The Sand-Reckoner

3

Archimedes kept to his agreement to meet the guardsman Straton by the naval quay that evening.

The rest of the family had accepted his decision not to follow his father's career as calmly as Philyra had. Arata, in fact, was relieved to find him searching for any work: she'd worried that he might not appreciate the necessity of earning money. She fussed about to ensure that he looked an aspiring royal engineer, and sent him off bathed and barbered and dressed in his new tunic and cloak. He tried to avoid the cloak- too hot for June! — but his mother draped it firmly about his shoulders. "It looks distinguished," she told him, "and you need to impress this man."

"He's only a soldier!" Archimedes protested. "He's just going to tell me who I should really talk to!"

"Even so!" Arata declared. "If he's impressed, he'll pass that on to his superior."

She wanted to send Marcus with him, too: a gentleman ought to have a slave in attendance. Archimedes was nervous, though, that they might meet the Tarentine mercenary Philonides again. He explained to his mother and sister what had happened at the docks.

Philyra listened to the account with indignant astonishment. She glanced at Marcus' impassive face, remembering the bruise on his side. "That's outrageous!" she exclaimed angrily. "We have a right to keep our own slave! You should have taken that stupid mercenary to a magistrate, and complained."

Archimedes just shrugged. "I wouldn't threaten a mercenary!" he said with feeling. "And courts are chancy places, especially with a war on. I don't know what sort of Italian Marcus is- do you?"

Philyra glanced at Marcus again, startled this time. She had never connected him in her mind with the great new power to the north. Yes, she'd known he was Italian, but there had always been wars in Italy, and in each war some prisoners always ended up in the Syracusan slave market. It had always been enough to call them simply "Italian" and assume that slavery had absorbed all the differences between them.

"Well, what sort of Italian are you?" she demanded bluntly.

Marcus' face was carefully blank. "I'm not a Roman," he muttered. "Roman citizens are never slaves." Then he added, in embarrassment, "Mistress."

"It doesn't matter what sort he is," said Arata resignedly. "If the question were raised in a court, we'd have endless trouble trying to prove anything at all. Better to avoid courts if we can." She clapped her hands and jerked her head at Marcus. He retreated back into the house, relieved.

Archimedes started for the door, but Arata caught his arm and drew him aside before he reached it. In a voice too low for the slaves to overhear she said, "My dear, have you given any thought as to whether we should sell Marcus?"

"No, of course not!" said Archimedes, surprised. "We don't have to sell him just because he's Italian!"

"Not that," whispered Arata, gesturing for him to keep his voice down. "We don't need four slaves, especially since your father sold the vineyard, and we can't afford to feed them. If we don't sell Marcus, it will have to be Chrestos. We couldn't sell Sosibia, not after so many years, and little Agatha- it wouldn't be right, my dear."

Archimedes hunched his shoulders unhappily. He understood now. His mother wanted him to start looking for a good buyer for one of the slaves right away. The decision about who to sell and where was his to make- because it would not be right to throw this sort of decision onto his father, not now, and the women lacked any authority in law.

He did not want to sell anybody. Marcus would hate to be sold, he thought absently. He would really hate it, no matter who the buyer. He liked Marcus, relied on Marcus: he could not possibly inflict such a humiliation upon him. But Chrestos- he could remember holding Chrestos as a newborn baby. How could he take money for a member of his family? Money wasn't worth it. He hated to think about money at the best of times.

"There's no hurry!" he protested at last. "The money I brought back from Alexandria will last us for a month or two, and after that, anything could happen. There's a lot of money in engineering. We could all become rich! It would be stupid to sell people if we don't need to."

Arata sighed. Some people might get rich from engineering, but she did not believe her son ever would. He was too unworldly, too softhearted. Like his father. She couldn't even complain at it: it was a quality she loved in them. She did not like postponing hard decisions, though, especially in such uncertain times. "If we wait until we're hungry," she pointed out quietly, "we'll have to take the first buyer that comes along. If we sell now we can choose a good home for him."

Archimedes squirmed uncomfortably. "Can't we at least wait to see if I get this job?" he pleaded.

His mother sighed again, resignedly this time. She did not want to sell any of the household slaves, either, and it was true they had a month or so of grace. She nodded, and her son gave a sigh of relief.

Philyra, who had hung in the doorway listening to this, went back into the courtyard of the house. Marcus was there, taking down his master's laundry. Philyra studied him for a minute, wondering, for the first time, what he had been before he was a slave. She had no clear memories of the time before he came to the household: he had always been there.

Earlier in the day she had indeed mentioned her suspicions of him to her brother. Archimedes had dismissed them at once. "Marcus?" he'd said. "Oh no! He thinks slaves who steal deserve the whip, not the stick, and he prides himself on his honesty. No, no, I'd trust Marcus with a fortune." Now he had backed that confidence with a refusal even to contemplate selling the slave.

But the problem was, he had trusted Marcus with a fortune, and she still couldn't imagine how that fortune could have disappeared in a year without dishonesty from someone. Archimedes' confidence merely made her feel guilty about her own suspicions.

The slave felt her eyes on him and turned toward her, his arms full of laundry, his face mild and inquiring. She noticed, as though for the first time, the crooked identation where his nose had been broken, and she wondered how that had happened, and when. "What sort of Italian are you?" she asked him again.

He let out a long slow breath, looking away from her. "Mistress…" he began, then gave a helpless jerk of one hand, flapping linen. "Mistress, I'm a slave, your brother's. You know that's true. Anything else I said might be a lie."

She gazed at him soberly. "When did you break your nose?"

He set his armful of laundry carefully down on an overturned washtub and turned back to the line for the last item. "Long time ago, mistress. Before I came to Sicily."

A soldier had broken it, during the first year of his slavery. He'd resisted the man's attempt to bugger him, and had been beaten senseless because of it. When he woke up, it was to find himself lying at the feet of the Campanian slave merchant who'd sold him to the soldier, and the soldier and slaver arguing about whether the soldier was due his money back. "Look what you've done to his face!" the slaver had complained. "Who's going to want him now?" Marcus had lain there, mouth full of blood, aching in every muscle, and hoped that nobody was going to want him now, because he didn't think he'd be able to resist so fiercely again; he'd give in, and make a whore of himself. He had been seventeen.

"Was it in battle?" asked Philyra.

Marcus shook his head. He folded the last tunic, set it down on top of the others, then picked the pile up. "Just a fight."

"But you were in a battle. You were enslaved after a battle."

"Yes," he agreed, meeting her eyes. "I was in a battle. We lost."

Philyra was silent a moment, thinking of the war to the north, and of the precariousness of the freedom of Syracuse. She shook her head, and Marcus took the gesture as dismissal. He gave a nod and set off upstairs with his pile of clean dry clothes.

It was dusk when Archimedes arrived at the sea gate. if the tarentine had indeed shared straton's shift, he'd taken himself elsewhere, and Straton was alone, leaning against the inside of the city wall, shield forced half over his chest, one leg resting on his slanted spear. He unfolded himself when he saw Archimedes and tipped his shield onto his back again. "There you are!" he said with relief. "When I asked around about your question, my captain got interested. He says they do need more engineers, for the army and the city both. He wants to talk to you. He's waiting for us at the Arethusa. That all right?"

Archimedes blinked and mentally thanked his mother for her fussing over the cloak. "F-fine!" he stammered hastily. Straton's captain was presumably the man in charge of the garrison of Syracuse while the rest of the army was away. He could, if he wanted, ensure that Archimedes was offered a job.

The Arethusa proved to be an inn on the promontory of Ortygia, near the freshwater spring of that name. Archimedes was not familiar with it- he had rarely ventured onto the citadel- but he noticed as they approached that it was a good inn. The building was large, faced with stone, and had probably been converted from an upper-class mansion. Its sign, which had some artistic pretensions, depicted the nymph Arethusa, spirit of the spring and patroness of the city, reclining among the reeds with the citadel of Ortygia in the background. Archimedes eyed her shapely nudity, and decided that yes, the inn probably did rent female company as well as sell food. He fingered the coins in his purse resignedly. The evening was clearly not going to be cheap, and he knew that he would be paying. He could not complain: treating Straton's captain to an evening's entertainment would oblige the man to be helpful.

Straton clumped into the inn's main room, spear over his shoulder, and gave his name to an obsequious waiter. Archimedes glanced warily at the painting of carousing centaurs on the wall and the silver-chased hanging lamps, and added another three obols to the likely bill. The waiter smirked and bobbed and led them to one of the inn's small private dining rooms. A short wiry man in his early thirties was already seated on the single couch, nibbling at a dish of olives; he rose politely when Archimedes and Straton appeared. Straton saluted; Archimedes extended his hand.

The captain smiled and shook hands. "You're the engineer?" he asked. "I'm Dionysios son of Chairephon, captain of the garrison of the Ortygia. I've already ordered- I hope that's all right?"

Dionysios was not wearing armor, though a red officer's cloak hung over the back of his couch and a sheathed sword over the arm. When Straton hesitated awkwardly in the doorway, his superior grinned at him. "We're both off duty, man," he said. "Make yourself comfortable."

Straton gave a sigh of relief, set his spear and shield against the wall beside the door, then dropped onto the free end of the couch and began to unfasten his baldric. Dionysios grinned again, this time with knowing sympathy for long hours standing guard, for sore feet, a stiff back, and boredom.

Archimedes, feeling much the odd one out, took the least comfortable place, in the middle of the couch between the two soldiers. The obsequious waiter bobbed about taking orders, then retreated.

"Straton tells me you've just come back from Alexandria and are looking to be of service to the city in the war," said Dionysios.

Archimedes nodded. "But," he added awkwardly, "I've found that I can't go up to Messana to join the army. When I got home- That is, my father's dying. I can't leave Syracuse until- You understand what I mean. If there's something I can do here in the city…" He trailed off with an uncertainty he did not feel. He had left his father to endure illness alone: he would stay with him now, until the end.

"Ah," said Dionysios. "I am sorry."

"Bad thing to come home to," said Straton sympathetically. "That, and the war."

Archimedes made an inarticulate noise of agreement.

There was a decent silence, and then the captain asked about Alexandria.

They talked about the city through the first course of the mealthe Museum, the scholars, the temples; the beauty of the courtesans. Straton was silent at first, nervous in his commanding officer's presence, but Dionysios was cheerful and relaxed, there was plenty of wine, and before long they were all three talking freely. Dionysios swirled the fragrant red in his wide-bowled cup and praised Egypt. "The House of Aphrodite," he said. "That's what they call Alexandria, isn't it? Everything that exists anywhere in the world is there, they say- everything anyone could desire. Money, power, tranquility, fame, learning, philosophy, temples, a good king, and women as beautiful as the goddesses who once came to Priam's son Paris to be judged. I'd love to go there!"

"It's the House of the Muses," agreed Archimedes warmly. "It draws the finest minds in the world as the Heraclean stone draws iron. I didn't want to leave it."

"But you've come back to Syracuse, because of the war?"

He nodded. "And because my father was ill."

Again there was a moment's silence, and this time Archimedes realized that it was more because of the mention of the war than from tact over his father's illness. The war was a subject that weighed heavy on the minds of the two soldiers, but one they did not want to discuss. Twelve years before, the Roman Republic had defeated an alliance consisting of all the Greek cities of Italy, half a dozen rebellious Italian tribes, and the army of the kingdom of Epirus across the Adriatic. The forces had been commanded by the brilliant and adventurous Epirot king, Pyrrhus, who was said to have been the finest general of the age. How could Syracuse alone succeed where such an alliance had failed? Her only hope of victory lay in the alliance with Carthage- and Carthage had always longed for her destruction. How could anyone discuss this war? What was there to say about a conflict where one's enemies were preferable to one's allies?

The waiter returned with a dish of broiled eel in beetroot saucethe main course- then filled up the wine cups and departed again. Dionysios helped himself to some fish. "Do you know anything about catapults?" he asked, finally getting down to the business that had brought them there.

Archimedes' earlier discomfort had melted: the company and talk had been almost easy enough to be Alexandrian, and the food was better. Sicilian cooking had always been famous as the finest in the Greek world. He scraped some fish onto his piece of bread, took a bite, and gave the answer that came most naturally to him. "The really interesting thing about them," he announced around his mouthful, "is how you make them bigger. The critical feature is the diameter of the bore in the peritrete. To increase the power of the throw you have to increase the other dimensions in proportion to the increase in the diameter of the bore. So it's the Delian problem in another form!"

Captain and guardsman stared confounded, and he realized that the company wasn't very Alexandrian after all. "The problem of how to construct a solid a given amount larger than a similar solid," he explained apologetically. "You, um, have to calculate the mean proportionals."

"What's Delian about that?" asked Straton.

"People started trying to do it when the priests of Apollo on Delos wanted to double the size of an altar."

"Don't you just double all the measurements?"

Archimedes gave him a look of astonishment. "No, of course not! Say you have a cube measuring two by two by two; that gives a volume of eight. Doubling the measurements to four will give you a volume of sixty-four- eight times as big. What you need-"

"What I meant," interrupted Dionysios pointedly, "was, do you know how to build catapults?"

"What's the peritrete, anyway?" asked Straton.

Archimedes looked from one of them to the other. "Do you know anything about catapults?" he asked.

"Not me!" declared Straton cheerfully.

"A little," said Dionysios. "The peritrete's the frame, Straton."

"The part the arms stick into?"

Archimedes dipped his finger in his wine and traced on the table the peritrete of a torsion catapult: two parallel boards separated by struts. He added the two pairs of boreholes, one at each end of the frame, with a column of twisted strings running from the top bore to the bottom. Each mass of strings gripped one arm, which extended from the frame so that the catapult looked rather like an immense bow, lying on its side and with a gap in the center to allow passage for the missile. A bowstring ran from the tip of one arm to the tip of the other, and a beam with a slide was fixed beneath the frame's center to hold the missile.

The two soldiers leaned over the table and examined the sketch. The waiter came back to refill the cups, regarded the blotted table with displeasure, but, at a glance from Dionysios, refrained from wiping it.

"So which is the critical feature?" asked Dionysios.

Archimedes tapped the boreholes. "All the force of the catapult comes from the strings," he said. "The twist in them is what makes the arms of the catapult spring forward after being drawn back. The thicker the column of strings, the more force they exert and the heavier the missile you can throw. The greater the diameter of the bore that holds the strings, the more powerful the catapult."

"And how powerful a catapult could you, personally, build?"

Archimedes hesitated, blinking. Dionysios' question seemed to have missed the point of his own explanation. "There's no theoretical limit!" he protested. "The most powerful catapult I've ever examined was a one-talenter in Egypt, but-"

"A one-talenter?" interrupted Dionysios eagerly. "You could build a one-talenter?" Stone-hurling catapults were classified by the weight of the missile they could throw. A talent- about sixty pounds- was officially a man's load, and the one-talenter was generally the most powerful catapult in a city arsenal. A few larger machines had been made from time to time, by exceptional engineers for great kings, but ordinarily even one-talenters were rare. Many cities had nothing heavier than a thirty-pounder.

"Of course!" agreed Archimedes. "Or one bigger than that- but you'd need special equipment to load and draw it."

Straton had been looking more and more uncomfortable; now he cleared his throat and said anxiously, "Sir- yesterday he said he'd never actually built any war machine."

Dionysios looked at Archimedes with surprise and indignation.

"You don't need to have actually built one to know how it's done!" Archimedes declared, defending himself against the unspoken charge of deception. "You just need to understand the mechanical principles. I do. It will take me a little bit longer than it would take a more experienced engineer, but I can produce a catapult that works."

Dionysios regarded him a moment longer, unconvinced.

"Look," said Archimedes, "you don't need to pay me anything until I've produced a working catapult."

Dionysios' eyebrows shot up. "A one-talent working catapult?" he asked.

"If that's what you want. If you have the wood and the strings for it. You know it will be big, don't you?"

"Obviously," agreed Dionysios. "The king has one at Messana, and it's nineteen feet across." He studied Archimedes a moment longer, very thoughtfully now: he was not sure whether he had found a treasure or a self-deluding fool. But there was no need to decide, if no money needed to change hands until a catapult was completed. He turned back to his food. "When the army went off to besiege Messana," he said, "King Hieron left one of his engineers- Eudaimon son of Kallikleshere in the city with orders to make sure that all the watchtowers in the city wall were equipped with their full complement of catapults. Mostly that's just meant renewing the strings, but there are quite a few new machines to be built as well. Some of the old ones are completely worn out, and some of the watchtowers were never supplied to begin with. Now, Eudaimon has had no trouble building the arrow-shooting catapults, but he's not so good with the stone-hurlers. Unfortunately, stone-hurlers- particularly big stone-hurlers- are what the king wants most. So if you can do some, you've got a job."

"I can build stone-hurlers," said Archimedes happily. "When do you want me to start?"

"Come to the King's house in the citadel tomorrow morning," replied Dionysios. "I'll introduce you to Leptines the Regent, and he'll approve your conditions of employment. I warn you, though: I am going to take you up on your offer, and recommend that you not be paid until the first catapult you build has been seen to work."

Archimedes smiled. "Thank you!" he exclaimed. He glanced down at his sketch on the tabletop, and felt a sudden thrill of excitement. A one-talent stone-hurler would need careful planning if it was not to be unwieldy. This was something new, something interesting. He wiped up the drawing with his napkin, dipped his finger in his wine cup again, and began to calculate.

The other two watched him a moment. Then Dionysios looked at Straton and raised his eyebrows.

Straton's answering look was glum.

"What's the matter?" asked the captain.

"I think I may have lost a bet," replied the soldier.

Dionysios looked at him, looked at the now deeply absorbed Archimedes, guessed the general nature of the bet- and laughed. "Never mind!" he said consolingly. "Your loss will be the city's gain- and they have flute girls in this place who could make you forget far worse griefs than that." He clapped his hands, and the waiter, who had been standing impatiently outside the door, entered to carry out the dishes and usher the flute girls in.

In the house near the lion fountain, Philyra was waiting up for her brother. Phidias in his sickroom fell early into a restless slumber; Arata settled on a mattress on the floor beside her husband, where she would wake easily if he needed her during the night. The slaves went to the hot upper room they shared at the back of the house. But Philyra went into the courtyard with the wide-necked lute her brother had given her, sat down on the bench beside the door, and began tentatively to pluck the strings.

Lutes were comparatively new instruments to the Greeks, unknown before the conquests of Alexander the Great. Philyra had seen them before, but never held one: one of her own was the best present she'd ever been given. This one was marvelously beautiful, with a round body of polished rosewood and a neck inlaid with shell. It had a deep, sweet tone, too.

Philyra plucked each of the eight strings in turn, then, with a breathless thrill, stopped them near the top of the neck and plucked them again. She was an accomplished player on the kithara, and knew how to raise the pitch of a string by stopping it on the crossboard with her finger- but for kitharists, such fingering was a virtuoso exercise and its use was limited. The lute promised a whole new dimension to music.

The whole family had always been musical. For as long as Philyra could remember, Arata and Phidias had played together almost every evening, he on the kithara, she on the lyre. Archimedes, as he grew older, had usually joined them on the auloi- the soft-voiced woodwind flutes that were played in pairs- and when Philyra in turn learned an instrument, she too had joined in the concerts. Sometimes the family had played for hours, late into the night, one offering a melody which the others would take up, alter, and pass back. It had often seemed to Philyra that music was an ideal world, that all the best things in the real world were there, but clearer, stronger, more poignant. There was her mother's steadiness, maintaining the balance and rhythm of their common life; there were her father's dreamy gentleness and his sudden tumultuous excitements. And there was her brother, not vague, as he so often was when you spoke to him, but fearfully, ruthlessly precise, and so deep and complicated that she had often struggled to follow him- though in the end he had always resolved his musical knots into an affectionate simplicity. When he left for Alexandria she had tried to play the auloi for a bit, because the strings had sounded so bereft without the flutes' voice to wind among them. But in the end she had gone back to her lyre and kithara. There was something disreputable about a girl playing the flute- and anyway, nobody could play like Medion.

She had missed him. She'd been angry that he hadn't come home when he was supposed to, and bitterly angry when their father fell ill- but now that he was back, the anger was already melting away. She hoped that he would soon return from his drink with the soldier, so that they could play some music together.

She experimented with the lute for perhaps an hour, then, tired by the intense concentration, put it away in her room and came back with her old kithara instead. Easily she plucked out a slow, soft tune with her left hand, while her right struck an occasional ripple of accompaniment with the plectrum.

"Remember once," sang Philyra, her low voice blending with the strings,

"Remember when,

I told you this holy word?

'The hour is fair, but fleet is the hour,

The hour outraces the swiftest bird.'

Look! It's scattered to earth, your flower."

She was very good, thought Marcus, standing at his window and listening to her. But that was no surprise. She'd played well before he left, and she'd had three years to get better.

Behind him, Chrestos was curled up on the pallet they shared, while Sosibia and her daughter shared another bed behind a curtain. But he could not sleep, so he stood there, gazing into the darkness of the courtyard below, and listening to the music.

When he had first entered the household, he had found the nightly concerts disturbing. In his own home there had not been much music. His mother had sung sometimes while she worked, and he and his brother had sung in the fields, but apart from that, music had been something one paid others to perform. He had bought some whenever he had money, because he loved it; now he could not afford it, and had it all the time, for nothing. At first he had resented his own pleasure in it: surely it degraded him that he enjoyed any aspect of his slavery? But he'd got used to music, accustomed to having it around, sensitive to its patterns and undertones. He'd almost forgotten what life was like without it.

Philyra sang on, her voice rising clear and sweet into the dark, old songs from the countryside, new songs from the royal courts, love songs and hymns to the gods. Marcus stood silently at the window, listening and watching the stars above the rooftops of Syracuse. After a while she stopped singing and simply played, passing the tune from right hand to left and back again, and he sat down against the bedroom wall, but kept listening, wondering why this ripple of notes should say so many things more than any human tongue.

At last Philyra stopped, yawned, and sat silent, her kithara upon her lap. Marcus stood up hurriedly, so that he could watch her go- but she did not. He understood then that the music had simply been to amuse herself while she waited for her brother to come home. He hesitated, nervous of approaching her. But what harm would it do for a household slave to advise her to go to bed? He turned from the window, crept out of the room- silently, so as not to disturb Sosibia- and down the stairs.

"Mistress?" he called, stepping out into the courtyard, and even in the dark he saw how she jumped.

"What do you want?" she called, guilt from her suspicions of him adding sharpness to her tone.

Marcus stopped a few feet away, faceless in the dark. "Mistress, don't wait up all night," he said gently. "Your brother may not be back for hours yet."

She made an exasperated noise. "He's bound to be back soon! He's been gone hours already."

"Probably he's treating this man to an evening's entertainment. That means he may not be back until after midnight. There's no reason for you to wait up. I'll open the door for him when he comes."

Night hid Philyra's frown, but not the suspicion in her voice when she said, "He never used to go out drinking until after midnight!"

Innocent! thought Marcus affectionately. To expect that he'd keep the same hours after three years on his own in a famously luxurious city! "He was often out late in Alexandria," he told her. "And tonight he'll have to go along with whatever the other fellow wants, to be sure of his help. It's probably a good sign that he's late: means there's something on offer."

For a moment, Philyra said nothing. She told herself that Marcus was implying that her brother had picked up expensive habits in Alexandria, and that this was exactly what Marcus would say, to account for the missing money. "What was he doing out late in Alexandria?" she asked at last, in a brittle voice. Truth or lies, she didn't really want to know, but it was unfair to go on suspecting Marcus without knowing what he had to say.

But Marcus answered at once, and mildly, "Nothing you need to worry about, mistress. He had a pack of friends, and they'd sit about drinking and talking and… making music, half the night. When there wasn't a lecture on next day, they'd see the sun up."

It still didn't sound like her brother. He'd never been inclined to either drinking or talking, and he'd never had any close friends. She tried to think of a question that would catch Marcus in a lie, but at that moment there was a quick rap on the house door.

Marcus opened it, and Archimedes stumbled in, smelling of wine.

He had not stayed at the Arethusa for the inevitable conclusion of the evening. His father's impending death had shriveled up desire, and whatever their other talents, the Arethusa's flute girls hadn't played the flute very well. It had set his teeth on edge to listen to them. In another situation, he might have offered to play himself and let them just dance, but to have made the offer then would have invited the lewdest of ribaldry. So he had done calculations until his companions were provided for, then excused himself with apologies, paid the reckoning, and come home.

"Can you fetch me a light?" he asked Marcus breathlessly, pushing the flute girls' wreath of wilted parsley farther back onto his head. "I need to write something down."

Philyra jumped up and hugged him, but he shook her off hastily. "Careful!" he exclaimed. "You'll smear it!"

Marcus gave a snort and hurried off.

"Smear what?" she demanded.

"Some calculations I was doing. Marcus! Is there anything to write with?"

"You were doing calculations?" Philyra asked in bewilderment.

He nodded; the gesture was revealed by the sudden glow of the lamp Marcus had just returned with. Archimedes held his left arm toward the light: it was covered with figures sketched in lampblack.

"Medion!" exclaimed Philyra in horror. "It's gone all over your new cloak!"

"Don't worry," he said reassuringly, "I can still read it."

Since Marcus had not brought anything to write with, Archimedes picked up the laundry board, found a lump of chalk, and began to copy the figures from his arm. "I'm going to have to correct most of these when I can look at a smaller catapult," he told the other two, busily writing. "I couldn't remember most of the dimensions to scale them up. But this should be close enough to let me order the wood, which will speed things up."

"You got the job," observed Marcus with satisfaction, and Archimedes nodded absently, frowning at his chalked calculations.

"I thought the man you were seeing tonight was just a soldier!" exclaimed Philyra.

"Oh," said her brother vaguely, "yes. But he asked about who I should talk to and his captain wanted to see me. They really do want engineers. I'm to build stone-hurlers, starting with a one-talent machine."

"What's the pay?" demanded Marcus.

"Uh? To be arranged. Nothing, until the first catapult is complete. But there doesn't seem to be anyone else in the city at the minute who can build big stone-hurlers, and the captain said that they're what the tyrant wants most, so I think it will be good. I'm seeing Leptines the Regent about it tomorrow morning."

"Oh, Medion!" cried Philyra, torn between delight and exasperation. "You must give me your cloak at once. You can't go see the regent all covered in lampblack!"

"You can't start doing laundry at this time of night!" protested Marcus.

Archimedes glanced up, finally realized that his sister had been waiting for him, and blinked. "Philyrion darling," he said firmly, "you should be in bed." Then he noticed the kithara she was clutching and added, "It's too late for music, too. But tomorrow night we can have a concert."

"To celebrate your new job!" said Philyra, happily dismissing the state of his cloak. "Mama and Papa will be so pleased!"

The following morning Archimedes reported his success to his parents; they were, as his sister had expected, pleased. After the first unanswerable questions about pay, however, Phidias asked wistfully, "And will it leave you much time for study?"

"I don't know," replied Archimedes awkwardly. He did not want to admit to his father that he foresaw scholarship squeezed to the edges of his life. "I think- I think not right at the moment, Papa. Because of the war. I will do everything I can to make sure I still have time to talk with you."

"Oi moi, the war!" sighed Phidias. "I pray that our king finds some way to get us out of it soon. It will be a bad war, my Archimedion, a very bad war. Our lovely city is like a dove in the pit with two fighting cocks. I am glad that I at least won't have to see what happens to her. My dear boy, you must look after your mother and your sister for me!"

Archimedes took his father's trembling hand. "I will," he promised solemnly. "But I hope, Papa, that King Hieron does find some way out of the war. They say he's a wise man: he may yet bring peace."

"He's ruled well," Phidias conceded- reluctantly, for he had always supported the city's turbulent longings for democracy. Even Hieron's enemies, however, had to admit that he had ruled well. He had come to power eleven years before in a bloodless military coup, and had since governed with moderation, humanity, and a strict regard for the law- much to the surprise of the citizens, who did not expect such behavior from a tyrant.

"Yes, I pray you're right," Phidias went on, then smiled at his son. "I am glad you're back," he said tenderly. "I was afraid to think what would happen to the house, left headless while the city was at war. You go devise some weapon to destroy our enemies, child. And make sure you get a good price for it!"

"Yes, Papa." Archimedes kissed his father's cheek, kissed his mother, who was attending the sick man, then went out into the courtyard.

Philyra was there, trying to clean his cloak. She had brushed it, beaten it, and poured boiling water on it, and had succeeded only in spreading the oily lampblack more widely. She rolled her eyes at her brother distractedly. "You're going to have to wear something else," she told him.

"It's too hot for a cloak anyway," he replied.

Marcus appeared at the foot of the stairs, carrying an old cloak of plain Egyptian linen. "That has wine stains on it!" snapped Philyra impatiently.

"But you can fold the edge over so they don't show," Marcus replied, suiting action to words.

Archimedes groaned, but spread out his arms and allowed his sister and his slave to drape the linen cloak around him, insisting only that the drape go under rather than over his right arm- "It's more dignified worn over both shoulders!" protested Philyra; "It's also hotter!" replied Archimedes. The other two stood back, assessing whether he looked fit to be presented to the king's father-in-law. Archimedes, however, looked at Marcus thoughtfully.

He had been debating whether to take Marcus along to help with the catapult-making. Marcus could undoubtedly be useful at it. He'd helped with the water-snails and with dozens of less successful machines: he knew how to follow technical instructions. He was strong, quick, and handy with a saw or a hammer. On the other hand- on the other hand, Marcus clearly still had some loyalties to the people the catapults were to be employed against. And the catapult-making would take him in and out of the military workshops and the arsenal- the most vulnerable and strategically important buildings in Syracuse. If someone lit a fire in them…

"Marcus," said Archimedes, "I want you to stay here and see if my mother wants anything done around the house."

The slave's face went blank. He had foreseen this problem, but he hadn't expected his master to have foreseen it as well. "You don't want me to come with you, sir?"

Archimedes shook his head. "You're not Samnite," he explained quietly.

Marcus stood for a moment frowning at him. He was not sure whether he felt relieved, because he was not required to construct devices that might injure his own people, or hurt, because his master thought him capable of treachery. He could feel Philyra's eyes on him, full of shocked accusation: did she really believe he'd be happy to see her city fall to Rome, her brother killed, and herself raped and enslaved? At last he said, "Sir, I swear that I would never do anything to injure this city or this house. May the gods destroy me in the worst way if I'm lying!"

"I believe you, since you swear it," said Archimedes. "But I think it would be better, all the same, if you stayed home."

Marcus hunched his shoulders. "Very well, sir."

Archimedes slapped him on the shoulder. The linen cloak, which was too short to hold its drape properly with the edge folded over, fell off. Archimedes redraped it again awkwardly and set off.

"He thinks you would betray the city!" exclaimed Philyra hotly, as soon as the door had closed behind him. "You have to tell me: what sort of an Italian are you?"

"What difference does it make?" growled Marcus. "I'm not a citizen anywhere. Anyway, what kind of claim does this city have on me to begin with? No one has ever pretended I came here of my own free will." He was a little surprised at his own honesty. "I've sworn I won't do anything to injure the city. Archimedes took my word for it. Isn't that good enough?"

"You know what sort of people the Romans have come to Sicily to help?" demanded Philyra.

Marcus again hunched his shoulders unhappily. The Romans had come to Sicily to help the city of Messana against Syracuse. Messana, however, was a robber state, the home of bandits. More than twenty years before, a group of Italian mercenaries, Campanians, had been posted to the city as a garrison by a previous tyrant of Syracuse; tempted by Messana's wealth, they had taken advantage of the chaos when the tyrant died to seize the city for themselves. They had murdered all the men and taken the women and children as their slaves. Calling themselves Mamertini- "the sons of Mars"- the Campanians had gone on to raid or exact protection money from the neighboring towns, all of which were under Syracusan protection. Syracuse had made war on the bandits sporadically, as Carthage and her own affairs allowed, but with little success- until Hieron rose to power. He had defeated the Mamertini in the field and laid siege to Messana itself. To save themselves, the Campanians had appealed to both the great powers of the West- to Carthage and to Rome.

Carthage had responded first. Always happy to frustrate Syracuse, she had sent a garrison to Messana. But the Carthaginian intervention had provoked a response from the new mistress of Italy. Rhegium, just across the straits from Messana, had fallen to Rome only six years before: Rome was unwilling to allow her African rival to control Messana. She sent her own expedition to the Mamertine city. The Mamertini preferred a Roman garrison to a Carthaginian one- they were Italians, too, after all- and expelled the Carthaginians. Syracuse, which had wanted nothing except to rid herself of a long-standing nuisance, suddenly found herself allied to Carthage and at war with Rome.

"I don't think the Romans should have come to Sicily," muttered Marcus. "It's a bad cause, a bad war. The Mamertini don't deserve any help." He looked back at Philyra's suspicious eyes and declared with sudden fervor, "Mistress, please believe me. I will never betray this house while I live."

Her suspicion changed to puzzled surprise, and he saw that he'd said the right thing, and smiled.

All through the walk to the Citadel, the linen cloak kept slipping. Like all cloaks, it had weights in its bottom corners to help it drape, but with the end folded over this was simply not enough. At the gates to the citadel Archimedes gave up, shook it out, and draped it around himself again, this time with the stains showing. He brushed ineffectually at the new dust patches collected on the walk, then strode through the gates, past the temple of Apollo, and on into the heart of the Ortygia.

King Hieron's house was not a palace. It was a large and elegant mansion, set in a leafy quarter of the citadel, near the Council House. It didn't even have any guards outside it, and Archimedes hesitated in the columned porch, wondering whether to knock on the door or wait for Dionysios outside. He glanced up and down the wide street. It was empty in the quiet morning light, so he knocked.

A middle-aged man in a red tunic opened the door at once and looked at him disapprovingly. "Your business?" he demanded.

"I, um," faltered Archimedes, "I was to see the regent this morning. Dionysios son of Chairephon told me to speak to him about a job. I'm a, um, engineer."

"Catapults," said the middle-aged man dismissively. "Your name is Archimedes? Very well, you're expected. Captain Dionysios is with the regent now, but they're busy. You'll have to wait."

Archimedes was ushered into the house and conducted to a vaulted anteroom which opened onto a garden. There were benches about the marble walls, and he sat down on one. The middle-aged man vanished back the way they'd come, leaving Archimedes to wonder if he was a doorkeeper; if so, he was a very abrupt and exalted one. But perhaps that's what slaves were like in kings' houses. Archimedes sighed and looked down at the marble floor. He scuffed it with a sandal, then took from his purse the piece of papyrus on which he'd fair-copied his calculations from the previous night, plus a few interesting thoughts he'd had that morning and wouldn't mind working on further. He wished he'd remembered to bring a pen and ink. He was looking about for something to use instead when he heard the sound of a flute.

Tenor aulos, he decided at once, set to the Lydian mode, playing a variation on a theme from an aria by Euripides. He listened to it intently for a couple of minutes: the player was good. The tune came to an end; there was a pause; and then the music began again, this time with a peculiarly breathy sound, skirting about the verge of dissonance. He grinned to himself: he recognized that sound. An aulos had a metal slide inside it which allowed a player to cover some of the fingerholes, and thus to play several different modes upon one instrument. This player had opened the slide which separated the fingerings of the Lydian and Hypolydian modes and was trying to play the notes between them. Archimedes had once tried the same thing himself. It had required some very tricky part-fingering, and it still hadn't worked.

He got to his feet and slouched out of the antechamber into the garden, following the music. He knew another way of playing those intermediate notes; he owed it to a fellow aulist to share it.

A passage led through a colonnade from the first garden into a second one. A fountain decorated with carved nymphs stood under a grape arbor, and there were roses flowering. On the edge of the fountain sat the flute player: a girl a year or two older than Philyra. Her black hair was caught up in a silver net, and she was wearing a rose-colored tunic fastened with a silver belt. But her hairnet was disarranged by the leather strap most aulists wore to support their cheeks during a long session, and she was so intent on her playing that she didn't notice Archimedes' arrival: she was a real aulist, not a decorative one. He wondered who she was. By her dress she was rich, but she was too young to be the king's wife and too old to be his daughter. Somebody's concubine, he decided. He coughed to attract her attention.

She lowered the aulos and frowned at him, irritated by the interruption. Her eyes were very black. In a moment, he thought, she'd tell him to go back to the public part of the house.

"It doesn't work," said Archimedes quickly. "But if you take your baritone aulos and set it to the Dorian mode, you can get the right effect if you avoid the B-flat."

Interest replaced the irritation in her eyes. She picked up a second aulos from the rim of the fountain beside her: it was an alto. "My other one's this," she said.

"Then set that to the Lydian mode and the tenor to the Dorian! But Lydian doesn't mix with Hypolydian, part-finger it as you will. When I tried it, I sounded even worse than you did."

She grinned. "Thank you for the compliment! Dorian's better?"

"Try it!"

"I will!" The girl shifted the slide on her tenor aulos, setting the instrument to the Dorian mode. She set the alto to the Lydian, raised both, and began to play the Euripides variation again. Her eyebrows rose; she played on to the end of the piece, shifting from one aulos to the other, from one mode to its neighbor, scattering the notes, bittersweet and sad. When she finished she set the flutes down and looked at him in surprised triumph. "You're right!" she exclaimed, and they grinned at each other.

Then she wiped the mouthpieces and asked, "Are you a professional?"

"What? Oh, flute player. No. I'm a mathematician." Then he bit his lip and corrected himself. "Engineer. I'm to see the regent about building some catapults."

"Catapults!" she exclaimed. "I wouldn't have expected someone who makes machines to be musical."

He shrugged. "Actually, it's a help. You have to tune them by ear."

"Catapults?"

"Mm, the strings. If a catapult's two sets of strings are out of tune with each other, the machine will shoot crooked when you come to fire it."

She laughed. "What do you do to tune them? Pluck them and tighten the key, like a lyre?"

"Exactly! Except you twist the strings, not the key. You have to use a windlass and wedges."

"I like that! The stringed instruments: lyre, kithara, harp, luteand catapult. I suppose big ones have a low pitch, and small ones a high pitch?" He nodded, and she laughed again. "Somebody should write a catapult chorus," she declared. "For scorpions, ten-pounders, and thirty-pounders." She raised the auloi to her mouth again and piped a mad dance on three widely separated notes.

Archimedes grinned. "A friend of mine is trying to build an air-powered catapult," he said. "It could play the flute part. But I'm afraid that all it does is go bang, very loudly, so maybe it should be percussion."

"Oh, no!" she exclaimed, lowering her auloi and putting a hand over her mouth. "An air-powered catapult? Where was this, Alexandria?"

He laughed in surprise. "Yes!"

"It would be! They'll do anything in Alexandria. Since you've been there, tell me: I heard somebody there has built a machine which allows you to play thirty auloi simultaneously. Do you know-"

But Archimedes had broken into delighted laughter. "That's Ktesibios!" he exclaimed. "The same friend who's making the air-powered catapult. He calls the instrument a water-aulos. I helped him with it!"

The girl pulled off her cheek strap and put her instrument down. Her hair, disturbed from its net, dropped black curls around her face. "Does it work?" she demanded. "The- the multiple aulos, I mean. I don't see how it possibly can!"

"It's not really thirty auloi," Archimedes told her. "It's thirty pipes, but they only play one note each. Each is a different length, see, like the reeds in a syrinx. To sound them, you press a key which opens a valve in the bottom of the pipe. Air is forced up into the pipe by the pressure of water in a tank underneath. That's why it's called a water-aulos. See, you have this inverted hemisphere submerged in the water, and two tubes which-"

"A water-aulos," repeated the girl, tasting the new word: hydraulis. "What does it sound like?"

"More like a syrinx than an aulos. Louder, though, and richer-toned- almost bell-like. It can be heard above a crowd. The Alexandrians put one in the theater. I told Ktesibios he ought to call it a water-syrinx, but he preferred his own name for it."

"You said you helped to make it?"

"Mostly I just helped Ktesibios tune the pipes. He never actually had any training in music, though he's the most astonishingly ingenious man. He's-"

"Could you build one?"

Archimedes blinked.

"Not now," said the girl quickly. "I know there's a war, and it's more important to build catapults. But afterward, if there is an afterward- could you build me a water-aulos?"

Archimedes blinked again. "I'd love to," he told her. "But they're complicated. They-"

"You couldn't?"

"I- not that. They take a long time to build. I couldn't afford to do one cheap. Ktesibios charged sixteen hundred drachmae for his."

The girl did not look at all disappointed. "My brother likes music," she said. "And he loves ingenious machines. I'm sure he'd be willing to pay sixteen hundred drachmae for a water-aulos if you could make one."

"Your brother?" asked Archimedes, with a sudden horrible feeling that he knew who this was.

"Ah," she said, and her straight black brows came down. "You didn't realize. King Hieron."

"No," he said, feeling numb, "I didn't realize." He studied her a moment: the silver belt, the fine tunic. But he could not concentrate on the expensive clothes. His eyes kept sliding back to her round face with its black curls and brilliant dark eyes, and her strong musician's hands. He added accusingly, "You don't look old enough."

"He's my half brother, actually," she said. All the animation had left her face and voice, and she sounded the bored aristocrat. "He was almost grown up when our father married my mother."

King Hieron was a bastard, the result of a wealthy Syracusan's youthful indiscretion: all Syracuse knew that. Archimedes guessed that this girl must be the rich man's legitimate daughter. She was not of his class at all. He shouldn't really be here, in the private part of the house, talking to her. Syracuse allowed women more freedom than many other Greek cities, but still, it was grossly improper for a young man to slip into any private house and chat with the owner's unmarried sister, unintroduced and unsupervised, and this girl was the daughter of a nobleman and sister of a king. But he pulled his stained cloak straight and told himself defiantly that he was a democrat. "I can build a water-aulos," he declared. "If your brother's willing to pay for it, I'd love to build you one. I prefer wind instruments to stringed ones anyway."

At this she smiled again, a long slow grin, and he knew he'd said the right thing, and grinned back. "What's your name?" she asked.

He had just opened his mouth to reply when the answer was shouted at him- "Archimedes son of Phidias!" — in a tone of shocked disapproval. He and the girl turned together, and found four men bearing down on them. One was Dionysios, one the exalted doorkeeper, one a middle-aged workman, and the fourth, from his purple cloak, had to be the regent Leptines.