The Sand-Reckoner

1

The box was full of sand. It was fine, glassy sand, almost white; it was moist, and had been flattened, then scraped smooth to produce a surface as level and firm as the finest parchment. But the sunlight, falling obliquely with the afternoon, glinted here and there on the edges of individual grains, catching on facets too small for the eye to distinguish. Innumerable facets, one would say- only each made a distinct point of brightness, and the young man looking at them suddenly found himself wondering if he could number them.

It was an old box. The olive-wood frame was scarred and battered, the dull bronze which bound the corners scratched here and there into new brightness. The young man set his hand against one of those scratched corners, calculating: the box was four finger-breadths high, but there was a groove for the lid, and the sand only filled it halfway. He did not need to measure the length and width: he had long before marked the rim with notches a finger-breadth apart, twenty-four down one side and sixteen down the other. He crouched over the box, which he had carefully placed in the quietest part of the ship's stern deck, out of the way of the sailors. Using one leg of the set of compasses he was holding, he began to scratch calculations in the sand. Say that ten grains of sand could fit in a poppy seed, and twenty-five poppy seeds could sit upon the breadth of a finger. There would then be six thousand by four thousand by five hundred grains of sand in the box. Six thousand by four thousand was two thousand four hundred myriads; multiply that by five hundred…

He blinked, frowning. His hands slipped nervelessly down his sides, and the point of the compasses scratched his shin. Absentmindedly, he rubbed at the scratch, then raised the compasses to his mouth and sucked their hinge while he continued to stare. This was an interesting problem: the number of grains of sand in the box was a bigger number than he could express. A myriad- ten thousand- was the largest number his language had a name for, and his system of writing contained no symbol for the indefinitely extendable zero. There was no way to write down a number greater than a myriad myriads. What term could he find for the inexpressible?

Start with what he knew. The largest expressible number was a myriad myriads. Very well, let that be a new unit. Myriad was written M, so this could be M with a line under it: M. How many of them did he need?

The blank white surface before him was suddenly covered in shadow, and behind him a voice said wearily, "Archimedes?"

The young man took his compasses out of his mouth and turned, beaming. He was thin, long-limbed, and angular, and the general effect as he twisted about was of a grasshopper preparing to jump. "It's a hundred and twenty myriads-of-myriads!" he exclaimed in triumph, brushing back a tangle of brown hair and regarding his interrupter with a pair of bright brown eyes.

The man behind him- a somewhat older, burly, black-haired man with a broken nose- gave an exasperated sigh. "Sir," he said, "we're coming into the harbor."

Archimedes didn't hear him; he had already turned back to the box of sand. There was no such thing as an inexpressibly large number! If a myriad-of-myriads could be a unit, why stop there? Once you reached a myriad-of-myriads myriads-of-myriads you could call that your new unit, and go on again! His mind soared over the exhilarating reaches of infinity. He put his compasses back into his mouth and bit them excitedly. "Marcus," he said eagerly, "what's the biggest number you can imagine? The number of grains of sand in Egypt- no, in the world! No! How many grains of sand would it take to fill the universe?"

"Can't say," replied Marcus shortly. "Sir, we're in Syracuse. In the Great Harbor. Where we disembark- remember? I need to pack the abacus."

Archimedes put his hands protectively over the tray of sandcalled by the same name as the more familiar reckoning machine- and looked around with dismay. He had come up to the ship's stern deck when the vessel had sighted the point of Plemmyrion and Marcus had started packing. Syracuse then had been only a patch of red and gold against green slopes; now a whole stretch of time seemed to have vanished into the sand, and Syracuse lay all around him. Here, in its harbor, the city- richest and mightiest of all the Greek cities of Sicily- appeared as nothing but walls. To his right loomed the citadel of Ortygia, a rocky promontory enclosed by massive battlements, and before him the seawall swept around in a long curve of gray to end in the tower-studded walls of the fort which commanded the approach from the marshes to the south. Two quinqueremes sat at the naval quays ready for sea, their sides feathered white with the triple banks of their shipped oars.

Archimedes shot a longing glance at the clear water of the harbor entrance behind the ship. There the Mediterranean stretched open and unbounded as far as the coast of Africa, brilliantly blue and hazy in the bright June afternoon. "Why the Great Harbor?" he asked unhappily. He was Syracusan-born, and the city's customs were as natural to him as its dialect. Merchant ships like the one on which he and Marcus were passengers usually put into Syracuse's Small Harbor, on the other side of the promontory of Ortygia. The Great Harbor belonged to the navy.

"There's a war on, sir," said Marcus patiently. He squatted down beside Archimedes and put out his hands for the box of sand.

Archimedes looked down sadly at the twelve billion grains of gleaming sand and his own scratched calculations. Of course. Syracuse was at war, and the Small Harbor was sealed off. All the traffic was forced into the Great Harbor, where the navy could keep an eye on it. He knew about the war: it was one of the reasons he had come home. The small farm his family owned lay to the north of the city, well beyond any possible zone of defense, and it was unlikely that there would be any income from it this year. His father was ill and could not practice his usual occupation as a teacher. Archimedes was the only son of the house, and supporting the family and protecting it through what was likely to be a very bad war was now his responsibility. It was time to give up mathematical games and find some real work. Walls, he thought miserably; unbreachable walls, closing in.

Slowly, he took his hands off the notched rim of the abacus. Marcus picked it up, found the lid, and closed the reckoning box away. He slid it into its canvas sack and walked off with it. Archimedes sighed and sat back, hands dangling over his knees. The compasses slipped from his limp fingers and impaled themselves in the deck. He stared at them blankly for a moment, then pulled up one side of the instrument and swept it around, scratching a circle in the rough wood. Let the area of the circle be K- No. He folded the compasses and pressed the cool double bar against his forehead. No more games.

In the cabin below, Marcus slipped the abacus swiftly into the space in the traveling chest he'd reserved for it, then forced the lid shut. A hundred and twenty myriads-of-myriads, he thought, expertly knotting a rope round the chest. Was that a real number?

It was certainly not a sensible one. He paused, however, and contemplated it a moment, as though it were a dubious bargain offered him by some unreliable shopkeeper. A hundred and twenty myriads-of-myriads! Was that the answer to that other new impossible question, "How many grains of sand would it take to fill the universe?"

Nobody but Archimedes would ask such an insane question. Nobody else would come up with such an incomprehensible answer. He had been a slave in the man's household since the young master was nine years old, and he still wasn't sure whether a calculation like that deserved admiration or contempt: both, probably. Just as well that the young lunatic was going to have to give up such questions and turn his mind to more practical matters…

Marcus stopped himself and turned his attention back to the chest, jerking at a knot to ease the sudden apprehension that tightened his throat. Practical matters like the war. For three years he and Archimedes had been away from Syracuse, and for two of those years he had been urging his master to go home. Now they were in the harbor, and he wished that they were anywhere else. Syracuse was at war with the Republic of Rome, and Marcus saw no way that the future could hold for him anything but grief.

There was not much sign of the war at the docks, except that everything was quieter than usual. Destruction was still remote, a matter of armies maneuvering far away, a ruinous storm whose approach could be watched, with dread, from a distance. In a concession to its threat, however, the customs official usual to peacetime had been joined at the quayside by two soldiers. The pattern of crimson letter sigmas on the round shields slung over their left shoulders declared them to be Syracusan, but Archimedes recognized neither of them. Syracuse was a big enough city that he couldn't know more than a fraction of his fellow citizens, but he still watched the men apprehensively. They might be foreign mercenaries, and mercenaries, as every citizen knew, had to be handled more cautiously than live scorpions. Under the previous government they would have given a beating to any citizen whose expression offended them. Things were much better under the present ruler, but only a fool would assume that the breed's basic character had altered. At least these men appeared to be Greeks rather than any unpredictable type of barbarian: the body armor they both wore was of the standard Greek type- a cuirass made from layers of linen glued together to form a stiff carapace with a fringe of plates about the hips- and the helmets pushed back on their heads were of the popular Attic Greek design, with hinged cheekpieces and no noseguard. It was impossible to tell more about their origin from their speech, however, for they said nothing, merely stood leaning on their spears, watching with bored expressions while the elderly customs official went about his business.

The customs official spoke first to the ship's captain, while the dozen passengers waited in a knot beside the gangplank. "You've come from Alexandria?" the official asked, and there was no doubt about his origin: he spoke in the broad Doric dialect of the city. Archimedes found himself smiling at the sound. The one thing he'd disliked about Alexandria was the way everyone had made fun of the way he talked. There were, after all, some good things about being home- and the best of them would be to see his family again. He wrapped his arms around himself in an effort to contain his impatience. He hadn't been able to tell his family what ship he would sail on or when it might arrive, and he was eager to surprise them.

The captain agreed that yes, the ship had come from Alexandria by way of Cyrene, and that the cargo was linen and glass and some spices. He produced the bill of lading, and the customs official began to go over it. Archimedes' attention wandered. A dead fish was floating in the water beside the ship. It lay on its side, its tail slightly raised. Live fish swam belly-down: why did dead ones always float on their sides? He imagined a piece of wood of approximately the same length and breadth as the fish. It would float on its side, too. What about a wider piece of wood, a box-shaped one? Would it float with one of its flat sides down, or with a long edge down?

The customs official had started to gossip with the captain. There was obviously going to be a long wait before the joyful reunion. Archimedes rubbed the dirty stone of the quay with a sandal, then squatted and pulled the set of compasses out of his belt. It was lucky he'd forgotten to give them to Marcus to be packed.

He was deep in the equilibrium of cuboids when a hand tapped him on the shoulder and a voice demanded, "Well?" He looked up from his sketches and realized that the customs official was talking to him. The two soldiers were staring and grinning and the sun was noticeably lower. Marcus was sitting patiently on the luggage chest at the foot of the gangplank, but all the other passengers had gone.

Archimedes sprang to his feet, his face hot with embarrassment. "What did you say?" he asked, still struggling to force floating cuboids from his mind.

"I asked you your name!" repeated the customs official in annoyance.

"Sorry. Archimedes, son of Phidias. I'm a citizen of Syracuse." He waved vaguely at Marcus. "That's my slave and my things."

The official softened at the discovery that he was dealing with a fellow citizen. Archimedes: an unusual name, particularly in a city where half the male population was named Hieron, Gelon, or Dionysios, after great leaders of the past. The name Phidias was vaguely familiar, though, linked with a couple of stories the official had heard of intellectual eccentricity. "Your father's the astronomer, isn't he?" he asked. "I've heard of him." He glanced at the geometrical figures scratched over the quay and snorted. "Got a son of his own breed, it seems. What were you doing in Alexandria?"

"Studying," said Archimedes, and swallowed a knot of irritation. It was no insult to be told that he was a true son of his father. "Studying mathematics."

One of the soldiers nudged the other and whispered something; the second man laughed. But the official ignored them. "You've come home because of the war?" he said approvingly, and when Archimedes nodded, he went on, still more approvingly, "That's the brave young fellow, to come back to fight for your city!"

Archimedes gave him a false smile. He was loyal to his city, as any man ought to be, but he had no intention of enlisting in the army if he could possibly avoid it. He was certain that he'd be far more use to Syracuse building war machines- and besides, he'd had the usual military training at school, and had comprehensively detested it. Drilling and hurling the javelin, wrestling and races in armor: exhaustion and blistered hands; humiliation by glamorous champions on the field and even more humiliating sexual advances in the bathhouse afterward. When his year with the state-issue javelin was finally over, he'd chopped the miserable weapon into sections and used the bits to make a surveying instrument. He had no plans to buy a new spear now. But he knew better than to disagree with a customs official.

The customs official smiled back obliviously and walked over to inspect Marcus and the luggage. "The slave's your own man?" he asked, calling the question over his shoulder. Marcus politely slid off the chest.

"Yes," Archimedes called back, relaxing. "My father bought him here in the city, years ago, and gave him to me when I left for Alexandria."

"There's no duty to pay on him, then. And the goods are your own, for your private use? Nothing you plan to sell?" The official looked them over with a practiced eye: a large, coffin-shaped chest of wood and leather, very battered, and a new wicker basket lashed to it with rope. The chest had undoubtedly carried all the owner's luggage out to Egypt, and the basket had been purchased for the inevitable surplus which had been discovered when it was time to come back. "What's in the basket?"

"A, um, machine," Archimedes said awkwardly.

The official glanced at him with a lift of the eybrows, and for the first time the two soldiers showed some interest. "Machine" in those days meant, first and foremost, "war machine."

"What sort of machine?" asked the official.

"It's for lifting water," said Archimedes, and the soldiers lost interest.

The nudger whispered again, and this time Archimedes overheard his comment: "What non-mathematicians would call a bucket!" His face went hot.

"You planning to sell it?"

"Well, um, not that one. That one's the prototype. It's just a model. I brought it back so I could show people how it works. If somebody wants one, I'll build a bigger one." He spread his arms in a vague indication of the real machine's unbucketlike size.

The customs official considered the concept of a prototype. He could not recollect ever having encountered one before. "It's not dutiable," he decided. "No need to trouble you, then. You're free to go." He nodded toward the nearest gate.

Marcus went to the foot of the luggage chest and picked it up. Archimedes looked round for a porter, saw none, and went to pick up the other end himself- just as Marcus got tired of waiting and set his end down. The soldiers once again nudged each other and laughed. Archimedes' face flushed once more. "Marcus!" he called irritably, bracing the heavy chest against his knee.

But at the name, the soldiers suddenly stopped laughing. "Marcus?" one repeated sharply. Archimedes thought he was the laugher, not the whisperer. He strode forward and stared across the luggage chest at the slave.

Marcus stared back impassively, his hands at his side. "It's what I'm what called," he said evenly.

"That's a Roman name," said the soldier, and it was an accusation.

Archimedes set his own end of the chest down, frowning with a mixture of alarm and disgust. It was clear that a Roman- even an enslaved one- could not be allowed to wander about the city as he pleased. On the other hand, no sensible person would seriously expect to find a Roman as a slave: slavery was the fate that the Romans were accustomed to impose upon others. "Marcus isn't Roman," he declared. "He's some other sort of Italian, from somewhere up north."

"Why does he have a Roman name?" returned the soldier, and Archimedes' alarm and disgust grew as he recognized the accent. Doric, but not Sicilian Doric: that way of swallowing the ends of words was distinct to Tarentum, the city which had once been Taras and the proudest of the Greek cities of southern Italy. A Tarentine in the service of Syracuse had probably fled his own city when the Romans conquered it, and could be relied upon to hate all things Roman. This particular soldier was obviously longing for Marcus to be Roman so that he could punish him.

"Can't help my name," said Marcus mildly. "Lots of Italians with Latin names these days. Comes of being conquered by Romans."

The soldier regarded him with narrowed eyes. "If you're not Roman, what are you?"

"Samnite," replied Marcus promptly. The Samnites had fought three wars with Rome, and rumor had it that despite three crushing defeats and complete subjugation they still hoped for an opportunity to fight a fourth. Not even a Tarentine could object to a Samnite.

This Tarentine, however, proved to be not merely vindictive but well informed. "If you were a Samnite, you'd call yourself Mamertus," he pointed out. "Why the Latin form of the name, if you speak Oscan?"

Archimedes had in fact noticed a Protean quality to Marcus' nationality in the past. The slaver who sold him had called him a Latin, but Marcus himself had sometimes claimed to be a Sabine and sometimes a Marsian. Archimedes had no idea what the truth was- but he knew Latins, Sabines, and Marsians were all part of the Roman alliance. Disgust was swallowed entirely by alarm: Marcus might well be sent to the state quarries for the duration of the war. Given the conditions under which quarry slaves were kept, he'd be lucky to leave them alive. "Marcus is a Samnite," he declared firmly. "And he's been in the family for years. My father bought him when I was nine years old. Do you think I'd smuggle an enemy into my own city? If you want to accuse me of something, do it in front of a magistrate."

The Tarentine gave Archimedes a hard look before turning his assessing stare back on Marcus; Marcus stared back with the same unruffled impassivity he had adopted from the first. The soldier shifted his grip on his spear and commanded, "Say, 'May the gods destroy Rome!' "

Marcus hesitated, then raised both hands to heaven and said loudly, "May the gods destroy Carthage, and grant victory to lovely Syracuse!"

The soldier whipped his spear up and around in a whistling blur; the shaft caught Marcus under his raised arm and knocked him sideways into Archimedes. Archimedes gave a yelp, nearly fell off the quay, and dropped to all fours, skinning his knee on the stones. Marcus fell on top of him with a grunt.

Archimedes was aware of a thick silence as he struggled to get back to his feet. He could feel Marcus on top of him shaking- whether with rage or with fear he couldn't guess. Then the slave's weight shifted and slid off, and Archimedes scrambled up. Marcus remained kneeling on the quay, right hand pressed against his left side where the spear shaft had caught him. Archimedes could feel blood trickling down his own shin. For a moment he was so angry he wanted to hit the soldier: what right did this foreigner have to knock him down on the docks of his own city? He took a deep breath and reminded himself that the soldier was a foreign mercenary, to be treated with great caution; that the soldier was armed and he wasn't; and that he did not want Marcus in trouble. "Why did you do that?" he demanded, struggling to swallow his rage. "He may not have said what you told him to, but he prayed for victory for the city!"

"He prayed for the destruction of Carthage," said the Tarentine. He was flushed now, and a bit breathless: he'd gone further than he'd meant to. Hitting slaves was one thing; knocking over freeborn citizens quite a different one. His comrade and the customs official were staring at him with distaste.

"Don't we all?" said Archimedes. Carthage had been the enemy of Syracuse since the city's founding nearly five centuries before.

"Carthage is our ally," said the soldier.

Archimedes was too astonished even to remember the caution needed with mercenaries. He looked from the Tarentine to the other soldier, then to the customs official. "Carthage?" he repeated disbelievingly.

The other soldier and the official looked embarrassed. "You hadn't heard?" said the official.

Archimedes shook his head numbly. He supposed that, in a way, it was a natural development. Carthage and Syracuse had long fought for the possession of Sicily, and the Carthaginians were undoubtedly as dismayed as the Syracusans by the intrusion onto the island of the rising power of Rome. Perhaps it was right that two old enemies should unite against a new common threat. But- Carthage! Carthage, which had tortured to death the entire male population of the city of Himera; Carthage, which worshiped gods that required her to burn her own children alive; Carthage, the crucifier, the deceiver, the enemy of the Greeks! "Has our tyrant really sworn a treaty with Carthage?" he asked.

"Our king," the Tarentine corrected him quickly. "He calls himself king now."

Archimedes just blinked. "Tyrant" to a Syracusan was the natural title for an absolute ruler: it implied no condemnation. If Syracuse's present tyrant wanted to call himself a king, that was his right, but it seemed a bit pointless.

"King Hieron hasn't sworn anything," the official said defensively.

"He's no fool," added the second soldier, for the first time speaking above a whisper and revealing his own accent- to Archimedes' relief- as the inimitable growl of the back streets of Syracuse. "If Carthage wants to help our shining city against Rome, she's welcome to, but King Hieron won't trust that sackarsed lot, and I say, well done! He's agreed to a joint military operation against the Romans, nothing more." He gave the Tarentine a disgusted look: it was quite clear that he thought that a prayer for the destruction of Carthage in no way deserved a blow.

Marcus grunted, and Archimedes remembered what he was supposed to be doing. "We hadn't heard anything about this alliance in Egypt," he said stiffly. "I'm sorry if Marcus offended you, but he thought he was praying for a Syracusan victory."

The official and the Syracusan soldier both nodded, accepting the explanation, relieved that Archimedes had tacitly agreed to forget the blow. The Tarentine, however, only scowled. Marcus might have prayed for a Syracusan victory, but he had not prayed for the destruction of Rome. The man's dark eyes returned to the slave, who was still kneeling on the quay, head bowed, rubbing his bruise. Something flickered behind the scowl: the desire to hurt and humiliate.

Archimedes too was keenly aware of his slave's evasion. He cleared his throat. "If you really believe that Marcus is a Roman- though why you should expect a Roman to be a slave I don't know- we can go see whoever is responsible for deciding about such things," he offered. "On the other hand…" He dug into his purse and pulled out two silver staters- two-drachma pieces, each one worth more than a day's wage for a mercenary. "It's getting late, and I want to go home to my family, not hang about the magistrates' courts." He held out the coins toward the Tarentine: they gleamed in his palm, fresh-minted silver stamped with the head of King Ptolemy of Egypt.

The Tarentine just stared at them. The Syracusan soldier, however, hurried over and took the coins with a grin. The customs official also hurried over, sucking his teeth. He looked a question at the Syracusan. The Syracusan grinned again and declared easily, "We'll split it three ways."

The Tarentine gave Archimedes a black glare. But the other two were happy to take the money and forget about Marcus, and he didn't quite dare override them. "You can't split two staters three ways!" he snapped instead.

Archimedes forced himself to smile, though the effort nearly choked him. "Of course you can," he said. "It makes eight obols each if you do. But here." He pulled out another coin, identical to the first two. "Good luck to the defenders of the city!"

The Tarentine snatched the coin with a look of pure hatred and strode off toward the nearest gate. His comrade shrugged, gave Archimedes a look of apology, and turned to the customs official with the other two staters. Archimedes limped over to Marcus.

"Are you hurt?" he asked.

Marcus rubbed his bruise once more, then shook his head. He got slowly to his feet, scowling. "May the gods destroy that Tarentine filth in the worst way!" he muttered. "Three staters tipped into the sewer!"

Archimedes slapped him across the face, anger and relief adding force to the blow. "You worthless lump!" he exclaimed in a vehement whisper. "You might have been sent to the quarries! Why didn't you say what he told you to?"

Marcus looked away, now rubbing the slap. "I'm not his slave," he declared.

"Sometimes I wish you weren't mine, either!"

"Sometimes I do too!" replied Marcus, looking back and meeting his master's eyes.

Archimedes let out his breath in a hiss. "Well, you nearly got away from me there, didn't you? That fellow wanted you chained up cutting stone till the war ended, whatever your god-hated nation is, and you certainly did everything you could to encourage him. Herakles! I should have let him take you! Why couldn't you have called him 'sir' and lowered your eyes when he talked to you, like a good slave?"

"I'm freeborn," said Marcus sullenly. "I've never crawled to you and your father, so why should I crawl to some Tarentine without a house or half acre to his name?"

"You and your free birth!" exclaimed Archimedes in disgust. "I'm freeborn and a citizen, and I don't quarrel with mercenaries." He was on the point of adding, "Anyway, I don't know why I should believe in your free birth, when you can't decide whether it's free Sabine or free Samnite!" when he noticed that the customs official was walking off while the remaining soldier stood listening. He swallowed the words. They were pointless, anyway. No one born to slavery could ever have been as obstinate, awkward, and proud as Marcus.

"There wouldn't have been any trouble if we'd been seen first," growled Marcus, still defending himself. "They wouldn't've had time for it. And we would have been first, if you hadn't been too busy drawing circles to pay attention." He glanced at the scuffed and scratched quayside and corrected himself: "Drawing cubes."

"Cuboids," said Archimedes wearily. He gazed at the half-eradicated drawings, then started, grabbed at his belt, and exclaimed, "I've lost my compasses!"

Marcus glanced around and quickly retrieved the compasses from the ground beside the luggage. Archimedes seized them gratefully and began checking them for damage.

"That thing looks sharp," said the Syracusan soldier, coming over. "Lucky you dropped it. If it'd been in your belt when Philonides knocked you down, it would've speared you. That leg all right?"

Archimedes blinked, then glanced at his grazed knee. It had stopped bleeding. "Yes," he said. He put the compasses through his belt.

The soldier snorted at this piece of folly, but offered to help with the luggage. The guardsman was, Archimedes noticed, about his own age, a wide-shouldered man with a close-clipped curly beard and a pleasant, shrewd-eyed face. For all his whispered jokes to his comrade earlier, he seemed genuinely to want to be friendly now. Archimedes accepted the offer.

With Marcus carrying one end of the chest, the soldier the other, and Archimedes trying rather ineffectually to help in the middle, they started toward the gate. "Thanks for the money," said the soldier. "My name's Straton, by the way, son of Metrodoros. When you come to enlist, mention me and I'll see that you're looked after."

Archimedes blinked again, then remembered the customs official's assumption that he'd returned to fight for his city. He was silent for a moment. He had no plans to enlist; on the other hand, some advice from a friendly source in the city garrison would be very welcome. "I, uh, wasn't planning to enlist, exactly," he said hesitantly. "I, uh, thought the king would want engineers. Do you know how I should go about asking for a job as one?"

Straton glanced at the wicker basket strapped to the chest- the big bucket! — and smiled to himself. "You know anything about catapults and siege engines?" he asked.

"Um, well," said Archimedes, "I never actually made one. But I know how they work."

Straton smiled again. "Well, you can talk to the king about it, of course," he said. "He might want people. I don't know."

Marcus laughed. The soldier's smile vanished, but he said nothing.

"Is King Hieron in the city now?" asked Archimedes earnestly.

King Hieron, Straton informed him, was off with the army, besieging the city of Messana. The man in charge in Syracuse was the king's father-in-law, Leptines. Straton wasn't sure whether Archimedes should approach Leptines or whether he'd do better to go north to Messana and speak to the king himself. He'd ask around. Would Archimedes like to meet him the following evening, for a drink? He'd be posted to the docks again all day, but his shift finished at dusk, and they could meet at the gate. Archimedes thanked him and accepted the invitation.

They had passed the gate by this time, and they set the heavy chest down in the narrow dirt street on the other side. "Where are you going?" Straton asked.

"Other side of the Achradina," Archimedes supplied at once. "Near the Lion Fountain."

"You don't want to carry this all that way," said Straton authoritatively. "Gelon the Baker down the road has a donkey he'd loan you for a few coppers."

Archimedes thanked him and went off to see about the donkey. Marcus started to sit down on the chest; Straton caught his arm. "Just a minute!" he said sharply.

The slave's face went blank, and he stood perfectly still, making no effort to retrieve his arm from the other's grip. The two men were much of a height, and they looked directly into each other's eyes. It was beginning to get dark, and behind them the new guard shift was closing the sea gate of Syracuse.

"I'm not Philonides," said the soldier quietly, "and I don't beat other men's slaves, but you deserve a thrashing. I don't care what sort of Italian you are, but just at the moment the city doesn't like any of your nation, and if we'd gone to a magistrate, you wouldn't have escaped without a beating at the least. Your master got you out of a nasty hole there- and in return you were insolent to him. I don't like to see a slave laugh at his master. Plenty of other people feel the same way, and some are like Philonides."

Marcus had relaxed as he realized that he was in trouble for his conduct rather than his nationality. "When did I laugh at my master?" he asked mildly.

Straton's hand tightened on the slave's arm. "When he said he wanted to be an army engineer."

"Oh, then!" replied Marcus calmly. "It was you I was laughing atsir."

Straton stared in offended surprise, and the corner of the slave's mouth twitched. He was beginning to enjoy this.

"You were laughing at him from the moment you set eyes on him," he said. "And when he said he'd never made a catapult, you made up your mind he doesn't know a thing about them, didn't you? Let me tell you this: if Archimedes makes catapults, and if King Hieron's half as clever as he's supposed to be, then whoever the king has making catapults at the moment is out of a job. Do you gamble?"

"Some," said Straton, puzzled now.

"Then I'll lay you a bet on it. Ten drachmae to the stater he gave you- no, make that twenty! I bet you that if my master becomes an engineer for the king, then whoever's in charge of whatever he's set to do will be demoted or unemployed within six months, and Archimedes will be offered his place."

"You have twenty drachmae?"

"I do. Before you decide about the bet, you want to hear how I got it?"

Straton stared suspiciously a moment, then gave a snort of concession. "All right." He let go of the slave's arm.

Marcus leaned back against the chest. "We went out to Alexandria three years ago. My master's father, Phidias, sold a vine-yard to pay for the trip: he'd been to Alexandria himself as a young man, and he wanted his son to enjoy the same opportunity. Archimedes did enjoy it, too- Herakles, he did! They have this big temple to the Muses in Alexandria, with a library-"

"I've heard of the Museum," said Straton with interest. "Myself, all I know is how to read, and that badly, but I've heard that the scholars of the Museum of Alexandria are the most learned men on earth."

"It's a lunatic asylum," said Marcus disgustedly. "Full of a lot of Greeks drunk on logic. My master raced in to join them like a lost lamb that's finally found its flock. Made a lot of friends, did geometry all day, sat up drinking and talk, talk, talking all night; didn't ever want to go home to Syracuse. You saw fit to tell me I deserve a thrashing for the way I talk to my master: let me tell you, I've earned the right to talk to him anyway I like! I could've stolen every copper he had and run off with it, anytime, and he wouldn't even have noticed until three days later. Instead, I looked after him and tried to make one drachma do the work of two. Phidias had given us money to last us a year- though with the prices they charge in Alexandria, it wouldn't have. First we spent that, and then we spent our return fare, and then we bartered and borrowed and sold bits and pieces, and then, after two years in the city, we were flat out of cash and in debt. I kept pointing this out to Archimedes until he finally paid attention and agreed to do some machine-making."

Marcus paused. "That's a common story, isn't it, apart from the geometry? Young man away from home for the first time, running wild in a big foreign city, faithful slave wringing his hands and saying, 'Oh, sir, remember your poor old father and come home!' All right, but here's where it gets uncommon. My master builds machines. Not ordinary machines, but machines so cunning and ingenious you could travel the world from one end to the other and never see anything like them. That's how we lasted two years in Alexandria: whenever we were short, he'd put something together and I'd go sell it. He'd been playing around with that for a while"- Marcus jerked his head at the wicker basket behind him- "but he'd never got around to seeing if anyone wanted a full-size one. Now he took it to a rich man we knew who'd recently acquired an estate in the Nile Delta and was looking for ways to improve his land. Zenodotos looked at the water-snail and fell in love- and that shows sense, because the water-snail is the most amazing machine Archimedes ever built, the most amazing machine I've ever seen in my life. Zenodotos instantly ordered eight of the things at thirty drachmae apiece. He agreed to supply us with all the equipment and the labor we needed to make them, as well as our keep while we were working and the traveling expenses to and from his estate."

"So we went up to his estate and set to work. When we got the first water-snail finished, people started coming around to have a look at it. Now in Egypt they've been irrigating since the creation of the world. They thought they knew everything there is to know about raising water- but nobody had ever seen anything like a water-snail. And everybody- I tell you, everybody- who had a bit of land in the Delta wanted one. I put the price up to forty drachmae, then to sixty, then to eighty: it made no difference. People still queued up to buy. And then, of course, the rich men in the queue didn't want to wait. They started coming up to me and slipping me a drachma and saying, 'See that your master does my order first.' That is where I got my money: selling the shavings of Archimedes' ingenuity."

"If it was so profitable, why aren't you still building water-snails?" asked Straton skeptically.

"Archimedes got bored with them," Marcus replied at once. "He always loses interest in his machines once he's got them working. He'd rather spend his time drawing circles- or, excuse me, cuboids. Of course, other people started making water-snails too, copying them from ours as well as they could. But still, everyone knew it was Archimedes' invention, and we were everyone's first choice of builder. We could have made a fortune, we could have! Instead, as soon as my master could afford to do geometry again, he found an enterprising fellow who was willing to pay a hundred drachmae for the design, handed our order book over to him, and went back to Alexandria to draw circles. I tell you, I could weep when I think of it. But listen! That's what happened the last time Archimedes applied himself to making machines. Now he's going to do it again. I'll bet on him against any engineer King Hieron ever heard of. You taking the bet?"

"Can I see this water-snail?"

Marcus grinned. "Certainly." Then, as the soldier moved toward the wicker basket, he added, "But I charge two obols for a demonstration."

Straton stopped in annoyance, one hand on the basket's lashings. "Your master lets you do that?"

"He lets me look after the money," said Marcus coolly. "Weren't you listening?"

Straton studied the slave a moment, then laughed. "All right!" he exclaimed. "I'm sorry I laughed at your master and insulted your loyalty. You're a good slave."

"I am not!" declared Marcus fiercely. "I'm freeborn, and not slave enough to forget it. But I'm honest. You taking that bet or not?"

"Twenty drachmae to a stater? Your master to be offered his predecessor's job within six months?"

"That's right."

Straton considered. It was an interesting bet. And despite what the slave had said, he thought he'd win. The slave was, after all, loyal to his master, and the master hadn't looked all that impressive to Straton. Ten to one was good odds. "All right," he agreed. "I'll take it."

Archimedes himself appeared as they were shaking hands on it. He was holding a torch, which flickered brightly in the growing dark, and a small boy followed him leading a donkey. Straton gave his new acquaintance the look of appraisal normally reserved for racehorses, and was reassured. No, this tall and gangling young man in a dirty linen tunic and shabby cloak did not look like a formidable genius. He was much in need of a haircut, a shave, and a bath, one knee was covered with blood while the other was dirty, and his face had a vague and vacant expression. The Egyptian stater, thought Straton, was pretty safe.

They loaded the chest on the donkey- the donkey was unhappy about it- and confirmed that they would meet next day. Archimedes handed the torch to Marcus, and the little party clip-clopped off down the road.

"What were you shaking hands about?" Archimedes asked his slave as they turned to climb the hill to the other side of the Achradina.

Marcus gave a self-satisfied smile. "I made a bet with that soldier. Going to get back that stater you gave him."

Archimedes looked at him with concern. "I hope you don't lose your money."

"Don't worry," said Marcus, "I won't."