Mr Hopkins’ Men: Reform and British Mathematics in the 19th Century by Alex D.D. Craik, Springer, London, 405 pp, £65 hardback, 2007, ISBN 978-1-84628-790-9; £25 paperback, 2008, ISBN 978-1-84800-132-9.
The Cambridge Mathematical Tripos examination of the mid-19th century and beyond was a fearsome ordeal, extending to forty-four and a half hours over eight days, and one’s success (or otherwise) in it branded one for life. If, like Augustus De Morgan, George Green or G.H. Hardy, you were ‘fourth wrangler’ (that is, you appeared fourth in order on the class list), then everyone knew it and you had to live with it evermore: no wonder that Hardy was so bitterly opposed to the Mathematical Tripos and campaigned to have it abolished.
The standard way of improving one’s chances in the examination was to be trained by one of the personal tutors who offered their services, and two of these in particular (William Hopkins and his student Edward Routh) have gone down in history for the notable successes that they achieved over many years. This book tells the story of Mr Hopkins and of ‘his men’, the students whom he taught during his career, including such luminaries as Arthur Cayley, George Gabriel Stokes, Peter Guthrie Tait and William Thomson (later Lord Kelvin). In one year alone he achieved no fewer than seven of the top nine wranglers.
The book is in two parts. The first half describes with great clarity the Cambridge scene, with particular reference to the various reforms that took place in mathematics teaching and examining during the early- and mid-19th century; included here are descriptions of the ‘student experience’ over forty years, a full discussion of the Mathematical Tripos, and a detailed biography of Hopkins. This section of the book concludes with a remarkable gallery of photographic portraits, which the author discovered in Trinity College library, of Hopkins’ wranglers over twenty-five years, including unusual images of those mentioned above in their student years. The second half of the book follows the later careers of these and other wranglers, and discusses a range of wider issues, such as the growth of a research community and the rapid expansion of universities and colleges in the British Isles and abroad. The book concludes with a lengthy reference section.
This is an unusual, well-written and well-researched book. So favourably has it been received that a paperback edition appeared within just a few months of the original publication. It can be warmly recommended.
Finding Moonshine: A Mathematician’s Journey Through Symmetry by Marcus du Sautoy, Fourth Estate, 2008, 376pp, £18.99, ISBN 0-007-214618.
As a young child Marcus wanted to be a spy and have his own gun. In order to fulfil his ambition, he set out to learn as many languages as possible, his goal being to join the Foreign Office. Mercifully, before he fulfilled his dream, his mathematics teacher singled him out and suggested certain books that revealed to him the beauty of the language of mathematics – a language of appreciable logic which immediately fascinated him.
In Finding Moonshine Marcus cleverly interweaves three strands; his own personal mathematical journey from a child to the present day, the historic journey of mathematical discoveries relating to symmetry and an account of his own struggles with the particular mathematical problem that is currently demanding much of his attention. The book begins on his 40th birthday in Israel and is written in monthly instalments with each strand being picked up and expanded upon in varying amounts each month.
The historic strand starts with Marcus and his young son searching in the British Museum for evidence of symmetry in Neolithic times. It contains a beautiful and in-depth look at the seventeen different tiling symmetries which are all found in the Alhambra and gives detailed accounts of the mathematics and lives of many mathematicians including Cardano, Tartaglia and Galois. Each of these is interspersed with quotations from their writings and other sources from the same period.
As ever with Marcus’ books, it will appeal to a wide audience. There is enough mathematical content, both present-day and historical, to make it of interest to the mathematician who knows this area well, but yet it is written in a style simple enough to be understood and appreciated by the amateur mathematician, student or sixth-former. The insights into the way Marcus and other pure mathematicians work and think is of particular relevance and usefulness to students and other would-be mathematicians. Does music help you to think mathematically? What is the point of writing your jumbled thoughts down and does this indeed help to clarify them and make sense of your mathematical musings? Throughout the book Marcus explains various strategies that can be used when working through seemingly impossible problems and when progress seems to be painfully slow.
The reader is challenged by questions such as: What is symmetry? Where is it found? Where does it come from? Answers are contained in the many autobiographical accounts. The reader is moved by joy and delight as the young Marcus reads mathematical texts for the first time, understands the fear of another proving what you have been wrestling with for months and desperately want to prove yourself and agonises with the PhD student who wonders at the seeming futility of such research. One highlight is a particularly sensitive and humorous account of the role John Conway plays in the search for the Monster. Again, insights into the life of this brilliant yet eccentric mathematician must only inspire and excite the reader.
Marcus may not have fulfilled his ambition to be a spy in the traditional sense but one cannot help noticing the many similarities between mathematicians seeking to unravel deep mysteries such as these and those involved in various forms of espionage.
How Round is your Circle? Where Engineering and Mathematics Meet by J. Bryant and C. Sangwin, Princeton University Press, 2008, cloth 352 pp, £17.95, $29.95, ISBN 978-0-691-13118-4.
If you want to know the answer to the title, or many other intriguing questions like "How do you make the first straightedge?", "How does a vernier work?" or "How do you measure an area using a coat hanger?" then you will be fascinated by this book. There are many aspects of engineering and mathematics covered, but it is also a book that will interest any mathematicians who like to get their hands dirty with real world problems. It is also for recreational mathematicians and historians of engineering and mathematics. The preface talks about why mathematicians should take the practical problems of engineering seriously since they present serious challenges when you leave the comfortable world of fictitious thin lines. They present such challenges in plenty with hands-on suggestions as well as theoretical back up. You could enjoy a good read, but you cannot fail to be drawn in to practical solutions and enjoy yourself. There are many popular mathematics books, on symmetry and groups for example, but most do not engage the reader as much as this one. There is some deep mathematics (Gröbner bases are not my field), but it is well written and you can skip this without a problem because before you know it you are in different territory.
As I read through the book, I was reminded how nineteenth century mathematicians were encouraged to build models and how this ceased with the death of Felix Klein. Bryant and Sangwin point out that many of the advances in that century were a product of solving real world problems. This is particularly evident in the discussion of using linkages from James Watt’s approximate method using a lemniscate type curve through other developments to Peaucellier’s cell using the concept of inversion in a circle. Modelling in the computer is briefly touched upon, but the book’s approach means that you also realise that problems like stopping such linkages from snagging can be the necessity that is the mother of invention. The recently reviewed novel A Certain Ambiguity (March 2008 Newsletter p. 25) has the Socratic teacher Nico pointing out that axioms about clouds and rain don’t tell you about rainbows. This book has many gems and rainbows.
The book will appeal to all recreational mathematicians also, not just because of the way it is written, but also because of the way puzzles, plane dissections and packing and the odd paperfolding or origami task are used to bring a point home. There are some new slants on leaning towers of dominoes and a very interesting final chapter on equilibrium with unstable polyhedra and unusual rolling pairs of slotted disks. I have very few criticisms on content, but on the latter they have missed David Singmaster’s analysis of this in Eureka and there is nothing on the Paul Schatz oloid. This has connections with mixing machinery and three dimensional linkages (otherwise covered admirably).
Space does not allow me to go into more detail, so I will just list a few topics: slide rules, Reuleaux’s rotors and drilling square holes, Galileo’s sector, the real-world need to trisect angles for sextants and quadrant marking (although they call Archimedes’ neusis construction Pascal’s trisector), how all approximations are rational, measuring areas with planimeters, scales and bridge curves. If you can’t find something to interest you here, then I don’t know what will.
More than one copy of this book should be in every school library too. It should help to inspire a new generation into mathematics or engineering as well as be accessible to the general reader to show how much mathematics has made the modern world.