by Jenny Woolf
2
‘You can do Arithmetic, I trust?’
Oxford Life
Yet what are all such gaieties to me
Whose thoughts are full of indices and surds?
–Double Acrostic, ‘There was an Ancient City, Stricken Down’
Carroll spent his life as an Oxford mathematics don, and his professional and recreational work in mathematics is part and parcel of the picture of him as a human being. Numbers – anything and everything about numbers – formed a thread running through his life and were an integral part of how he saw the world.
The sparse recorded family memories of his childhood show that his love of numbers dates from his early childhood. Collingwood describes him, when a little boy, showing his father a book of logarithms with the request to ‘Please explain’. His father told him that he was too young to understand about logarithms, yet he insisted ‘But, please explain!’1
As an adolescent, he was concerned with various quirky issues relating to measurement of time. In the family magazine The Rectory Umbrella, he ponders whether a clock that has stopped completely is, logically, better than a clock that is five minutes slow, since the stopped clock will always tell the right time twice a day. ‘You might go on to ask, “How am I to know when eight o’clock does come?”’ he jokes. The answer was, he said, that ‘… when eight o’clock comes, your clock is right. … keep your eye fixed on your clock and the very moment it is right, it will be eight o’clock. “But –” you say. “There, that’ll do, reader. …”’!2
He was also preoccupied about when exactly a day changes into the next day on its passage around the Earth. He went on to propose a fixed international date line, an idea finally taken up in 1884. He was a persistent young man, and it seems he must have dwelt very often on this matter of when a day begins, for Collingwood wryly recalled many years later that the difficulty of answering this apparently simple question had ‘cast a gloom over many a pleasant party’.3
Carroll lived up to his early promise and showed a good deal of mathematical ability during his schooldays. At the end of his first year at Rugby School, for instance, he came first in mathematics in the lower fifth form, and in his second year he won second prize for mathematics in the annual general examination and first prize in the mathematics division for the half. In 1848, his mathematics tutor at Rugby, Robert Mayor, told Carroll’s father, ‘I have not had a more promising boy at his age since I came to Rugby.’4
The next stage was obviously university. Christ Church, Oxford, had been his father’s college, and his father was keen for his oldest son to follow in his footsteps. Accordingly, just before his 19th birthday, in January 1851, Carroll went up to Oxford to further his mathematical studies.
The city which first met his youthful gaze was a very different place from the sprawling and in parts rather dingy built-up area of today. Mid-nineteenth century Oxford, although wealthy and imposing, was decidedly rural and compact, circular in form and measuring about 3 miles across. It consisted of two spacious streets which intersected in the centre of town, and boasted a large market place and magnificent stone bridge. Above all, the whole area was physically dominated by the colleges, ‘several of which stand in the streets and give the city an air of magnificence’, as the Revd James Barclay observed in his gazetteer of 1810.
Margaret Fletcher, the founder of the Catholic Women’s League, was the daughter of an Oxford don, and spent her childhood in the city. She does not seem to have met Carroll, even though she was there when he was in Oxford, but her autobiography gives a fascinating view of the city in the mid-nineteenth century. She recollected that it was ‘countrified and quiet … almost innocent of traffic … with grass growing in the streets in the Long Vacation’. There was an almost stately atmosphere, with women moving slowly in their rustling, voluminous crinolines, and dons walking together in pairs ‘to distant horizons, so intent on their discussions that they seemed unconscious of their surroundings’.5
The railway had only just arrived to connect Oxford to other parts of Britain. Of course the line which carried the filthy, clanking engines did not run close to the colleges, but had been directed to the other side of the river, well away from the ancient centre. When Carroll took his long, donnish walks across the surrounding countryside, he would have climbed the dusty road up to Headington Hill, and turned at the top to see a magical, vista of ancient towers rising from the meadows below: a fairy-tale view which had hardly changed in centuries. He might have exchanged a few words with the local children who brought their cows to feed on the road’s grass verges, or picked the wild flowers that grew in the hedgerows. Less pleasantly, at certain times of year, he might have caught the all-pervading stink of the fish lying stranded and dying in the water meadows around the colleges as the annual river floods receded for the summer. He might also have worried about the cholera that was ravaging many of the unsanitary old houses which still lined the streets.
No diaries and few letters exist to give his very first impressions of the place, but he obviously did well in his studies. He obtained a first class honours degree in mathematics in 1854, and in 1855 he was appointed as a mathematics lecturer, moving in as a resident of the college and starting to acquire his own students. On 24 April 1855, the 23-year old triumphantly recorded in his diary that ‘Leighton sent me £5 fee for last term’s coaching – this is the first earned money I ever received – the first that I can fairly call mine.’
He had every reason to feel pleased with himself, but disillusion began to set in before he had been long in his new job. Grand, wealthy and aristocratic it may have been, but Christ Church was snobbish and somewhat old-fashioned in its attitudes. What is more, its mathematical record had been undistinguished for many years. Indeed, it was very different from any modern university college. A bustling hive of academic excellence Christ Church may now be, but that was not the case in the early and mid-nineteenth century.
As the college most favoured by the hunting, shooting aristocracy, the Christ Church of Lewis Carroll’s late teens and early twenties was dirty, noisy and teeming with the undergraduates’ dogs. Carroll remembered years later how there had been around 70 dogs running about the place day and night.6 No pampered pets, these dogs were as undisciplined as their masters often were, and the old dean’s verger was often found outside the chapel with a stout whip, with which he flogged dogs that tried to follow their masters into chapel.
Old Dean Gaisford, who was running the place when Carroll arrived, was a classical scholar who did not interest himself much in college affairs. Many of the wealthy undergraduates and their families saw university simply as a rite of passage, a place they had to go before they got on with their real tasks of running their country estates or working in the colonies. They were preoccupied with horses and their social lives and they found studying tedious and tiresome. Although they were kept under strict discipline of a type we would think more appropriate for schoolchildren, there was a good deal of wild behaviour and vandalism. On one occasion they are reported to have vandalized the college buildings, burning some of the old benches and tables, and on another occasion they caused an explosion which created a small ravine in the quad.
This was not the kind of atmosphere which Carroll particularly enjoyed, and it was not long before the studious young lecturer was writing in his diary that he was ‘weary of lecturing, and discouraged … it is thankless, uphill work, goading unwilling men to learning they have no taste for.’7 However, he persevered. After gaining his MA in 1857, he became a Senior Student, (or Fellow as it was known in other colleges) in 1858. He thereby gained a small income, but most crucially, he was then entitled to stay at the college for the rest of his life, provided that he fulfilled certain conditions. The two most important of these were that he should remain celibate, and that he should be ordained a minister of the Church of England before four further years had elapsed. Both these conditions, of course, had a major effect on his life.
The cur
riculum which Carroll was expected to teach was routine, although it covered a good deal of ground. Some interesting new mathematical work was being done on the continent of Europe at the time, but little news of it reached Christ Church, and there is no sign that Carroll was aware of it. His method was to cling to old approaches and refine them in detail. Mostly, he led the undergraduates through elementary geometry and algebra, writing treatises and textbooks on topics which now attract little attention.
In fact, some of his critics maintain that he never did anything of note in his mathematical career, and the mathematician Warren Weaver has even pointed out several gaps in his basic knowledge.8 Yet his professional work was not undistinguished. In the realm of ordinary mathematics he made some minor discoveries in determinants, and Weaver concedes that his interest in logic was leading him into interesting directions when he died. In 1894 he developed a proof on the interpretation of conditionals which uses a ‘truth table’, a concept which did not come into general use much before 1920. In this Carroll had made a real discovery, and yet so uninterested was he in the bigger picture that he did not publicise it at all. Furthermore, the Carroll Diagram he developed is a superior and more flexible version of the Venn Diagram, and it has now, in the 21st century, been introduced into the British school curriculum.
Despite his feelings of boredom and discouragement, too, several of the textbooks he created for his students went into numerous editions and did well. Whether the students enjoyed working from the textbooks was another matter. Carroll’s notable and lifelong difficulty in ‘seeing the wood for the trees’ often manifested itself as a love of hopeless complexity. It is a trait particularly demonstrated by his Guide to the Mathematical Student, published in 1864.
This book consists of a list of mathematical topics together with suggestions for the order in which they should be studied. It is arranged in 26 subject areas, which are sub-divided into over 400 topics. References to over 2,000 examples are listed, and he intended that the student using it ‘should turn to the Syllabus for each reference, and work two or three examples in the subject there indicated … of course passing over all references to subjects he has not read’. At the end of each day’s work, the student was supposed to mark the point that he had reached. It was an extremely good idea in theory, but in practice it was just the sort of finicky and methodical project that Carroll himself delighted in but others often found hard to take. The book must have represented countless hours of hard work; but though it is entitled ‘Part I’ no further parts were ever written – or requested.
Unfortunately, Carroll’s love of elaborate rules and fixed plans did not help make his teaching attractive to the undergraduates. In mathematical logic, he liked to follow every step of an argument from the beginning to the end with strict precision, an approach that has been described as using a sledgehammer to crack a walnut-sized problem. This painstaking methodology also made it hard for him to keep up with the amount of work he had to do, and his diary shows that in the early days he sometimes felt overwhelmed by the tasks he had set himself.
His greatest difficulties, however, seem to have been in managing his undergraduates. These may have been quite serious, and are hinted at in the biography that H L Thompson wrote of Dean Liddell in 1899. Dean Liddell, Alice’s father, had arrived to take charge at the college soon after Carroll began his lecturing job, the old dean, Dean Gaisford, having died in June 1855. It seems that Liddell’s arrival was not eagerly anticipated (Carroll recorded unenthusiastically in his diary on 7 June that ‘The Times announces that Liddell of Westminster is to be the new Dean: the selection does not seem to have given much satisfaction in the college.’).
In describing the period directly after Liddell’s arrival at Christ Church, Thompson notes that the new dean was much occupied and bothered about ‘a tutor’ who could not control his men.9 There is ambiguity about whether Carroll was officially a tutor at the time (though he does refer to himself as one), but reading his existing diary entries for the few months after the dean arrived, it is obvious even from this unemotional record that he could not keep control of the men, and his terse recollections have a slightly nightmarish quality.
In fact, Carroll’s own record of the very beginning of this period is no longer available. His diary from September to December 1855, the very first months of Dean Liddell’s incumbency, survived his death, but went missing while in the care of his family. So nobody now knows the exact details of what happened to him during those first weeks; though the record after the diary resumes is bad enough.10
In January 1856, when the next surviving diary volume begins, Carroll was returning to Christ Church after the Christmas vacation, and he summoned 60 men to attend a meeting about his Mathematical Lecture. He recorded that only 23 turned up. He told the rest of the men to report to his rooms, but none of them did. He sent for them individually, to no avail. The next day he called on the dean, and it was agreed that the idle men should be reported to him. On 25 January, when Carroll sent for the men again, they came; but they were clearly reluctant, and several of them insolently failed to turn up for his lectures in the next few days. Clearly, he had a problem; but at this point his personal self-censor came into play. On 29 January, he wrote that, ‘In future, I shall record all matters connected with the Mathematical Lecture in a separate book.’ And that is the last we hear of it. The separate book was not preserved by his family.
The sorry truth is that he was never able to control unruly groups. He had tried a little teaching to classes of poor children, and enjoyed it at first, but even there quickly became discouraged. He did not approve of beating and abusing pupils, which was the fashion at the time, and his gentleness may have seemed like softness to them. His relatively high-pitched voice and his speech hesitation probably added to the difficulty of dominating groups either of children or adults.
At Christ Church, he could not avoid dealing with rude, rebellious and snobbish young men. He was not much older, and considerably less wealthy, than many of them, and his stammer provoked laughter and teasing even among his own contemporaries. But he kept his feelings to himself, coping with his unpopularity by being curt and remote with his undergraduates, so that, consequently, his teaching was described as ‘dull as ditchwater’. According to a persistent rumour, some undergraduates even raised a petition in protest at being taught by him, and even as late as 1931 his unpopularity was not forgotten. Sir Herbert Maxwell, a Conservative MP, wrote to The Times in that year to recall that from nearly 70 years earlier he remembered ‘the lean, dark-haired person of Charles Lutwidge Dodgson’ and the ‘singularly dry and perfunctory manner in which he imparted instruction to us, never betraying the slightest personal interest in matters that were of deep concern to us.’11
It seems puzzling that someone so imaginative, communicative and original could have so failed to inspire. Perhaps Carroll’s shyness and acute sensitivity caused him to defend himself with the donnish dryness and remoteness which eventually came to characterize his public image. Perhaps he just decided he didn’t care about thick-headed undergraduates, for to the few who were interested in learning he was unstinting in his help. His peculiar and imaginative humour would have been better suited to a more supportive environment, or a more studious atmosphere, for, just as with the storytelling, he blossomed when faced with a receptive and appreciative audience.
Fortunately, he was to do extremely well as a teacher when, in the 1880s and 1890s, after his retirement, he began to teach logic at Oxford High School for Girls. There, he had girls listening to him – and girls who wanted to learn, too: very different from jeering male undergraduates. Ethel Rowell, who later became a mathematician, recalled that when Carroll stood at the desk in the sixth form room and prepared to address the class, she thought he looked very tall, serious and formidable. As he continued, though, he relaxed. In the all-female atmosphere, his facts became more fanciful and his fancies became more fantastic, she recalled. After some time, he spotted her
talent and enthusiasm, and she was delighted when he finally offered her private tuition. ‘As the subject opened out, I found great delight in this, my first real experience of the patterned intricacies of abstract thought,’ she wrote. But more than this, she added, ‘he bestowed on me another gift of aspect more gracious. He gave me a sense of my own personal dignity. He was so punctilious, so courteous, so considerate, so scrupulous not to offend, that he made me feel that I counted.’12
Another student in the Oxford classes remembered that ‘the girls adored him, he entertained them with written games on the blackboard. He was perfect with children, and there were always tribes of little girls attached to him. He made everyone laugh.’13
Just as he enjoyed teaching a receptive and polite audience all his life, so Carroll’s personal curiosity about the magic and puzzles of numbers themselves never failed. Many of his letters to other dons or intellectuals are concerned with mathematical matters, and joking references to mathematics occur in many letters which he wrote to people who had no professional interest in the subject at all. In 1878, for example, his friend Edith Denman mentioned to him that she liked figure-drawing better than he did. His reply was characteristic:
There is a rashness, which I can only deplore, in your assertion that I cannot be as fond of figure-drawing as yourself! The point cannot be satisfactorily settled till we have measured the two fondnesses by the same unit. Now the unit of pleasure (which I suggested years ago, and which Society hasn’t yet adopted!) is ‘the pleasure felt in eating one penny-bun in one minute.’
Please to estimate the pleasure which you get from an hour of figure-drawing, using that as a unit, and then we can compare numbers: my number is 235. Trying to settle it without a unit is like arguing about two rooms, each saying, ‘I’m sure this room is the hottest!’ without ever referring to a thermometer …14