The Three-Body Problem

by Catherine Shaw

  Then yesterday morning, I found that the best antidote to unusual feelings of lassitude was activity, as I had absolutely no time to indulge in them from the moment I arose, and ended by quite forgetting them. Indeed, I had promised the oldest group a surprise in their arithmetic class today, as yesterday they not only complained quite bitterly about the boredom of the sums I had set them, but worked them correctly one and all, so that they clearly have nothing further to learn in that domain, and senseless repetition would engender only loathing, it seems to me.

  I racked my brains and looked in my books in vain. Finally, I decided that the best solution would be to search if I could not find a better book, so I donned my wraps and ventured out into the cold, to a bookshop where I often go, to ask advice of the knowledgeable bookseller. ‘Why, don’t you read The Monthly Packet, miss?’ he enquired of me in astonishment, when I had laid my difficulties before him. ‘There’s a mathematician, Mr Lewis Carroll, used to publish puzzles for young people there; a better way to teach reasoning while enjoying oneself can scarce exist.’ He dug out a large pile of old and dusty copies for me, and I carried them home forthwith and turned the pages until I found the puzzles he spoke of.

  I had never seen The Monthly Packet before. Dear me, it is very edifying – perhaps a little too edifying at times, so many morals are concealed within its pages, which are essentially addressed to young ladies. But the puzzles, when I found them, turned out to be most delightful; there is a whole story in sections called A Tangled Tale, and each section contains a puzzle. Some of them seem so difficult I can’t even begin to think how to solve them myself yet. After having read through them at length, I decided to begin at the beginning, and copied out the first ‘Knot’ in the Tangle from the April 1880 number. When the girls arrived, I sent my darling advanced class – consisting of Emily, aged thirteen, and Rose, eleven – into my sitting room with the paper, with instructions not to come out until they had solved it between them. They disappeared with a flutter of frocks and ribbons, and I set regular sums to the middle group and began counting dried peas with the little ones, who are still learning that if Violet has four peas and Mary takes three of them, then when Violet opens her mouth in an O of protest to wail that it is not fair, because she has only one pea left, this means that she has solved the subtraction correctly and has earned a kiss, much to her surprise.

  The first knot is so entertaining, that I am sending it to you bodily; as the author says, if you have a headache, it will distract you from it, and if you do not, it will give you one.

  A Tangled Tale, by Mr Lewis Carroll

  Knot One: Excelsior

  Goblin, lead them up and down

  The ruddy glow of sunset was already fading into the sombre shadows of night, when two travellers might have been observed swiftly – at a pace of six miles in the hour – descending the rugged side of a mountain; the younger bounding from crag to crag with the agility of a fawn, while his companion, whose aged limbs seemed ill at ease in the heavy chain armour habitually worn by tourists in that district, toiled on painfully at his side.

  As is always the case under such circumstances, the younger knight was the first to break the silence.

  ‘A goodly pace, I trow!’ he exclaimed. ‘We sped not thus in the ascent!’

  ‘Goodly, indeed!’ the other echoed with a groan. ‘We clomb it but at three miles in the hour.’

  ‘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!’

  ‘’Twas three hours past high noon when we left our hostelry,’ the young man said musingly. ‘We shall scarce be back by supper-time. Perchance mine host will roundly deny us all food!’

  ‘He will chide our tardy return,’ was the grave reply, ‘and such a rebuke will be meet.’

  ‘A brave conceit!’ cried the other with a merry laugh. ‘And should we bid him bring us yet another course, I trow his answer will be tart!’

  ‘We shall but get our desserts,’ sighed the elder knight, who had never seen a joke in his life, and was somewhat displeased at his companion’s untimely levity. ’Twill be nine of the clock,’ he added in an undertone, ‘by the time we regain our hostelry. Full many a mile shall we have plodded this day!’

  ‘How many? How many?’ cried the eager youth, ever athirst for knowledge.

  The old man was silent.

  ‘Tell me,’ he answered, after a moment’s thought, ‘what time it was when we stood together on yonder peak. Not exact to the minute!’ he added hastily, reading a protest in the young man’s face. ‘An thy guess be within one poor half-hour of the mark, ’tis all I ask of thy mother’s son! Then will I tell thee, true to the last inch, how far we shall have trudged betwixt three and nine of the clock.’

  A groan was the young man’s only reply; while his convulsed features and the deep wrinkles that chased each other across his manly brow, revealed the abyss of arithmetical agony into which one chance question had plunged him.

  It doesn’t seem clear, somehow, when you first read the story, where the problem lies. I wondered what the girls would make of it, and left the sitting room door open a little, to hear their discussions. At first there was silence, as each one read through the problem separately.

  ‘I can’t make head or tail of it,’ I heard Rose’s adorable voice complaining.

  Although there is only a year or two between them, Rose seems much younger than Emily; a dimpled child still, with a long fall of honey-coloured hair held back from her face by a wide pink ribbon. ‘Whatever does it mean?’ she went on, rather crossly. ‘What is “clomb”? What is “trow”?’

  ‘“Clomb” means “climbed”, you silly,’ said the ever reasonable Emily. ‘And “trow” means … well, it doesn’t mean anything really. Don’t pay any attention to it.’

  ‘But I don’t see what one is to do,’ continued Rose.

  ‘Oh really, the men are walking from three to nine o’clock, and we want to know how far they went, and when they got to the top of the hill,’ said her friend.

  ‘But what do they do after they walk to the top of the hill?’ Rose’s voice became more and more plaintive.

  ‘Why, they come down and back again!’

  ‘How silly! Whatever for?’

  ‘They took a walk, for goodness’ sake! Now Rose, we have to find out how long they walked, seeing that they went three miles an hour climbing, four miles an hour on level ground and six miles an hour descending.’

  There was a moment’s silence.

  ‘Oh,’ said Rose, ‘well, if they walked for one hour before climbing the hill, and then climbed up for two hours and down for one hour … why then … no, it doesn’t make sense.’

  ‘Yes, it does,’ Emily helped her. ‘If they walked on level ground for one hour, and came back on it for one hour, then … they had four hours to go up and down hill and so … they spent twice as much time climbing as descending. That means that they must have spent two thirds of the time going up, and one third coming down. Two thirds of the four hours makes … makes two thirds of two hundred and forty minutes, oh bother, I hate doing this; one third eighty, that’s one hour and twenty minutes, oh do help me Rose! Two thirds makes one hundred and sixty, that’s two hours and forty minutes. So they took two hours and forty minutes to go up, and one hour and twenty to come down. Then that means they reached the top at six-forty.’

  ‘All right,’ said Rose with some pleasure. ‘Then we’re done. But it’s a silly problem. Why does it say “within half an hour”?’

  ‘Oh, you are a goose! Don’t you remember that we started by guessing they walked one hour on the level? Why, we don’t know that – we don’t know how much they … OH – I see – I don’t think it matters! That must be the answer. Let’s suppose the hill began at t
he very door. Come now, tell me how much time they spent going up and how much coming down.’

  ‘That’s easy; they climbed for four hours and came down for two,’ said Rose, ‘so they got to the top at seven.’

  Emily scrutinised the puzzle once again. ‘And what if they did the other thing – spent the whole time on level ground?’

  ‘And what about the hill?’

  ‘Perhaps the hill was small – so small they could jump right over it in one second!’

  ‘Oh, that’s silly,’ said Rose, ‘they wouldn’t be panting so.’

  ‘Well, perhaps they’re easily tired. Anyway, they’d get to the top at six o’clock, of course, so whatever happens, they must get there between six and seven. The answer’s six-thirty!’

  ‘I don’t think I understood it really,’ Rose was beginning, looking doubtful and tripping over the ample petticoats she always wore, but Emily seized her arm and pulled her triumphantly back to the schoolroom, where she stood in front of me, her pre-Raphaelite face framed in the sheath of dark hair which fell over her shoulders, earnest and all unaware of her loveliness, and recounted her solution to the problem with pride.

  This triumph of teaching had a surprising consequence. Today, when five o’clock arrived and the girls were putting on their wraps and looking out of the window to see who came to fetch them, I saw to my surprise that Emily’s mother had accompanied her governess. I have only met her once before, in September, when Mrs Squires left and Emily’s mother, Mrs Burke-Jones, came to tell me that Emily was very fond of the day-school and wished to continue on with me. She was very kind then. Today, she told me with a smile that Emily had quite astonished her uncle Mr Morrison, Mrs Burke-Jones’s brother, last night at table, by setting him yesterday’s puzzle and then explaining her solution to him! It turns out that Mrs Burke-Jones’s brother is a mathematician, and personally acquainted with Mr Lewis Carroll, the author of the puzzles (who goes under another name in real life, where he is an Oxford Don, it seems). He told his sister (somewhat to her surprised dismay, I seem to guess) that Emily had inherited his own delight in mathematics, and that she could consider, if it continued, going on with her studies as long as she liked, even to university. Yes – Mrs Burke-Jones informed me that there is a college for ladies, right here at the University of Cambridge – two of them, in fact! Of course, the ladies may not take degrees, but they may study and have tutors and follow the classes and even pass the examinations – good heavens! Emily jumped with joy at the prospect, while her mother looked doubtful and said, ‘Fortunately, she is only thirteen, there is plenty of time to think about this.’

  At any rate – this is the most exciting thing – Mrs Burke-Jones has invited me to a dinner party next Saturday, of ten people at her house! It is the first such invitation I have received since living in Cambridge, though I have been to tea a few times with the governesses of some of the girls. The famous brother will be present, as he expressed a wish to meet the teacher who provides such delightful lessons to his niece, and various other friends and colleagues of the family. I know nothing as yet about Mr Burke-Jones, but shall surely find out more on Saturday. I did learn that Emily has a brother of eleven, and that the two children will have dinner in the nursery, but will be allowed to come down and greet the guests. I shall write to you afterwards with every detail!

  Goodnight, my dearest twin

  Vanessa

  Cambridge, Sunday, March 4th, 1888

  Dearest Dora,

  Last night was the great event! I never imagined myself in the midst of such a lively group – and nearly all mathematicians. There were six mathematicians altogether, four bachelors and two married gentlemen with their wives. Most surprisingly of all – one of them was neither more nor less than my Mr Weatherburn from upstairs! We were equally surprised to see each other there.

  But let me tell you everything in order.

  I arrived punctually in front of Mrs Burke-Jones’s lovely house. The windows were all lit up festively, and it looked very tempting; light spilt out onto the walk, and the air was so bright and crisp you could see your breath spiralling away. Some people had already arrived. As I waited, feeling too shy to ring, a carriage drew up and an elderly couple alighted. Their driver came around to open for them, and guided them up the walk to where Mrs Burke-Jones’s kindly maid was standing at the open door. I meekly followed up behind them, all nervous, but she welcomed me just as kindly as she did them, and led us into the drawing room, where she announced ‘Professor and Mrs Cayley and Miss Duncan’. I know now that the gentlemen is Professor Arthur Cayley, a very senior and much respected professor at the university, with an important Chair whose name escapes me. The others treated him almost like a king. Inside the drawing room, I was all prepared to be quite paralysed, not knowing anybody, but immediately Mrs Burke-Jones took me under her wing, and introduced me to a young lady of about my own age or even younger, but very much more at ease than myself, called Miss Chisholm, and two further gentlemen called Mr Wentworth and Mr Morrison. Following my arrival, there appeared yet another bachelor, Mr Young, then a married couple, Mr and Mrs Beddoes, and finally, last of all, Mr Weatherburn, who greeted everyone warmly, as he clearly knows them all well, and then looked extremely surprised as he was brought round and introduced to me. I noticed that he is afflicted with a slight stammer when shy.

  ‘Dear me, b-but we are already very nearly acquainted!’ he exclaimed in surprise, levelling his brown eyes upon me with the same intentness I had noted in them, the other day, in the hall.

  ‘How so?’ said Mrs Burke-Jones, surprised.

  ‘Why, my rooms are in the same house as Miss D-Duncan’s,’ he explained, and added that he knew I held a class there each afternoon, but had had no idea that ‘Morrison’s niece’ was one of the fortunate pupils.

  Mrs Burke-Jones was as nice as could possibly be.

  ‘Oh, Miss Duncan does wonders,’ she cried effusively, ‘Emily does so love her lessons, and the mathematical parts more than anything!’

  Whence a short discussion with Mr Weatherburn about my interests and methods, in which I showed myself to be woefully ignorant, alas, and was deeply grateful to have heard tell of Mr Lewis Carroll at all, for it was certainly the only thing he mentioned that was the least bit familiar to me!

  I floundered on, my dear Dora, blushing greatly, and felt most relieved when Mr Weatherburn moved away from the subject of mathematics to other topics. The guests soon gathered into one or two larger groups, and the conversation, seemingly quite naturally, turned to the death of their poor colleague, Mr Akers. As the newspaper article I sent you hinted, nobody seemed to like him at all. They did make some attempt to speak of him more or less with the respect due to the deceased, but even so, I suspect a certain dose of snideness lay behind some of the remarks. Mr Beddoes in particular seemed to sneer subtly behind his apparent admiration, when he spoke of his colleague’s work. But his remarks elicited only knowing smiles from his colleagues, although they freely discussed the character of the poor defunct gentleman. ‘He was quite insufferable,’ appeared to reflect the general opinion; Miss Chisholm even put in that ‘a sarcastic smile was part of his permanent apparel.’ Mr Weatherburn did not say anything at all, as the others traded remarks and observations, and for some little time the question of the nature of his death was politely avoided, but suddenly someone could not resist any more, and directly put the question which must have been tickling all of them from the beginning, to judge by myself!

  ‘Well, Weatherburn, you dined with him on the fatal evening, so what did you actually talk about, eh?’

  ‘Yes, uh, well,’ said Mr Weatherburn, whose speech is slow and careful, perhaps because of the stammer, ‘he seemed to want to talk about an absolutely remarkable idea he had conceived lately. He was quite excited, I recall.’

  ‘A remarkable idea? What kind of idea?’ came a chorus of excited cries from the throats of all the mathematicians present.

  ‘Apparently he had b
een working on the n-body problem. He didn’t want to get into details, however. The closest I got was when he pulled a piece of paper from his breast pocket and scribbled the solution down onto it, to show me, saying that it provided a complete and incredibly original solution to the differential equations of the n-body problem. But he shoved the paper away, naturally, before I could really inspect it closely. He told me that he had already written a rough manuscript of his proof of convergence of the series. Indeed, I quite believe he asked me to dine mainly from a need to vent his irrepressible feelings of triumph. He seemed particularly concerned with the possible reaction of Professor C-C-Crawford.’

  ‘Ah yes, I’m not surprised. I believe Crawford’s been working on the n-body problem in secret for several months now, ever since it was set as the main problem of the Birthday Competition organised by King Oscar of Sweden, though he won’t admit it,’ remarked Mr Wentworth.

  ‘Except for making elliptical allusions to it after he’s had a few glasses!’ added Mr Morrison.

  ‘Well, Crawford certainly has been working hard lately,’ remarked Mr Young. ‘My word, he’s been almost invisible for over a week now, shut in his rooms working from morning till night. Normally, he wouldn’t miss an evening at Mrs Burke-Jones’s for the world.’

  ‘Crawford’s been saying he’s working on the n-body problem – really?’ cried Mr Beddoes, looking aghast. ‘But that’s – that’s quite impossible! Utterly impossible. Ha!’ There was a moment’s pause, and then he added in a different tone, somewhat aggressively, ‘It’s far too difficult for a mathematician like Crawford. Why, the man has ideas, but no rigour! How can he hope to compare with a genius like young Poincaré in France?’

  ‘Does anyone know what became of the paper Mr Akers had in his breast pocket – the one he showed Mr Weatherburn at table?’ enquired Miss Chisholm suddenly.

  It was exactly the question I had been longing to put, but I really didn’t dare insert even a word into a conversation about things I knew nothing of. I must say that my heart warmed to her because of her forthrightness. I noticed that every time she spoke, Mr Wentworth, Mr Morrison and Mr Young all looked at her with favour.