Charles Lutwidge Dodgson (1832-1898) is the real name of Lewis Carroll, the author of *Alice's Adventures in Wonderland, Through the Looking Glass* and of other such books. He wrote these for Alice Liddell, the daughter of Henry Liddell, the dean of Christ Church in Oxford and he became forever famous as `the man who wrote *Alice*' and for a long time this was the only way he was remembered. Yet he was also a mathematician with very original ideas and a creditable photographer. These aspects were only properly recognized since the 1980s. His diaries and his collected publications were becoming available since the 1990s and with that material, more studies of his non-fiction work was possible. This book gives a survey of what is known so far.

The book starts with a (short and in some respect only partial) biography, concentrating on the mathematician in him. Introduced by his father (a country parson) to mathematics as a young child, he excelled in this subject at school. Later, just like his father, he studied at Christ Church in Oxford and got a master degree in mathematics. While studying he gave private lessons to students. He was ordained a deacon when he was 29, but never became a priest. When Liddell became the dean of Christ Church, Dodgson was appointed as 'Master of the House' which gave him a reasonable income but also a large teaching load. He starts publishing his pamphlets as aids for teaching, and some work on the evaluation of determinants. He took on photographing as a hobby. He became rather good at it, using it as an art form, rather than as just a way to catch reality. Urged by Alice Liddell to write up the stories he told during their boat trips, he started working on *Alice's Adventures* which was published under his pen name Lewis Carroll. Teaching was his main occupation besides his writing and the photography. He became (reluctantly) Curator of the Christ Church Common Room for ten years when he was 50. This was a time-consuming burden requiring management and many decisions to take. In that context he could use his already existing interest in voting systems. He died a week before his 66th birthday.

The next chapters are written by specialists and discuss in more detail Dodgson's contributions to different mathematical subjects. The first one deals with geometry. Dodgson was teaching geometry following Euclid's books as it was usual in those days. However there were some new ideas, among others by Sylvester, criticizing Euclid's approach. Euclid's arguments were not always waterproof in a mathematical sense, and there was the emerging hyperbolic geometry of Lobachevsky. Dodgson was defending however Euclid in his booklet *Euclid and his Modern Rivals*. The discussion about the parallel postulate involved either infinitely long lines or infinitely small quantities. So he tried to replace it by a finite alternative using an "obvious" property about areas outside and inside an hexagon inscribed in a circle. He thought of non-Euclidean geometry as nonsensical.

In the next chapter on algebra, it is explained what his condensation method for the evaluation of determinants is. This may be his most useful original contribution to mathematics. He was also opposed to the name 'matrix' with the meaning we give it today. He called it a 'block' because a matrix refers to the mould, rather than the object, which is the mould filled with numbers. To denote the elements in a matrix, which we denote by a letter with two indices, he had his own strange notation. His work on determinants was published but (like many of his mathematical publications) remained largely unnoticed for a long time.

Logic has been one of Dodgson's favourite subjects and he wrote several texts about it. In those days, as we still have today, there existed several proposals for an approach to, and the notation of, formal logic. The one from Boole was, and still is, a very useful one. Dodgson adhered the formal approach which was not easily accepted by classical logicians, and required a lot of dispute. He developed several tools to deal with logic problems: a method with diagrams, he had his own formal notation, a method of trees, and an algorithm to solve syllogisms. Bertrand Russell once said about Dodgson' work that it was brilliant but largely useless.

Dodgson also wrote several texts on voting systems. The problem of cycles (where each candidate can win from the next candidate in a cyclic way) was known several centuries before, but Dodgson was probably not aware of that. He detected it on his own and made proposals for a correct voting system, for assigning seats to parties in a political election, or for a correct outcome of a tournament. He wrote about these problems in terms of game theory, an approach that John Nash would bring to a culmination only much later.

It is clear that the author of the *Alice* books would also be interested in recreational mathematics, puzzles, riddles, and games. Many examples are discussed as well the way in which Dodgson solved them. He also had techniques to remember dates and numerical data, and techniques to check divisibility which could be smuggled into a number game.

It is strange that Dodgson was so little recognized for his mathematical work. Most of his, sometimes original, approaches were only discovered at the end of the 20th century. He was not the best salesman for his results. Perhaps he didn't take it seriously enough, and maybe he was too quarrelsome (sarcasm indeed happened sometimes), or perhaps he was rather obscure when sticking to his own notation and ideas. He had for example his own symbolic notation for the trigonometric functions. He also was stammering a bit, but that did not seem to be a serious hinder to him. Moreover his contributions are very diverse. He did not have a single field in which he became the renowned expert and finally perhaps he was also in the shadow of Lewis Carroll. The legend goes that Queen Victoria, charmed by reading *Alice's Adventures in Wonderland*, asked for his next book and promptly received his *An Elementary Treatise on Determinants*. This story is however a hoax. He may not have received during his life the recognition by mathematicians he deserved, but nevertheless this book has an extensive chapter discussing his mathematical legacy in geometry, trigonometry, algebra, logic, voting, probability, and cryptology as it became clear only recently.

The book ends with a complete bibliography listing all his publications and a list of references for additional reading. With that this book completes a survey of the life and work of a (perhaps somewhat dull) mathematician that is in many ways the opposite (but in as many ways also the complement) of the light-hearted Lewis Carroll that authored the books witnessing of such a rich fantasy dedicated to a little girl called Alice. A man well known as Lewis Carroll, yet perhaps too long underestimated as Charles Dodgson. This nicely edited book with many illustrations and written by experts on the subject will certainly help to turn the tide by adjusting the image and allow you to form a proper opinion about Charles Dodgson.