The Sand-Reckoner

8

Phidias' illness had taken a turn for the worse again. He slept most of the time, and it was hard to rouse him; when he did wake, he was often confused and could not understand where he was or what was wanted of him. To Archimedes' grief, he did not seem to appreciate even that the catapult had passed its trial and that his son was in a position to provide for the family. Hieron's personal physician had indeed come to visit him, but apart from leaving a drug which Phidias could take if he was in any pain, had done nothing the family's own doctor hadn't done already. "There is no hope of a cure," he had said.

Archimedes could not stop himself from hoping anyway. Every morning and evening he would go into the sickroom to see his father. He would try to start a conversation, and when that failed he would simply sit, doing calculations or playing music while Phidias slept.

On the morning two days after the dinner party, the day of the demonstration, he went into the sickroom as usual and found his father asleep. He sat down on the couch, took the skeletal hand in his own, and brushed back the thin white hair. "Papa?" he said. Phidias woke and smiled up at him in silence.

"I'm going down to the docks now," he told his father. "I'm doing a demonstration of mechanics for the king."

The brittle hand clenched suddenly on his own. "Don't go away!" Phidias begged.

"It's only for an hour or two," said Archimedes.

"Don't go away to Alexandria, please, Medion!"

"Papa! I'm not, I won't. I'm just doing a demonstration at the docks. I'll come home and check on you afterward."

"Don't go away again, please!" whispered Phidias, as though he hadn't heard; then, more softly still, "Look after your mother and sister for me."

"I will, Papa," Archimedes said. "I promise."

He stayed where he was for a few more minutes, and eventually the tight grip on his hand relaxed, and his father went back to sleep. He stood up very gently so as not to wake him, then stood looking down at the yellow face soberly. Was it imagination, or was there a translucent quality to the skin, a gasp to the shallow breath, which had not been there before?

Arata came in. Archimedes had invited her to come watch the demonstration, and she had put on her best gown preparatory to going, but at the sight of her husband's face she pulled her chair out from the wall and sat down to keep watch over him. "I don't want to leave him this morning," she told her son. "You take Philyra."

Archimedes did not protest. He said only, "Send Chrestos to fetch me if… if he asks for me, or if anything happens. I don't care about the king: I'll come."

Arata nodded, and Archimedes bent to kiss her forehead, then went out into the courtyard.

Philyra was already waiting, bright-eyed and impatient in her best tunic and cloak. Archimedes thought she needn't have bothered with the tunic, since it was invisible apart from the border around the hem: Philyra was respectably swathed in cream-colored wool from head to foot, and her face was already pink with the heat- unless it was excitement. Marcus and young Agatha, waiting beside her, both looked considerably more comfortable in plain linen tunics. Agatha was going because it was genteel to have a lady's maid, and Marcus was carrying a basket with some refreshments.

"Medion!" exclaimed Philyra, "You're not wearing that cloak!" It was the linen one.

"I'm not going to be able to wear any cloak to do the demonstration," objected Archimedes. "You can't haul on a rope in a cloak. So I thought…"

Philyra shook her head firmly. Marcus, grinning, set down the basket, ran upstairs, and came back with the yellow cloak. Archimedes swore under his breath, but put the thing on, and the party set out.

When they were approaching the docks, the streets grew crowded, with large numbers of people jostling along in the same direction as themselves. Archimedes eyed them dubiously. "Is something happening?" he asked a plump waterseller.

"Haven't you heard?" replied the waterseller. "One of the king's engineers thinks he can move a ship single-handed."

"But…" said Archimedes, blinking. "Are all these people coming to see that?"

" 'Course," said the waterseller reprovingly. "Should be quite a sight."

"But- but how do they all know?" asked Archimedes.

"It was posted in the marketplace," replied the waterseller. "What's it to you?"

"I'm the engineer," said Archimedes bemusedly, wondering who had posted it.

"So you're Archimedes son of Phidias!" exclaimed the water-seller, looking him up and down disappointedly. "I thought you'd be older."

Philyra gave a laugh of delighted astonishment and took her brother's arm. "Medion, you're famous!"

The quayside, when they reached it, was crowded with people standing about talking, eating, and drinking and pointing out to one another the ship Archimedes had selected. This was by no means the largest ship in the king's fleet, but it was nonetheless indisputably a ship: a fat single-masted transport about seventy feet long. It had been drawn up out of the water and its sides curved upward from the stone slipway twice as high as a man. Philyra stopped when she saw it, stared a moment, then looked at her brother anxiously. So did Marcus. They had both accepted Archimedes' assurances that his system would work, but now they were faced with an object larger than their house, the project suddenly appeared totally impossible.

"Can you really move it?" asked Philyra.

But he was surprised that she could doubt it. "Oh yes!" he exclaimed. "It only weighs about twelve hundred talents unloaded, and I've given myself a mechanical advantage of fifteen hundred. I'll show you!"

The area about the ship was in the process of being roped off to protect it from the crowd, but the sailors doing the roping recognized Archimedes and allowed his party through. He had just begun explaining the system to Philyra when there was a blare of trumpets, and they looked up to see the king arriving. First came a file of guards, led by an officer on horseback. The shields slung over their backs gleamed with color, and their helmets and spear points shone in the sun. Behind them came the king, riding upon a magnificent white horse and robed in purple. He was accompanied by Kallippos, on a tall bay, then followed by trumpeters, and by a covered litter carried by eight slaves. The crowd cheered and applauded, and- eventually- made way for him. Philyra clutched Archimedes' arm with excitement as the royal procession drew to a halt before them.

The litter was set down, and the passengers climbed out: first the queen, purple-cloaked like her husband; then the little boy, Gelon, also in purple and looking hot. Lastly a dark-haired girl climbed out and stood a moment straightening a cloak of very fine crimson cotton worked with gold stars. Archimedes stood up straighter, grinning with pleasure. So Delia had come to see his demonstration! She was even prettier than he remembered. He tried to meet her eyes, wondering how he could thank her for her message. But when he finally caught her gaze, she returned his smile only with a cold flat stare.

Philyra had no real idea who the girl in red was, but she thought she would float up in the air with pride when the whole royal party came over to shake hands with her brother. She was aware of the watching crowd talking about them, pointing Archimedes out to one another as the son of Phidias the Astronomer, the Alexandrian-educated engineer who had offered to do something impossible.

Kallippos shook hands with Archimedes very brusquely, then at once strode off to inspect the system of pulleys, leaving Archimedes glancing nervously after him.

Queen Philistis smiled graciously at Philyra when Archimedes introduced her. "I believe we've met before," she said. "You won prizes for music at your school, didn't you, child? Your whole family is much gifted by the Muses, it seems."

Philyra blushed. She had indeed won prizes for music, and the queen had handed them out, but she hadn't expected Philistis to remember.

Delia merely gave Philyra a look of black-eyed disdain. Under the disdain, she was troubled. When she had first noticed that Archimedes had a girl on his arm, she had suffered a perplexing moment of indignation- followed by relief as she noticed the strong family resemblance between the two, and remembered that he had a sister. Such feelings were, she knew, entirely inappropriate- no, lunatic! It didn't matter if Archimedes had a girl or a boy or half a dozen strumpets. He was nothing to her, and that was how she wanted it. She transferred the disdainful look to him, and he blinked in confusion.

"And that's the ship you're going to move, is it?" asked the king. "Herakles!"

As Philyra had, he surveyed the height and length of it, then looked at the gangling young man beside him. The disparity between the two seemed insurmountable. The king silently approved his decision to have the time of the demonstration posted in the marketplace. If the fellow failed, as seemed likely, the public nature of that failure would make his own forgiveness of it appear more magnanimous, and strengthen his hold on the man. Of course, it would also make the failure more humiliating- but that couldn't be helped, and failure had sharp teeth whether or not anyone else saw it bite.

The little boy, Gelon, also stared at the ship, then at Archimedes. He did not usually like going to public functions with his mother, but when his father had explained what this one involved, he'd been eager to come. "You're going to move that all by yourself?" he asked.

Archimedes grinned and tugged his cloak straight. "Certainly."

"You must be strong!" said Gelon admiringly.

"I don't need to be," said Archimedes happily. "That's the point. There are two ways to move something heavy. One is to be very strong, the other is to use a machine. Do you see those pulleys?"

A spider web of rope ran between the front of the nearest ship shed and the stone mooring posts on the quay: running through pulleys attached to pulleys, reversed around tackle blocks, attached to other pulleys, run about the axles of toothed wheels, reversed again, and attached to more pulleys. Kallippos was standing by the mooring posts, counting them.

"That's my machine," said Archimedes. "Do you know how a pulley works?"

"You pull on it," said Gelon authoritatively.

"That's right. You pull on a rope which travels twice as far as the load moves, so it takes you half as much effort. By using enough pulleys you can move any load with any effort. But maybe we should see first whether strength will move the ship. Lord King, since you've brought along so many of your guardsmen, perhaps they'd like to push?"

Hieron had brought some thirty guardsmen along with him, under the command of Dionysios. (Archimedes looked for Straton among them, but for once did not find him.) The men were perfectly happy to set down their spears, brace themselves against the ship's sides, and push. Faces crimson with effort, feet skidding on the slipway, they struggled for a while without success, then gave up. The watching crowd groaned in sympathy. Archimedes' grin broadened. "Dionysios!" he called. "Can I give you and your men a ride?"

Dionysios looked starkly disbelieving, and the guardsmen shook their heads pityingly. But when Archimedes hurried over to the ship and pulled down the boarding ladder, they hauled themselves aboard. Dionysios went last. He looked at Archimedes, began to say something, then shook his head and climbed in after his men.

"Me too!" shouted little Gelon, running down onto the slipway. When Hieron nodded his consent, Archimedes helped the child onto the ladder. Dionysios caught the little boy's hand when he was halfway up and lifted him the rest of the way. Gelon ran at once to the ship's prow and climbed onto the figurehead to wave to his father and mother.

Archimedes took a deep breath, then went to the thick rope which emerged from the pulleys and fastened it to the ring he had fixed securely to the ship's keel. He gestured for Marcus to follow him and made his way to the place where the other, thinner end of the rope emerged from its long and convoluted passage. He could feel the crowd watching him; closer at hand, the engineer Kallippos was staring at him, face tight with the same indefinable expression it had worn when they last parted. He tried to ignore them all, and took off his cloak; the sweat evaporated from his bare arms and damp tunic with a sudden delicious coolness. He handed the heavy folds of the yellow wool to Marcus.

"Is this going to work?" whispered Marcus nervously.

Archimedes looked at the anxiety on his face, and for the first time felt a quiver of doubt. He looked at the ship, at the spider-web of rope between it and himself, mentally reviewing his mechanical advantage. It was sound, it should work. It should- but what if a pulley jammed? What if a rope thread jammed a wheel, or a tooth on one broke? Things did break. Had he made enough allowance for the weight of the rope itself?

Everyone was watching him. Oh, Apollo, if he failed, with everyone watching him…

"It's going to work," he told Marcus, with all the resolution he could muster. It should. There was a stool he'd sat on while he was working the system out; he went over and pulled it out from the shade of the ship shed, into the bright sun where everyone could see it, and sat down. "Just coil the rope up when I hand it to you," he ordered Marcus, and took hold of the rope.

Sitting down was bravado, really; it would have been easier standing up. He had allowed for an effort of one talent, but as he started pulling he suspected that he had not made enough allowance for the weight of the rope itself. Still, he might have to dig his heels in, but he could do it. Hand over hand, he slowly but steadily drew the rope; back and forth the rope wove through the pulleys, reducing the load again and again by the distance it traveled until it was commensurate with his effort.

The ship shuddered on the slipway, then began to glide forward. It did not jerk or pitch, but moved so smoothly that at first the watching crowd just murmured, uncertain whether it really was moving at all. Then, unsteadily at first, from a few throats, but growing, came a roar of delighted wonder. Beside him, Archimedes heard Marcus laughing. Seven tons of ship and thirty men were drawn up by a single pair of hands and the power of one mind.

Archimedes drew the ship up as far as the ship shed, then dropped the rope and stood up. The crowd was still cheering. He turned toward them: a sea of faces, with a purple patch before them that was the king. His arms were trembling from the strain of pulling, and he felt suddenly dizzy. Nobody had ever cheered for him before. He had expected to feel triumph, but he was suddenly afraid. Under this acclaim he felt exposed, freakish. It was not really such an exceptional thing. The principles had always been there, unchangeable as stars. He had simply applied them. "O Apollo!" he whispered, as though he were genuinely begging the god for help.

Marcus caught his shoulder. "Wave to them!" he whispered, and Archimedes waved: the cheers redoubled. He shook his head angrily.

"Sir," said Marcus, "your cloak."

Archimedes shook his head again and began walking back toward the king without it.

As he drew closer, he noticed first his sister's face. Philyra's cloak had fallen off her head and one arm, her hair was tousled, and she was radiant. Then, next to her, he saw Delia, still applauding, her eyes glowing with pride. His irrational dread suddenly lifted, and he grinned back at both of them. Philyra gathered up her skirts and ran over to him, laughing. "Medion!" she exclaimed, flinging her arms around him. "It was unbelievable!"

He put an arm around her, but said nothing, and kept walking until he was facing the king.

Hieron's face too had lit with sheer delight, and as soon as Archimedes was within reach, he seized one surprised hand in both his own and shook it. "You really could move the earth, couldn't you?" he asked, grinning.

"With another world to stand on," replied Archimedes, "anyone could."

The king laughed, still shaking the hand. Then his eyes flicked to the system of pulleys and he let go. "Can I try it?" he asked.

Archimedes blinked and looked back at the ship, from which the guardsmen were now leaping. "It will have to be pushed down the slipway the hard way," he said apologetically. "And I'll have to, uh, move some of the wheels."

Hieron at once turned to the guardsmen: "Dionysios!" he shouted. "Get some volunteers and push it back down again! I'm going to pull it up this time!"

"I want to!" shouted little Gelon, running to his father.

"You can help me," conceded the king, scooping the boy up. "Come on, Archimechanic, you can tell us where to pull."

The ship was moved up and down the slipway so many times that eventually the foreman of the shipyard came up and begged the king not to wear out the keel of a perfectly good vessel. The king moved it; Dionysios moved it; people fought their way through the crowd to take turns hauling on the rope. Archimedes explained the principle of the pulley so many times he lost count. It wasn't until some time had passed that he realized that he had not seen Kallippos since he first took hold of the rope. He glanced around, looking for the engineerand noticed Chrestos, just arrived flushed and out of breath at the edge of the crowd. Archimedes stared in consternation, then thrust his way through the startled mob to where the slave was standing.

"What's happened?" he demanded. "Did my mother send you?"

The boy was so out of breath from running that he couldn't speak, but he nodded.

"This is your slave?" asked Hieron quietly.

Archimedes stared at him blankly: he had not noticed the king following him. Then he nodded. "I asked my mother to send him," he said, "if my father…"

"She says…" panted Chrestos, "come… as fast as you can."

The world went cold, even in the hot sun. Time seemed to slow down.

"You may borrow my horse," said the king.

Archimedes met the king's eyes, and felt a spasm of wild gratitude at the way his situation had been understood without explanation. "I can't ride," he choked out through a tight throat. "I'll run. Lord, my sister…" He wasn't even sure exactly where she was; she'd been beside him, but he realized now that she'd gone off some time before with Marcus and Agatha. Probably she was sitting down in the shade somewhere, but where? She could not run, not in that thick cloak and long tunic, but she should come home now too, if their father was… She should not be abandoned on the docks.

"I will see to it that your sister gets home as quickly as possible," Hieron said evenly.

"Thank you!" exclaimed Archimedes passionately. He turned and began pushing his way through the crowd that had trailed eddylike after the king. As soon as he had a space of clear cobbles before him, he broke into a run.

Philyra was sitting inside one of the ship sheds on a coil of rope, disconsolately eating the picnic she'd expected to share with her brother. Outside the noise of the crowd bubbled on, festive but with a wildness to it. She felt as though her life had suddenly become dislocated from everything it had been before. She told herself firmly that it was good, it was wonderful that Archimedes was really going to succeed in his new career, that there was no reason for the apprehension that had tightened her stomach and taken away her appetite for the food. But her first exhiliration and pride were irrevocably gone. Things were going to be different now, and she was realizing that she'd liked them as they were.

A soldier came into the ship shed, then stopped abruptly. Philyra grabbed the hot cloak she'd taken off when she sat down, relieved at the way Marcus at once jumped up and stood between her and the soldier.

"Is this lady the daughter of Phidias the Astronomer?" asked the soldier, speaking correctly to Marcus rather than addressing an unmarried girl directly.

Marcus nodded warily.

"Please come with me," said the soldier.

Philyra hurriedly draped the cloak around herself while the slaves heaped the food back into the basket, and they followed the soldier out onto the sunlit quay.

The ship was being eased back into the water and the crowd was beginning to disperse. The soldier led them over to a crimson-cloaked officer and saluted. "This is the lady, sir!" he said, and Philyra modestly held a corner of her cloak up before her face. The officer was the one who'd come to the house once, the captain of the Ortygia garrison- Dionysios, that was the name. "The king wishes to speak with you, lady," he told her, his tone respectful. "Please come with me."

Philyra glanced nervously about, looking for her brother. He was nowhere to be seen. Beside her, Marcus was scowling.

King Hieron was standing beside his white charger. His son was in the saddle, looking pleased with himself, while his wife and the lady in red- the king's sister, someone had said- waited by the litter. Hieron came forward when Philyra was led up and inclined his head graciously. "Lady," he said solemnly, "I am very sorry to be the bearer of bad news. Your brother has been summoned back to his house: it seems that your father's illness has taken a sudden turn for the worse."

Philyra dropped her veil, forgetting modesty, and stared at Hieron in shock.

"I have promised him that I would see to it that you were conveyed home as quickly as possible," the king went on. "And my inestimable wife has kindly offered to take you in her litter. If you and your slave girl would get in, she will drop you off at home on her way back to our own house."

Philyra swallowed, looking at the queen. Philistis came over and took her hands graciously. "I am so sorry that you should receive such terrible news in public," said the queen sincerely.

Philyra bobbed her head, remembered her manners, and vaguely murmured, "Thank you, O Queen." She went over to the litter and climbed in. Agatha followed her, trembling, and then the queen and the king's sister.

Marcus watched the slaves pick the litter up and set off. He felt sick with apprehension, whether for Phidias or Philyra he could not say. No one was paying any attention to him; the king mounted his horse behind his son, the soldiers fell into line, and the royal party set off for the Ortygia. Marcus settled the food basket on his arm and began to walk away. He went slowly at first, but as he left the docks his steps grew longer and longer, and when he reached the house in the Achradina he was running.

Hieron reached his own house before Marcus could make the longer journey on foot. As soon as he did, the king turned to his doorkeeper. "Kallippos needs to speak to me," he said. "Find him for me, and tell him so."

Before the chief engineer could be found, however, Delia returned with the queen, and went at once to see her brother.

Hieron had retreated to the library, where Delia found him reading. He looked up quickly when she came in, then put his scroll aside and moved his feet so that she could sit down. "Did they arrive in time?" he asked.

Delia nodded. "But he wasn't conscious," she added. "They had their own doctor there, and he said it might be hours, and might be any minute. The… wife of Phidias came out to thank us for bringing her daughter home. Philistis offered her any help they wanted, on your behalf, and she thanked us, but said they didn't need any help."

Hieron snorted. "Well," he said, after a minute, "I'm glad they were in time." He picked up his book again.

"What are you going to do about Archimedes?" demanded Delia in a low voice.

He set the book down again. "Keep him," he said fiercely. "Keep him if I possibly can, no matter how much he costs. Zeus! You saw it. It was a game to him, moving that ship: when he understood how the rest of the world regarded it, he was shocked. He's as good as an extra army to any city lucky enough to own him."

"But what are you going to do?"

He shook his head. "I don't know. I've always thought the legends make King Minos sound a most appalling fool, but at the moment I can find some sympathy for the man. He had the most ingenious mind in the world at his disposal, and he didn't want to lose it. So he locked up its owner in a tower. It didn't work, but I can understand why he was tempted to it!"

"You're not planning to lock Archimedes up!" cried Delia. It was more a command than a question.

"Herakles!" exclaimed Hieron, looking at his sister in surprise. "Not if you're going to strangle me if I do."

Delia flushed. Her protectiveness surprised her as well. But that morning she had watched Archimedes do the impossible, and she had forgotten all caution in the wave of delighted pride. Surely she was entitled to feel proud, since she had discovered him? And to feel responsible as well, if her brother's notice threatened him. "You're not, are you?" she asked more quietly.

"No, I am not," said Hieron. "Minos was a fool. You don't get people to work for you by locking them up in towers, particularly when they're a great deal cleverer than you are yourself. Daedalus, you remember, simply devised an impossible means of escape and flew away. I don't think Archimedes could fly, but after today I wouldn't like to bet that he couldn't if he really applied his mind to it."

Delia relaxed. "You worried me," she complained, and at last took the offered space on the couch.

Hieron was gazing at her thoughtfully. "You like him," he stated.

She blushed again. "I discovered him," she said. "I… feel responsible. I don't want him hurt."

Hieron nodded, as though this made perfect sense. "I promise you, I won't hurt him. To tell the truth, I think it would offend the gods if I did. It would be like smashing a priceless work of art. I've never seen anything like him."

"I will not take orders from him," said a voice in the doorway, and they both looked up to see Kallippos standing there. The royal engineer was disheveled and sweat-stained, and his feet were covered with dust: he had been walking. He glared angrily at Hieron. Delia jumped nervously to her feet.

Hieron simply smiled. "Kallippos, my friend," he said, "I'm glad you've come. Shall we go into the dining room and have a cup of chilled wine?"

"I won't take orders from him," repeated Kallippos, as though Hieron had not spoken. "I'm not Eudaimon, King. I don't just copy, I think. I won't let somebody else do my thinking for me. I'm too old and my family's too good to stand being that man's subordinate. I resign."

"I was afraid you were going to say that," said Hieron. "Now, my friend-"

"You arranged it!" shouted Kallippos furiously. "You invited him to do something impossible, and asked me to say he couldn't. Well, I said it: I don't deny it. And I was wrong. But I am not going to take orders from some flute boy from a mud house in the back streets of the Achradina!"

"I don't ask you to," said Hieron.

"Hah!" sneered the engineer. "You may make his position officially equal to mine, but we both know you intend him to be my superior."

"I have no intention of appointing Archimedes son of Phidias to the position of royal engineer," declared the king. "May the gods destroy me if I do."

Kallippos stared for a moment in astonishment, then shouted, "Then you're out of your mind! You saw what that boy did! Do you think I could have done that? I couldn't even have done the catapult!"

"My friend!" protested Hieron. "You are the finest engineer in the city's employ, and if you left, I could not replace you. For you to resign now, when we are threatened with all the terrors of siege, would be a disaster for the whole of Syracuse. How can you contemplate such a thing? Archimedes is young and inexperienced. I know your quality, and I never expected you to work under him. Before the demonstration I thought it might be possible to appoint him as engineer with a rank equal to yours. Now I see that this is quite impossible. I repeat, I am not going to appoint him to a salaried position at all."

Kallippos opened his mouth to speak, then shook himself. "King," he said, trying again. "Don't you understand that he's better than I am?"

"My friend," said Hieron, "I know perfectly well that he has Apollo and all the Muses taking turns to breathe into his ear. But his natural home is Alexandria, and any job I gave him he would eventually come to regard as a prison. So I am not going to give him a job. For what he makes for the city, he will be paid, and generously, but what he actually does will be up to him: that will please him far more than any position I could offer. He is not, and never was, your rival. You are an engineer, and a very good one: he is a mathematician who happens to make machines occasionally. All I want you to do is to join me in asking him to assist in constructions for the good of the city where we judge he has a contribution to make. Now, do you want to come into the dining room, wash your feet, and have a cup of chilled wine?"

Kallippos stared at Hieron for another long minute. Then he made a slow, snorting noise, half laugh, half sigh, wholly relief. Delia saw that he had not in the least wanted to resign, but had felt that he had no other option. "Yes," he said now, starting to smile. "Yes, O King. Thank you."

Delia watched the two men go out, then sat back down on the couch, heavily. She knew her brother well enough to understand that Hieron had not said quite what Kallippos thought he had. Hieron had known that Kallippos was too proud to agree to become another man's subordinate- especially when the other man was younger and of a less distinguished family. Now he had arranged matters so that Kallippos would be content to ask Archimedes to "assist" with particular problems- and, no doubt, take all the "advice" he was given. Eudaimon, too, had been brought "in hand." There remained only Archimedes himself to bring under the yoke- and that would not be done in the way she'd feared. She should have realized that her brother would never do anything so crude as bind a man to an iniquitous contract of employment. The kind of chains he preferred were both subtler and stronger, forged in a gray area between manipulation and beneficence, put on with gifts and received with gratitude. But what sort of chains he might find for Archimedes she could not guess.

Phidias died at about four in the afternoon, without regaining consciousness. Arata had watched him all morning with growing concern, and at noon, when his breath seemed to be failing, sent for her children. All through the long hot afternoon the family sat together about the bedside, while Phidias' breath stopped, then started, then stopped again. When the end finally came, they did not at first recognize it, and waited for some time for the frail gasping to resume. Eventually it became clear that it would not. Archimedes covered his father's face, and the women of the household began to beat their breasts and raise the high-pitched ritual keening.

Archimedes went out into the courtyard, splashed his face with some water, and sat down against the wall, hands dangling limply from his upfolded knees. He was not sure what he believed about an afterlife. Like most educated Greeks, he found the stories his own people told about the gods and the Underworld totally incredible, but to replace those stories he had only contradictory reports from the teachings of the philosophers. The soul was the true Platonic Form, immortal and unchangeable, struggling through the shadow play of the world, reborn many times until it could find its way back to the God who had made it. The soul of the Wise was king, and by virtue might attain eternal union with the Good. The soul was a handful of atoms, born with the body, disintegrating with that body's death, and the gods lived apart from the world and had no interest in it. What was he to believe?

It hadn't mattered much, before.

After a while, he went upstairs and took out his abacus and compasses. He drew a circle in the sand: that was immortal and unchangeable. Its end was its beginning, and it defined the total of all angles. The ratio of its circumference to its diameter was forever the same number: three and a fraction. What that fraction was, though, was impossible to calculate. Less than a seventh. Try to define it further, and it slipped away from you, more precise than your measurement, infinitely extendable, infinitely variable. Like the soul. Like the soul, it could not be comprehended by reason.

That thought was comforting.

He inscribed a square in the circle, then an octagon, and began calculating in earnest.

When Arata came up about three hours later, she found her son crouched over the abacus, sucking the hinge of his compasses. Scratched into the sand was a multi-sided polygon, circumscribed by a circle and filled with a tangle of superimposed reckonings.

"Dearest," she said gently, "the neighbors have started to arrive."

It was traditional for friends and neighbors to pay their respects to the dead as soon as possible, and for the family to greet them dressed in black, with hair cut short in mourning. Arata's hair was freshly cropped, and she had draped about herself a black cloak bought many years before for her mother's funeral and worn infrequently since. Philyra, too, was dressed in mourning; even the slaves were prepared. But Archimedes still wore the good tunic he had put on that morning, and his hair hung in tangles over his forehead. Faced with his mother's summons, however, he merely took the compasses out of his mouth and said, "It's more than ten seventy-firsts and less than a seventh."

Even if the evening light hadn't shown clearly the dry trails of tears across his cheeks, Arata would have known better than to mistake his absorption for lack of feeling. She crouched down beside him as quietly as though he were a wild animal she was trying not to startle. "What is?" she asked.

He gestured with the compasses at a point on the diagram where the circle's circumference was cut by its diameter. The letter [Phi] had been written in the angle between them. "That." There was a silence, and then he said, "People often say it's three and a seventh, but it's not. It's not a rational number at all. If I could draw more sides to the polygon, I could approximate it more closely, but no one can ever calculate it absolutely. It goes on forever."

Arata regarded the circle and the scratched figures. Phidias would have understood them. That thought had just become painful. "Why does it matter?" she asked.

He stared blindly at the circle. "Some things do go on forever," he whispered. "If some part of us wasn't like them, would we be able to understand that?"

At that she saw the reason for his calculations, and, strangely, found comfort in them. Her husband too had loved and believed in these infinite things, and now he was with them. She put an arm about her son's shoulders, and for a moment they were both quite still. Then Arata sighed. "Dearest," she said resolutely, "you're the head of the family now. You must change and come down and greet the neighbors."

Archimedes dropped the compasses and put his hands over his face. He did not want to speak to anyone.

"You must," Arata insisted. "He was always so proud of you. Let everyone see that he left a son who honors him."

Archimedes nodded, pulled himself to his feet, and went with her. The black cloak she found for him had been his father's. Putting it on made him shudder.

Several of the neighbors, alerted by the commotion earlier in the day, were already gathered in the courtyard. Archimedes greeted them courteously, and they responded with condolences, then went to pay their respects to the body. Phidias, washed, dressed in his best clothing, and garlanded with herbs and flowers, lay on the sickroom couch facing the door, eyes closed, one thin hand clutching a honey cake as an offering to the guardian of the realms of the dead. Archimedes gazed at the corpse with a curious sense of indifference. This formal object had nothing to do with the astronomer, the solver of puzzles, the musician who had brought him up.

Philyra had already seated herself at the head of the couch and begun playing a dirge upon the kithara; as the women of the neighborhood arrived they sat down next to her and joined in, either singing or simply keening, so that the room filled with the thin moaning of grief. Arata sat down on a chair next to the couch, but made no sound, and covered her head.

Archimedes wondered if there were more people he should inform of the death. Phidias had been an only child, but Arata had a brother, and there were friends. Should he ask his mother about it? It seemed better not to disturb her. What about the funeral? In this hot weather it would have to take place next day. He supposed he should be arranging wood and incense for the pyre, and seeing about a funeral feast. Did he have money for it all? Presumably the shopkeepers would give him credit.

It seemed unbelievably strange to be worrying about such things, with his father lying there dead.

He went back into the courtyard, and was relieved to see Marcus returning from the public fountain with a heavy amphora of the water the visitors would need to ritually purify themselves from the contact with death. "Marcus," he whispered, hurrying over to him, "who should we send to about this?"

"Your mother's already informed everyone," said Marcus. Archimedes blushed, ashamed that Arata should have had to worry.

The visitors kept arriving all evening. When it began to grow dark, the slaves found torches and set them up in the courtyard and by the door. They had just been lit when Archimedes became aware of a commotion in the street outside- and then Hieron came through the open door, followed by his secretary. The unexpected appearance of the lord of the city caused an alarmed ruffle in the now-crowded courtyard, but Hieron ignored the stir and walked straight to Archimedes. "My condolences," he said, shaking hands. "You have lost a father who was one of the best men in the city, and your grief must be great."

Archimedes blinked, fiercely pleased by such a public declaration from such a source. The neighborhood had always liked Phidias- but it had always laughed at him, too. "Thank you," he replied. "I do grieve for him, very much."

"It would be your shame if you did not," said Hieron.

Like any other mourner, he went on to the sickroom to view the body; when he entered, the women were so startled that they stopped keening, and there was a sudden, profound, reverberating silence. Once again, Hieron ignored the effect he produced, and he bowed his head respectfully to the dead. "Phidias, farewell!" he said. "I always regretted that I could not study with you longer. May the earth be light upon you!" Then he went up to Arata, who was still sitting veiled beside her husband's body. "Good lady," he said, "your loss is great. But I trust that the outstanding promise shown by your son is some comfort to you."

Arata was utterly speechless. She clutched her cloak tightly against her breast and nodded wordlessly. Hieron nodded back, in farewell, and withdrew.

Back in the courtyard, he turned again to Archimedes. "Please," he said, "allow me to show the esteem in which I held your father, and the respect I have for you, and permit me to provide for the funeral. If you agree, my slaves and the resources of my house are at your disposal."

"I, uh," stammered Archimedes, almost as speechless as his mother. "I, uh- thank you."

Hieron smiled. "Good. Just tell my secretary Nikostratos here what you want, and he'll see that it's arranged for you." He gently directed Archimedes toward the secretary with a pat on the arm, and turned to go. Then he turned back again. "Oh," he added, "it struck me that you haven't yet been paid for that astounding catapult you built. I'm ashamed that I can't possibly pay as much as such a fine machine is worth, but Nikostratos has something for it. I wish you joy!" With that he gave his hands a perfunctory ceremonial wash in the water which stood ready by the narrow door, then stepped out into the night.

Archimedes looked at the secretary. Nikostratos, a bland-faced, nondescript man in his thirties, burdened with a heavy satchel, looked back. "Do you wish to tell me what arrangements you want now, sir?" he asked.

"Uh- yes," said Archimedes, keenly aware of the astonished neighbors. "Um- I suppose we should go into the dining room."

Marcus fetched lamps for the dining room, then stood listening while the secretary jotted down the requirements for the funeral. He added up the bill in his own mind as they proceeded: wood, incense, wine and cakes for a hundred- Archimedes said sixty at first, but the secretary thought this too mean. It was not going to come to less than twenty-five drachmae, Marcus concluded, and would probably be considerably more. The king was not going to save money by paying for a funeral and skimping on a catapult. And Marcus doubted that Hieron meant to skimp on the catapult either, despite his words about not paying as much as it was worth. He just wished he knew why the king of Syracuse was putting himself out to flatter and conciliate a catapult engineer.

When the funeral requirements had been fixed, the secretary took out an olive-wood box, which he set down before Archimedes. "The money for the catapult," he stated. "Can I ask you to sign for it?"

Archimedes looked at it vaguely and asked, "How much is it?"

"Two hundred and fifty drachmae," replied the secretary matter-of-factly. He pulled a ledger out of his satchel.

Archimedes stared, then lifted the lid from the box. New-minted silver which had been packed to the rim spilled out onto the dining table. He shook his head. "It was supposed to be fifty!" he protested. "And the king said that-"

"I was instructed to say that if the catapult were priced according to its value, it should be a thousand," said Nikostratos.

Archimedes stared at him for a long moment in silence. Then he looked down and picked up one of the coins which had fallen onto the table. Hieron's face, diademed and smiling, had been stamped in profile on the obverse. He studied it. A number of things he had seen and heard without really paying attention fell into place. He had always known that he was exceptional as a mathematician, but he'd thought that at mechanics, which were merely a hobby to him, he was only ordinarily good. But he realized now that Epimeles had not been flattering him: the Welcomer really was the best catapult built in Syracuse in twenty years. That pivot- that was something nobody had thought of before. The reason the workshop slaves had laughed was that he hadn't realized that. Eudaimon had been not merely short-tempered, but jealous. Kallippos had really believed that it was impossible to move a ship single-handed.

He was the best engineer in the city, and what his mind and hands could shape was so powerful that the king himself was now trying to cultivate him. This piece of silver, shining in his hand, was a tribute to his power. It was deeply satisfying, but at the same time frightening. The Roman army might soon be arriving to lay siege to Syracuse, and his own abilities would be in the first line of defense against it. The danger at once seemed much closer, and much more real.

He took fifty drachmae out of the box, then pushed the box itself back toward Nikostratos. "Tell the king I thank him for his generous offer," he said, "but I will take the price agreed, and no more."

Nikostratos was genuinely surprised- a strange sight, in such a dry man. He tried to push the box back. "This is the sum the king instructed me to pay you," he protested. "He won't want it back!"

Archimedes shook his head, "I am Syracusan; I don't need to be paid extra to defend Syracuse. I will take the agreed price for the catapult because my family needs it, but I will not profit from my city's urgent need by taking more."

The secretary stared. Archimedes took the ledger out of his hands and found the entry- "To Archimedes son of Phidias, for the one-talent catapult for the Hexapylon, 250 dr." He crossed out "250 dr" and wrote above it, "50 dr., as agreed," and signed his name.

Nikostratos suddenly smiled widely. "The gods have favored Syracuse," he said quietly. He took back his ledger and the olive-wood box, and put both away. Still smiling, he murmured his good nights and departed.

Archimedes looked at Marcus, who still stood watching by the door. "I suppose you disapprove of that?" he said challengingly.

But Marcus grinned widely and shook his head. "Not me," he said. "If a man isn't willing to fight for his own city, he deserves slavery."

And you, Marcus thought to himself, have just refused to be bought.