The Three-Body Problem

by Catherine Shaw

  By this time, the train was drawing into Cambridge, and I put all exciting thoughts of becoming the new scandal of the town by introducing unheard-of modern schooling methods out of my thoughts, and concentrated myself on what I should say as I arrived in the court, and how I could impress upon the judge that he must listen to me, no matter what point the trial had reached, and even if Arthur had already been condemned, or for that matter, already hanged.

  I opened my valise and extracted the whole bundle of papers comprising the evidence I had put together.

  ‘Good,’ said Mrs Burke-Jones, ‘now, I shall take your valise home with me – and here, you shall take this leather bag for your papers …’ She tipped her own things out of it, and placed them in the picnic basket, with gestures that for all their haste never ceased to be charmingly precise and ladylike, ‘… and my dear, let me look at you. Here, perhaps this comb – no, let me do it, you have no mirror.’

  She unpinned and removed my hat, and combed my hair carefully, replacing a pin or two. Then she sat back and looked at me critically.

  ‘Here, my dear,’ she said, ‘you shall wear my hat.’ And before I could say a word, she had removed it from her hair and balanced it carefully on mine.

  It was truly a lovely hat, the kind which is very expensive and very simple, and belong only to ladies who can afford a great many hats. It was in black velvet. I felt it was really too beautiful for my dress.

  ‘Not at all,’ she said, ‘quite the contrary. Your lovely figure and your way of walking, together with the hat, lend a great deal of elegant simplicity to the dress. I cannot help you much, in this difficult and crucial moment, but I can do this small thing: make sure that your appearance will help you impress the judge properly. Now, we have arrived. I will get you a hansom.’

  Out we rushed, and although everyone in the train wanted to flag a hansom, Mrs Burke-Jones’s simple, distinguished gesture was the first to succeed. The top-hatted driver stopped in front of her with eager respect, tipped his hat and opened his door. She ushered me in, and gave him a bill and the address of the courthouse.

  ‘Thank you, thank you so much,’ I called as he whipped up his horses.

  ‘I wish you courage,’ she called back.

  ‘Mother, please can we not go to the courthouse too?’ cried Emily as I drove away.

  ‘Certainly not! It is not a place for children!’ replied her mother, and swept her little brood down the street, as my coachman and I turned the corner.

  ‘You in a hurry, ma’am?’ he called smartly down to me.

  ‘Oh, yes, the biggest possible one!’ I told him.

  Suddenly, all my fear and apprehension seemed to melt away. I felt as though time had stopped, and could not resume its steady march until I reached the courthouse and released it. We trotted along sharply enough, my driver taking care to pass others along the streets, abusing them vigorously the while, until finally, he drew up in front of the imposing entrance, hopped down and opened the door for me. It was just after six o’clock.

  ‘Here you are, miss,’ he said, ‘it’s all paid for, hurry along now.’

  I added his gentlemanliness, and his kind eagerness to help me, to all the other generous and beautiful gestures I had encountered in my long quest, to be remembered and treasured later. Storing it in my heart, I jumped out and ran up the steps to the stately entrance, calling back, ‘I hope I meet you again someday!’

  I pushed the doors, and entering the hall, addressed myself hastily to the clerk at the desk.

  ‘Can you tell me, please, about the trial of R. vs Weatherburn? Is it over yet?’

  ‘No, miss, the jury’s out, been out this quarter of an hour. A painful long speech counsel for the defence give. Most people was sleeping. The jury’s not expected to take long. The judge held the trial up a bit today over midday – some say he was waiting for another witness – but finally he called for the closing statements.’

  ‘I am the witness he was waiting for, and I have only just arrived,’ I told him. ‘Can you take an urgent message to the judge, immediately, and show me the way to the public gallery?’

  ‘Oh, miss – it may be too late now,’ he said, looking very doubtful. But he pushed paper and pen toward me, and indicated a door behind him with his thumb. As fast as I was able, I penned a few words to the judge, blotted them, handed them to the clerk and dashed into the public gallery.

  I waited there for a few dreadful minutes. The jury was still out, and the members of the public were murmuring to each other. They appeared to think that the outcome of the trial was a foregone conclusion, and that the jury would be returning at any moment. I looked over at Arthur, and felt my heart gripped with agonising tenderness. He looked like a man who has quietly abandoned the turmoil of living; he did not see me, did not lift his eyes even a single time, nor feel my burning gaze, but remained as still as a man who has already quietly accepted death and defeat. It was easy to see that, like the smirking public in the gallery, he was in no doubt at all of his conviction, and that he had not conserved even the slightest vestige of hope; his was not a fighting nature, and his reaction to the blows of Fate was one of withdrawal and silent despair. My heart seemed to stop, watching him.

  Quite suddenly, at the very back of the courtroom, two doors opened simultaneously. From the left-hand one issued a bailiff, who proceeded to hold the door and usher in the twelve members of the jury. One by one, they took their seats in the jury box. From the other issued a very young, smartly dressed messenger boy, carrying my message. He brought it respectfully to the judge and handed it to him with a bow and some murmured words.

  I held my breath. The judge read my message. He looked up. He looked over at the jury. He looked at the public. He appeared to reflect. The foreman of the jury watched him patiently, waiting to be called upon to give the verdict. The judge finally turned to him.

  ‘Members of the jury,’ he said, ‘have you reached a verdict?’

  ‘Yes, my Lord,’ replied the foreman.

  Every person in the courtroom knew what the judge’s next sentence should be, and the foreman’s answer. I found I could not breathe. I restrained the impulse to leap to my feet, to call out, and concentrated myself upon the judge, willing him to say something different. He opened his mouth.

  ‘Members of the jury,’ he said, ‘you have worked hard and long on this case. Now you have finally finished that work, and I am going to make a very unusual request. I am going to ask that you hold back the decision you have reached, and that we listen to the statement of one final witness. This witness has only just arrived from abroad. I would like to tell you that I had decided to exclude the testimony of this witness if she did not arrive by five o’clock today, as every other witness had been heard, and the closure of the trial could not be indefinitely postponed. However, she is now here, and it appears that she carries such evidence as may help to avert a very grave miscarriage of justice. I therefore propose to counsel for the prosecution and the defence, and to you, members of the jury, to hear this witness. She is a witness who has already been called to testify during this trial. After hearing her complete testimony, we may either adjourn the trial until tomorrow, if the prosecution would like to seek answers to the newly arisen testimony, or else, prosecution and defence may briefly renew their closing statements, and you, members of the jury, may deliberate once again. I would like to know if MISS VANESSA DUNCAN is presently in the court?’

  ‘Yes, I am,’ I said ringingly, standing up in the middle of the public gallery.

  ‘Then, in spite of your extremely unorthodox proceedings, Miss Duncan, I would like to invite you to step into the witness box,’ said Mr Justice Penrose, quite benevolently.

  I left the public gallery through the same door by which I had entered it, and asked the clerk to guide me into the courtroom proper, as a witness. He did so, and I walked down the alley and stepped into the box, encouraged by the consciousness of the documents I held in their leather bag, and of the quiet ele
gance of Mrs Burke-Jones’s hat.

  ‘Miss Duncan,’ said the judge, ‘you have already been questioned and cross-questioned in this court. However, you say here that the information you bring is entirely new. The situation we find ourselves in at this moment is an extremely unusual one, and therefore I am willing to follow an unusual proceeding. I invite you to simply tell your tale in your own words, subject to the objections of counsel if you rely too heavily on suppositions or hearsay in any part of your testimony.’

  ‘Thank you, my Lord,’ I said, trying to control my voice, which wavered somewhat with sudden nervousness. And I began to speak.

  ‘I would like to tell the members of the jury, and everyone else in this courtroom, a great many facts about the murders of the three professors of mathematics, Mr Akers, Mr Beddoes, and Mr Crawford. I believe that I have been able to retrace the full sequence of events leading to their deaths, and I have done my very best to substantiate each of my statements with concrete evidence. I would like, if I may, to recount all the events leading up to the murders, although this may take some extra time.’

  ‘Please do,’ nodded the judge.

  ‘I will begin, then, with King Oscar II of Sweden, and the announcement of his Birthday Competition. This announcement appeared in volume 7 of the mathematical journal Acta Mathematica, published in 1885–1886. I have an English translation of it here.’

  I opened Mrs Burke-Jones’s leather bag and extracted the translation of the announcement that Mr Morrison had made for me at a gay tea party which seemed long ago. I handed it to the judge, who looked it over, and handed it down to the lawyers, who then handed it to the jury.

  ‘As you may see, the closing date for submission of entries to the competition was June 1st, 1888; just four days ago,’ I said, ‘and the prize, an award of money and a golden medal, to say nothing of the great honour involved, is considerable. The main subject of the competition is known as the n-body problem, where n is any number of bodies or particles subject to the laws of physics known as Newton’s laws. Newton himself solved the problem of the behaviour of such bodies when there are only two of them, but the problem of three or more bodies has not, until now, been completely solved.

  ‘Now, the world’s foremost experts on this problem, and on the other problems posed in the announcement of the competition, are not British, but French and German mathematicians. I have often heard the name of a certain Henri Poincaré as being one of the candidates in the competition most expected to obtain astonishing new results. However, the competition was, as you may see from the announcement, open to every mathematician, everywhere. And here in Cambridge, a group of three mathematicians, all more or less specialists in subjects related to these, decided to join forces and work together, in complete secrecy, to see if they could not pool their different capacities to produce a solution. These three mathematicians were Mr Akers, Mr Beddoes and Mr Crawford.

  ‘At this point, I would like to produce a second piece of evidence: Mr Akers’ personal agenda. I collected this agenda from Mr Akers’ sister, presently living in Belgium, to whom the police had transmitted her brother’s personal effects. In this agenda, on the dates of October 18th, December 13th and February 14th, we find the following notation: ABC 2 p.m. Similar notation appears for a date occurring after Mr Akers’ death, the 17th of April. Note that the group called ABC met regularly every two months, on a Tuesday afternoon.

  ‘Clearly ABC stands for Akers, Beddoes and Crawford, and the purpose of these regular meetings was to collaborate and combine efforts in the direction of solving at least part of the n-body problem in time for King Oscar’s Birthday Competition. Recall that Mrs Wiggins testified that a few people occasionally met in Mr Crawford’s rooms; she specifically remembered clearing away traces of a whisky-drinking party of three in Mr Crawford’s rooms around mid-February. These were certainly the traces of the ABC meeting of February 14th. As for the choice of Tuesday afternoon, it can easily be checked that this must have been the best choice of day and hour in the week, in order for their separate teaching and tutoring schedules to combine to allow each of them a free portion of the afternoon.

  ‘Having established the link between these three mathematicians, I would like to add that they decided to keep their efforts a secret, perhaps in order to avoid any public disappointment should the outcome appear unsatisfactory. However, they were but human, and it was difficult for them to keep their activities entirely secret. It is natural for a mathematician to wish to share his triumph when he produces a particularly good idea, and at least some mathematicians of my acquaintance were vaguely aware that both Mr Akers and Mr Crawford were working on the n-body problem. Mr Beddoes, however, was extremely discreet. In my presence, having been told that Mr Akers had talked about the n-body problem to Mr Weatherburn, and that Mr Crawford appeared to be secretly working on the question, he reacted with shocked annoyance, which he then hastily explained away by making the ridiculous observation that those two mathematicians were quite incompetent to deal with such a problem. I remember the scene very clearly, although at the time, I was not aware that his annoyance was a natural reaction to the realisation that his two colleagues were each, separately, throwing out vague hints which contradicted the promise of secrecy they had made to each other. I do not know of anyone who was actually explicitly told about the joint effort, or the regular collaboration meetings, but it seems clear that they occurred.

  ‘Let me now address the events of February 14th, and the murder of Mr Akers. An ABC meeting took place on that day at 2 p.m., and according to Mrs Wiggins’ testimony, it took place in Mr Crawford’s rooms.

  ‘We shall never know exactly what happened during the meeting, as the only three witnesses are now dead. However, judging by subsequent events, the meeting must have run more or less as follows. At some preceding meeting, Mr Crawford, who had the reputation of being a brilliant and inventive mathematician, although lacking in rigour, must have put forth the germ of an excellent idea towards the solution of the n-body problem. All three mathematicians would have spent the next two months working on this idea or ideas of their own, hoping to find some new element to be submitted to the subsequent meeting. What actually happened is that some time before the February 14th meeting, a most extraordinary idea occurred to Mr Akers; an idea of an essentially computational nature, with explicit formulae, which could complete Mr Crawford’s idea and bring it to fruition. Explicitly, he believed that he could adapt Mr Crawford’s idea to obtain a complete solution of the three-body problem, the first fundamentally unknown case of the general problem with n bodies for any number n. Before the meeting, Mr Akers had verified and developed his idea to a certain extent, and sketched it in a carelessly written manuscript of several pages. Then, fired by the enthusiasm of his own discovery, he came to underestimate the importance of Mr Crawford’s contribution, and to consider that he had solved the problem by himself. I believe that he attended the ABC meeting with the intention of announcing to his colleagues that he had solved the three-body problem, and that he intended to pursue his own researches to their conclusion and submit a manuscript independently to the competition.

  ‘Mr Beddoes and Mr Crawford must have been very angry, and argued with him. Mathematicians tend to be extremely prickly about the possession of ideas, and it is quite probable that if Mr Beddoes and Mr Crawford had been entirely convinced that Mr Akers’ idea was absolutely independent and original, they would have been ready to congratulate him. However, from whatever he let fall, they must have perceived that his idea was not so much a new direction, as a brilliant way of making Mr Crawford’s idea work. Then I believe a serious quarrel occurred, during which Mr Crawford downed an entire half-bottle of whisky, as he tended to do in moments of extreme stress or excitement, observed by both Mr Akers and Mr Beddoes.

  ‘Mr Akers left Mr Crawford’s rooms, and went to the library, his mind filled with his discovery to the exclusion of moral considerations. Mr Crawford remained in his rooms and bega
n to reflect hard on his own idea, no doubt feeling that if Mr Akers had been able to make it work, he might have a chance at arriving at the same result, although his talents lay in a different direction. For his part, Mr Beddoes must have considered Mr Akers’ behaviour quite simply unacceptable, and he must have determined within himself to prevent him from continuing his work alone, and to compel him to share it. At any rate, he betook himself to the mathematics library of the university, and there, he saw his colleague Mr Akers, talking quietly among the bookshelves to Mr Weatherburn. Mr Beddoes heard how he was unable to keep the triumph of his discovery to himself, and he heard him invite Mr Weatherburn to dinner at the Irish pub, no doubt, as Mr Beddoes thought, to gloat. He determined to slip into Mr Akers’ rooms in college during the dinner, and search them to find any trace of written work expressing the idea which he felt that Mr Akers had no right to keep to himself.

  ‘It must have been rather difficult for him to locate the manuscript, of whose existence he may not even have been completely certain; Mr Akers’ papers were always in a great state of disarray, and his handwriting was difficult. Mr Beddoes must have searched among them at some length, finding it necessary to decipher several papers before reaching the conclusion that they concerned mathematics irrelevant to the n-body problem.

  ‘While Mr Beddoes was searching Mr Akers’ rooms, the latter was having dinner with Mr Weatherburn. During dinner, unable to contain his pride and delight in his original and brilliant discovery, he began once again to refer to it to his dinner companion, even going so far as to pull a scrap of paper from his pocket, and write down the most important formula upon it. He then, however, thought better of revealing so much to a third party, and pressed the paper back into his waistcoat pocket. I have it here, and submit it to you as my third piece of evidence. I obtained it from Mr Akers’ next of kin, his sister living in Belgium, to whom it was sent together with the rest of his personal effects.