The Sand-Reckoner Read online

  The Sand-Reckoner

  Gillian Bradshaw

  Gillian Bradshaw

  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 Archim
edes 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?"