The Three-Body Problem Read online

  Mr Mittag-Leffler is very famous here, and I soon was able to discover that he resides in a villa in Djursholm, a pretty town on the outskirts of Stockholm. Although a professor at the University of Stockholm, his offices lie and his main work is done in his lovely home where he has already collected one of the greatest mathematical libraries in the world. All his work as an editor of the journal Acta Mathematica is done from his home where the manuscripts were to be sent. I wrote down the address carefully: Auravägen 17, Djursholm, and I showed the paper to the driver, who set out on a brisk trot through the gracious streets of the city.

  A city? It is an archipelago, truly; they call it the city of twenty-four thousand islands. It seemed that we constantly crossed water, and the sun was on the horizon by the time we drew up in front of Mr Mittag-Leffler’s imposing villa. The large building is dominated by a round tower in one corner, stretching nobly upwards, which very nearly gives it the aspect of a small castle.

  I paid the man and alighted from the cab, and taking the children by the hand, I began to walk up the wide path leading to the stately entrance. My knees trembled beneath me, and Emily and Robert were silent with wonderment, knowing or feeling that I was reaching my heart’s destination. We stood long in front of the heavy main door, as I tried to control the dreadful knocking in my chest. The sun had now sunk altogether below the horizon, and the entire sky was drowned in deepest blue, though no stars were yet visible. I raised my hand to the large bell, and rang.

  After a short wait, the door was opened by a kindly lady. Her surprise was extreme upon perceiving us; truly we must have had the aspect of three waifs, having travelled so long, eaten so little, and – worst of all – having had so very little time to make ourselves presentable. This morning, I had put on my nice grey dress for the first time, having desired all these last days to keep it fresh and clean for this very moment, but the endless day of travel had somewhat removed its bloom; as for Emily, her lovely white dress with its many flounces was desperately crumpled and wilted, as she had not thought to bring another on her impetuous departure. Poor little Robert looked weary and disordered. We all three straightened up, however, in front of the plump servant, and put on our very best airs of distinction and pride. I addressed her in English.

  ‘We come from England, and must see Professor Gösta Mittag-Leffler,’ I began.

  I do not believe she spoke a single word of English, for only the last words produced some reaction in her rounded features. She looked extremely doubtful, but clearly she was not in the habit of turning away any visitors of the illustrious professor, no matter how unimpressive. She ushered us into a small waiting room near the door, beckoned a maid who stood in the hallway to keep an eye upon us, and bustled away, arousing in my breast the fierce hope that I may, after all, find the professor at home and disposed at least to speak with me.

  It was not long, indeed, before Professor Mittag-Leffler himself descended, and entered the modest waiting room to greet us. He was a hale, energetic gentleman in his forties, imposing and yet extremely kind. I saw immediately that he would be forthright and courageous in his views, and that in spite of his strict and ceremonious appearance, he was quite prepared to listen to whatever I chose to tell him. Perhaps there was even a twinkle of amusement in his eyes, at the sight of the motley crew we represented, with Robert as inseparable as ever from his cherished locomotive.

  He addressed me in nearly impeccable English. ‘Please tell me in what way I may be of use to you?’

  I was moved by his kind reception, but too overwhelmed by a sense of desperate urgency to answer with the ceremony he clearly deserved and expected. I had risen upon his entrance, and now, as he advanced towards me, extending his hand politely, I seized it impulsively in mine.

  ‘I am here to beg you for an immense, an unheard-of favour,’ I began immediately. ‘It is a question of life, death and murder!’

  His face blanched somewhat, and I perceived he thought me mad. I continued as hastily as I could.

  ‘I come from Cambridge, England, sir,’ I told him. ‘Three mathematicians have been murdered there within the last months.’

  ‘Ah, yes,’ he murmured, his brow clearing. ‘I have heard of the dreadful spate of deaths at the University of Cambridge. It is truly terrible, and I regret that a young lady like yourself should be in any way concerned with such events. Yet I fail to see how I can possibly be of service to you.’

  ‘I have come to you directly from England,’ I told him, ‘because a man has been accused, wrongly accused of the murders, and I believe you and you alone hold the key to the truth.’

  ‘I?’ He was utterly taken aback, most completely amazed by my words. ‘But I cannot possibly have the slightest idea, Miss …’

  ‘Duncan …’

  ‘Miss Duncan, about the identity of the author of the terrible Cambridge murders!’

  ‘Professor Mittag-Leffler,’ I said to him, with all the earnestness I could muster, ‘you do not know, you cannot possibly know, how great a role was played by the n-body problem, and King Oscar’s Birthday Competition, in motivating the murders.’

  I saw that he became more and more amazed; he remained silent for a long time, and when he spoke, he seemed genuinely shocked and saddened by my words.

  ‘Who could have believed such a thing?’ he said softly. ‘If your words are true, I will regret having participated in organising the competition for the remainder of my life.’

  ‘No, please do not say that!’ I said. ‘No evil can be attributed to the existence of the competition. I have come to you, because as I said, I believe you may well have something which will provide the final proof against the murderer.’

  I saw that now, he began to understand.

  ‘Are you referring to the manuscripts submitted to the competition?’ he enquired directly. ‘Are you suggesting that one of them may contain the clue to which you refer?’

  ‘Precisely,’ I told him.

  He reflected for a moment. ‘The manuscripts are secret and anonymous,’ he observed.

  ‘Anonymous!’ This came as a startling surprise to me. ‘Anonymous! You mean you do not know the authors?’

  ‘No, I do not know the authors,’ he replied. ‘The rules stated that each manuscript should be accompanied only by an epigraph.’

  ‘Of course. I saw that in the announcement of the competition. But there were also the names in sealed envelopes marked with the epigraphs. I thought you would open them! Otherwise, how can you attribute a prize?’

  ‘The manuscripts will be read anonymously and judged on their merits,’ he said. ‘When the winning manuscript is selected, the envelope with the corresponding epigraph, and only that one, is to be opened by King Oscar himself, and the name of the author published.’

  ‘And the other names will never be revealed or known?’

  ‘Never. That would be against the rules decided on and approved by His Majesty.’

  My mind leapt and twisted, seeking some opening, some way of avoiding the obstacle which thus arose before me. I decided that he could come to no decision about the extent to which he must transgress the rules in the cause of justice, unless he knew something more of the situation.

  ‘It is my belief that a mathematician from Cambridge submitted a memoir to the competition, containing a complete solution to the n-body problem,’ I told him. His eyes flashed with a purely mathematical interest.

  ‘Really!’ he exclaimed. ‘This is a marvellous and unexpected development!’ But his face then darkened somewhat. ‘Yet something is wrong. I opened each and every one of the submitted memoirs today, in the presence of my colleague Edvard Phragmén, who is staying here, and I did not perceive any manuscript at all coming from England.’

  It was my turn to be taken aback. ‘But you must have!’ I said pleadingly. ‘I do not know where it was actually posted from, but I cannot believe it doesn’t exist. Are you sure it cannot have escaped your notice, buried within the large pile of manuscripts you ex
amined today?’

  ‘I have not examined such a very large pile,’ he replied, ‘there were but twelve in all. And not a single one in English.’

  ‘What languages are they written in?’ I enquired faintly, engulfed by a wave of dismay.

  ‘French, or German, or both,’ he answered.

  ‘Both?’

  ‘Yes, a couple of manuscripts arrived in a double version, in the two languages, written out in different hands.’

  A light began to shine within me.

  ‘Could not an English mathematician have had his manuscript translated into French and German and copied out by others, so as to hide his identity forever in the event of not winning the prize?’ I said.

  ‘Well, it is not impossible, of course,’ he answered thoughtfully.

  ‘I believe we may be able to tell, only from looking at the manuscripts, if they correspond to the memoir I mean,’ I told him. Feverishly, I set my valise flat upon the floor, unbuckled it and extracted the now much-fingered manuscript of Mr Beddoes, and from within its pages, the famous paper written by Mr Akers.

  ‘Please look at these,’ I told him. ‘They are rough forms of the complete solution of the n-body problem which I believe must be given in one of the memoirs you opened today. Surely, by examining each of the twelve submissions, you will be able to tell whether one of them contains the mathematics corresponding to what is written here.’

  He grasped the papers I held out to him, sat down abruptly in a comfortable armchair, and bent over them, concentrating intently, pushing up his small round spectacles, turning the pages, murmuring to himself. Mr Akers had been a disorderly man, but Mr Beddoes’ neat, regular handwriting was easy to follow, and I saw that the professor was fascinated by what he read, and that the ideas expressed there rang a bell within him, like the echoes of thoughts which he might have had but never did.

  We waited for some time in complete silence. Even Robert hardly moved, simply rolling his little train back and forth silently over the cloth-covered table, and lifting his large eyes occasionally to the illustrious Professor’s face. After ten or fifteen minutes, Professor Mittag-Leffler looked up from his reading, a surprised and confused expression on his face.

  ‘What I read here is truly remarkable,’ he said. ‘The manuscript contains the germs of at least two excellent ideas. I do not perceive any actual error in the reasoning lightly sketched here. And yet, my intuition tells me that such methods cannot, should not be able to provide the result. It seems incredible to me. But a mathematician’s intuition, while a splendid guide, should not be trusted absolutely; I have been surprised before. If this is a new example of such a surprise, it is a truly marvellous one, and will almost unquestionably win the competition.’

  ‘But, Professor, what I have shown you here is not a memoir submitted to the competition,’ I reminded him gently. ‘It is merely a brief sketch. It remains to see if the work was completed and submitted with full details.’

  ‘You are right,’ he said, ‘and we can examine the manuscripts and determine if that is the case quite quickly. Please allow me to invite you and your children to accompany me to my study.’ He looked at me briefly, and added ‘Although these cannot possibly be your children, my dear young lady. But I do not presume to ask why they have accompanied you here. Let us go.’

  We moved down a long and admirably decorated hallway, and encountering a maidservant, he spoke to her in Swedish.

  ‘I have asked her to call for Phragmén to come and join us,’ he told me. ‘I strongly wish to have his opinion on the manuscript which interests you.’ We arrived at the room which the professor called his office, although many more, if not most, of the rooms in this splendid villa were obviously devoted to the pursuit of mathematics. There, on his desk, neatly piled, lay the twelve memoirs he had opened on that day. Next to them lay a sheet on which he had carefully inscribed the title of each manuscript and the epigraph with which it was signed in lieu of a name.

  ‘I will tell you, in secret,’ he said with a slight smile, ‘that one of our candidates has, probably unwittingly, broken the rule, and sent a signed letter together with his epigraphed manuscript. However, had he not done so, I would have known him by his handwriting. It is number nine, the extraordinary Henri Poincaré,’ and he slipped one of the manuscripts out of the pile with a tender, caressing movement. ‘I do not need to have read it to know that it is bursting with the ideas of a genius,’ he said, his voice soft and vibrant with respect. He replaced the manuscript in its place, and extracted another from some way above it. ‘I believe there is some chance that the bilingual manuscript numbered seven may have a relation to the papers you have just shown me.’

  He took both the French and the German versions of the manuscript from the pile, and laid them before me. The titles were as follows:

  Über die Integration der Differentialgleichungen, welche die Bewegungen eines Systems von Punkten bestimmen,

  Sur I’intégration des équations différentielles qui déterminent les mouvements d’un système de points matériels,

  and the epigraphs read

  Nur schrittweise gelangt man zum Ziel.

  Pour parvenir au sommet, il faut marcher pas à pas.

  The professor took them up in his hands. ‘On the integration of the differential equations which determine the movements of a system of material points,’ he translated; ‘to rise to the top, one must advance step by step.’ He set Mr Beddoes’ notes upon the desk, open to the page which appeared to contain the central result, and laying the French manuscript upon the table next to it, he began to turn the pages slowly, looking over the statements and formulae and comparing the two manuscripts.

  ‘This is the one,’ he said, his voice vibrating somewhat with excitement and tension. ‘If you look here, you will see the key formula, and around it, the rest of the argument contained here. It is unmistakable.’

  I looked where he pointed, and immediately recognised the very formula, now so familiar, which appeared on the paper scribbled by Mr Akers at his very last dinner. The Professor continued to compare the two manuscripts, nodding his head and indicating to me the similarities.

  ‘The French manuscript is much longer, and contains many details and computations,’ he said. ‘Indeed, it hardly corresponds to the opinion expressed in the original announcement of the competition, that Mr Lejeune-Dirichlet’s proof, at least, was not based on long and complex calculations. Yet at the root of these calculations, there lies a stroke of genius, if the result is true.’

  Something in his tone caught my attention. ‘Do you doubt its validity?’ I asked him.

  ‘I … don’t … know,’ he answered, slowly. ‘I myself have thought long and hard about this very problem. As I told you before, I was absolutely convinced that such methods as those used here could have no chance of solving it. And yet, I desire only to be pleasantly surprised. The manuscript must be read and checked carefully in every detail. I myself will work on it, and my associates also.’

  In his deep passion and interest for the work at hand, Professor Mittag-Leffler had entirely forgotten that I myself was driven onwards by a very different question. I hardly dared to ask him something he had already told me was expressly forbidden, but one thought of Arthur, and the extreme danger he was running at that very moment, persuaded me.

  ‘Professor Mittag-Leffler,’ I began humbly, ‘I must ask you, I must beg you to open the sealed envelope which accompanied this manuscript. It is imperative to discover the author.’

  ‘It is impossible,’ he answered. ‘The King’s wishes cannot be lightly disdained. The sealed envelopes are to be handed to him personally for safe-keeping until his birthday, next January.’

  ‘January!’ I cried horrified. ‘It is far too late! A man’s life is at stake, Professor. He who has been accused of the Cambridge murders stands to lose his life – and he is innocent!’

  ‘And you believe that you know the author of this manuscript?’

  ‘I be
lieve it is one of two people,’ I told him. ‘I must know if I am right, and if so, which of them it is. The guilt or innocence of not one, but two people depend upon it.’

  ‘Can you not tell it by the handwriting, then?’ he asked.

  ‘I wish I could. But if he had his manuscript translated, and posted from Europe, then they would not be in his handwriting, would they?’

  ‘If he had them translated professionally,’ replied the professor, ‘then, although the languages themselves would be written correctly, the mathematics would probably be expressed in a somewhat peculiar manner, as the typical idiom is foreign to any but a mathematician.’

  He took up the two manuscripts, and began perusing them more closely.

  ‘It is hard to tell,’ he said, ‘for these two languages are not my own. But I do seem to detect some peculiar expressions in both languages. It is not absolutely impossible that they were translated from the English by someone with a perfect knowledge of the languages, but an imperfect one of the mathematical discourse. I cannot be absolutely sure.’

  At that moment, there came a discreet knock on the door, and a young man entered, wearing the selfsame earnest but ardent expression on his face that I was becoming used to seeing on those of my various mathematical acquaintances. The professor welcomed him, and introduced us to each other briefly. But the young Dr Phragmén had eyes only for the mathematics.

  ‘Are you looking at the manuscripts, Professor?’ he asked, his voice quite vibrating with eagerness. ‘Have you come across something particular?’

  ‘Indeed yes,’ cried the professor, thrusting the anonymous manuscript number seven in front of the face of his surprised associate. ‘Miss Duncan has called the central result of this paper to my attention, and I must say that at first sight it appears so astonishing as to be nearly unbelievable! Have a look at the main theorem. Why, this author claims to show a closed formula for the series in the case of the perturbative three-body problem, and deduces that the series describing the movements of the bodies must then converge!’