‘I have been considering the matter,’ she informed him, after a little while, around a mouthful, ‘and in all cases, where the angle is ninety degrees, as I suppose you meant to draw it, the length of the long side must be a number which, multiplied by itself, is equal to the lengths of the two shorter sides, each multiplied by themselves, added.’ She swallowed noisily, and licked her chops clean. ‘Quite an interesting little observation. How did you come to make it?’

‘I never did,’ Temeraire muttered, ‘it is the Pythagorean theorem; everyone who is educated knows it. Laurence taught it to me,’ he added, making himself even more miserable.

‘Hmh,’ the other dragon said, rather haughtily, and flew away.

But she reappeared at Temeraire's cave the next morning, uninvited, and poked him awake with her nose, saying, ‘Perhaps you would be interested to learn that there is a formula which I have invented, which can invariably calculate the power of any sum? What does Pythagoras have to say to that?’

‘You did not invent it,’ Temeraire said, irritable at having been woken up early, with so empty a day to be faced. ‘That is the binomial theorem, Yang Hui made it a very long time ago,’ and he put his head under his wing and tried to lose himself again in sleep.

He thought that would be the end of it, but four days later, while he lay by his lake, the strange dragon once again landed beside him. She was bristling furiously and her words tumbled over one another as she rushed, ‘There, I have just worked out something quite new: the prime number coming in a particular position, for instance the tenth prime, is always very near the value of that position, multiplied by the exponent one must put on the number p to get that same value – the number p,’ she added, ‘being a very curious number, which I have also discovered, and named after myself.’

‘Certainly not,’ Temeraire said, rousing with smug contempt when he had made sense of what she was talking about. ‘It is not p, it is e; you are talking of the natural logarithm. And as for the rest about prime numbers, it is all nonsense. Consider the prime fifteen—’ and then he paused, working out the value in his head.

‘You see,’ she said, triumphantly, and after working out another two-dozen examples, Temeraire was forced to admit that the irritating stranger might indeed be correct.

‘And you needn't tell me that this Pythagoras invented it first,’ the other dragon added, with her chest puffed out, ‘or Yang Hui, because I have inquired, and no-one has ever heard of either of them. They do not live in any of the coverts or anywhere around breeding grounds, so you may keep your tricks. Who ever heard of a dragon named anything like Yang Hui; such nonsense.’