You just have to bear in mind that the unit of calculation is the square root of minus one.’

‘But that’s just it. It doesn’t exist. Any number, whether it’s positive or negative, gives a positive when it’s squared. So there can’t be such a thing as a real number that’s the square root of something negative.’

‘Quite right; but why shouldn’t you try to apply the operation of square root calculation to a negative number anyway? It can’t produce a real value, of course, and that’s why the result is called imaginary. It’s as if you were to say: someone always used to sit here, so let’s put out a chair for him today; and, even if he’s died in the meantime, let’s act as though he was going to turn up.’

‘But how can you do that if you know for certain, mathematically for certain, that it’s impossible?’

‘You do it anyway, as if it wasn’t the case. You’ll have some degree of success. In what way are the so-called irrational numbers different? A division that never comes to an end, a fraction whose value never, ever comes out how ever long you spend on the calculation? And what do you think about parallel lines meeting in infinity? I reckon that if you were too conscientious there would be no maths at all.’

‘You’re right about that. If you think of it that way it’s strange enough. But the curious thing is that, in spite of this, you really can calculate with such imaginary or otherwise impossible values, and you have a tangible result in the end!’

‘Well, in order for that to happen, the imaginary factors must cancel one another out in the course of the calculation.’

‘Yes, yes; I know what you’re saying. But isn’t there still something very strange about it all? How should I put it? Just think about it for a moment: in that kind of calculation you have very solid figures at the beginning, which can represent metres or weights or something similarly tangible, and which are at least real numbers. And there are real numbers at the end of the calculation as well. But they’re connected to one another by something that doesn’t exist. Isn’t that like a bridge consisting only of the first and last pillars, and yet you walk over it as securely as though it was all there? For me there’s something dizzying about a calculation like that; as if it goes off God knows where for part of the way. But the really uncanny thing about it is the strength that exists in such a calculation, holding you so firmly that you land safely in the end.’

Beineberg grinned: ‘You’re starting to sound like our chaplain: “... You see an apple - that’s light vibrations and the eye and so on - and you reach out your hand to steal it - that’s the muscles and the nerves that set it in motion. - But between the two there is something that produces the one from the other - and that is the immortal soul, which has sinned in the process ... yes - yes - none of your actions is explicable without the soul, which plays upon you as it might on the keys of a piano ... !”’ And he imitated the intonation with which the cleric liked to deliver that old simile. - ‘By the way, the whole business doesn’t interest me that much.’

‘I thought it would interest you, of all people. At least, I thought of you straight away because - if it really is so inexplicable — it’s almost a confirmation of your belief.’

‘Why shouldn’t it be inexplicable? I think it’s entirely possible that the inventors of mathematics were stumbling over their own feet when they were doing this. Because why should whatever lies outside our intellect not be allowed a bit of fun with that same intellect? I’m not going to go into that, though, because these things lead nowhere.’

 

The same day Törless had asked his maths master if he could visit him to have some passages from his last lecture explained.

The next day, at lunchtime, he climbed the stairs to the master’s little apartment.

He now had a quite new respect for mathematics, since it seemed all of a sudden to have turned unexpectedly from a dead chore into something very much alive. And because of that respect he felt a kind of envy for his teacher, who must surely be familiar with all those relationships, and who carried his knowledge of them around with him wherever he went, like the key to a locked garden. Apart from this, though, Törless was also spurred on by a rather hesitant curiosity. He had never been in the room of an adult young man, and he was excited to learn how the life of another, knowledgeable and yet settled person might look, at least as far as one could tell from his outward surroundings.

He was normally shy and reticent with his teachers, and thought that for this reason he did not enjoy their particular affection. So his request seemed to him, as he held his breath with excitement outside the door, to be quite daring, less concerned with receiving enlightenment - because even now he quietly doubted that he would receive it — than with the opportunity to cast a glance behind the teacher, so to speak, and into his daily cohabitation with mathematics.

He was led into the study. It was a long room with a single window; a desk scattered with ink stains stood close to the window, and by the wall there was a sofa covered with a green ribbed fabric, scratchy and tasselled. Above the sofa hung a faded mortarboard and a number of brown, darkened photographs, in visiting-card format, from university days. On the oval table with its x-shaped feet, whose supposedly graceful flourishes looked like an unhappy attempt at elegance, lay a pipe and some coarse, leafy shag. The whole room smelled of cheap tobacco.

Törless had barely absorbed these impressions and become aware of a certain unease in himself, as though he was touching something disagreeable, when his teacher entered the room.

He was a young man of thirty at most, fair-haired and nervous and a very capable mathematician, who had already delivered several important treatises to the Academy.

He immediately sat down at his desk, rummaged around for a moment in the papers scattered around the place (it later occurred to Törless that he had immediately sought refuge in them), cleaned his pince-nez with his handkerchief, laid one leg over the other and looked expectantly at Törless.

Törless had now started to look at him as well. He noticed a pair of rough, white woollen socks and registered that above them the rims of the teacher’s long Johns had been blackened by boot polish.

On the other hand his handkerchief looked white and elegant, and his tie was stitched together, but made up for this by being splendidly gaudy and chequered like a painter’s palette.

Instinctively, Törless felt further repelled by these little observations; he could hardly hope that such a person might really possess any significant knowledge, when quite clearly none of it could be detected in his personal appearance or anywhere in his surroundings.