Gillian Bradshaw: The Sand-Reckoner

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…