The cannons arrived from the Khan, and Khalid spent part of every day out on the bluff over the curve in the river, shooting them off with old jalil and Paxtakor; but by far the bulk of his time was spent thinking about optics and proposing tests to Iwang. Iwang returned to his shop, blowing thick glass balls with cut sides, mirrors concave and convex, and big, perfectly polished triangular rods, which were for him objects of almost religious reverence. He and Khalid spent afternoon hours in the old man's study with the door closed, having made a little hole in the south wall letting in a chink of light. They put the prism over the hole, and its straight rainbow shone on the walls or a screen they set up. Iwang said there were seven colours, Khalid six, as he called Iwang's purple and lavender two parts of the same colour. They argued endlessly about everything they saw, at least at first. Iwang made diagrams of their arrangement that gave the precise angles each band of colour bent as it went through the prism. They held up glass balls and wondered why the light did not fractionate in these balls as it did in the prism, when everyone could see that a sky full of minuscule clear balls, that is to say raindrops, hit by low afternoon sunlight, created the towering rainbows that hung east of Samarqand after a rainshower had passed. Many a time when black storms had passed over the city, Bahram stood outside with the two older men observing some truly beautiful rainbows, often double rainbows, a lighter one arched over the brighter one; and sometimes even a third very faint one above that. Eventually Iwang worked up a law of refraction which he assured Khalid would account for all the colours. 'The primary rainbow is produced by a refraction as the light enters the raindrop, an internal reflection at the back surfaces, and a second refraction out of the raindrop. The secondary bow is created by light reflected two or three times inside the raindrops. Now look you, each colour has its own index of refraction, and so to bounce around inside the raindrop is to separate each colour out from the others, with them appearing to the eye always in their correct sequence, reversed in the secondary because there is an extra bounce making it upside down, as in my drawing here, see?'

'So if raindrops were crystalline, there would be no rainbows.'

'That's right, yes. That's snow. If there was only reflection, the sky might sparkle everywhere with white light, as if filled with mirrors. Sometimes you see that in a snowstorm, too. But the roundness of raindrops means there is a steady change in the angle of incidence between zero and ninety degrees, and that spreads the different rays to an observer here, who must always stand at an angle from forty to fortytwo degrees off from the incoming sunlight. The secondary one appears when the angle is between fifty and a half degrees and fifty four and a half. See, the geometry predicts the angles, and out here we measure, using this wonderful sky viewer Bahram found for you at the Chinese caravanserai, and it confirms, as precisely as hand can hold, the mathematical prediction!'

'Well, of course,' Khalid said, 'but that's circular reasoning. You get your angles of incidence by observation of a prism, then confirm the angles in the sky by more observation.'

'But one was colours on the wall, the other rainbows in the sky!'

'As above, so below.' This of course was a truism of the alchemists, so there was a dark edge to Khalid's comment.

The current rainbow was waning as a cloud in the west blocked the sun. The two old men did not notice, however, absorbed as they were in their discussion. Bahram alone was left to enjoy the vibrant colours arcing across the sky, Allah's gift to show that he would never again drown the world. The two men jabbed fingers at Iwang's chalkboard and Khalid's sky device.

'It's leaving,' Bahram said, and they looked up, slightly annoyed to be interrupted. While the rainbow had been bright, the sky under it had been distinctly lighter than the sky over it; now the inside and outside were the same shade of slate blue again.

The rainbow left the world, and they squelched back to the compound, Khalid cheering up with every step, many of them right into puddles, as he was still staring at Iwang's chalkboard.

'So – so – well. I must admit, it is as neat as a proof of Euclid. Two refractions, two or three reflections – rain and sun, an observer to see – and there you have it! The rainbow!'

'And light itself, divisible into a banding of colours,' Iwang mused, travelling all together out of the sun. So bright it is! And when it strikes anything at all, it bounces off and into an eye, if there be an eye to see it, and whatever part of the band, hmm, how would that work… are the surfaces of the world all variously rounded, if you could but look at them close enough…'

'It's a wonder things don't change colour when you move,' Bahram said, and the other two went silent, until Khalid started laughing.

'Another mystery! Allah preserve us! They will just keep coming for ever, until we are one with God.'

This thought appeared to please him immensely.

He set up a permanent dark room in the compound, all boarded and draped until it was much darker than his study had been, with shuttered chinks in the east wall that could let in small shafts of light, and many a morning he was in there with assistants, running in and out, arranging demonstrations one way or another. One he was pleased enough with to invite the scholars of the Sher Dor Madressa to witness it, because it so neatly refuted Ibn Rashd's contention that white light was whole, and the colours created by a prism an effect of the glass. If this were so, Khalid argued, then light twice bent would change colour twice. To test this, his assistants allowed sunlight in through the wall, and a first prism's array was spread across a screen in the centre of the room. Khalid himself opened an aperture in the screen small enough only to allow the red part of the little rainbow through it, into a draped closet where it immediately encountered another prism, directing it onto another screen inside the closet.

'Now, if the bend of refraction caused by itself the change in colour, surely the red band would change at this second refraction. But look: it remains red. Each of the colours holds when put through a second prism.'

He moved the aperture slowly from colour to colour, to demonstrate. His guests crowded around the door of the closet, examining the results closely.

'What does this mean?' one asked.

'Well, this you must help me with, or ask Iwang. I am no philosopher, myself. But I think it proves the change in colour is not just a matter of bending in itself. I think it shows sunlight, white light if you will, or full light, or simply sunlight, is composed of all the individual colours travelling together.'

The witnesses nodded. Khalid ordered the room opened up, and they retired blinking into the sun to have coffee and cakes.

'This is wonderful,' Zahhar, one of Sher Dor's senior mathematicians said, 'very illuminating, so to speak. But what does it tell us about light, do you think? What is light?'

Khalid shrugged. 'God knows, but not men. I think only that we have clarified (so to speak) some of light's behaviour. And that behaviour has a geometrical aspect. It seems regulated by number, you see. As do so many things in this world. Allah appears to like mathematics, as you yourself have often said, Zahhar. As for the substance of light, what a mystery! It moves quickly, how quickly we do not know; it would be good to find out. And it is hot, as we know by the sun. And it will cross a void, if indeed there is any such thing as a void in this world, in a way that sound will not. It could be that the Hindus are right and there is another element besides earth, fire, air and water, an ether so fine we do not see it, that fills the universe to a plenum and is the medium of movement. Perhaps little corpuscles, bouncing off whatever they strike, as in a mirror, but usually less directly. Depending on what it strikes, a particular colour band is reflected into the eye. Perhaps.' He shrugged. 'It is a mystery.'