The Shadow at Noon

The longest day of the year found them on a hill in the middle of nowhere, staring at a stick.

It was a good stick. Teo had chosen it with great care three days earlier, from a grey, bleached tree by the side of the Maker road... a straight, smooth branch, cut to exactly the length of his own forearm, from his elbow to the tip of his longest finger. He had planted it upright in a crack in a flat slab of road-stone at the top of the hill, and checked it with a plumb line made of a string and a pebble until it stood perfectly straight. Then he had sat down beside it and waited.

Bram was sitting on the other side of it. He had been sitting there for an hour. He did not seem to mind.

Mira, on the other hand, was pacing.

"Explain it again," she said.

"I've explained it three times."

"Then a fourth won't hurt you." She stopped and folded her arms. "You're sitting on a hill in the Ashlands staring at a stick on the longest day of the year, and you say it's going to tell you how big the world is. Explain it. Slowly. As if I were a Shelf girl who grew up thinking the world was a flat plate with the Flame in the middle."

Teo sighed.

He looked at the stick. It was still an hour or more before noon. Its shadow lay long and thin across the grey stone, pointing west of north, and shrinking slowly as the sun climbed.

"There was a Maker," he said. "Or no... not a Maker. Someone older. Much older. A man who lived before the Makers, so long ago that the Makers themselves thought of him as ancient. His name was Eratosthenes."

"That's a ridiculous name."

"Hesk said that, too." Teo smiled faintly. "He lived in a city called Alexandria, by a great sea, and he was in charge of a library there. The biggest one in the world. And one day, somebody told him a strange thing. They said that in a town called Syene, far to the south, on the longest day of the year, at noon, the sun shone straight down to the very bottom of a deep well. You could look down the well and see the sun reflected in the water at the bottom. No shadow on the walls at all." He paused. "That meant that at that moment, in Syene, the sun was exactly overhead. Straight up."

Mira frowned.

"So?"

"So, in Alexandria, on the same day, at the same time, it wasn't. Eratosthenes put a stick in the ground — a stick like this one — and it threw a shadow. A small one. Which meant that in Alexandria, the sun was a little bit to one side of straight up. Not overhead." Teo held up his hand, fingers spread. "Now. If the world were flat — like your Shelf plate — then the sun would be the same in both places. It's so far away that its light comes down in straight lines, all parallel, like rain falling with no wind. On a flat world, if the sun is straight overhead in Syene, it would be straight overhead in Alexandria too. No shadow anywhere."

"But there was a shadow."

"There was a shadow. And the only way that could happen — the only way, if the light is falling straight — is if the ground itself is curved. If Alexandria is tilted a little bit away from the sun, compared to Syene. Because the world is round." Teo picked up a stone and drew a circle in the dust. "Like this. Here's Syene, at the top. The sun shines straight down the well. And here's Alexandria, a little way round the curve. The light comes in the same straight line... but the ground is tilted. So the stick throws a shadow."

Mira crouched down and looked at the circle.

"...And the size of the shadow tells you how much the ground is tilted," she said slowly.

Teo looked at her in surprise.

"Yes. Exactly that. Eratosthenes measured the angle of his shadow. It was about a fiftieth of a full circle. Which meant that Alexandria was a fiftieth of the way around the world from Syene." He drew a line in the circle, a narrow slice like a piece of pie. "So all he needed was the distance between the two cities. He had people who measured it by walking... he hired men to pace it out, step by step. Five thousand of their units. And fifty times five thousand..."

"Is the distance all the way around," Mira finished. She stared at the circle in the dust. "He measured the whole world. With a stick. And a shadow. And somebody walking."

"Two hundred and fifty thousand of his units. The Leaf said that later, when the Makers measured it properly, he turned out to be very close. Within a few parts in a hundred, perhaps, depending on how long his units really were." Teo shrugged. "Two thousand years before the Makers flew. With a stick."

Mira sat back on her heels and was silent for a long while.

"And Nell," she said at last. "That's what your note was. You asked her to put a stick in the ground in Calder Reach. Today. At noon."

"The same length as her forearm. I told her to measure her own arm against the stick, so we'd know." Teo looked at the shadow, which had grown noticeably shorter. "Then when we get back, I'll know the shadow there, and the shadow here, and the difference between them..."

"...is how much the ground has curved between Calder Reach and this hill," Mira said. "And then all you need is the distance."

They both turned and looked at Bram.

Bram, who had been sitting by the stick in perfect silence for an hour and a half, reached into his coat and drew out a long, thin cord, and held it up. It was covered in knots. Hundreds of them. Big ones, small ones, knots tied in clusters of ten and clusters of a hundred.

"Paces," he said simply.

Teo had asked him on the first day out of the Fens. He had explained what he wanted in a few sentences, and Bram had listened, and nodded, and since then, every step he had taken on the Maker road, walking beside the mule, he had counted. Every hundred paces, a small knot. Every thousand, a big one. He had done it for three weeks without once complaining or losing his place. Teo had checked.

There was a problem, of course. There always was.

"It isn't a straight line," Teo said, frowning at the knotted cord. "That's the trouble. We came down the river first, south. Then east along the road, then the road bent south again. For this to work, Calder Reach and this hill have to be... one above the other. North and south of each other, on the same line, like Syene and Alexandria. Because the shadow only tells you about the curve between north and south."

"Why only north and south?" Mira asked.

"Because the sun at noon is always due south of us. Here, this far north. The shadow at noon always points north. So it only measures how far north or south you've come, not how far east." Teo rubbed his forehead. "If you walked a thousand miles due east, the noon shadow would be exactly the same length at the end. Only the time of noon would change."

He sighed.

"So I need to know how many of Bram's paces were going south, and how many were going east. And I don't have a map, or a compass, or anything except the sun and my memory of which way the road bent."

"Then estimate," said Bram unexpectedly.

Teo looked at him.

"Estimate," the big man repeated, in his slow, deep voice. "Fen farmers do it every year. How much seed for a field. How much hay for a winter. You don't know. You guess. Then you see how wrong you were, and next year, you guess better."

Teo opened his mouth to argue... and closed it again.

'He's right. Eratosthenes guessed, too. His walkers didn't walk a perfect line. The town wasn't exactly south. And he still got close.'

'It doesn't have to be perfect. It just has to be honest about how wrong it might be.'

The shadow was very short now.

Teo knelt beside the stick with a stub of chalk and a strip of smooth bark. The sun was high and white and fierce, almost directly above them... almost, but not quite. The shadow of the stick was a small, dark sliver lying on the stone, pointing north. He watched it. It shrank, very slowly. Shrank. Shrank...

And then, for a long, long moment, it did not move at all.

"Now," he said quietly.

He marked the tip of the shadow on the stone with chalk. Then he measured it against the stick, carefully, the way Hesk had taught him to measure everything... twice, and then a third time, because the second measurement did not quite match the first. He wrote the numbers on the bark. Then he sat back on his heels and let out a long breath.

Far away, three weeks and a few hundred miles behind them, at the foot of a great grey wall, a ten-year-old girl was kneeling beside a stick on the roof of a burned-out chandlery, marking the tip of its shadow with chalk.

Or at least, he hoped she was.

'...If she remembered. If the cousin gave her the note. If it isn't raining in Calder Reach. If the stick didn't fall over. If Hesk didn't help.'

Mira was watching him.

"Do you think she did it?" she asked.

Teo looked down at the little white chalk mark on the grey stone.

"I don't know," he said. "But if she did... when we get home, I'm going to know how big the world is."

He tucked the strip of bark carefully inside his shirt, next to the brass button and Hesk's map. Then he stood up, and pulled the stick out of the crack in the stone, and looked east along the endless grey road.

On the horizon, so far away that it might have been a cloud, or a range of hills, or nothing at all, something was glittering in the noon sun.