The Sand-Reckoner

2

The Achradina was an old quarter. The first Greeks to colonize Syracuse had settled on the promontory of Ortygia, and Ortygia- a grand region of temples and public buildings prudently fortified and garrisoned- was still the seat of the government. The Achradina had formed early, though, when the houses and shops of the growing city overflowed the crowded citadel and sprawled messily along the shore. When the city grew again, in wealth and power, the New Town had been laid out inland for the rich, while the Tyche quarter, a straggle of buildings along the road north, had become the place of settlement for the poor. The Achradina belonged to the old middle class. Narrow-streeted and dirty, bordered by the walls which guarded the city from attack by sea, it was the heart of Syracuse- dark, crooked, and full of secret delights.

Archimedes crossed it joyfully. A city-state customarily inspired in her citizens the most intense and passionate sense of patriotism and civic pride, and though Archimedes had always been something of a misfit in his own city, still every dusty cross-roads seemed to him to be shining with the glory of Syracuse. Each step, too, took him closer to home. Eagerly he marked off each familiar landmark: the small park with its bedraggled plane trees; the baker's shop around the corner where the family had bought its bread; the public fountain with its statue of a lion, which had supplied water for the household. A scent of herbs and roast meat wafted from the cookshop down the road, where he'd often run to fetch an evening meal if for some reason one hadn't been prepared at home. Nikomachos' house; Euphanes the Butcher's shop, with his house on top of it… and finally, there it was. Archimedes stopped in the street and gazed in silence at the front of featureless mud brick and the weathered wood of the single door. His chest began to hurt and his eyes stung. At one time this house had defined what was meant by a house. It had been the only house that mattered, the center of the universe, that container for everything that was important in his small world. And it was still true that all the people he loved most were behind that door.

He wished they lived in Alexandria.

Marcus raised the torch and likewise gazed at the house, remembering the first time he'd seen it, when Phidias had led him back from the slave market in chains. Not home, he reminded himself fiercely, though uncertain why he was denying the joy that hovered at the edge of his awareness. Just the house where I am a slave. For a moment he remembered his own home in the hills of central Italy, his own parents. He shoved them quickly from his mind: probably dead now, anyway. Some of the bricks on Phidias' house were crumbling, he noted, and the roof needed retiling. Not surprising. He himself had been the only man in the household, unless you counted the masters, and you couldn't count them, not when it came to retiling roofs. The place must have been running down. He had work ahead of him.

Gelon the Baker's son, who'd come to look after his father's donkey, shuffled his feet and asked, "Is this it?"

They unloaded the donkey, set the chest down, and sent the baker's son home with his father's beast, handing the child the torch to light his way. In the soft summer darkness Archimedes took a deep breath and knocked upon the door.

There was a long silence. Archimedes knocked again, and at last the door cracked open and a woman peered anxiously out, the lines in her worn face deeply shadowed in the light of the lamp she held. "Sosibia!" cried Archimedes, breaking into an enormous smile, and the housekeeper gaped, then screamed, "Medion!" — the diminutive ending of his proper name, his family nickname, which he hadn't heard for three years.

The reunion was as noisy and as joyful as anything Archimedes had pictured to himself. His mother, Arata, came running and flung her arms around him. His sister, Philyra, seized him as soon as his mother let go. "You've grown up!" he told her, and held her at arms length to admire her. She had been thirteen when he left: now sixteen and a young woman, she had not, in fact, changed much. Still straight and thin, gawky and bright-eyed, with her unruly brown hair pulled into a knot behind her head. She knocked his hands away so she could hug him.

"You haven't!" she replied. "You're as much a mess as ever!" Sosibia and Sosibia's two children hovered grinning and exclaiming in the background. But there was an absence. "Where's Papa?" asked Archimedes, and the noise died.

"He's too ill to stand," Philyra said, into the sudden quiet. "He hasn't been able to get out of bed for months." Her voice was heavy with reproach. For months she had tended their father and watched him waste away, while Archimedes, the darling and only son, lingered in Alexandria.

Archimedes stared at her, stricken. He had known that his father was ill. For a couple of months that awareness had lurked in the back of his mind, putting a wash of anxiety over all his preparations for coming home. But despite that, he had expected to find his father much as he had left him. He would have reckoned illness to be a persistent cough, a bad back, chronic indigestion. He had not expected a deforming monster to have moved into the house and pinned his father to his bed.

"I'm sorry, my darling," said his mother gently. She had always been the family peacemaker, the voice of calm practicality. She was shorter than her children, wide-hipped, broad-browed; there was more gray in her brown hair than her son remembered. "I'm afraid it will be a great shock for you to see him. You can't have realized how ill he is. But I thank the gods you're safely home at last."

"Where is he?" asked Archimedes in a hoarse whisper.

Phidias's sickbed had been placed in the room Archimedes remembered as his mother's workroom. It was at the opposite end of the small courtyard which opened onto the street and formed the center of the house. The stairways to the bedrooms on the upper floors were steep and narrow, and a ground-floor room was much more convenient for an invalid. When Archimedes crossed to the old workroom, he found a lamp lit and his father sitting up and looking eagerly toward the door: he had heard the commotion, and was impatiently awaiting his son's appearance. Archimedes faltered on the threshold. Phidias had always been tall and thin; now he was skeletal. The whites of his eyes had turned yellow, and they stared out of deep hollows; his skin, too, was yellowish, and crumpled and dry. Most of his hair had fallen out, and what remained was white. When he stretched out his arms toward his son, his hands trembled.

Archimedes went through the door in a rush, dropped to his knees beside the bed, and threw his arms about his father's emaciated frame. "I'm sorry!" he choked. "I didn't… if I'd known…"

"Archimedion mine!" exclaimed Phidias, and folded his stick-like arms about his son. "Thank the gods you're home!"

"Oh, Papa!" cried Archimedes, and burst into tears.

In the courtyard, Marcus dragged the luggage in out of the street and shut the door. When he turned back to the house, Sosibia caught his shoulders and kissed him lightly on the cheek. "You're welcome back, too!" she said softly. "I wish it were to a happier house."

He looked at her in surprise, touched despite himself. He and Sosibia had never been friends. When they first met, her chief concern had been to make it clear that though he might have been purchased to replace the household's previous manservant, she had no intention of allowing him to take the dead man's place in her bed. Marcus had not at first understood her (he had been eighteen at the time, fresh from Italy and almost Greekless), but when at last he did, he in turn had made it clear that the very idea of sleeping with a plain, fortyish household slave disgusted him. This unanimity about sleeping arrangements had not, obviously, led to any feelings of goodwill, and for years they feuded, Sosibia sneering at Marcus as a crude barbarian, and Marcus disdaining Sosibia as a servile old woman. Now she was welcoming him back. "Well," he said gruffly, "it's good to be here."

There was a silence, and then he nodded to the two children, who stood behind their mother, watching- Chrestos, a boy of fifteen, and thirteen-year-old Agatha. "You two have grown," he commented. Another reason not to be welcome, he thought privately. Four adult slaves was too many for one middle-class household: now that Marcus was back, it was quite likely that Chrestos would be sold. But Sosibia had not acknowledged this uncomfortable prospect, so he ignored it as well. "When we came up to the house, I thought to myself that there'd be a lot of work waiting for me," he said instead. "I'd forgotten we have another man about the place now."

Chrestos grinned. "Welcome home, Marcus," he said. "You're welcome to do my work if you want to!"

His little sister laughed, slipped forward suddenly, and gave Marcus a shy kiss on the cheek. "Welcome home!" she whispered.

Not home, Marcus reminded himself, but some part of him was still glad. The first year of his slavery had been a nightmare he still sweated to remember, but that nightmare had ended in this house, and he had waked again to a world governed by sane rules. "It's good to be back," he repeated gruffly.

There was another silence, and then Marcus jerked his head at the doorway on the other side of the small courtyard and asked, "Is the old man dying?"

Sosibia hesitated, then made a gesture against evil and nodded. "Jaundice," she said resignedly. "He can't eat now, poor man: he lives on barley broth and a little honeyed wine. It won't be much longer."

Marcus reflected on Phidias. A kind man; an honest, hardworking citizen; a loving husband and father. A good master. He might resent the man for that last, but it was not Phidias' fault that he himself was a slave. "I'm sorry," he said sincerely; then added, in a harsh voice, "The gods made us mortal. It will come to us all."

"He has lived well," said Sosibia. "I pray the earth receives him kindly."

Archimedes stayed with his father for half an hour, retreating when the dying man fell asleep. He himself had no interest in anything more that night. Sosibia and his mother made up the bed in his old room, and he lay down and sought oblivion in sleep.

He woke early next morning, and lay for a while looking at the patterns the rising sun made upon the wall beside his bed. The window shutter was of wickerwork woven in a crisscross pattern and covered the whitewashed plaster with bars and triangles of orange light. As the sun rose higher, the light grew paler and the triangles shifted and widened. They slid from the wall onto his bed, where they lay in bright profusion over the sheet, like tiles of fresh ivory.

His eyes stung. In Alexandria he had purchased a game for his father, a set of ivory tiles cut into squares and triangles. They could be fitted together to form a square, or arranged to make a ship, a sword, a tree, or any of a hundred other figures. The puzzle was a geometer's delight. He'd liked it, so he'd been sure his father would too. Gifts given to his father now were destined for the grave. That fact was an enormity so devastating that it left him feeling as if half his soul had been removed.

When Archimedes was growing up, Phidias had been the one person who really understood. It had often seemed to Archimedes that everyone else had a blind spot in the middle of the head. They could look at a triangle, a circle, a cube- but they couldn't see it. Explain it to them, and they couldn't understand. Explain the explanation, and they stared and wondered aloud why you thought that was so wonderful. Yet it was, unspeakably wonderful. There was a world there, a world without material existence but luminous with pure reason, and they couldn't see it! Only Phidias had seen it. He had shown it to Archimedes, taught him its ways and its rules, and joined in with all his exclamations of amazement. As Archimedes grew older, they had gone on to explore the other world together. They had been co-conspirators, laughing together over an abacus, arguing together about axioms and proofs. They had walked together into the hills on clear nights, to observe the rising and setting of the stars and take sightings of the altitude of the moon. Only the two of them, out of all Syracuse, were at home in the invisible world. Others- even the closest and best loved of others- were forever outsiders.

It had been Phidias who suggested that Archimedes go to Alexandria. "I went when I was your age," he said, "and I heard Euclid himself lecture. You must go." He had sold a vineyard he could not afford the loss of, parted with a slave he could not easily do without, so that his son could study mathematics at the world's greatest center of learning. And Alexandria had been everything Phidias had promised- but it had been more, as well. For the first time, Archimedes had found others who understood, and some of them were young men of about his own age. It was the first time in his life he hadn't felt like a freak; the first time he'd been free to open up his mind outside his own house. So he had tossed it open to embrace the sky, and the ideas came in- thronging, pushing for attention, scattering, battling, seething, dancing together. He had felt like a fish raised in a garden pond, suddenly discovering the vastness of the sea. It was a liberation more intoxicating than anything he had ever imagined.

At the end of the first year, Phidias had started to write letters asking, "When are you coming home?" and Archimedes had not known how to answer them. Instead he had written to tell his father about Aristarchos' hypothesis that the earth went around the sun, about Conon's work on eclipses, about the Delian problem, and the attempts made by various geometers to square the circle. And Phidias had replied in kind- amazed and enthusiastic, full of arguments and proofs- but somewhere in his every letter, there the question would be again: "When are you coming home?" Archimedes had known- oh, with perfect clarity! — that his father missed him terribly; that Phidias now had no one to share his ideas with, no one to understand him. But he had not wanted to go home.

Then, early in the spring, Phidias' last letter: "A war has started with Rome, and I am not well. I have had to stop teaching. Archimedion dearest, you must come home. Your mother and your sister need you." Your mother and your sister. Phidias himself had needed Archimedes long before, but for himself he had made no demands- only asked that one pleading question, persistently ignored.

This time the question was a command, and could not be ignored. Archimedes had slowly and reluctantly gone about the business of selling the furniture he'd bought in Alexandria, finding another tenant for the rooms he rented, disposing of a few machines he'd built and some of the tools he'd bought to make them. He had welcomed every delay. When his ship finally set sail for Syracuse, he had wept to see Alexandria dwindle behind him. Those tears seemed shallow now: the grief that lay before him would be much deeper.

The door to his room opened, and Philyra looked around it. She saw that he was awake, and came in.

Philyra was almost seven years younger than her brother, but tended to behave as though she were seven years older. She was a confident, outspoken, no-nonsense sort of girl; she'd been popular at her school and was well thought of in the neighborhood. She was very fond of her brother, but found him hopelessly vague and dreamy, much in need of firm management. She now advanced on him purposefully with a length of yellow cloth folded over her arm- towel or blanket or clothes, he wasn't sure. He sat up in bed, drawing up his long legs to make space for her, and she sat down and regarded him critically. He became uncomfortably aware that he was naked under the sheet, that his bare skin was blotched with fleabites, that his jaw and neck were scruffy with unshaved beard and his hair was dull with dirt. In daylight, too, he could see more clearly how she'd changed since he saw her last, how her body had filled out and rounded. Here in the house she was wearing only a light linen tunic, and it clung to her breasts revealingly. He felt abruptly embarrassed in front of her.

"When did you last have a bath?" asked Philyra, wrinkling her nose.

"You can't bathe on ships," he replied defensively.

Philyra sighed. "Well, you'll have to go to the bathhouse in the New Town as soon as you've had breakfast. You look perfectly disreputable! Do you have any clean clothes?"

He cleared his throat unhappily and didn't answer. "I didn't realize Papa was so ill," he said instead. "How long…"

"Since October," she said coolly. "He wrote to you then, but I guess you didn't get the letter until after the winter."

Ships did not sail on the Mediterranean between October and April; even if Archimedes had received his father's letter in late autumn, there would have been no way for him to come home until the sea-lanes opened again. But the thought of Phidias lying ill all winter while Archimedes enjoyed himself in Alexandria appalled him.

"I didn't get it until the end of April," he said miserably. "And even then I thought I had time to tie up all my business in Alexandria. All he said in it was 'A war's started and I'm not well.' I thought it just meant he wanted me home to help teach his students until he was better."

"He thought he would recover," said Philyra, and suddenly her eyes filled with tears. "He had a fever with jaundice, but Mama had it too, and she got better. We thought he was getting better, too. Only he wasn't, and this spring…"

Archimedes reached out and touched her shoulder, and she lost her no-nonsense composure, dropped her bundle, flung herself into his arms, and wept. "It's been horrible!" she cried passionately. "He gets worse and worse, and there's nothing we can do!"

"I'm sorry," he said helplessly. "I wish I'd been here."

"He wanted you," sobbed Philyra. "He kept sending Chrestos down to the harbor to see if there were any ships from Alexandria, and sometimes there would be, but you were never on them. And sometimes he'd say you must be dead, that your ship had sunk or that you'd died in Alexandria, and he'd weep for you and tell us all to go into mourning. That was the worst of all. Why didn't you come back last year?"

"I'm sorry!" he said again wretchedly, tears rising to his own eyes. "Philyra, I swear I would have if I'd known."

"I know," she said, swallowing her sobs. "I know." She patted him on the back, as though he were the one who had broken down, then drew away, wiping her eyes. Nothing could be done about death, and she was determined to bear the grief with all the dignity she could command. She picked up her bundle again and spread it out on the bed: it turned out to be a new cloak, woven of fine yellow wool, and a linen tunic with a yellow spiral pattern down the sides. "I made these for you last year," said Philyra. "You don't have any clean clothes, do you?"

"I don't think so," he admitted, tracing the pattern with one slow finger. It was a series of double spirals, all centered on a line from shoulder to knee; from each central point a line circled outward one turn, then twisted about and ran into the center of the succeeding spiral. An interesting pattern. A line constructed tangent to both spiral A and spiral B would…

Philyra firmly removed his hand from the pattern; he looked up and blinked at her in surprise. "It's to wear," she told him. "Not to do geometry on."

"Oh," he said. "Yes." After a moment, he remembered that the clothes were a present and added, "Thank you. I like them very much."

She shook her head in mock despair. "Ai, Medion! You haven't changed at all!"

He wasn't sure what to make of that, and she smiled again at his bewilderment and brushed back a stray lock of his dirty hair. "Now," she went on, businesslike and hopeful, "do you have any money? We've been running out. We had to sell some blankets and pots to pay the doctor."

Archimedes shrugged. Most of the water-snail's earnings had vanished in Alexandria. But there was a little left, and a little more from the odds and ends he'd sold on leaving the city. "I have a bit," he said. "About a hundred drachmae, I think- Marcus knows exactly."

"A hundred drachmae!" she exclaimed eagerly. "That's good! I was afraid we'd have to go around to Papa's old pupils at once, and beg them to take up mathematics again. But a hundred drachmae will buy us a couple of months' grace."

Archimedes cleared his throat and shifted nervously. "I'm not going to teach," he declared.

She stared at him in exasperation. "Medion, you can't make a living from geometry!"

"I know that!" he protested. "I'm going to get a job as an army engineer." He launched into the arguments he had carefully prepared beforehand. "With a war on, the city must need catapults, and the tyrant must be willing to pay for them. There's more money in machines than in teaching. And I'm good with machines, you know I am. That irrigation device I built last summer earned more in two months than Papa ever earned in a year. Besides, shouldn't I help defend the city if I can? I'm going to see somebody about it this evening."

At this she smiled, encouragingly rather than with conviction. She had heard about the water-snail from his letters home, but she rather doubted that it was as successful as he claimed, and as for catapultswell, the king already had engineers who could build them, so why would he want somebody new and completely untried? Even if he did, it seemed unlikely that anyone would get rich from it. Her brother had made lots of machines as they grew up together, and many of them hadn't worked at all. Machine-making didn't seem to her as reliable a source of income as teaching mathematics. Still, she liked his machines. As a small girl she had sat quietly watching as he built them, and listened to his explanations with solemn attention. As far as she was concerned, her brother's constructions were the most wonderful of toys, whether they worked or not. She would be very pleased if he could make a living from them. It was worth a try, anyway- and the household now had a hundred drachmae and a couple of months before the money ran out.

Archimedes saw that she had accepted his plan, and felt a twinge of dread, as though another gate had closed in the walls which surrounded him. He had decided in a rare moment of practical planning that there were three things he was good at- pure mathematics, mechanics, and playing the flute. To earn a living he must put one or another of those skills to use. Music was something personal, something he did for himself and his friends; it seemed profane to play to order. As for pure mathematics, as Philyra had pointed out, he couldn't make a living simply from doing geometry, and as for teaching it, he'd occasionally been called in to help his father in the past, and he was uncomfortably aware that he was bad at it. The students never understood things which seemed to him glaringly obvious, and his impatient explanations only confused them. So machine-making it would have to be.

He was dreading it. Building a new machine was fun- he liked seeing the need set out as neatly as a geometrical proposition, and devising an apparatus that would satisfy it; enjoyed the complete absorption in the task, the complex coordination between his hands and his mind, and the unarguable solid reality of the finished solution. But once a machine had been built, to make another of the same type, and then another and another and another- it was boring. No, worse: it was a stifling prison where the soul's wings atrophied and died. Pure mathematics was light and air and delicious freedom, and he loved it more than anything on earth. But he was not a nobleman, and couldn't afford to devote himself to pure mathematics without a thought for sordid considerations of gain. He had a family to support. The invisible world could no longer be his homeland, but only a place he visited when he had the time.

He would have no company on his visits to it, either; none. He would be alone- as his father had been, these three years past. With a spasm of fresh pain, he supposed that Fate was just.

Then he remembered the war. In Alexandria it had been hard to believe in it; here in Syracuse it loomed larger, far more threatening. Lines from an old song wandered through his mind:

Let none of humankind ever once say what chance tomorrow will bring, nor, seeing a happy man, that contentment will stay for, swift as a dragonfly's wing swifter again comes change.

"You get dressed," ordered Philyra, patting his hand. "I'll go talk to Marcus about getting your other things washed."

Marcus was taking a bath when Philyra found him. Private houses did not generally have bathrooms, and bathhouses in that age were for citizens only: Marcus was washing in the courtyard with a sponge and a bucket. It was not uncommon for even the free men of a household to walk about naked indoors, and a slave's nudity was nothing to worry about, but Philyra hung back awkwardly, waiting at the foot of the stairs until Marcus had finished. She felt uncomfortable about him. She too was aware that one of the household slaves would probably be sold, and she was hoping that one would be Marcus. She had always sided with Sosibia in the household feud, and regarded Marcus as an awkward barbarian. Besides, after a three-year absence he felt like a stranger. She could contemplate selling him; she couldn't bear the thought of inflicting that fate on any of the others. She noticed now that though Marcus had a shocking bruise on his left side, and though he was just as flea-bitten as her brother, he looked sleek and fit. That would mean a good price, but still her lips narrowed with disapproval. Marcus had been sent to Alexandria to look after Archimedes, but had returned glowing with health, while his master's ribs were like a washboard.

An inconvenient fair-mindedness, however, reminded her that Archimedes had always been thin, and Marcus sturdy. And when Archimedes was doing geometry he would forget to eat unless you set his food on top of the abacus- and sometimes even then he'd simply move it out of the way and carry on calculating. It was probably unfair to blame Marcus too much for the state his master had come home in.

Marcus poured the remains of the bucket of water over his head, shook himself, and picked up his tunic. Philyra pushed herself out of the doorway into the sunny courtyard. "Marcus!" she said sharply. "Where's my brother's luggage?"

Marcus jumped and pulled his tunic hastily over his head before replying. He felt as awkward with Philyra as she with him. She'd been a schoolgirl when he'd left the house; now she was a young woman. "There," he said, indicating the chest in the corner of the yard. "But I wouldn't open it, mistress."

"Why not?" she demanded. "I can't believe the things in it are clean! It's going to be a good drying day." Indeed, the air was already hot: anything washed now would be dry well before the evening.

He shrugged. "It has presents in it," he said. "One of them is for you." His eyes lingered momentarily on the front of her tunic. She became aware of how it was clinging, and hitched it up, reddening.

"But I just told him I was going to see to his things!" she protested. "And he didn't say anything about presents."

Marcus snorted. "Would you expect him to think of a thing like that?"

No, she wouldn't. Archimedes probably remembered the presents, and must know that they were in the same chest as the clothes. But he would not put the two things together, and know that it would spoil her surprise if she opened the chest. She made an exasperated noise, Marcus grinned, and something swung into balance between them: they were both members of the same household, both aware of the tastes and foibles of the same small collection of people.

"There's no hurry, is there?" he asked.

There was not, not really. She just wanted everything to be settled: her brother indisputably home, in his own room as he should be, with the chest reduced from traveling chest to clothes chest. She walked over to the luggage and glared at it resentfully. "What's in the basket?" she asked.

"The famous water-snail," replied Marcus, grinning again. "We can unpack that, if you like." He went over to the chest and began to untie the rope.

"Won't he want to show it to me himself?" she asked doubtfully.

"No," replied Marcus, undoing another knot. Suddenly he wanted very much to show her the water-snail, to impress her. "We made thirty-two of the things in Egypt, and he's sick of the sight of them. But it's an amazing machine. Here, let me show you!" He drew the rope off the basket, hauling the coils under the chest. Philyra leaned against the courtyard wall with her arms crossed, trying to look uninterested, though in fact she was acutely curious. Marcus was sharply aware of the way her stance cast one thin hip into linen-hung relief. Too thin, he told himself- like her father and her brotherbut somehow prettier than such an angular girl should be. Perhaps it was the brightness of her eyes. Not that it had any bearing on him: he was as much her brother's property as the machine he was unpacking. Still, where was the harm in showing a pretty girl a machine?

He untied the knot that secured the basket's lid, opened the basket, and lifted out from a nest of straw a wooden cylinder. It was about a cubit long- the distance from a man's elbow to his fingertipsand the outside was made of planks bound, barrel-like, with iron hoops. The interior held a complicated structure smeared with pitch. A stand was fixed to the cylinder's core with a pin, so that the whole thing could turn like a wheel.

"The Egyptians usually lift water with a machine called a water-drum," said Marcus, turning the cylinder in his hands. "A sort of wheel with eight buckets around the rim. A full-sized one moves a lot of water, but it's heavy to turn- it needs a couple of men to shift it. Your brother started off with one of them, and finished up with this. The real machines we built were bigger, of course- they stood about as tall as a man- but otherwise they were exactly like this. As you can see, this still has eight inlets"- he showed her the eight openings in the cylinder's base- "but they're not buckets. They're tubes." He stuck a finger into one, and she saw that it was indeed a kind of tube, and that it ran up around the core at an angle. "They coil right around the cylinder several times and come out here, at the top." He slapped the top edge of the cylinder, which was identical to the bottom. "Each one is a bit like a snail shell, which is why it's called a snail. They're made of strips of willow, stuck onto the core with pitch and closed over with planks. I don't know how he fixed on the angle they spiral at, but it's very important: a lot of the people who tried to copy it got it wrong, and then it didn't work. Now, what you do to work it…" Marcus glanced about, fixed on a large amphora of water sitting in the corner of the courtyard, and hurried over to it, holding the water-snail. He set the machine down, fetched the bucket he'd used for his bath, and poured some water from the amphora into the bucket. Then he set the bucket in a dip in the courtyard, balancing it with some loose stones so that it stood at an angle on its side, and placed a laundry footboard before it to make a platform. "It has to sit at an angle," he explained to Philyra. "The exact angle is important, too- that's another thing people who copied it used to get wrong. This one is right if the stand's straight." He set the foot of the water-snail in the water of the bucket and the head on the platform. "Now all you have to do is turn it." He gestured for her to oblige.

Philyra hitched the hem of her tunic away from her feet and crouched down beside him. She put one hand on the wooden cylinder and began to turn it slowly; it revolved easily on its stand. Water ran into the tubes at the foot of the snail. She kept turning, and presently water ran out of snail's head. She kept turning the machine gently, watching it: water ran in, ran down the tubes, and…

"It's running uphill!" she exclaimed, shocked. She took her hand off the machine as though it had burned her.

Marcus grinned. "Quick!" he said. "It takes most people a bit longer to realize that. Some need to have it pointed out to them. But it doesn't- not exactly. Watch more closely."

Philyra turned the machine again. Water ran into a tube; as the tube went up, the water ran down, into the spiral, and then along it as it turned on. She laughed delightedly.

Marcus grinned back at her. "It runs downhill all the way up," he said.

"Sometimes," said Philyra, "I think my brother is a mistake of Nature. He shouldn't have been born a human being at all; he should have been an attendant spirit in the workshops of the gods. I suppose a full-sized one of these is very much easier to turn than a water-drum?"

" 'Course it is," agreed Marcus. "You don't need two men; you don't even need one. A child can operate it- because all you have to turn is the snail itself: the water just runs downhill." He sat back on his heels and gazed lovingly at the machine. "We had people queueing up to buy it," he told her. "We could have made a fortune!"

"I thought you did!" Philyra said in surprise. "More in two months than my father earns in a year, my brother said."

Marcus shook his head sadly. "Eighteen hundred eighty drachmae. Enough to pay our debts and live well in Alexandria for a year. But we had orders for another thirty of the machines- at eighty drachmae apiece! — and every expectation of more again. He preferred to do geometry."

Philyra stared at the water-snail and swallowed. She could not imagine eighteen hundred and eighty drachmae all together in one lump; still less could she imagine spending such a sum. The rent from the family's small farm was three hundred drachmae a year- less, now that the vineyard had been sold- and Phidias' teaching had brought in perhaps as much again. The water-snail had earned not just more than her father's salary but three times the household's entire annual income- and Archimedes had spent it all, except for a hundred drachmae.

Marcus understood her sudden silence and wished he'd kept his mouth shut. He shifted uncomfortably. "Alexandria's expensive," he said defensively. "And there was the debt, and the fare back." There'd been a woman, too, who had accounted for quite a lot of the money, but he had no intention of mentioning her to Archimedes' sister. "Your brother wasn't as extravagant as it seems," he finished instead- which was certainly true, given Alexandrian prices, to say nothing of the woman's. "Besides, there's a hundred and sixty drachmae left."

"A hundred and sixty?" asked Philyra suspiciously. "He told me a hundred."

Marcus shrugged and grinned again. "You expect him to keep track of money?"

This time she did not smile. Instead, she gave him a cool, assessing stare. "You were keeping track of it for him, were you?"

For a moment, he didn't understand. Then his face darkened. "I haven't taken a copper of it!" he declared indignantly. "You can ask him."

"If he wasn't keeping track of it, what good would that do?"

Philyra, watching his face, saw the anger in it suddenly dwindle into sullen impassivity. It was as though something else drained out with it- a sense of freedom, an identity. She suddenly regretted her suspicion. And yet- eighteen hundred and eighty drachmae! She didn't see how such a huge sum of money could have just vanished. Her vague, dreamy brother was easy prey for any cheat.

"I never took a copper of his money," Marcus repeated sourly. "You can ask him."

Bitterly, he remembered how he and his master had returned to Alexandria from making water-snails in the Delta. When the riverboat docked, Archimedes had leaped off and gone straight to the Museum, leaving Marcus to take the luggage back to their lodgings. The luggage- and the box containing eighteen hundred and eighty drachmae. A lot of money. Enough to buy Marcus passage on a ship back to Italy, and to pay for a pair of oxen, some sheep, and a year's rent on a little farm once he got there. He'd been painfully aware, as he trudged along with the heavy chest, how easy it would be to get away. It wouldn't even have been as though he were leaving his master stranded: Archimedes could always have gone back and made a few more water-snails. In the end what held him back was not the honesty on which he had always prided himself, but despair. The events that had enslaved him- the lost battle, the dead men- were still there, ineradicable and absolute. He could not go home, and there seemed little point in going anywhere else. His slavery, which until then he had always thought of as something imposed upon him contrary to his true nature, suddenly revealed itself as the inescapable condition on which he held his life.

He recognized now that he was putting off the girl with a slave's defense- My master hasn't complained, so you have no right to- and he stood up angrily, swept up the water-snail, and carried it back to its basket. Philyra followed him, her expression still a mixture of suspicion and apology. "Maybe I will ask him," she said.

"You do that," growled Marcus, tipping the last of the water out of the snail onto the dirt of the yard.

"In the meantime," said Philyra, drawing herself up, "take all the dirty things out of the chest and put them ready for washing. Just leave the other things in it, for my brother to sort out."

"Yes, mistress," said Marcus bitterly. He turned his back on her and ostentatiously began to put the snail away. But he sensed it when she left, and turned to watch her. She walked with a straight, stiff step, back straight and head with its knot of untidy hair held high, and she went directly to the room at the end of the courtyard where her father was dying. His resentment vanished, leaving only sadness. Her father was ill, her mother undoubtedly distracted by caring for him. She was trying hard to be a prudent and sensible guardian of the house and not a burden upon it: if he'd been free he would have applauded her for it. She was young and ignorant. It was not her fault he was a slave.

Archimedes stumbled downstairs a few minutes later. He was dressed in the new tunic, which, unbelted and crooked, he contrived to make almost as disreputable as the one he'd taken off the day before. He blinked at the heap of dirty laundry beside the chest as though it were the fragments of something that had broken and he were trying to work out what.

"I told your sister not to unpack the chest herself, because it had presents in it," said Marcus quickly. "The presents are still there."

"Oh," said Archimedes, but as though the words hadn't registered.

He looked, Marcus thought, even vaguer and more preoccupied than usual. "Do you want to take the presents out and give them to your family?" he suggested pointedly. "Your sister's in a hurry to shift the chest."

"Oh," said Archimedes again. He came over and stared into the chest. Marcus had already sorted the presents into one corner: a jar of myrrh for Arata, a lute for Philyra, and a box full of ivory tiles for Phidias.

Archimedes bent over and picked up the box. Like its contents, it was of ivory, and it had been decorated with a picture of the god Apollo and the nine Muses, sketched with a fine red line. He remembered looking at it in the shop, assembling the pieces of the puzzle, and smiling as he imagined his father's delight in doing the same. Phidias would not play with the puzzle now. He was too tired, too ill, too busy dying. One more puzzle abandoned- and there had been many, many other puzzles Phidias had been too busy and too tired to solve during the course of his life. He had needed to earn money for the household, bread for the children. He had needed to be a citizen, a husband, a father before he could be a mathematician and astronomer. Archimedes had profited from it. Now he numbly regarded the empty half of himself, an unpayable debt passed on.

Marcus saw his face fall slack and empty, like the face of an idiot, and was concerned. He touched his master's elbow. "You can still give it to him, sir," he said. "It's a good present for an invalid."

Archimedes began to cry soundlessly. He raised his head and stared blindly at Marcus. "He's dying."

"So I was told," replied Marcus evenly.

"I should have come back last year."

That was what Marcus had said at the time. Now he shrugged and said only, "You're back now. Sir, he dies after a good life, with all his family about him. No man can ask the gods for more."

"He lived on scraps all his life!" Archimedes replied fiercely. "Bits and pieces, hours snatched here and there, nothing! Oh, Apollo! Pegasus, hitched to a plow! Why should the soul have wings, if it's never allowed to fly?"

This made no sense whatever to Marcus. "Sir!" he said sharply. "Bear it like a man!"

Archimedes gave him a look of astonished incomprehension, as though Marcus had addressed him in some unidentifiable foreign tongue. But he stopped crying and wiped his face on his bare arm. He glanced at the door at the far end of the courtyard, then sighed and walked toward it, the box in his hand. Marcus picked up the jar of perfume and the lute and followed him.

Arata and Philyra were both in the sickroom, finishing the work of preparing the invalid for the day. When Philyra saw the lute in Marcus' hands her face went utterly still, but her eyes awoke into a sudden intense life. Archimedes glanced back at his slave and jerked his head, and Marcus bowed and handed the jar of myrrh to Arata, then bowed again and offered the lute to Philyra. Her face flushed as she took it, and her hands curved over the sounding board with a fiercely possessive tenderness. She looked at her brother and breathed, "Medion!" — half in protest, half in adoration. But Archimedes was not looking at her.

Phidias had slowly levered himself to a sitting position to accept his own present. He took the ivory box in his trembling hands and studied the picture on the lid. "Apollo and the sweet Muses," he observed softly. "Which one is Urania?"

Archimedes indicated her silently. Urania, Muse of Astronomy, stood at Apollo's elbow, pointing at something which lay on the low table in front of the god- the puzzle, probably. Her diaphanous draperies were identical to those of her eight sisters, but she was distinguishable from them by her crown of stars.

Phidias smiled. "Next to the god," he said quietly. "Just where she should be." He looked up at his son, his yellowed eyes still full of his smile- looked in the luxurious confidence that here at last he would be understood. "She's beautiful, isn't she?" he asked.

"Yes," whispered Archimedes, the expected understanding going through him in a warm flood. "Yes, she is."