The Sand-Reckoner

by Gillian Bradshaw

  Eudaimon paid no attention to what his rival was doing until Archimedes had been working on the catapult for four days and was ready to balance the stock on the stand. Then the chief catapult engineer came up and watched in silence as the beam- thick enough for a ship's mainmast, and still only partially finished- was suspended above its tripodal stand by a system of ropes and lowered carefully. When Archimedes signaled the workmen to stop lowering and secure their ropes, however, Eudaimon stiffened. With the beam dangling just above the pin, Archimedes began to thread about it the first of his aiming devices.

  "What's that?" asked Eudaimon harshly.

  Archimedes glanced at him- a process that involved turning his whole body, since his eye was still bandaged- then went on threading his pulleys. "It's to help it pivot," he said.

  "There's nothing like that on the fifty-pounders at the Euryalus fort!" snapped Eudaimon. He sounded affronted by it.

  "Isn't there?" said Archimedes, mildly surprised. "How do they pivot, then?"

  "Didn't you look?" said Eudaimon.

  Archimedes shook his head. Biting his tongue with concentration, he threaded a rope around a pulley set into the stand, looped it through the attachment on the stock, and fixed it back on the stand, to a windlass. Only when he'd made it fast did he realize that Eudaimon hadn't answered his question, and look back.

  Eudaimon was still standing behind him, staring at him with a mixture of shock and outrage. "What's the matter?" asked Archimedes.

  "You didn't look at the fifty-pounders in the Euryalus?" asked the chief catapult engineer.

  "No," said Archimedes. "It's a long walk out there, and I found a machine I liked much closer."

  "But they're the closest in size to what you're trying to build!"

  "Yes," said Archimedes, "but I'd still have to scale them up, and it's just as easy to scale up a fifteen-pounder. How do they pivot?"

  There was a silence. Then the workshop foreman, Epimeles- a big, slow, soft-spoken man in his forties- said, "They don't. To aim them you have get a few strong lads to move the stand."

  "Well, that's stupid!" observed Archimedes. He began threading his second pulley. There would be one on either side of the catapult. The operator would turn a windlass on the side required and use a third windlass to adjust the elevation.

  He paid no attention when one of the workmen sniggered, but looked up sharply at the sound of a blow and a cry of pain. He was just in time to see Eudaimon striding off and one of the workman clutching his ear. Archimedes dropped his rope and dashed after the chief. Eudaimon stopped abruptly and spun about, his seamed face black with anger.

  "You had no business hitting that man!" Archimedes told him furiously.

  "I will not be laughed at in my own workshop by my own slaves!" Eudaimon shouted back.

  "They're not your own slaves, they're the city's. You had no business hitting him! And anyway, what was it to you? It's not as though you'd made those fifty-pounders!"

  "I'm in charge here!" declared Eudaimon. "I can have that fellow flogged if I like. Maybe I do like. Elymos! Come here!"

  The man he had struck stepped back in alarm, and the other workmen stared at the chief in horror.

  "You don't dare!" cried Archimedes in outrage. "I won't let you!" He turned to the foreman. "You run up the road and tell the regent about this!"

  "Do you think Leptines wants to be bothered with a squabble in the workshop?" said Eudaimon.

  "He will if he has any decency!" replied Archimedes. "He's in charge, and nobody should allow people to go about flogging people when they haven't done anything wrong!"

  "I will tell the regent," said the foreman decisively, and turned to go.

  The foreman was as much a slave as the rest of the workmen, but he was a valuable, experienced, and trusted slave, and his word carried some weight even in the king's house. Eudaimon started in alarm and ordered, "Stop!"

  Epimeles turned back and looked at Eudaimon levelly. "Sir," he said, "you and… this gentleman are both authorized to use the workshop. If you say Elymos is to be punished, and he says he is not, surely it's for our master to tell us which one of you to obey?"

  "I am in charge!" grated Eudaimon.

  "In that case the regent will tell us to obey you and flog Elymos," said the foreman quietly.

  There was another silence, and then Eudaimon said, "I never gave any such order." He glared at them all. "You all know that! I never gave any such order." He turned on his heel and walked off.

  The foreman let out his breath slowly. Elymos gave a whistle of relief and sat down, and his friends thumped him on the shoulder. Archimedes thought of thumping the slave's shoulder too, but refrained: he was aware that the threat of flogging had been made only because of him.

  "Are you all right?" he asked instead, coming over.

  Elymos nodded and grinned up at him. "Thank you, sir," he said. "I'll remember how you stood up for me."

  "You shouldn't have laughed," Epimeles told him sternly, coming over as well.

  Elymos ducked his head appeasingly: Eudaimon might order floggings, but Epimeles was the person who was really in charge of the workshop. "Couldn't help it! It was funny!" protested Elymos.

  "But it wasn't even his fault that those fifty-pounders can't pivot," said Archimedes. "He didn't build them."

  At this Elymos laughed again, more loudly this time. "That makes it even funnier!"

  Some of the other workmen laughed as well. Archimedes stared, perplexed, and they nudged one another and giggled among themselves. Archimedes realized that the laughter was directed at himself, and flushed. He went back to his catapult and began rethreading the ropes in hurt silence. People had always laughed, were always laughing, at him. He got lost in his geometry and didn't notice things, or he got excited about things they didn't understand, and they laughed. Even slaves he had defended were laughing at him.

  Elymos leaped up and followed him. "Oh, sir, don't be offended!" he said. "It's just a workshop joke, that's all."

  "Well, I don't get it!" said Archimedes angrily.

  The slave sniggered again, then, at a sharp glance, looked solemn. "Sir, I couldn't explain it. Not to you. Jokes are never funny if you explain them. But please don't be offended, sir. It's just a…a slaves' joke, that's all." He hurriedly took the third rope and tried to thread it around a pulley.

  "Not that one!" Archimedes told him hastily. "That goes on top. No- no, leave it! Go fetch me the chalk, if you want to be helpful!"

  The foreman, Epimeles, watched for a little while as the massive beam was set down upon the joint pin in its stand. Archimedes had calculated the approximate area of equilibrium and ordered a series of holes drilled along it. The stock was found to balance best upon the middle one: Epimeles smiled. He watched for a minute longer as the huge machine pivoted left and right in response to the windlassesthen sighed and reluctantly left the building. He had a long walk before him.

  It was dusk when Epimeles got back to the Ortygia, but he did not go directly to the barracks next to the workshops, where he and the other workmen lived. Instead he went to the king's house and knocked upon the door.

  Agathon opened it- that was his job, after all- and regarded the workshop foreman with displeasure. "Your business?" he demanded.

  "Came to show you something," replied Epimeles calmly.

  Agathon snorted and invited him in.

  The doorkeeper had a lodge beside the door, a small but comfortable room with a couch and a carpet and a stone water cooler against the interior wall. Epimeles sat down on the end of the couch with a sigh of relief and began rubbing his calves. "Walked up to the Euryalus and back this afternoon," he commented. "I could do with a cup of wine."

  Agathon looked even more disapproving than usual, but took a jar from beside the wall, poured some into two cups, and added some water, fresh and chill from the stone. "Why should I be interested that you were up at the Euryalus?" he asked, sipping.

  Epimeles drank most of his wine off at
once, then set the cup down. "Because I went up there for that engineer you told us tolook after," he said. "I found this." He opened the small sack he'd been carrying and brought out a coil of fine cord. It was divided into sections by a series of regular knots, which had been dyed red or black.

  Agathon inspected it straight-faced, then said, "There's something peculiar about a fort having a measuring line, is there?"

  Epimeles pulled a second measuring line out of the sack. It was seemingly identical to the first, but older, fraying a little and discolored. He stretched the two cords out side by side, and it was immediately apparent that they weren't identical after all: the new cord's divisions were shorter than the old one's. "This one's mine," said Epimeles, touching the old cord. "It's accurate."

  Agathon looked at the two cords expressionlessly.

  "You know that when you're building a catapult, the essential thing is to make all the parts stand in exactly the right proportion to the diameter of the bore?" coaxed Epimeles. "You get a catapult that works, and you measure it, and then you either reproduce it exactly or scale it up or down."

  "I believe I'd heard that," said Agathon. He did not, in fact, know very much about catapults, but he had no intention of admitting it- and he understood enough to grasp the implication of the cord. "You're suggesting that Eudaimon left this"- he touched the new measuring cord- "at the Euryalus, so that anyone who took measurements of the machines there would get the wrong figures, and any catapult built in imitation wouldn't work?"

  Epimeles nodded. "See," he said, "the two fifty-pounders up at the Euryalus are the biggest catapults in the city at the moment. Eudaimon assumed that Archimedes would measure them, then guess at the increases necessary to make them throw the extra ten pounds: it's the way he'd have gone about designing a one-talenter himself. This afternoon it came out that Archimedes couldn't be bothered to walk up to the Euryalus and took his measurements from some little fifteen-pounder close by. Eudaimon was…" The foreman hesitated, picking his words, then said, "… outraged, shocked, and disappointed. When I saw that, I thought I'd go up to the Euryalus and see what he'd been up to- and sure enough, I found this, in the storeroom where the gear's kept. The lads at the fort all agree that it was just where their old one was, but that it's new, and they don't know how it got there. But they remember Eudaimon coming up there in the afternoon four days ago."

  "I see," said Agathon grimly.

  It wasn't evidence to convict a man of treason: they both knew that. But it could be an underminer, a question mark, a stone in the shoe. It could hurt Eudaimon.

  Epimeles shifted the cord toward the doorkeeper. "I thought you should look after it."

  Agathon nodded thoughtfully and picked up the false measuring cord. He began winding it about his hand. "I'm surprised you went all the way up to the Euryalus to look for it," he said. The fortress lay at the extreme point of the city wall, six miles from the Ortygia.

  At that Epimeles grinned. "I would have gone twice as far if it'd help get your lad put in charge of catapults. It will, won't it?"

  Agathon looked up in surprise.

  "Well, you know he's good!" said Epimeles, surprised at the question on his face. "You told us to look after him and make sure nobody interfered with his one-talenter, and we realized why pretty quickly. He's so good he doesn't even realize how good he is. That one-talenter- you know what he's done with it? The little fifteen-pounder he copied can be pivoted, of course, so he thought up a system with windlasses so that his will pivot as well. When I told him that the fifty-pounders at the Euryalus don't pivot, he just looked surprised and said, 'Well, that's stupid!' "

  Epimeles laughed. Agathon looked at him sourly and asked, "Is it?"

  "People will say so now, won't they? But nobody ever used to expect anything bigger than a forty-pounder to pivot. Archimedes has just invented an entirely new system for aiming big machines- and he doesn't even realize! It was easier for him to design it than it was to walk up to the Euryalus and have a look at how other people did it. Some of the lads laughed about that, and he didn't even understand why. Zeus! I almost feel sorry for Eudaimon. He's never built a catapult that wasn't copied piece by piece from another catapult, and when he can't get definitive measurements- and on the big machines, each one is a bit different- he guesses and he struggles and he runs all over the city trying to find out what the right figure is. Archimedes sits down and scribbles for half an hour and has the perfect number there in his hand. Zeus!" he said again. "Eudaimon's like some little local athletics teacher who trains hard every year and toils to come third or fourth in the city games- and he's trying to race against a fellow who could take the crown at Olympia and barely raise a sweat. He's not good enough to compete in the same event. He's not even good enough to realize that!"

  "So he cheats," said Agathon sourly.

  " 'Course he does," agreed Epimeles. "Mind you, he would against any opponent, and I can't entirely blame him. When he loses this job, where will he go? He's got family, too, depending on him."

  "You're almost sorry for him?"

  The foreman looked down. "No," he said, quietly, "I am sorry for him. But I don't want him in charge. Nobody likes building catapults that are feeble, or kick over, or can't shoot straight. That one-talenter, now- that will be a real Zeus, a hurler of thunderbolts. You can feel it when you look at it. It sort of pulls the whole workshop in around it like a whirlpool and it makes my hair stand up to touch it." He paused, then added, "And don't worry. Nobody's going to hurt that machine now. The lads and I will see to that."

  "Has Archimedes asked you to guard it?"

  Epimeles looked offended. "You think we need him to ask us? A divine thing like that? That catapult is our work as well! But no, he hasn't asked us. I don't think he's even noticed that he's putting Eudaimon out of a job, and it's never occurred to him that Eudaimon would ruin the catapult to hurt him. He doesn't notice Eudaimon much. He doesn't notice people much anyway, and when it's a person he doesn't like he notices him even less. He's pleasant enough when he does notice, though, and he treats the lads decently. I'll have no trouble working with him." He grinned at the prospect, and finished his cup of wine. "Will you show that"- he gestured at the measuring cord- "to the regent?"

  Agathon sucked his teeth thoughtfully for a minute, then shook his head. He had a low opinion of Leptines. "I'll wait for the master to come home," he said. "He'll be very interested."

  5

  The catapult was completed in the middle of the morning four days later. It crouched in the center of the workshop like a predatory insect: a long low stock like an abdomen perched upon the three-legged stand, and at the far end, the great bowlike arms stretched wide like a praying mantis striking. The single eye of the aperture between those arms had the unblinking stare of death. When Archimedes winched back the string- an arm-thick leather cable- it gave a groan like a giant waking; when he released it, the clap of the ironclad arms against the iron heel plates was like the shattering of mountains. The workmen cheered and stroked the beast's bronze-plated back and wooden sides.

  Archimedes had expected the machine to be finished that morning, but still he stood back and contemplated it with delight: his first catapult. "It's a beauty," he told Epimeles.

  "The finest I've ever seen," the foreman agreed. Archimedes looked at him in surprise: he knew that Epimeles had been in the workshops for over twenty years, and he hadn't thought the man was given to flattery. Then he looked back at the one-talenter and grinned: best in twenty years or not, it was a beauty.

  "Well," he said, and picked up the cloak he had brought along that morning in the expectation of another visit to the king's house. "I'll go tell the regent it's done, shall I? And ask him where he wants it, and when he wants the trial. But…" He dug in his purse. "Why don't you and the lads buy yourselves a drink to celebrate?"

  "Thank you, sir- not yet," said Epimeles at once. "After the trials, sir, would be better."

  Disappointed, Archimedes put his
money back in the purse: he suspected that for all the flattery, Epimeles wasn't certain the machine would work. He sighed and walked off a bit disconsolately.

  "What was wrong with a drink to celebrate?" asked Elymos, who was fond of wine.

  "The gods hate arrogance," replied Epimeles. "We haven't got it safely through its trial yet. You want somebody to tamper with it while we're busy drinking?" He patted the huge machine with loving awe.

  Archimedes recovered his good humor on the walk to the king's house. The week just past had been thoroughly enjoyable. It had been fun making the one-talenter, and things were well at home: his father actually seemed to have recovered a little. Perhaps it was just not having to worry about when his son would return, but Phidias was sitting up in bed, drinking barley broth three times a day, and taking an interest in things. He listened to the music the rest of the family played for him, he discussed Alexandria with his son, he even played a bit with the puzzle. Archimedes decided that it would help again when he himself got a salaried position as a royal engineer; it would take another burden off his father's mind. Well, that should now happen as soon as the catapult had proved itself.

  And now- now he would see Delia again. Archimedes fingered the small package he'd stowed in a fold of his cloak, the new cheek strap and the old one, and walked faster.

  He had no serious expectation that there could be anything between himself and the king's sister. But he had no expectations about anything: he was living in the present and trying not to think of the future, which held at best a life of drudgery and at worst the horrors of defeat in war. Delia was a pretty girl. She was clever, she had made him laugh, and she played the aulos very well. Today he would see her again and give her a gift: what more could he ask for? He began to whistle an old song as he walked, letting the words run through his mind: