The Three-Body Problem (25 page)

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 believe 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!’

‘What?’ responded the clearly astounded Doctor. ‘A complete solution to the perturbative three-body problem? But this is more than we dared hope for in the best of cases!’

His amazement and rejoicing were such that I could not remain quiet, although it would certainly have been more seemly to do so.

‘Is it so very important, then? What has he proven?’ I asked.

‘Oh, yes, it is of capital importance!’ cried the enthusiastic young doctor, stabbing at the famous formula with his finger. ‘He has given a formula for the mysterious series in terms of known analytic functions, and deduced from this that the classical series describing the motion of the bodies converges, that is, has a real value at any given time, rather than a meaningless, infinite value. That means that in this case, what we call the perturbative three-body problem, that is where one of the bodies is very large compared to the other two, like a star and two planets – our own Earth and Jupiter, as it may be – one is able to predict the orbits of the planets, instead of having no idea whether they will not end up by drifting away through space.’

‘Good heavens,’ I exclaimed. ‘I thought it was well known
that the Earth orbits regularly about the sun. You don’t mean that without the solution given here, we might have to fear its departing at any moment?’

‘Well, no, the nature of the series does tell us that the Earth will certainly continue its orbit for many years yet – but not so very many! We have no guarantee that in a million years it will still be doing the same!’

‘Oh,’ I said with a tinge of disappointment. It is perhaps natural for citizens of a country as stable and peaceful as Sweden to feel threatened by the prospect of turmoil a million years from now, but I myself was concerned with far more immediate circumstances. I still wished nothing but to know the name of the author of the fated manuscript. Yet I feared to insist upon seeing it, for I was afraid to hear a reiteration of the professor’s previous refusal. My mind was seared by the image of Arthur, waiting in the dock, silent and withdrawn, scarcely interested in the battle over his destiny waged around him by lawyers, judge and jury – the outcome of which could – would, probably, send him to his death – and the professor was thinking about planets! I tried to speak, to tell him what was in my mind, but tears collected in my eyes, and spilt over. Perceiving them, Professor Mittag-Leffler immediately became distressed. He took several rapid steps about the room, thinking intensely.

‘I know what worries you, Miss Duncan,’ he said. ‘And yet – I cannot do what you wish. But wait! Do not give way to despair. There may be a solution.’

‘Please tell me what it is,’ I begged him, trying in vain to
control my voice which wavered desperately, while Emily and Robert approached me and wrapped their arms around me tightly, looking at the professor with their large eyes full of severity and distrust, like little wild cubs in a lair, suspecting the creature creeping about outside of being a threatening predator.

‘I see only one thing to do,’ he said in measured tones. ‘We cannot open the envelopes because the King has forbidden it. The only one who can go against these orders is the King himself. We must petition him with our request.’

‘Shall we see the King?’ asked Emily with breathless respect. For myself, I felt as though my very heart was pressed in from all directions with fear. I imagined that the King would certainly refuse a request so puny, as compared with his royal concerns. Even more, I feared that we would have to endure a great delay, while our petition was made with all the proper ceremony.

‘The matter is desperately urgent,’ I told the professor. ‘The trial has been going on for two weeks already, and the judgement may be pronounced at any moment, any moment at all. It may even have happened today, for aught I know. There is no time to lose.’

‘My relations with the King are close,’ he said. ‘I will send a messenger to the Palace now, with a message to be delivered as soon as he rises tomorrow. I will express the urgency of the situation, and we will go to the Palace immediately tomorrow morning, so as to be already present, should he send for us. If all goes as you wish it, and as I wish it also, I do not hesitate to say, I myself will
provide you with conveyance to the station, and with the tickets you need to travel home again. This seems but a minor service which I can render to justice, in the name of mathematics. Let me now have you shown to bedrooms for the night. I beg you will repose yourself as much as possible; I will have you called at six o’clock, that we may be ready for every eventuality.’

I saw that he understood my feelings, and that there was no need for me to attempt to express them; I saw, also, that he was doing everything that he possibly could to aid me, and that going against the express wishes of the King, in however trivial a matter, appeared absolutely impossible to him – even now, even when it concerned a question of life or death! I pressed my teeth together, not to allow my anguished impatience to burst out (oh, the idea that the very envelope I so desired to open lay within the very house, and we could have seen inside it in a moment! How I longed, but did not dare, to suggest steaming it secretly open over the kettle and then sealing it up again … ) and thanked him with as much calmness as I could muster. He led us ceremoniously and kindly to the hall, and summoned the buxom lady who had bade us enter earlier on, and spoke to her in Swedish; she showed us to the beautifully furnished adjacent guest rooms where we are now. Taking charge of the children, she swept them away, pausing only to say ‘bad, bad’ to me with a motherly smile, from which I concluded, not that the children were misbehaving, but that they were to be washed. She took charge of my valise, also, and I undressed and fell into bed. But thoughts whirled too strongly in my brain, and
I finally admitted to myself, Dora dear, that I should not find sleep before I had committed everything to paper, for it has become such a habit with me, during these long, dreadful weeks, that I can no longer do without it, and it somehow relieves my anguish and momentarily restores hope to me. Now that I have given you a complete account of the events of this crucial day, I shall return to bed, and try to sleep, and not to think too much about the fact that perhaps, tomorrow morning, I shall find myself pleading for Arthur, no longer with charwomen, children, policemen, lawyers and mathematicians, but with a king!

Please pray for me, as always

Vanessa

Oh, my dear Dora,

What a day this has been! I have learnt much, and reality has superseded my foolish imaginings about royalty.

As Professor Mittag-Leffler had promised, we were called at six o’clock. I was brought tea in bed, and then shown to a large bathroom wherein a steaming bath had been placed, together with large towels and every luxury. I made a detailed toilette, for I detected more than mere politeness in such gracious treatment; I understood that it also contained a component of careful planning in view of our royal reception, in which I admit I could still hardly believe.

When I took up my grey dress, I saw that it had been steamed and pressed during the night. Once I had put it on,
however, I still hesitated to descend, for my hair was wet and I could not do it up. But the kind housekeeper soon reappeared, and towelled my hair kindly, and fluffed it with her fingers, and brushed it with a brush, and bade me with gestures come down to breakfast, and that she would take care of my hair later. It was already beginning to dry and wave thickly over my shoulders, and I felt a little ashamed, as though I were descending in negligee, but I must needs go, so I went.

I was delighted to see Emily and Robert already installed at the well-laden table before me, happily eating toast with jam, large aprons wrapped around them, laughing together, although their eyes were still small and their cheeks rosy with sleep. The Professor was conversing with them most cheerfully, and he bade me join the meal, and in his kindness and understanding, which I shall never forget, immediately addressed my deepest concerns.

‘It is now seven o’clock,’ he told me, looking at a beautiful silver watch he extracted from his pocket, ‘the King will receive the message in one hour. By then, we shall already be in the Palace, and his response will be conveyed to us immediately.’

He paused to pass me the various crystal pots and covered dishes and to see that I took a sufficient quantity of each, and then continued.

‘I have made the acquaintance of these two delightful children, and am now much more familiar with the full circumstances of your journey and your double quest. I am filled with admiration, and wish to support you in every
possible way, for I perceive that you are moved to very daring acts by the simple perception of injustice.’