'Here? The place is practically elvish!' said Ponder. 'The Archchancellor was right.'

All three explorers stood quietly. The Librarian's nostrils flared. Rincewind sniffed, very cautiously.

'Seems okay to me,' he said, at last.

Then a man in black entered the room. He came in quickly, opening the door no more than necessary, in a kind of aggressive sidle, and stopped in astonishment. Then his hand flew to his belt and he drew a thin, businesslike sword.

He saw the Librarian. He stopped. And then it was really all over, because the Librarian could unfold his arm very fast and, importantly, there was a fist like a sledgehammer on the end of it.

As the dark figure slid down the wall, the crystal sphere in Rincewind's hand said: 'I believe I now have enough information. I advise departure from this place at a convenient opportunity and in any case before this gentleman awakes.' Hex?' said Ponder.

'Yes. Let me repeat my advice. Lack of absence from this place will undoubtedly result in metal entering the body.'

But you're talking via a crystal ball! Magic doesn't work here!'

'Don't argue with a voice saying "run away"!' said Rincewind. 'That's good advice! You don't question it! Let's get out of here!'

He looked at the Librarian, who was sniffing along the bookshelves with a puzzled expression.

Rincewind had a sense for the universe's tendency to go wrong. He didn't leap to conclusions, he plunged headlong towards them.

'You've brought us out through a one-way door, haven't you ...' he said.

'Oook!'

'Well, how long will it take to find the way in?'

The Librarian shrugged and returned his attention to the shelves.

'Leave now,' said the crystal Hex. 'Return later. The owner of the house will be useful. But leave before Sir Francis Walsingham wakes up, because otherwise he will kill you. Steal his purse from him first. You will need money. For one thing, you will need to pay someone to give the Librarian a shave.'

'Oook?'

The concept of L-space, short for 'Library-space', occurs in several of the Discworld novels. An early example occurs in Lords and Ladies, a story that is mostly about elvish evil. We are told that Ponder Stibbons is Reader in Invisible Writings, and this phrase deserves (and gets) an explanation: The study of invisible writings was a new discipline made available by the discovery of the bidirectional nature of Library-space. The thaumic mathematics are complex, but boil down to the fact that all books, everywhere, affect all other books. This is obvious: books inspire other books written in the future, and cite books written in the past. But the General Theory12 of L-space suggests that, in that case, the contents of books as yet unwritten can be deduced from books now in existence.

L-space is a typical example of the Discworld habit of taking a metaphorical concept and making it real. The concept here is known as 'phase space', and it was introduced by the French mathematician Henri Poincare about a hundred years ago to open up the possibility of applying geometrical reasoning to dynamics. Poincare's metaphor has now invaded the whole of science, if not beyond, and we will make good use of it in our discussion of the role of narrativium in evolution of the mind.

Poincare was the archetypal absent-minded academic -no, come to think of it he was 'presentminded somewhere else', namely in his mathematics, and it's easy to understand why. He was probably the most naturally gifted mathematician of the nineteenth century. If you had a mind like his, you'd spend most of your time somewhere else too, revelling in the beauty of the mathiverse.

Poincare ranged over almost all of mathematics, and he wrote several best-selling popular science books, too. In one piece of research which single-handedly created a new 'qualitative'

way of thinking about dynamics, he pointed out that when you are studying some physical system that can exist in a variety of different states, then it may be a good idea to consider the states that it could be in, but isn’t as well as the particular state in which it is. By doing that, you set up a context that lets you understand what the system is doing, and why. This context is the

'phase space' of the system. Each possible state can be thought of as a point in that phase space.

As time passes, the state changes, so this representative point traces out a curve, the trajectory of the system. The rule that determines the successive steps in the trajectory is the dynamic of the system. In most areas of physics, the dynamic is completely determined, once and for all, but we can extend, this terminology to cases where the rule involves possible choices. A good example is a game. Now the phase space is the space of possible positions, the dynamic is the rules of the game and a trajectory is a legal sequence of moves by the players.

The formal setting and terminology for phase spaces is not as important, for us, as the viewpoint that they encourage. For example, you might wonder why the surface of a pool of water, in the absence of' wind or other disturbances, is flat. It just sits there, flat; it isn't even doing anything.

But you start to make progress immediately if you ask the question 'what would happen if it wasn't flat?' For instance, why can't the water be piled up into a hump in the middle of the pond?

Well, imagine that it was. Imagine that you can control the position of every molecule of water, and that you pile it up in this way, miraculously keeping every molecule just where you've placed it. Then, you let go. What would happen? The heap of water would collapse, and waves would slosh across the pool until everything settled down to that nice, flat surface that we've learned to expect. Again, suppose you arranged the water so that there was a big dip in the middle. Then as soon as you let go, water would move in from the sides to fill the dip.

Mathematically, this idea can be formalised in terms of the space of all possible shapes for the water's surface. 'Possible' here doesn't mean physically possible: the only shape you'll ever see in the real world, barring disturbances, is a flat surface. 'Possible' means 'conceptually possible'. So we can set up this space of all possible shapes for the surface as a simple mathematical construct, and this is the phase space for the problem. Each 'point' -location -in phase space represents a conceivable shape for the surface. Just one of those points, one state, represents 'flat'.

Having defined the appropriate phase space, the next step is to understand the dynamic: the way that the natural flow of water under gravity affects the possible surfaces of the pool. In this case, there is a simple principle that solves the whole problem: the idea that water flows so as to make its total energy as small as possible. If you put the water into some particular state, like that piled-up hump, and then let go, the surface will follow the 'energy gradient' downhill, until it finds the lowest possible energy. Then (after some sloshing around which slowly subsides because of friction) it will remain at rest in this lowest-energy state.

The energy in this problem is 'potential energy', determined by gravity. The potential energy of a mass of water is equal to its height above some arbitrary reference level, multiplied by the mass concerned. Suppose that the water is not flat. Then some parts are higher up than others. So we can transfer some water from the high level to the lower one, by flattening a hump and filling a dip. When we do that, the water involved moves downwards, so the total energy decreases.

Conclusion: if the surface is not flat, then the energy is not as small as possible. Or, to put it the other way round: the minimum energy configuration occurs when the surface is flat.