The pole stood in the courtyard of the Library, and for a year the copyist had been told to leave it alone.
It was a plain pole of cedar, two metres tall, set upright with a plumb line and a bowl of water to prove the ground was level. At every noon a boy was sent to scratch the end of its shadow on the paving. In winter the shadow stretched across half the yard. In spring it shortened. Eratosthenes read the scratches like a ledger and said nothing about what they were for.
On the longest day of the year, at noon, the shadow was the shortest it would be all year. The copyist pressed a thumb into the small dark patch at the foot of the pole and looked up for approval.
“Short,” said Eratosthenes. “Not none. Measure it.”
The copyist measured, and then, because Eratosthenes was waiting, did the thing he disliked: he said what it meant. “If the sun were straight above us, there would be no shadow at all. There is some. So the sun is a little off the vertical here.”
“How little?”
It came to about a quarter of a metre, one part in eight of the pole's height. Eratosthenes drew it in the dust: the pole, the shadow, and the line from the tip of the shadow to the top of the pole. “That line is a sunbeam. It leans from the pole by this angle. A fiftieth part of a full turn.” He said this as if he had known it all along, which he had not; he had measured it twice that morning to be sure.
· · ·
The other half of the story lay far to the south, at the bottom of a well. Travellers from the south had said, for as long as anyone remembered, that at Syene, on this same day, at this same hour, the sun lit the water at the very bottom of a deep well. No shadow on the wall. A pole there would cast none.
“So at Syene the sun is straight overhead,” said Eratosthenes. “Here it is a fiftieth of a circle away from straight overhead. Same sun, same hour, two towns.”
The copyist saw the next step before he was told it, and was ashamed of how pleased it made him. “The sun's rays arrive side by side, because it is so far away. Then the difference in the angle is not about the sun at all. It is the bend of the ground between the towns.”
“Say it as a number.”
“If the gap between us is a fiftieth of a turn of the whole Earth, then the whole Earth is fifty of these gaps.”
Eratosthenes nodded once. The distance, he said, was five thousand stadia; the pace-counters of the kings had walked it, and the oldest of them had complained about every one of those stadia. He wrote the sum on the wall in charcoal, in letters so small the copyist had to lean in: fifty times five thousand.
“A quarter of a million,” said the copyist.
“Stadia. Do not say miles. Whoever reads this later will not know which stadion I meant, and I am not certain myself.” He rubbed out the last zero, looked at it, and wrote it in again.
· · ·
That evening the copyist went back to the courtyard alone. The scratches were still there, a row of small marks going back to the winter, each one a noon. He stood where the pole had been and tried to feel the ground curve under his feet, and could not. That was the point of the whole business, he decided. The Earth was big enough that you needed two towns and a long walk to catch it bending.