The Three-Body Problem Page 4

  ‘Thanks to Newton and his Law, we know the answer to that; it is because the pull of gravity is exactly proportional to the inverse square of the distance between the bodies. But don’t trouble your head with that – suffice it to say that thanks to this, we on Earth do not fall to the sun, nor does the moon fall upon us! At any rate, before Newton, Kepler determined the form of the orbit and found that it is an ellipse rather than a perfect circle, and will continue endlessly in the same manner. That is the two-body problem, with a large and a small body. Now, suppose you have the sun and two planets. That is a three-body problem, in the rather special case which really does occur in our solar system, where one is extremely large and two of them are relatively small. What do you think would happen?’

  ‘Well, wouldn’t the two planets just go on orbiting in ellipses around the sun, as they do in our solar system?’ said Emily.

  ‘You are almost correct! But not quite,’ cried her uncle, ‘because you imagine that each of your two little planets has a relation of gravity only with the sun, and acts exactly as though it were alone with the sun, and you forget the tiny influence of each planet upon the other! Small though they are, they pull about on each other and cause tiny distortions in the shape of the ellipses, and it becomes nearly impossible to find what the exact nature of the paths they trace will be as time goes on. You see – take this little planet going around this star. If the other planet wasn’t there, it would go like this: round and round in a stable ellipse, for ever. But now add the other planet. What happens is that when the first planet goes once around the star, its ellipse is deformed a tiny bit by the influence of the gravity of the other planet, so that it doesn’t quite, quite get back to where it started from. The difference is minuscule – if we were talking about the influence of the other planets in our solar system on the Earth, why we make an ellipse around the sun once a year exactly, and the deformation is probably a matter of a few inches or so. Now, the planet will orbit around the star again, in an ellipse very similar to the old one, but not quite identical. And again, it won’t come back exactly to its starting point. This will keep happening and happening, so that instead of getting one neatly drawn ellipse again and again, you get a spiral of ellipses, each one a little different from the previous one.’

  The marble in Mr Morrison’s hand began to move around the ball in a spiral which grew progressively more distorted and wild.

  ‘And the question is,’ he continued eagerly, ‘what if, due to the tiny deformations of the ellipses over time, they finally end up spiralling away like mad things, and perhaps even breaking loose altogether and hurtling off uncontrolled into space! It must happen eventually – even in our very own solar system! No, don’t bother to look worried – the calculations show that it isn’t going to happen for a very, very great many years. So you have plenty of time to study mathematics and learn about the n-body problem.’

  ‘So that is the influence the planets have on each other,’ I remarked thoughtfully. ‘It really seems to describe the way in which the relations between human beings come to distort the direct and pure relations between each individual and the Divine.’

  ‘It does, awfully!’ he answered. ‘Well said. Now that you mention it, I seem to know rather a lot of people who are in the process of drifting away from their research – which I suppose could be considered the mathematician’s relationship with the Divine – for reasons of jealousy and resentment, or such. Mathematicians do tend to go a little mad sometimes. Perhaps it’s a result of all that concentration.’ He took up the journal, and continued his translation where he had left off.

  This problem, whose solution would considerably extend our knowledge with respect to the system of the world – this odd expression is French for the solar system, with the sun and the planets – appears to be solvable using the analytic methods we already have at our disposal; at least we may suppose this, since LEJEUNE-DIRICHLET communicated, shortly before his death, to a geometer amongst his friends, that he had discovered a method to integrate the differential equations of mechanics, and applying this method, he had succeeded in giving an absolutely rigorous proof of the stability of our planetary system. Unfortunately, we know nothing of this method, unless it is that the theory of infinitely small oscillations appears to have served as the starting point for his discovery. We may, however, assume almost with certainty that this method was not based on long and complicated computations, but on the development of one simple fundamental idea, which we may reasonably hope may be rediscovered by dint of deep and persevering work. In the case, however, where the proposed problem cannot be solved before the date of the competition, the prize could be attributed in recompense for work in which some other problem in mechanics was treated in the indicated manner and completely solved.

  ‘Ah, now here is an interesting thing,’ Mr Morrison told us. ‘Dirichlet told a mysterious friend that he had solved a fundamental problem, and thereupon immediately died, nearly thirty years ago, leaving no clue as to his method. How thoughtless of him.’

  ‘Could we not identify the friend?’ I began hopefully.

  ‘Oh, actually, he has been identified; indeed, he has identified himself, and quite arrogantly at that. It is the German mathematician Kronecker, a great rival of Weierstrass. He claims that what Dirichlet told him is ill-represented in this paragraph. But for all that he says, he clearly has no idea or no clear memory of what Dirichlet’s method might have been. He is more likely to be protesting out of anger that he was not included in the commission. Now come the three other problems set for the competition, but they are less interesting.’

  2. Mr FUCHS proved in several of his memoirs that there exist uniform functions of two variables, which are connected, by the way in which they are generated, to ultraelliptic functions, but are more general than these, and which could probably acquire great importance in analysis, if their theory were developed further.

  We propose to obtain, in explicit form, the functions whose existence has been proved by Mr FUCHS, in a sufficiently general case, so that one can recognise and study their most essential properties.

  3. The study of functions defined by a sufficiently general differential equation of the first order whose first term is an integral and rational polynomial with respect to the variable, the function and its first derivative.

  MM. Briot and Bouquet opened the way to such a study in their memoir on this subject (Journal de l’École Polytechnique, cahier 36, pag. 133–198). The geometers familiar with the results discovered by these authors also know that their work is far from having exhausted the difficult and important subject which they were the first to investigate. It appears probable that new research undertaken in the same direction could lead to propositions of great interest to analysis.

  4. We know what light was shed on the general theory of algebraic equations by the study of those special equations arising from the division of the circle into equal parts, and the division by an integer of the argument of the elliptic functions. The remarkable transcendental number obtained by expressing the module of the theory of elliptic functions by the quotient of the periods similarly leads to the modular equations, which have been the source of absolutely new notions, and to results of great importance such as the solution of the fifth degree equation. But this transcendental number is just the first term, the simplest special case of an infinite series of new functions which M. POINCARÉ has introduced to science under the name of Fuchsian functions, and applied with success to the integration of linear differential equations of arbitrary order. These functions, which play a role of manifest importance in Analysis, have not yet been considered from the point of view of algebra, as the transcendental associated to the theory of elliptic functions, of which they are the generalisation. We propose to fill this lacuna and to obtain new equations analogous to modular equations, by studying, even in just a special case, the formation and the properties of the algebraic relations relating two Fuchsian functions when they have a com
mon group.

  ‘For those in the know,’ Mr Morrison told us, ‘this Henri Poincaré is considered sure to win the competition. He is a kind of genius, and all of his work is exactly round about the questions proposed above, all four of them, really; he might try his hand at whichever he pleases. He is much admired in Sweden, look – he has published two articles in this very volume. He was formerly a student of Hermite, the commissioner from Paris.’ And he turned some pages, and showed me the first mathematical article in the volume, written in French by this very H. Poincaré, and whose title was precisely ‘On a theorem of Mr Fuchs’, before concluding his translation of the announcement.

  In the case where none of the memoirs presented for the competition on one of the proposed subjects would be found worthy of the prize, this can be attributed to a memoir submitted to the competition, which contains a complete solution of an important question of the theory of functions, which is not one of those proposed by the commission.

  The memoirs submitted to the competition must be equipped with an epigraph and with the name and address of the author in a sealed envelope addressed to the Chief Editor of Acta Mathematica before the 1st of June 1888.

  The memoir for which HIS MAJESTY will deign to attribute the prize, as well as that or those memoirs which the commission will consider worthy of an honourable mention, will be inserted into Acta Mathematica, and none of them must be published beforehand.

  The memoirs may be redacted in whatever language the author wishes, but as the members of the commission belong to three different countries, the author must provide a French translation together with his original manuscript if the memoir is not already written in French. If no such translation is included, the author must accept that the commission has one made for its own use.

  The Chief Editor.

  ‘The 1st of June, why that is in just three months!’ remarked Emily. ‘Are you submitting a memoir to the competition, Uncle Charles?’

  ‘I? Why no, absolutely not,’ he exclaimed. ‘I know next to nothing about the questions posed here. There are not many people in England nowadays who would be capable of seriously solving them, although if there are any at all, they would be right here in Cambridge.’

  ‘Really? Do you know them? Are they doing it?’ she asked.

  ‘No one has actually declared that he is setting about it,’ he said thoughtfully. ‘But you know, that doesn’t prove much, does it? After all, it’s quite imaginable that a person might keep the whole thing silent, to avoid embarrassment in case of failure, and yet submit a secret manuscript nonetheless.’

  ‘And if someone were doing it secretly, who could it be?’

  ‘Strike me pink if that isn’t exactly what Akers was thinking of doing,’ he said suddenly. ‘Why yes, wasn’t my friend Weatherburn telling us that Akers had said he had found a solution on the night before he died? The poor fellow, what hard luck for him – perhaps he was finally on the brink of the fame and recognition he always dreamt of.’

  ‘Did he, poor man? And why didn’t he get it?’

  ‘Oh, Akers was a good mathematician, he had a good, swift brain, but he lacked something which could have made him really great. He didn’t have a fundamental grasp of the larger nature of things. It was as though you put him in front of a puzzle, Emily, and he would grasp two pieces and try to put them together, and if they didn’t fit, he would try another and yet another, very quickly and with a sharp eye, so that eventually he would put quite a number together, and yet somehow he would have no idea of what the picture puzzle was actually showing. It’s difficult to explain.’

  ‘But do you think he might have found a solution to the first problem anyway?’

  ‘Why not? Perhaps it was there for anyone to see, and just needed a sudden, blinding vision to find it. Perhaps he found it by dint of “deep and persevering work”, or even by doing the kind of “long and complicated calculations” he was not supposed to need! Unless he wrote something down, we shall never know now; why it’s just as bad as Dirichlet.’

  ‘But he did write something,’ I observed. ‘He had a paper with a formula in his breast pocket, and he even told Mr Weatherburn that he had written a rough manuscript!’

  ‘Oh! Has anybody searched through the papers he left in his rooms?’ Emily squeaked, jumping up and down eagerly.

  ‘Oh yes, naturally, his notes and papers have been gone through carefully and inspected and organised, by the police, and also by mathematicians, I should think. Nobody appears to have found anything like a manuscript containing a complete solution to the n-body problem – if they had, we would certainly know about it by now.’

  ‘Perhaps he already sent it in?’

  ‘Unlikely, if he told Weatherburn the day before his death that it was just a rough manuscript – and he would have had plenty of time before the 1st of June to better it.’

  ‘If only we had the breast-pocket paper he showed Mr Weatherburn,’ I put in, ‘surely it would help!’

  He looked at me musingly. ‘You are right, really, Miss Duncan. Imagine – what you are saying might actually turn out to be very important! The personal effects he had on him at his death are probably still in the police station, as his death is still being investigated. I wonder if they have found that paper; I wonder if anybody has been to ask. I shall go down to the station tomorrow and enquire about it myself.’

  ‘Oh, how exciting,’ cried Emily. ‘Imagine if you find it – then you could solve the problem and win the prize, and you could give the medal to Miss Duncan as a gift.’

  ‘Emily!’ Mr Morrison was quite shocked. ‘One doesn’t steal other people’s ideas.’

  ‘Really? Can ideas be stolen?’ she responded in surprise.

  ‘Oh yes, ideas are more valuable than property for a mathematician. He would far prefer to lose his money or belongings than his ideas.’

  ‘Well, but here it would be from a man who’s dead!’

  ‘You may steal from a man’s memory, Emily,’ he answered. He looked intensely serious, and I felt deeply impressed. I will not soon forget how he spoke.

  For men like these, ideas seem greater, more real, more meaningful, more desirable, than all the treasures which have made men dream since the beginning of time. It is a moving thing.

  I send a great many kisses to everyone,

  Your loving

  Vanessa

  Cambridge, Tuesday, March 20th, 1888

  Dear Dora,

  The delight at receiving, finally, a long letter from you almost outweighed my feelings at your sad news. So Mr Edwards is leaving for India. No wonder after he learnt it, he hesitated for so long to come and see you. It must have been difficult for him to face the necessity of giving you news that he knew must grieve you. And thus, he himself is reluctant to go, and disappointed in himself for not having succeeded more brilliantly in his studies, thereby leaving only this option open to himself. Oh, Dora … many ladies marry civil servants and join them in India. It is quite frequent, so you must not think that everything is necessarily over. But I understand that you could not even think of such a thing now, when you still know him so little. You would have needed a long courtship, as anyone would, and now you will have only letters. Surely you will soon be one of the most written-to creatures in the country! And you will have his leaves to look forward to. I dearly hope that you will find that knowing what happened, however sad, is far better than not knowing, and your taste for life will return with the springtime.

  I was very much hoping, I must confess, to have some exciting news to be able to continue my mathematical tale; I awaited the results of Mr Morrison’s visit to the police with great interest. But alas, Emily has told me that he returned empty-handed, as all of Mr Akers’ personal effects have already been transferred to his closest kin, who is a woman living somewhere on the Continent. The police showed Mr Morrison a list, and it seems that not only was there a paper in his pockets, but even a great many bits of paper, all covered with mathematical scribbling, a
s well as the usual assortment of keys, coins, diary and so on. At any rate, it is all gone now.

  Emily also showed me a newspaper clipping from a few days ago, which her uncle gave her; I had not seen it.

  DEATH OF MATHEMATICIAN REMAINS MYSTERIOUS

  The murder of Dr Geoffrey Akers, Fellow of St John’s College in Pure Mathematics, remains unexplained. The police have but a single, seemingly inexplicable clue. Upon asking themselves whether the murderer may not have been a thief, they examined his rooms completely to see if anything had been taken. The investigation apparently proved inconclusive. The room showed signs of having been thoroughly searched; it was very disarranged, and the drawers were standing open, but nothing of value had disappeared. ‘’E probably made the mess himself, ’e did,’ said Mrs Wiggins, the bedder. ‘It could ’ardly be messier than it already was. Nought but dirty papers and cigar ends. Mr Akers had nothing of value in his rooms anyway, unless they wanted to steal his old clothes.’ It is conceivable that the murder could have been perpetrated by a disappointed thief, who had been hoping for better.

  So, so! This may explain the absence of any manuscript solving the n-body problem. As for the famous bit of paper, either it has been sent off to his next of kin, or else he may have simply thrown it away … or else … the gruesome thought cannot be avoided … perhaps the very selfsame person who struck the dreadful blow then slipped his hand slyly into the dead man’s pocket, searching there …

  Oh, Dora, what am I saying? It sounds as though – instead of a mere thief – the murderer could be a mathematician, killing in order to steal the idea? That very idea which Mr Morrison spoke of as being more valuable than money or belongings?

  What a terrible train of thought! And yet, the more I consider it, the more I feel that it must be so. He was killed in his rooms at the university. Why should a stranger have made his way there? Oh dear. I wish I could speak to someone about this. Next Thursday I return to tea at Emily’s. Perhaps Mr Morrison will step upstairs. Dare I ask him what he thinks? But what if it should be he? No, this is ridiculous. I must stop!