Mathematical legacy

This chapter re-examines Dodgson’s achievements during his lifetime, and how some of them resurfaced in the 20th century. The chapter is divided into subject areas: geometry (including his alternative version of Euclid’s parallel postulate), trigonometry (for which he devised a new set of symbols), algebra (his work on determinants), logic (where the work of Bartley and others has led to a significant re-evaluation of Dodgson’s involvement), voting theory (on which he was in contact with a number of senior politicians), and probability, and concludes with a section on ciphers and cryptology. There has been a growing interest by scholars in Dodgson’s serious mathematical work since the third quarter of the 20th century, particularly as more of it has been published.

Recreational mathematics

Dodgson was interested in puzzles and games throughout his life. This chapter concerns a wide range of recreational games and puzzles that he used to inform and entertain his child-friends and his colleagues. Among the topics included are problems involving time, probability, geometry, algebra, and arithmetic, and there are sections on calculating the days of the week, memorizing numbers, testing for divisibility, and on a wide range of other topics. Dodgson compiled puzzles for some of his young friends, and some of these were published in his lifetime (including some that he included in letters to magazines). Others were found in his papers after his death and published subsequently.

Algebra

This chapter is concerned with a single textbook written by Dodgson in 1867—An Elementary Treatise on Determinants. His book concerned the study of determinants, an algebraic construct that had come to the fore during the 19th century, and presented a new method, called the method of condensation, for evaluating them; this method was due to Dodgson and was the subject of his only refereed journal paper. The book also applied his results to the solution of simultaneous linear equations, and included the proof of a theorem describing when a system of linear equations is consistent. The book’s impact and influence are discussed in the concluding section of this chapter.

A mathematical life

This chapter sets the scene for the rest of the book by outlining the main aspects of Dodgson’s mathematical life and activities. It is divided into his early years as the son of an Anglican clergyman, his time as an undergraduate studying Classics and mathematics at Oxford, his life as an Oxford lecturer at Christ Church, his life outside mathematics (including his interest in photography and his writing of children’s books), his mathematical writings and college life, and his final years. It concludes with a brief discussion of Dodgson as a mathematician in the context of Victorian mathematics in England at the time.

Voting

Dodgson became interested in methods of voting while becoming involved with college appointments, and became one of its most celebrated experts, following in the footsteps of Condorcet and Borda. Critical of standard methods of voting such as first-past-the-post, single-transferable-vote, etc., he presented examples to show their weaknesses. He also proposed a system of proportional representation, and communicated this to newspapers and to Parliamentarians of the time. His Principles of Parliamentary Representation discusses the assignment of seats to multi-member districts and to parties within each district to parties. It is only recently that his contributions to the theory of voting have been recognized.

Geometry

This chapter presents Dodgson’s geometrical work, and in particular his interest in Euclid’s Elements. It describes the contents and structure of this work and considers the various Victorian editions, and looks at how Dodgson used the Elements in his teaching – both face-to-face and in his writing for students. Following a discussion of the Euclid debate, prevalent at the time, the chapter describes his book Euclid and his Modern Rivals, in which Dodgson defended Euclid against other more modern texts. There is then a discussion of the ‘parallel postulate’ of Euclid, and a mention of other geometrical interests that Dodgson had.

Mathematical bibliography

This is the most detailed list of Dodgson’s mathematical writings and publications that currently exists. Information is given of each work’s date, publisher, and other relevant details.

Logic

This chapter discusses Dodgson’s work on syllogisms (a topic that can be traced back to Aristotle and Ancient Greece) and how to solve them systematically using a marked board and some counters. His method is explained in detail in this chapter. Dodgson introduced it in his Game of Logic, which he used to teach syllogisms to children, and which he then developed in his Symbolic Logic, Part I. The rest of the chapter is concerned with further work that Dodgson carried out, but which was not published at the time because of his premature death at the age of 65.