“See, you can use any number for a base, I used XII because it’s handy. XII becomes 1-0, see, and XIII is I-I, then when you get up to XXIV—”

“Sh.”

Mede studied the slate. Finally he said, “Will you remember this?” and at Ganil’s nod, wiped the neat crowded figures off the slate with a rub of his sleeve. “I hadn’t realized that one could use any base. . . . But look, use the base X, I’ll tell you why in a minute, and here’s a device that makes it easier. Now X will be 10, and XI will be II, but for XII, write this,” and he wrote on the slate, 12.

Ganil stared at the figure. At last he said in a peculiar struggling voice, “Isn’t that one of the black numbers?”

“Yes, it is. All you’ve done, Ganil, is to come at the black numbers by the back door.”

Ganil sat beside him, silent.

“What’s CXX times MCC?” Mede inquired.

“The tables don’t go that high.”

“Watch.” Mede wrote on the slate:

and then as Ganil watched,

Another long pause. “Three Nothings . . . XII times itself . . . Give me the slate,” Ganil muttered. Then after a silence broken only by the patter of rain and the squeak of chalk on slate, “What’s the black number for VIII?”

By twilight of that cold Altarday they had gone as far as Mede could take Ganil. Indeed Ganil had gone farther than Mede could follow him. “You must meet Yin,” the fair man said. “He can teach you what you need. Yin works with angles, triangles, measurements. He can measure the distance between any two points, points you can’t reach, using his triangles. He is a great Learner. Numbers are the heart of this knowledge, the language of it.”

“And my own language.”

“Yes, it is. Not mine. I don’t love numbers for themselves. I want to use them. To explain things . . . For instance, if you throw a ball, what makes the ball move?”

“Your throwing it.” Ganil grinned. White as a sheet—much whiter than Mede’s sheets—his head ringing with sixteen straight hours of mathematics minus meals or sleep, he had lost all fear, all humility. His smile was that of a king come home from exile.

“Fine,” said Mede. “Why does it keep on moving?”

“Because . . . because the air holds it up?”

“Then why does it ever fall? Why does it follow a curve? What kind of curve is it? Do you see how I need your numbers?” It was Mede who now looked like a king, an angry king with an empire too immense to control. “And they talk about Mysteries,” he snorted, “in their little shuttered shops! —Here, come on, let’s get some dinner and go see Yin.”