The Three-Body Problem (4 page)

And who should come into the nursery just at that very moment, as we were finishing our tea, but Mr Morrison! He sat down on a low stool, stretching his legs out in front of him, and said we seemed to be having a far better time than the grown-ups at proper tea down below, and that he’d be dashed if he didn’t prefer to stay with us. Emily played the fool, teasing him in all kinds of ways and saying he would do nothing of the kind, until he laid a bet with her that he would not only come along next time she had a tea party, but bring his friends. Dear me – I do hope he was merely joking.

‘I would be so grateful,’ I asked him, ‘if you would tell me something about the Birthday Competition I heard mentioned the other evening. Was it not a celebration of some king’s birthday? What king could possibly wish to celebrate his birthday with mathematics?’

‘Why, by all means,’ he answered eagerly. ‘Our benefactor
is King Oscar II of Sweden, of the Bernadotte family. He studied mathematics in depth while at the University of Uppsala, and has a great fondness for the subject, as well as a close friendship with Sweden’s leading mathematician, Gösta Mittag-Leffler. The Birthday Competition was his own suggestion, and I believe that rather than using mathematics to celebrate his own anniversary, he entertains the hope that the illustrious date, surely to be accompanied by pomp and festivities of all sorts, may shed some glory onto at least one member of the horde of unknown but devoted researchers scattered all about Europe, and illuminate the one and only Swedish mathematical journal. Furthermore, in posing a specific problem as the subject of the competition papers, he hopes to motivate such work as may possibly produce a solution.’

‘And is it possible to tell me what the subject of the competition is?’

‘Why, I have a couple of volumes of
Acta Mathematica
in my room below,’ he said, ‘the announcement of the competition appeared there some two or three years ago, and I may well have it.’ And ignoring my remonstrances and expostulations that he not disturb himself, he sped away to investigate, and soon came back with the volume in hand, to show me a page so unintelligible to me, that he might not have taken the trouble! To begin with, not only the announcement of the competition, but the entire volume, is written partly in French and partly in German, without a single English word between its two covers. The volume begins with the announcement of the competition,
written in columns with the left-hand one German and the right-hand one in French – it looks rather as though French were a briefer language than German, with great gaps between the French paragraphs, so as to make them begin at the same levels as the corresponding German ones. I could not understand much of it, although many of the French words such as ‘
anniversaire
’ and ‘
mathématiques
’ certainly do appear familiar. But Mr Morrison sat himself at Emily’s school desk, and taking her quill in hand and drawing a bit of paper towards him, began to translate it for me, transferring his gaze from the page to his writing to my face, and interspersing his translation with all manner of interesting remarks and explanations, so that I could not be bored for a single moment, although the text is not only quite long but almost impossible to comprehend without the aid of friendly explanations!

‘Must all mathematicians, then,’ I enquired, ‘necessarily be familiar with the French and German languages?’ Emily had crept near us and was listening with interest as her uncle translated.

‘Oh,’ he responded with a slight blush, ‘not to speak, so to speak. We merely need to read the languages, and even then, only to read mathematics in the languages. It is far easier than trying to read a novel; why, at worst, we need only look as far as the next formula, which is written in a kind of international language, equally intelligible to everybody.’ And he showed me a formula on the first page of the article following the announcement of the competition, which said something to the effect that
x dx
+
y dy
= 0 has for general integral
y
=

‘Weierstrass is really the most venerable and famous of German mathematicians of today,’ Mr Morrison told us, ‘the father of them all, in some way, something like Professor Cayley here in Cambridge, who should have been on the commission, they say, had an Englishman been included at all. Do you know, Miss Duncan, that Mr Weierstrass is quite famous for having produced not only mathematical ‘sons’, but also a ‘daughter’? Yes, the famous Sonya Kovalevskaya was his student, she who two years ago won the Bordin Prize from the French Academy of Sciences with a paper so impressive that they doubled the prize money to recompense it as it deserved. She now holds an extraordinary professorship in Stockholm, is an editor of the very journal I am holding in my hands, and advises Mittag-Leffler, I believe, on the organisation of the Birthday Competition.’

I was amazed. Germany and Sweden appear to be countries with wonderful ideas about ladies who wish to study, and England appears to lag far behind (particularly if one judges by the ideas expressed in
The Monthly Packet,
which greatly stress the value of obedience and docility). I wonder if I shall ever have the good fortune to visit such a country.

‘What on earth does all that mean?’ asked Emily curiously.

‘Well, let me show you,’ said her uncle, warming to his task of explanation. He cast about her schoolroom, and moving to the shelves where various toys were gathered, took up a ball and a box of marbles and sat with them upon the floor.

‘Now,’ he told her, ‘you know what gravity is, don’t you? You know that objects fall to the floor because they are attracted by the gravitational force of the Earth, which is very large compared to themselves.’

‘Well,’ she observed guardedly, ‘I know that an apple fell upon Newton’s head.’

He shouted with laughter. ‘Indeed, nobody can grow up in Cambridge without knowing that! And there is truth to it, you know, and to the notion that that event sparked the whole theory of gravity in his brilliant mind. Ah, he was
our great genius, unequalled in the last hundred and fifty years. He understood that if you have a giant body, like the sun, for instance,’ and he set the ball upon the floor, ‘and a smaller body moving near it,’ he suited the action to the word with a marble, ‘it will enter the sphere of the sun’s gravity and begin to orbit around and around, unable to escape the power of the sun.’

‘Why does it not fall to the sun, just as a marble falls to the earth?’ asked Emily in surprise. ‘A marble does not orbit – how queer that would be, to see it flying around and around.’

‘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.