by Jenny Woolf
Although he is not known to have socialized with progressive mathematicians, nor belonged to any of the mathematical societies of his time, numbers were always on his mind. He dwelt on them in idle moments and during wakeful nights, and nearly always had a little mathematical problem or idea on the go. Many of these ideas crossed the boundaries between mathematics and play, and his friend and colleague T B Strong noted that he was particularly interested in transforming predictability into surprise; he loved curiosities and apparent contradictions and double meanings.
Strong remembered how interested and perplexed Carroll was in the idea that a sum worked out accurately with figures should fail when it came into contact with details of fact. As an illustration of this, Strong quoted a problem about building a wall, which Carroll sometimes asked his students to explain. The problem goes something like, ‘If it takes ten men five days to build a wall 100 yards long, how long would it take 10,000 men?’ When the mathematical answer was worked out and given, Carroll would retort that it was mathematically correct, yet it was not true. In real life, he would say, most of the workmen would not have got within a mile of the wall.
Carroll’s natural tendency to playfulness came increasingly to the fore as he grew older. By the time he was in his late forties, he had mostly given up trying to inspire his undergraduates, but he decided to try and communicate some of his feelings about mathematics to the general public in a palatable form. His book Euclid and His Modern Rivals (1879) is probably the most extraordinary piece of mathematical literature ever written. In it, Carroll investigates the increasingly beleaguered cause of Euclidian geometry in the form of an opinionated play.
By the late 1870s, Euclidian geometry was badly out of date, but it was still widely taught and Carroll was devoted to it. Euclid of Alexandria lived around 300 BC. Versions of his Elements of Geometry had served mankind well for 2,000 years, but by the late 19th century contemporary mathematicians were realizing that his system had flaws. New approaches to mathematics – and to geometry – were surfacing throughout Europe, but they were passing Carroll by.
Euclid’s ‘rivals’ of the play’s title were, in fact, merely alternative ways of interpreting the original Euclidian texts compared with the traditional way of doing it. So the book would not have been of any use to someone wishing to examine new approaches to geometry, although of course it covered the various methods of teaching Euclidian geometry, and it also devoted serious attention to the technical matter of equivalents to Euclid’s Fifth Postulate: If a straight line crossing two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if extended indefinitely, meet on that side on which are the angles less than the two right angles.
The subject matter may seem as dry as the biscuit which the White Queen presented to Alice, but Carroll’s treatment of the topic is delightful. In his book, he casts Euclid as a ghost who visits the wryly-named examiner Minos, and discusses with him the worth of each ‘rival’ interpretation compared with his own original. Euclid disappears at the end of Act I. ‘“If you had any slow music handy, I would vanish to it: as it is –” Vanishes without slow music.’ In Chapter 2, Herr Niemand (‘Mr Nobody’) enters in the train of his own meerschaum’s cloud of smoke, to enter into some donnish jokily mathematical discussions.
The play was a brave and interesting effort, although it could not hope to hold back the tide of modern mathematical thinking. It is probably unperformable – except for mathematicians – and its original approach was no doubt meant to catch a reader’s attention rather than hold a real live audience.
There is more mathematical humour in many of Carroll’s notes and pamphlets, such as one he wrote about his thoughts on running the Christ Church Common Room. Here, he commented that the consumption of Madeira wine over the past year had been zero, and added that:
after careful calculation, I estimate that, if this rate of consumption be steadily maintained, our present stock will last us an infinite number of years. And although there may be something monotonous and dreary in the prospect of such vast cycles spent in drinking second class Madeira, we may yet cheer ourselves with the thought of how economically it can be done.15
Even in the driest expositions, his sense of humour would flicker through. One chapter of a later, unpublished manuscript entitled ‘Grills’, offers his thoughts on the geometry of crossing parallel lines. Carroll probably intended to use the chapter in a book he was planning, and omitted it because there are some mistakes in the calculations, but the studious reader may also be startled to come across a section in the manuscript which reads: ‘I grant you that the Diagram is a ghastly one – as ghastly, perhaps, as is to be met with in the whole range of Pure Mathematics. Still you know, it was my duty to draw it. England expects every man to do his duty. And I love England. The conclusion is obvious.’16
As Carroll grew older, his attention turned away from mathematics and towards the study of symbolic logic – a means of checking the validity of arguments based on syllogisms. These days symbolic logic has been largely dropped in favour of modern logic, which is used to define the foundations of mathematics. Yet symbolic logic remains an interesting mental discipline in its own right. It is Ancient Greek logic (which Carroll had studied as a young man as part of his classical education) expressed in mathematical symbols. Syllogisms consist of two or more ‘premises’ or statements, which lead to conclusions, one of the more famous examples being that presented by Aristotle in the 4th century BC, in which the two premises:
All Greeks are men
and
All men are mortal make it possible to conclude that
All Greeks are Mortal
However, in symbolic logic, just as in mathematics, Carroll tended to be interested in issues of specific detail, rather than the overall picture. Carroll and T. B. Strong corresponded at great length about a number of logical issues, but Strong complained that his attempts to raise the subject of the relation of words and things was a failure, as Carroll simply declined to write upon it.
One of Carroll’s great interests in later life was in developing logic as an entertainment for young people, and he spent a great deal of time on this. He wanted to help his readers to increase their powers of clear thought, he said, to see their way through arguments and make sense of what they read in the newspapers and what they heard from the pulpit. ‘Sift your reasons well,’ he urged, ‘and make sure they prove your conclusions.’17 In this worthy cause, he developed a Game of Logic (1886) which was designed specially for older children. He used it both with what he referred to as his ‘child-friends’ and also when he took logic classes at Oxford High School for Girls.
The Game of Logic has been reprinted many times, and is available in modern editions, but Carroll took exceptional pains to make the original volume attractive and enticing. It was bound in bright red, a colour which he felt particularly appealed to the young, and its title swirls across the front in gold. A small envelope inside contains a small board and attractively coloured counters. He had canvassed his lady friends on the most suitable colours for these counters, and had finally decided on dusky pink and grey. (It is a colour combination he particularly liked, since he also yearned to dress his actress friend Isa Bowman in those shades, which he considered more ‘ladylike’ than the brighter outfits which she favoured herself.)
The design on the game’s board resembles a squared-off Venn Diagram – and in fact is an improvement upon it, having eight regions to the Venn’s seven. The syllogisms are expressed in words, with the options to transfer them to symbols later. The game was intended to aid a young person learning with a teacher, so the instructions need to be read carefully. But once the original idea is grasped, the game is interesting, and it is made more enjoyable by the comic or ridiculous premises which Carroll deliberately used to add a dash of entertainment for his young audience. For instance,
No bald person needs a hair-brush
No lizards have
hair
All wise men walk on their feet
All unwise men walk on their hands
From the premises
All dragons are uncanny
All Scotchmen are canny
he derived the reassuring conclusion that
All dragons are not-Scotchmen
and
All Scotchmen are not-dragons
‘Symbolic Logic, Part I’ takes the study of logic to more advanced levels, and some of the examples Carroll offers here are positively baroque:
No shark ever doubts that it is well fitted out
A fish, that cannot dance a minuet, is contemptible
No fish is quite certain that it is well fitted out, unless it has three rows of teeth
All fishes, except sharks, are kind to children
No heavy fish can dance a minuet
A fish with three rows of teeth is not to be despised.
Carroll probably got the idea of using entertaining and comical mathematical examples from his own childhood. At school, he had used Francis Walkingame’s The Tutor’s Assistant, a very popular work which had originally been published in 1751. Although constantly revised, the 1840s editions of the book (which Carroll would have used) contained many little Georgian dramas, such as:
A man overtaking a maid driving a flock of geese, said to her, ‘How do you do, sweetheart? Where are you going with these 30 geese?’ ‘No, sir,’ said she, ‘I have not 30; but if I had as many more, half as many more, and 5 geese besides, I should have 30.’
It is a resounding put-down to the man’s chat-up line, and there are more mathematical maids on the following page of the book, where:
A gentleman going into a garden, meets with some ladies, and says to them, ‘Good morning to you 10 fair maids.’ ‘Sir, you mistake,’ answered one of them, ‘we are not 10, but if we were twice as many more as we are, should be as many above 10 as we are now under.’
Carroll went a step further than Walkingame in the 1880s by creating a group of mathematical short stories, collectively entitled A Tangled Tale, in which each tale encompasses a mathematical problem. These stories feature, among others, two boys and their teacher, Balbus, (‘The Stammerer’in Latin) and the teenage Clara and her eccentric aunt, Mad Mathesis. The surreal flavour of the stories can be conveyed by a couple of quotations, as when two knights in the first story, ‘Excelsior’ are toiling up a hill:
And on the dead level, our pace is –?’ the younger suggested, for he was weak in statistics, and left all such details to his aged companion.
Four miles in the hour,’ the other wearily replied. ‘Not an ounce more,’ he added, with that love of metaphor so common in old age, ‘and not a farthing less!’
In the story ‘Oughts and Crosses’, Clara explains to her aunt that she’s told the little ones at tea-time that:
The more noise you make the less jam you will have, and vice versa. I thought they wouldn’t know what vice-versa meant, so I explained it to them. I said ‘If you make an infinite noise, you’ll get no jam; and if you make no noise, you’ll get an infinite lot of jam.’ …
Carroll’s stories appeared between April 1880 and November 1884 in The Monthly Packet periodical. He invited readers to send in solutions, which they did under a variety of imaginative pseudonyms. He then commented on the solutions and their reasonings, showing a characteristically deft ability to pinpoint the woolliness of some of the replies: ‘… it is interesting to know that the question ‘answers itself’ and I am sure it does the question great credit; still, I fear I cannot enter it on the list of winners, as this competition is only open to human beings …’.18
After writing ten issues of A Tangled Tale, he quit. He seems to have become bored with the series, although he took his leave elegantly and politely, claiming that his characters were ‘neither distinctly in my life (like those I now address) nor yet (like Alice and the Mock Turtle) distinctly out of it.’19
His puzzles by no means always involved mathematics. ‘Puzzles from Wonderland’ appeared in 1870 in Aunt Judy’s, a children’s magazine, and it offers riddles set within rhymes. Some are very easy, and have something in common with crossword puzzle clues, or even Victorian cracker jokes, such as:
Dreaming of apples on a wall,
And dreaming often, dear
I dreamed that, if I counted all, –
How many would appear?
The answer, of course, is ‘ten’, since he dreams ‘of/ten’.
More challenging are some of the probability problems from Curiosa Mathematica Part II: Pillow Problems, of 1893, a work which was aimed at filling the time on sleepless nights. One such problem was the following:
A bag contains one counter, known to be either white or black. A white counter is put in, the bag shaken and a counter drawn out, which proves to be white. What is now the chance of drawing a white counter?20
More intriguing is the matter of the number ’42’. Carroll was supposed to be particularly interested in that number and, for years, mathematicians have been finding examples of it throughout his works. Some of the ‘findings’ are obvious, such as when the King of Hearts announces in the Alice in Wonderland court scene, ‘Rule Forty-two. All persons more than a mile high to leave the court.’ In Carroll’s long ghost poem ‘Phantasmagoria’, the narrator reveals his age in the first verse as 42, and in ‘The Hunting of the Snark’, the only rule quoted is No 42. Furthermore, the baker in that poem, who was famed for ‘a number of things’ had ‘forty-two boxes all carefully packed’.
Certain ‘Findings’ of 42 involve the most tortuous number-play. Others, though, are both accessible and genuinely thought-provoking. In his book Lewis Carroll in Numberland, the mathematician Robin Wilson goes so far as to say that Carroll seems to have had an obsession with the number 42. Alice in Wonderland has 42 illustrations, he observes, and the trial title page for Through the Looking-Glass shows that Carroll also anticipated having 42 illustrations for this (although in fact he ended up with 50).
Also in Wonderland, Alice recites some apparently nonsensical times tables after she has followed the White Rabbit underground, and these turn out to have an unexpected link with 42: ‘“Four times five is twelve” says Alice “and four times six is thirteen and four times seven is – oh dear! I shall never get to twenty at that rate!”’ It sounds like nonsense, but in fact Carroll was counting each calculation on a different base. Therefore, 4 × 5 equals 12 on base 18; 4 × 6 = 13 on base 21; and so on. It goes all the way up to 4 × 12 = 19 (on base 39). And sure enough, Alice never does get to 20 by continuing this way, because 4 × 13 = 20 does not work on base 42.
Perhaps the most extraordinary use of the number 42 is when the White Queen says,’”Now, I’ll give you something to believe. I’m just one hundred and one, five months and a day.”’ Edward Wakeling, editor of Carroll’s diaries, and also a mathematician, has worked out that since the story was specifically set on 4 November 1859, when Alice was exactly seven-and-a-half (Alice Liddell’s birthday was 4 May 1852), then the White Queen, taking account of leap years, was exactly 37,044 days old at the time when she spoke. The Red Queen, being a member of the same chess set, was, of course the same age as the White Queen, giving the two queens a total combined age of 74,088 days – the sum of 42 × 42 × 42!
As well as playing with numbers, Carroll loved playing with words. He was extremely clever at saying exactly what he meant, yet not meaning what he appeared to say. Consequently, his writing often repays close examination to tease out the real meaning – and so, it seems, did his conversation. If a mother offered her hideous baby for admiration, he once remarked, one should exclaim in an admiring tone ‘Now there’s a baby!’ and all honour would be satisfied.
Carroll was also particularly fond of acrostics, a kind of word-play which was very popular with Victorians, apparently including Queen Victoria herself. In its simplest form, the acrostic poem is based on a word or phrase, known as a column-word or column-phrase. The first letter, syllable
or word of each line of a simple acrostic-poem spells out the column-word or column-phrase. A good example of a simple Carroll acrostic is an inscription he wrote in a book called Holiday House which he bought as a gift for Lorina, Alice and Edith Liddell for Christmas 1861. Here are the first eight lines of his verse, the initial letters of which spell out the word LORINA:
Little maidens, when you look
On this little story-book,
Reading with attentive eye
Its enticing history,
Never think that hours of play
Are your only HOLIDAY …
Carroll sometimes seemed almost to be inspired by the limitations of the acrostic form, and several of his acrostics, like that of ALICE PLEASANCE LIDDELL in ‘A Boat Beneath a Sunny Sky’ (Through The Looking-Glass), are also precise and evocative poems in their own right.21 However, he also relished the task of creating more technically difficult acrostics, and some of these are very clever – and not a little obscure. They include double acrostics in which each verse is a riddle and the puzzler must figure out the two ‘column-words’ which relate to the entire poem and whose letters in order begin and end each riddle’s solution.
A little known example is ‘A Day in the Country’, which Carroll wrote in 1866, a year after Alice in Wonderland first appeared. It was probably done to amuse country friends whom he was visiting, for Carroll liked to create poems to entertain people he liked. Since it tells the story of a failed photographic session, he may have been photographing his friends, too.
The poem (given in full below) has never been fully solved, although some of the clues have been deciphered. Note that the first two verses give the two vertical column-words, each of which has eleven letters. The first column word (in this case PORTMANTEAU) gives the initial letters of the horizontal solution-words, the second column word (in this case PHOTOGRAPHY) gives the final letters of these horizontal words. The other verses give clues to the meaning of the horizontal solution-words. These horizontal words can be of any length, as long as they start and finish with the appropriate letter.