US$62.50, ISBN 978-0-88385-340-5. Any odd composite integer is the subtraction of two squares, which can sometimes help you in looking for fast factorisations. I guess we all know that, but it’s the sort of thing you only really know after reading books such as these. The first, Those Fascinating Numbers, is a sequential tour through the positive integers. Each integer covered is related to number theoretic properties. So, (opening randomly) you might be told that 1031 is the fifth number such that the decimal number 111…11 (1031 times) is prime, and, most importantly, you are directed on to the number 19, which has the same property and is one of nine numbers known with it. Cross-referencing is everything in this sort of book and here it is meticulous and helpful. This is a first rate resource for teachers and should certainly interest researchers. Its list of numbers begins with 1, and the first integer not specifically appearing is, of course, the first patently uninteresting number – 95 (see 106 which is also absent). An average student reading this book would acquire some mastery of the arcane terms of elementary number theory – amicable numbers, pseudoprimes, horse numbers, vampire numbers (?) … together with some grasp of the elementary number-theoretic functions. In the nature of things it is easier to stumble on something here than search for it but just what you stumble on gives a real feel for the sort of thing that mathematicians have considered or found to be considerable; also the index takes you, not to the page, but to the number. I loved this book and am glad I have it to play with. It would be fun to organise a competition for the most obscure mathematics book described as ’an introduction’. I seem to possess many introductions that never bloomed into full-blooded relationships. On that basis, Biscuits of Number Theory is superbly named, though the intention of the book is probably a little different from that of a standard introduction. The cover shows, well, not Leibniz, but Fibonacci, and he seems to be nibbling a biscuit in the shape of 987 – a number mentioned in the other book but only in a very technical context. Biscuits starts with a quote from the American philosopher and biscuit salesman Garrison Keillor, who once observed that all the children in Lake Wobegon were above average. All the articles in the book are too; because that is what the book is – a collection of accessible and even profound essays on number theory gleaned from a wide variety of writers and journals – everyone from Euler to Quine, plus many recent popular expositions, mainly from American sources. Another great resource for teachers, this is an invigorating and generally undemanding excursion into surprise. The book is arranged into seven parts: Arithmetic, Primes, Irrationality and Continued Fractions, Sums of Squares, Fibonacci Numbers, Number Theoretic Functions and Elliptic Curves, the editors prefacing each with an informative introduction to the articles they have selected. I really loved the articles on elliptic functions – I never seemed to be very receptive to anything that I did not have to instantly write down in my own mathematics education, and the background information on the subject and its development that the articles here contained was very interesting to me (though perhaps well-known to a specialist). I noticed the article ’Great Moments of the Riemann Zeta Function’ mentioned the contribution of Nina Snaith, who gave a fine LMS Popular Lecture on it this year. That article also describes the chance meeting between Dyson and Montgomery which brought random matrices into the weaponry of those working on the Riemann hypothesis and it gave me a clearer idea of what else one might want in the first book on fascinating numbers – some reference to number patterns appearing in the natural sciences up to and including the numerology of Eddington and Dirac. Still, you can’t have it all (as so many different theorems find so many ways to tell us). These are first-rate books. They will live long on our library shelves and be actually used throughout their lives. Bob Lockhart
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