Everyone who reads this will know of Lewis Carroll the author of the two children’s books Alice’s Adventures in Wonderland, and Through the Looking-Glass. A few will also know that he was really Charles L. Dodgson, and fewer still that he was a Lecturer in Mathematics at Oxford University. This is not a coffee table book about Lewis Carroll, but a scholarly textbook containing the latest research outlining and assessing his contribution to mathematics. Normally, this reviewer has a jaundiced view of ‘edited by’ textbooks, but not this one. This is very well researched and very well written. All the authors are well qualified to write about particular aspects of Dodgson’s mathematics, in fact Professor Abeles’ chapter does a particularly good job, providing an excellent and critical summary of his mathematical works.
As befits this kind of scholarly History of Mathematics textbook, the biographical account of Dodgson’s general life is largely confined to Chapter 1. This is written by Robin Wilson and can almost be read by the non-mathematical reader, but not quite. It provides something to whet the appetite for the more detailed specialist later chapters. It also introduces the style whereby the prose is peppered with quotes and excerpts from Dodgson’s published works and diaries. These quotations put the prose in context and provide valuable insight into the style and mind of Charles L. Dodgson. It is in this chapter we learn his character, and although he was an excellent mathematician and a good communicator, it is not all perfect. There was a side to him that mitigated against recognition as a professional mathematician: his tendency to be a bit reluctant to change, conservative with a small ‘c’, and his lack of interest in publishing in the standard mathematical literature. The remainder and substantial part of the book is devoted to his mathematics, namely geometry, algebra, logic, voting systems and recreational mathematics. The final two chapters summarise his legacy, and there’s a comprehensive and annotated bibliography.
Since the time of the Greeks, geometry meant the geometry of Euclid derived from five axioms. However, in Dodgson’s time there was a revolution with the recent developments in non-Euclidean geometry in the early and mid-19th ^{ }century. In schools the curriculum was of course still Euclid and Dodgson certainly followed this tradition. To be fair to Dodgson, Euclid was the school diet for the next hundred years too, so he was hardly out of line, nevertheless the university curriculum was beginning to reflect the new geometry, and although Dodgson was exposed to hyperbolic geometry he continued to teach and investigate the niceties of Euclidean geometry, in particular the fifth (or parallel) postulate and whether it could be proved from the other postulates. Of course, with Dodgson’s interest in mathematical puzzles, he devoted some of his geometrical attention in this direction too.
Dodgson’s main algebraic interest was in the field of determinants, also a recent mathematical development. Perhaps his most substantial book, published just two years after Alice in Wonderland, was his treatise on determinants: An Elementary Treatise on Determinants with their Application to Simultaneous Linear Equations and Algebraic Geometry published in 1867. In this book, he gives a new method to expand determinants called condensation whereby an n x n determinant is expressed as a sum of (n – 1) x (n – 1) determinants. The method is demonstrated here for the cases n = 3 and n = 4. It was not taken up as other methods were preferred, but interestingly, recent developments in condensation algorithms has given it a new lease of life. Dodgson was not interested in publishing through the normal journals, and he seems to have overlooked his audience when he composed this textbook on determinants, it is over formal and more or less useless to students in its use of bespoke notation and terminology. As Lewis Carroll, his popularity was and continues to be immense, but all this was left behind it seems when he wrote under his real name, this textbook on determinants is all but unreadable.
Logic was a later interest for Dodgson, and this is a pity as he left what promised to be innovative work unfinished when he died in 1898. Dodgson as Lewis Carroll loved word play of course, and the marriage of this with meaning and implication was a natural field for him. However, his alternative to Venn diagrams to display union and intersection of sets has been lost to time, which is a shame as in many ways it is superior. Perhaps it is another consequence of Dodgson not publishing in the formal way.
In his time, Dodgson tried to change the UK voting system from the still used first-by-the-post method to something fairer. He failed, but he had a lot to say about alternative voting systems. All kinds of different election methods are covered and much of Dodgson’s work in this area has great currency. Sports tournaments are also described here, most clearly demonstrated by the tennis tournament where the concept of seeding helps to prevent unfairness such as two favourites meeting in the first round. Proportional representation and single transferrable votes are dealt with in some depth, none can be said to be entirely fair which is probably why the UK voting system is still stuck in the dark ages; there is always an excuse not to change. Most of the different voting systems pre-date Dodgson, but his insights and suggestions are intriguing.
Perhaps the least surprising area for Dodgson is his exploration of recreational mathematics. This has obvious links with teaching as puzzles were often used by Dodgson to retain the interest of students, as well as to entertain the more general public. He publicised various ‘well known’ number puzzles that start with ‘think of a number’ then end with him telling, seemingly by magic, what the number was. Most of these depended on the properties of our number system, but these of course are obscure to the general public. Dodgson carefully calculated how they worked using algebra. He published a number of so-called Pillow Problems that were usually based on probability, starting with drawing and replacing black or white balls out of a bag. Most were meticulously solved of course, though some had no solution or were ill defined in some way. This does seem out of keeping with the usually precise and capable Dodgson and surprised this reviewer.
Dodgson was principally a teacher, and though he did resign his lectureship to concentrate on writing, he never really lost the desire to teach. He helped the young understand through his games and puzzles. Of course, he is much better known for his children’s literature that has much better survived the test of time.