The correction with Beto's name and Danya's name came out on a Friday. Beto made his mother buy three copies. Danya pretended not to care, and read it on the notice board twice when nobody was looking.

The Friday after that, he read the back page first.

From the Real World had a picture of a bathtub this week. A bearded man was leaping out of it, dripping, with one arm in the air. Underneath it said:

Archimedes was asked by King Hiero to find out if his new gold crown was pure gold. In his bath he saw the water rise and shouted "Eureka!", which means "I have found it!" He ran home naked through the streets of Syracuse, measured the water the crown spilled over, and caught the goldsmith cheating.

And beside it, on the Honest Page, in neat black capitals: TRUE.

"Says who?" said Danya, out loud, to the notice board.

"Says the paper," said Lan.

Lan was seven. She had come up behind him with a skipping rope round her neck like a scarf. She had lived on the Green for a month and had made a friend of everyone on it except the printer's cat, who was working on it.

"I know what that means," she said, pointing at the picture. "Eureka. It means you found it. My teacher says it when she finds her glasses."

"That bit's true," said Danya. "It's a real Greek word."

"So what's wrong?"

"Nothing, maybe." Danya took out his notebook. On the inside of the cover, in small letters, he had written two words so far: FELL and DROPPED. "Except this is the third one. Every week somebody sends in a famous story and somebody marks it True. And nobody ever says how they know."

"I'll find out!" said Lan. She was already untangling the rope. "I promise. Before supper."

"You don't know who it is."

"I'll go first," said Lan. "That's how you find out."

The print shop was hot. The printer had the press running and both windows open, and the paper went thunk, thunk, thunk into the tray.

"Third week running," the printer said, before Danya could speak. The printer nodded at the box by the door. "Go on. It's in there."

It was. Under a note about a choir practice, there was a slip torn from a notebook.

Archimedes. Mark it True. Every school knows it.

No name. Pencil first, lightly, then inked over, carefully, every letter.

Lan held it up to the window, the way Danya had shown her. She could see the grey under the black.

"It's a ghost," she said. "A pencil ghost."

"Same person," said Danya. "Three weeks."

"Who checked that?" said the printer, from the press. She said it the way Danya said it. She was laughing at him, a little, but kindly.

"Me," said Danya. "I will."

"Then the big book's at the back," said the printer. "Somebody has turned it to the bath."

The big leather book lay open on the back table. On the page was the same picture as the paper, only older and browner: a bearded man, a bath, a splash. Beside it was a small picture of a gold crown on a cushion.

The Crown, said the heading.

Lan put her nose almost on the page. "It smells of soap," she said.

It did, very faintly. Danya bent closer.

The hammer rang once. A spark jumped off the anvil and did not go out. It waited. They followed.

The spark came under the print shop door and skipped across the floor. It jumped on the crown in the picture, and then it went to the green curtain and waited, flickering.

The spark slid under the green curtain into the drying room, and the drying room went back much further than the shop did.

"I'll go first!" said Lan, and did.

The corridor was the same long corridor, with pages pegged up overhead. But this time the floor was damp, and somewhere a long way back there was a sound of dripping, steady and slow, like a tap nobody had turned off.

"What exactly does it say?" said a voice.

A young bear in a striped scarf and a blue cardigan sat on her stool by the first door, with her red pencil and a pot of glue. She had put her feet up on a box to keep them dry.

"Claim," said Danya. "This is Lan."

"I know what that means," said Lan. "Claim. It means you say something is yours."

"Close," said Claim. "It means you say something is true. And then somebody checks." She held out the red pencil. "What exactly does your paper say?"

Danya read it out. Claim listened with her eyes shut.

"Say the claim in one sentence," she said.

"I'll go first! Archimedes caught a cheating goldsmith in his bath," said Lan.

"That's one sentence," said Claim. "It isn't one claim. Underline the exact words. How many things happen?"

Danya took a strip of paper and wrote them out, one per line.

1. Archimedes worked out how things float and push water aside.

2. King Hiero asked him to test a gold crown.

3. He thought of the answer in his bath, shouted "Eureka" and ran home naked.

4. He measured the water the crown spilled, and caught the goldsmith.

"Four," said Lan. "That's a lot of things for one bath."

"What exactly does each say? Any of them could be true," said Claim, "and the others not. Fell, or hit? Bath, or idea? Those are different claims."

"The picture is the bath," said Danya slowly. "Every picture is the bath. Nobody draws him doing sums."

"Underline the exact words. The bath is the claim the story leans on," said Claim. "Underline it twice."

Danya did. Lan watched the red line go across, and then across again.

"I could have done that," she said.

"Who checked that? Next time you do it," said Danya, and gave her the pencil.

They walked. The years on the cards beside the doors went down and down. There was a cartoon with bubbles coming out of the bath. A film of a man in a towel running down a street. An old schoolbook with the picture in every chapter. A painting from hundreds of years ago, with a very clean Archimedes and very clean water.

At each door Danya asked, "Where did they get it from?" And at each door, if they looked hard, there was a little note that pointed further back.

At one door, a painted ceiling from an old palace showed Archimedes in a marble bath as big as a pond, with three servants holding towels and a king watching from a balcony.

"A king watching," said Danya. "In the bath?"

"Where did they get it from?" said Trace, appearing at his elbow as if he had been walking beside them all along. "Look in the corner."

In the corner of the ceiling, very small, was a painted scroll with Latin on it. Trace read it the way other people read a bus timetable.

"The painter got it from a book," he said. "And the book got it from an older book. A copy is not a source. The king on the balcony is the painter's own idea. It made a better ceiling."

"So the painter stretched it," said Lan.

"The painter made a picture," said Trace. "Pictures always add things. A towel, a balcony, a king. That's fine, as long as nobody takes the picture for a witness."

Lan looked up at the painted king for a long time.

"He looks cross," she said.

"He thinks his goldsmith cheated him," said Danya. "I'd be cross too."

"Keep going back," said Trace. "Who wrote it down first? The dripping is louder down there."

Then the cards stopped saying ordinary years. They said AD, and then, a long way further, BC.

They followed the dripping.

Trace walked ahead with his notebook in his cardigan pocket. There was a damp patch on one knee of his trousers, where he had knelt to read a low card.

The last door had no picture, only a long shelf of old scrolls and, on a stand, a book in Latin. A card beside it said:

Vitruvius, a Roman builder. Ten books about architecture. Book Nine tells the story of the crown.

Trace read the translation pinned beside the stand. It said that Hiero had given gold to a craftsman to make a crown. That Hiero suspected some of the gold had been swapped for silver. That Archimedes, getting into a bath, saw the water run over the side as he sank in, and leapt out, and ran home naked, crying Eureka. And that he then measured how much water a lump of gold spilled, and a lump of silver, and the crown.

"Show me the source for the rest. That's the whole story?" said Danya. "That's all of it. Bath, run, spilled water."

"Check the date," said Trace.

"When did Vitruvius write?"

Trace flipped back through his notebook. "Older isn't the same as first. But this is the first we have. Vitruvius was writing about two hundred years after Archimedes died."

"Says who, though? And Archimedes' own books?" said Danya. "Does he say it?"

"Keep going back," said Trace. He pointed to a door beside the stand, very low, with no card at all. "That's his room. Go and look."

Lan did not go and look. She was standing in front of the stand, with her hands on her hips.

"I know what that means," she said. "It's the oldest. That's the oldest door in the whole corridor. So it's true. Mark it True. Come on, quick!"

"Who was there?"

An old bear with round glasses and a long blue scarf was sitting on a stool beside the scrolls. Nobody had seen him come in. He was eating a fig, very slowly.

"Weigh," said Danya.

"Who was there?" said Weigh again, to Lan. "In the bath. When the water ran over."

"Archimedes," said Lan.

"And who wrote it down?"

"Vitruvius."

"Was Vitruvius there?"

Lan opened her mouth. Then she looked at the card. About two hundred years after.

"No," she said. "He wasn't born."

"How long after did they say it?" said Weigh.

"Two hundred years," said Lan. Her face had gone red. "But it's the oldest one. It's the oldest one we've got."

"It is," said Weigh. He finished the fig and folded the paper it had come in. "The oldest telling we have. That counts for something. But a good witness can still be a long way off, and this one is two hundred years off. Weigh it. Don't count it. The oldest door in the corridor is not the same as being in the room."

Lan scowled at the stand for a long time.

"I promised I'd find out before supper," she said.

"You found out something," said Danya. "You found out where it starts."

"That's not the same as true."

"No," said Danya. "But it's not nothing either."

Lan thought about that. Then she unwound the rope from round her neck and wound it the other way, which was what she did when she had changed her mind.

"Fine," she said. "What's in the little room?"

The little room was cooler than the corridor. On a low table lay a pile of books, translated from Greek, with drawings of circles and floating shapes in the margins.

"Archimedes," said Trace. "His own work. What he actually wrote."

Danya turned the pages carefully. There was a book called On Floating Bodies. It said that a thing put into water is pushed upwards, and that a floating thing pushes aside its own weight of water. It had diagrams of floating shapes, like halves of balls, tipping and coming upright.

"That's claim one," said Danya. "He really did work out how things float. It's in his own book."

"Who was there?" said Weigh, who had come to the door.

"He was," said Danya. "He wrote it himself. That's the best witness there is."

"And the crown?"

Danya looked through every book on the table. Lan helped, holding them open with both hands. There were circles and spirals and a book about counting grains of sand.

"It isn't here," said Danya at last. "There's no crown. There's no bath. There's no running. He wrote about floating, and he never tells the story."

"Then weigh that," said Weigh. "His idea is his. The bath is somebody else's story about him."

Lan sat down on the floor.

"So the story is wrong?"

"Weigh it. I didn't say that," said Weigh. "I said weigh it. Nobody knows if he was in a bath. Maybe he was. Two hundred years is a long time for a story to be passed along. Things get added. Running naked through the streets is the kind of thing that gets added, because people like to tell it."

"I like to tell it, I promise," said Lan.

"So do I," said Weigh. "That's the trouble."

"Keep going back, but there's a side door," said Trace. "You'll like this one. Keep going back is not the only way. Sometimes you go sideways."

He led them back up the corridor a little, to a narrow door with a card that said 1586.

"Same year as Stevin," said Danya at once. "In Delft. The lead balls."

"Check the date," said Trace, pleased. "Same year. Different man. Look who it is."

Inside, on a table, was a little set of scales, very neat, with a long arm and two pans. One pan hung down into a glass of water. Next to it was a short pamphlet in Italian, and a card in English:

Galileo Galilei, twenty-two years old. A little paper about a little balance.

Danya read the translation. Galileo said he had read the story of the crown and the spilled water. He said the water spilled by a crown would be so little that it would be very hard to measure it well enough to catch a cheat. And then he described a better way. You hang the crown on a balance and weigh it in air. Then you weigh it again with the crown hanging in water. Gold and silver lose different amounts of weight in water. A careful balance can tell the difference.

At the end, Galileo said he thought it was probable that this was the way Archimedes really did it, because it was accurate, and because it used things Archimedes had proved himself.

Lan was looking at the little scales with her nose almost touching the glass.

"I could do that," she said. "With Bà Tư's kitchen scales."

"How does a balance catch a cheat?" said Danya. He liked to know how a thing worked before he trusted it, the way Beto did.

"Weigh the man, too," said Weigh, from the doorway. "Galileo was a young teacher with no money. He wanted a way that anyone careful could repeat. Ask what he could check."

Danya read the pamphlet again. "Gold is heavier for its size than silver," he said slowly. "So a lump of gold is smaller than a lump of silver that weighs the same. In water, the bigger lump gets pushed up more. So it loses more weight."

"So if the crown has silver hidden in it," said Lan, "it's a bit too big. And it loses a bit too much in the water."

"And the balance shows it," said Danya. "Even if the difference is tiny. A balance is good at tiny."

"I know what that means," said Lan. "It means you don't have to measure spilled water with a teaspoon."

"How do we know? Galileo didn't believe the spilled water," said Danya.

"Fell, or hit? Spilled, or weighed? Galileo thought it was too small to measure," said Claim. She had followed them, with the glue pot. "What exactly does the paper say?"

"It says he measured the water the crown spilled," said Danya. "And caught the goldsmith."

"Then that claim," said Claim, "is the one Galileo argued with."

Danya wrote Galileo's name on the second column of his notebook, in pencil, very small.

"Galileo again," he said. "Two weeks running."

"Whoever is famous is not always right. Famous people collect stories," said Weigh. "And they are sometimes the ones who check them."

Danya sat on the damp floor and looked at his four claims.

1. Archimedes worked out how things float and push water aside.

In his own book. True.

2. King Hiero asked him to test a gold crown.

Only Vitruvius said so. Two hundred years later.

3. The bath, the shout, the running.

Only Vitruvius. Two hundred years later.

4. He measured the spilled water.

Vitruvius said so. Galileo thought it would not work well.

"How old was Archimedes?" said Lan. "When he died?"

"Weigh the man, too," said Weigh. "Born about 287 BC. Died about 212 BC."

Danya did the sum. 287 take away 212. "About seventy-five," he said.

"And how long from Archimedes dying to Galileo's little balance?" said Weigh.

Danya started to do the sum and stopped. "BC and AD," he said. "That's hard."

"Check the date. There's no year zero," said Trace. "One BC goes straight to one AD. So you add them and take one away."

212 add 1586 take away 1. Danya wrote it out carefully.

"Seventeen hundred and ninety-seven years," he said. "Nearly eighteen hundred years before anyone asked if the spilled water would really work."

"Come on, quick, that can't be right! Eighteen hundred years of people telling the bath," said Lan, "and nobody checking the water?"

"Galileo checked," said Danya. "When he was twenty-two."

Claim was underlining probably in her copy of Galileo's pamphlet.

"So," she said. "Are the claims true, stretched, made up, or nobody knows?"

Danya thought about it for a long time. He was getting used to thinking for a long time.

"One is true," he said. "Two and three are Nobody knows. Nobody who was there wrote them down. Four is stretched. It might be how he did it. Galileo thought it probably wasn't."

"Were they lying?" said Lan. "Vitruvius?"

"Did he think it was true?" said Weigh.

"Probably."

"Then he wasn't lying," said Weigh. "He was telling a story about a clever man, two hundred years later, the way people told it. People like a better story. A bath is a better story than a balance."

"Stretching is honest," said Trace, "when you say that you did it."

"That's what the Honest Page is for," said Danya, and stood up.

He wrote the correction out on a clean page.

From the Real World — a correction.

Archimedes worked out how things float and push water aside. He wrote it down himself, in On Floating Bodies. TRUE.

He thought of it in his bath, shouted "Eureka" and ran home naked. NOBODY KNOWS. The first telling we have is by Vitruvius, about two hundred years after Archimedes died. Archimedes' own books never tell the story.

He caught the goldsmith by measuring the water the crown spilled. STRETCHED. That is Vitruvius's version. In 1586 Galileo argued the spilled water would be too little to measure, and that a balance in water was probably the real method.

He showed it to Claim. She underlined first telling we have and nodded.

He showed it to Trace. Trace checked the dates and the sum with no year zero and said, "Yes. That's the oldest door. Good."

He showed it to Weigh. Weigh read it twice, slowly.

"Weighed," he said. "Not counted. And not taken just because it was oldest."

Lan read it last, with her lips moving.

"You didn't put the naked running," she said.

"I did. It's under Nobody knows."

"Good," said Lan. "Because it's the best bit. And nobody knows."

At the bottom, in pencil, Danya added a line: Try Galileo's balance with kitchen scales and a bowl of water. Then he inked over the pencil so the printer could read it. He did not notice himself do it.

On the inside of the cover, under FELL and DROPPED, he wrote a third word: EUREKA.

The spark slid back under the curtain, and the drying room was just a cupboard again.

The big book was open at the bathtub. The floor of the print shop was perfectly dry. Lan sniffed the page, and it smelled only of old paper now.

The printer read the correction at the press, with her finger under each line.

"Three weeks," she said. "Three corrections. People will start writing in to check you."

"Good," said Danya. "They should."

"I promise I'll find the ghost. And the note in the box?" said Lan. "The pencil ghost?"

"Still no name," said the printer. "But you two are getting close. I can tell. You've got the look."

The spark dimmed to a warm dot and settled back on the anvil. The Green was just the Green again.

Back on the Green

Bà Tư's kitchen scales were the old kind, with a dial and a hook underneath. Lan borrowed them, and a bowl of water, and a spoon, and Bà Tư came too, to make sure nothing was dropped in the soup.

They hung the spoon from the hook by a thread and read the dial. Then they lowered the spoon into the bowl, still hanging, and read the dial again.

"It got lighter!" said Lan. "The water's holding it up!"

"It reads lighter because the water is pushing up on it," said Danya. "Archimedes wrote that down himself."

"In his own book?"

"In his own book."

Lan looked at the spoon in the water for a long time.

"Then that part's true," she said. "I checked it. I promise."

"You did," said Danya.

"And the bath?"

Danya thought of the long corridor, and the dripping from the very back, and the old bear eating a fig.

"The bath," he said, "is a very good story."