CHARLES DODGSON

By: Melissa Abeyta

March 17, 2005

The following is a quote from Alice’s Adventures in Wonderland.

“Then you should say what you mean,” the March Hare went on.

“I do,” Alice hastily replied; “at least I mean what I say, that’s the same thing, you know.”

“Not the same thing a bit!” said the Hatter. “Why, you might just as well say that “I see what I eat” is the same thing as “I eat what I see!” (St Andrews quote website)

The March Hare and the Mad Hatter are two of the many interesting and entertaining characters brought to life by Lewis Carroll. But who exactly is Lewis Carroll? Lewis Carroll is nothing more than a pseudonym that Charles Dodgson used when writing children’s literature and other literary works. Charles was also a lecturer of mathematics and a mathematician. Most people, including myself, were surprised to find out that the person who invented such creative characters was also a mathematician. Finding out that Lewis Carroll was really Charles Dodgson was not the only thing that shocked me. Charles led a very interesting life and made several contributions to both the literary world and the mathematical world. I will attempt to give you a background on how he came to be a mathematician, his contributions to the mathematics, his mathematical critics and of course, it would not be a complete paper without talking about Charles’ other half (that of Lewis Carroll).

Charles Dodgson had an early love for mathematics. His father’s favorite subject was mathematics and Charles wanted to be like his father. Early on in his education he excelled in the subject and received many rewards for his achievements. [St Andrews website]

Charles wanted to study at Christ Church College in Oxford, just as his father had, but he had to wait as situations prevented him from starting. In May 1850, he could not start studying at Oxford when he wanted because he had “to wait until accommodation could be arranged,” (St Andrews website). During this time he went back home and stayed with his parents. In January 1851, Charles returned to Oxford to stay with a friend while he attended college, however the sudden death of his mother sent him back home. On his third attempt he was able to start college. He wanted to work hard in the hopes that he might win scholarships and be financially independent. “He was able to mix social and cultural activities with hard work at mathematics and in November 1851 he was awarded a Boulter Scholarship worth 20 pounds a year,” (St Andrews website). Charles went on to receive a First class in mathematics and a Second class in classics in December 1852. Due to his achievements, he was awarded a Fellowship of 25 pounds a year for life. [St Andrews website]

Charles’ sights were now set on becoming a lecturer in mathematics. In order to obtain such a position he needed to do well in the senior scholarship competition. He began to take on students (unofficially) and let his own studies slide. He failed to receive the scholarship in 1855. His schedule was incredibly hectic. Upon failing another scholarship examination, he returned to his parents’ home. In the summer of 1855, Charles taught at his father’s school. He returned to Oxford in October of that year and finally obtained the position he had long waited for: a mathematics lecturer. Charles would remain at Christ Church College in Oxford until 1881, where he lectured on mathematics, and wrote guides for his students. [St Andrews website]

Charles finally achieved his goal of becoming a teacher, however it turns out he was not that good at it. “His lectures were what one graduate called ‘dull as ditchwater’, and his pupils collectively wrote a letter to Dean Liddell asking to be transferred to another tutor,” (Thomas, 1996, p. 95). His lectures were boring and were attended by few. “He noted ruefully that his first lecture had been attended by nine students and his last (twenty-five years later) by two,” (Dictionary of Scientific Biography, 1971, p. 137). His inability in becoming an effective lecturer might be due to an incurable stammer he had since birth (Cohen, 1995).

Charles seemingly followed the path of his father in most aspects except religion. Charles grew up in a strict Christian household and his father wanted him to become a priest (St Andrews website). Charles took deacon’s orders in 1861, but did not take the next step in becoming a priest (St Andrews website). “As time went on he found it harder to attempt the view that non-Christian were condemned and, as a man of great honesty, would therefore find the oaths he would be required to swear to become a priest unacceptable,” (St Andrews website).

Charles’ study and love for mathematics allowed him to make several contributions to the math world, with most of them dealing with improving teaching. In 1860, Charles wrote A Syllabus of Plane Algebraical Geometry, Systematically Arranged, with Formal Definitions, Postulates, and Axioms. This book was an attempt to translate some of Euclid into algebraical terms (Cohen, 1995). That same year he also published Notes on the First Two books of Euclid, Designed for Candidates for Responsions. Charles “was keenly aware of the defects in available texts on the subject, and he dutifully persisted, keeping in mind that every undergraduate studying mathematics had to cover the first six books of Euclid,” (Cohen, 1995, p. 254). Charles’ main efforts were to help students prepare for tests. In researching Charles I found out that he was very knowledgeable in mathematics and wanted to help students learn.

Charles went on to publish two more mathematical papers in 1861: The Formulae of Plane Trigonometry, Printed with Symbols (Instead of Words) to Express the “Gionometrical Ratios”, and Notes on the First Part of Algebra. In the first paper “he invented new symbols to represent trigonometrical functions,” (Cohen, 1995, p.254). In other sources however, they do not make a big deal of these new symbols saying that although they are original, they are not “particularly meritorious,” (Dictionary of Scientific Biography, 1971, p. 137).

Charles wrote several more mathematical books and papers. His writings dealt more with geometry than any other part of mathematics (Cohen, 1995). He was fascinated with Euclid and wrote at least 12 pieces about him (Cohen, 1995). The book “on which his reputation as a mathematician largely rests” was Euclid and His Modern Rivals (1879) (The 1911 Edition Encyclopedia website). The book is a comedy “about a mathematics lecturer, Minos, in whose dreams Euclid debates his original Elements with such modernizers as Legendre and J.M. Wilson and, naturally, routs the opposition,” (Dictionary of Scientific Biography, 1971, p. 137). Charles dedicated the book to the memory of Euclid (Cornell website). Some people thought the book was an attack on non-Euclidean geometry however; Charles was really attacking the changing method of teaching classical geometry (Dictionary of Scientific Biography, 1971).

Charles received several favorable reviews on the book and it sold out in six weeks (Cohen, 1995). “Vanity Fair (April 12, 1879) judged it “absolutely refreshing” and believed it “a book marvellous for the labour contained in it, and still more marvellous for the brightness of the humour with which the ponderous stuff of geometry is handled…The English Mechanic (May 2) insisted, however, that Charles put the case for Euclid well: “On the whole…we regard our author as having triumphantly proved that, so far, no work has been produced which is comparable with Euclid’s immortal Elements, as an introduction to geometry for beginners,” (Cohen, 1995, p. 385). The book also showed the world a humorous, playful side to Charles. “After the ghost of Euclid makes his first appearance he leaves saying:- “…that concludes our present interview; so we will meet again when you have reviewed my Modern Rivals one by one. If you had any slow music handy, I would vanish to it: as it is…[vanishes without slow music],” (St Andrews website).

There was a discrepancy in my research as to how many acts were in the book. A biography book written by Morton N. Cohen states that it had four acts however, the Dictionary of Scientific Biography stated it had five acts. In searching the web I found a scanned copy of the 2nd edition of the book on the Cornell University website and it had four acts. Charles ended up releasing a second edition six years after the original book had been published, so it is possible that the number of acts changed. In searching World Cat at a local library I found that there were no original copies of the 1st or 2nd edition of the book in the state of Colorado. The University of Denver had a re-released copy of the 2nd edition and it had four acts, matching the biography book and the Cornell University website. My research has led me to believe that the 2nd edition had four acts, however it might be possible that the 1st edition had more, although I do not know this for sure.

Another contribution Charles attempted to make involved the squaring the circle problem that has been around since approximately 1650 B.C. “In 1875, C. L. Dodgson began working on a computationally simple approximation method for would-be circle squarers that would convince them of the futility of their attempts,” (Abeles, 1993, p. 151). His work on the problem continued until 1893 (Abeles, 1993). “Some would-be circle squarers were able to derive reasonable approximations for pi from their constructions, while others argued for abysmally poor ones. Probably the chief recipient of failed quadrature attempts in the 19th century was Augustus DeMorgan. When DeMorgan died, Charles Dodgson took up the burden of refuting the circle-squarers’ faulty arguments. Dodgson never finished the work, but enough of it remains in manuscript form to demonstrate that his approach was unusual. He introduced a method to compute approximations for pi that was efficient and simpler than the prevailing method using Machin’s and Gregory’s series, and he provided an intuitively appealing setting that demonstrated the strong connection between the early geometric attempts at quadrature and the newer analytic ones,” (Abeles, 1993, p. 153).

“Dodgson intended to call his pamphlet “The Limits of Circle-Squaring: Simple Facts for Circle-Squarers.” Four undated proof sheets containing two theorems undoubtedly intended for an early chapter are in the Warren Weaver Collection in the Harry Ransom Humanities Research Center at the University of Texas. The introductory chapter, written in 1882, was printed in a limited edition in The Lewis Carroll Centenary, Special Edition (1932) from the manuscript in the Parrish Collection in the Princeton University library. Only in the Parrish Collection do we find the theorems linking this approximation method with the Euclidean constructions that would have enabled a misguided circle-squarer to apprehend his own errors,” (Abeles, 1993, p. 154). Charles was trying to provide “a “do-it-yourself-kit” approach for the mathematical dilettante,” (Abeles, 1993, p. 158). “Rooted in Euclidean geometry, a common background for al late-nineteenth-century British circle-squarers, the method could produce a relatively accurate and efficient approximation that was intuitively sensible,” (Abeles, 1993, p. 158). As stated earlier, the article was never published but his work on the topic was innovative and helpful (Abeles, 1993).

Charles was a mathematical logician as well and wanted to increase understanding by treating it as a game (St Andrews website). In 1887 he published The Game of Logic and in 1896 he published Symbolic Logic Part I. In the later piece of work, Charles modified the Venn diagram that was developed by John Venn in 1881 (St Andrews website). He made the boundaries linear and introduced “colored counters that could be moved around to signify class contents – a very simple and effective device,” (Dictionary of Scientific Biography, 1971, p. 137). Although his changes created certain advantages, his method has been forgotten (St Andrews website).

In 1959 and 1970, William Warren Bartley found missing pieces to Charles’ work on symbolic logic and published this information in 1977. Bartley claims that this newly found “material shows specific achievements which can be credited to Dodgson,” (St Andrews website). The following list contains the achievements:

- By 1896 Dodgson had developed a mechanical test of validity for a large part of the logic of terms, an achievement usually credited to Leopold Lowenheim [in 1915]. (St Andrews website)

- As early as 1894 Dodgson used truth tables for the solution of specific logic problems. The application of truth tables and matrices did not come into general use until 1920. (St Andrews website)

- By 1896 Dodgson had developed the method of trees for determining some validity, which bears a resemblance to the trees frequently employed by contemporary logicians. (St Andrews website)

Charles proved to be a “prolific composer of innocent looking problems in logic and paradox, some of which were to engage the attention of professional logicians until well in to the twentieth century,” (Dictionary of Scientific Biography, 1971, p. 138).

Even though he made contributions to the mathematics world he was criticized. He was mainly criticized for not paying attention as to what was going on in the mathematical world. “The pity is that his talents were inhibited by ignorance and introversion, for he made no attempt to keep abreast of contemporary advances in mathematics and logic or to discuss his ideas with other academics,” (Dictionary of Scientific Biography, 1971, p. 137). He was also criticized for not realizing a stronger connection between logic and mathematics. “His casual realization of the connections between symbolic logic and mathematics might have become vivid and fruitful had he been properly acquainted with what had already been done in the area,” (Dictionary of Scientific Biography, 1971, p. 138). Francine F. Abeles, who seemed to be a strong supporter of Charles, writing several works about him and his work commented on his unwillingness to know the details. “Not wanting to involve himself in the details of each argument, he set out to devise a method whereby each novice circle-squarer could convince himself that his method was flawed,” (Abeles, 1993, p. 153). It is unclear to me as to how you can prove or disprove something without knowing all the details.