The Sand-Reckoner

by Gillian Bradshaw

Agathon gave a sudden snort of laughter. "True son of his father, there. Phidias is supposed to have claimed that Euclid's Elements is a greater book than Homer's Iliad, and to have sacrificed to the gods in gratitude for some mathematical observation of the stars."

  "You know something about him?"

  "Most of Syracuse has heard of Phidias the Astronomer. Bit of an eccentric, bit of a reputation, see? Teaches, too: only man in the city who teaches advanced mathematics. The master studied with him for a little, oh, must be fifteen, twenty years ago now."

  Delia stared. To Agathon, "the master" was always and exclusively Hieron. "I didn't know that!" she exclaimed.

  "Why should you?" asked Agathon. "Long time ago, before he bought me, even. But the master's said a couple of times he wished he'd had more time to study mathematics with Phidias. He only had a couple of months, see? Then your father stopped paying the fees, and the master went into the army. I doubt Phidias even remembers him."

  Delia nodded: she'd long known the story of how her father had paid for his bastard's education, but only until the boy was seventeen. Hieron had then been a year too young to join the army, but he'd been thrust into it anyway, and left to make his own way in the world- with spectacular results. "So why does Hieron regret not having studied longer?" she asked. "Was Phidias a very good teacher?"

  "Don't think so," said Agathon. "No, it's that mathematics is useful to kings. War engines, surveying, building, navigation…" Agathon trailed off, staring at Delia, then lost his disapproving look completely and uncrossed his arms. "Very well!" he exclaimed. "You're right: he'd be interested in Archimedes son of Phidias. If that fellow's confidence is well founded, he's valuable."

  Delia nodded.

  "I'll see what I can find out," said Agathon. Then he looked at Delia again and asked, "And?"

  He'd done it again. Delia sighed. "How much would you trust Eudaimon?" she asked.

  "Ah," said Agathon, face relaxing to something as close to geniality as it was ever likely to attain. "You mean, do I think he's going to try to sabotage your dusty musician's one-talent catapult?"

  Delia didn't answer for a moment. To suggest that Eudaimon would deliberately disable a machine which was potentially of great value to the defense of the threatened city was to accuse him of treason. "I don't know him very well," she said at last, humbly. "Father's been cursing him ever since Hieron went away, he's obviously furious to have a rival, and I don't like him- that's all."

  Agathon shrugged. "He's a man that's worked all his life and never been much good at what he works at. He's the worst of the master's engineers, which is why he's here and not at Messana. He's bitter and he's tired, he's getting old, and he's clinging to his position by his fingernails. He doesn't want some Alexandrian-educated mathematical flute player traipsing in and showing him up- that much is certain. And I think he'll convince himself that that catapult would fail anyway, but still want to make sure it does. Yes, I think if he gets the opportunity to sabotage that catapult, he'll take it. You want me to ensure he doesn't get that opportunity."

  "Isn't that what Hieron would want you to do?" she asked innocently.

  Agathon gave another snort of laughter. "You and the master!" he said affectionately. "I don't know where it comes from. Can't be from your mothers, because you don't share 'em, but it can't be from your father, because he was a fool."

  Delia smiled and got to her feet. "Can you do it?" she asked eagerly. "Without actually accusing Eudaimon of anything, I mean."

  "Oh, yes!" said Agathon comfortably. "Few words in the ear of the foreman at the workshop. He knows who I am. He'll keep an eye on the catapult and Eudaimon both, and report anything suspicious. You want me to have a word with the regent as well?"

  Delia nodded. "But," she added nervously, "don't tell him I…"

  "…have any interest in wine-stained players of the aulos. No."

  "It would be misunderstood," said Delia, blushing.

  "I hope so," said Agathon, with a return of the disapproving look. "I certainly hope that what he understood would be mistaken."

  Dionysios son of Chairephon helped Archimedes from the king's house as far as the temple of Athena on the main road. There he stopped. "I'm headed for the barracks," he said, gesturing to the left, "but I think you'd better go home and lie down for a bit," gesturing right, toward the Achradina. "Eudaimon caught you pretty hard."

  "He's a bugger-arsed idiot!" said Archimedes, with deep feeling. "By Apollo! Builds catapults and doesn't even know what a cube root is! Who's the real engineer for Syracuse?"

  "Kallippos," said Dionysios at once. "A gentleman of good family and better skill. But he's with the king at Messana. The king thought Eudaimon could manage what there was to do here in the city, but there was more to be done than we realized. Wait here- I'll call your friend Straton and tell him to help you home."

  Archimedes shook his head- cautiously, because his eye hurt if he moved it suddenly- and began turning his lump of wet leather to find a cool spot in it. "I'd rather go to the workshop and order my wood," he said. The leather suddenly unfolded and revealed itself as a long wide strap. Archimedes blinked at it: it was a shape he knew well. "Oh," he said blankly. "She's spoiled her cheek strap." Then he realized that he now had an excuse to see her again- to give her a new cheek strapand despite the ache in his eye, he beamed. He folded the leather carefully and set it tenderly back in place.

  "Do you really play the aulos?" asked Dionysios curiously.

  "Of course I do!" said Archimedes, surprised. "You don't think the king's sister would have spoken to me for two seconds otherwise, do you?"

  "I hoped not," replied Dionysios, relieved that his new associate realized when a girl was out of reach. "And, my friend, you should not have spoken to her at all. When I saw you chatting with her there in the garden, I not only expected you to be sent off at once, but I thought I was likely to be in trouble myself for inviting you there in the first place. Nai by Zeus, you were lucky you confined yourself to flutes! Well, if you're really going to the workshop, I can show you the way: it's right next to the barracks."

  The royal catapult workshop was a big dirt-floored barn near the tip of the Ortygia promontory, secured by the same enclosing wall as the barracks of the Ortygia garrison. It was full of beams and presses and saws, and a forge jutted from one wall. The sides were stacked high with timber and iron, bronze and copper, oily boxes of sinew and of women's hair- the last the most favored material for stringing a catapult, a source of sorrow to slave girls and useful income to poor women. About a dozen people were busily at work about the room, some clustered around an arrow-firing catapult which stood half assembled in the center of the building, others making catapult bolts and heel plates. There was a smell of sawdust, glue, charcoal, and hot metal. Archimedes stopped in the doorway and took a deep breath of the scent, then smiled: it was a good smell, the smell of making. He wished Dionysios joy and strode forward eagerly to find the foreman and place his own order for wood.

  Marcus spent most of that day digging out the latrines- a job too heavy for young Chrestos to manage easily on his own, and thus one which had been postponed since the beginning of the summer. In the Sicilian heat the delay had made the job even fouler than it would have been ordinarily, but he set about it stoically, and carted the night soil off on a borrowed donkey.

  At evening, when he returned from dumping the last load, he found his master in the sickroom, just arrived back, cloakless, with an aulist's cheek strap tied over one eye, but enormously cheerful. A knot of unease that had been situated somewhere between his shoulders unraveled. He was only too keenly aware of what would happen to the household slaves if the young master failed to get a job.

  Archimedes was happily telling the assembled family about the royal catapult workshop when Marcus came silently to the door. "They weren't very helpful this morning. They just pointed me at the stores and left me to my own devices. I thought that was fine- you should see the stores! Top-quality Epirot
oak in any thickness you like, and a dozen sorts of glue! But about noon the king's doorkeeper came in to check that I had everything I needed, and after that they realized I was official. After that they set about doing anything I asked. It's amazing how much it speeds things up. I thought it was going to take me a month to make this catapult, and I was cursing the pay. But with this sort of help I can do it in a week."

  "But how much is the pay?" asked Philyra anxiously. Marcus looked at her approvingly: it was what he was very anxious to know himself, but hadn't quite dared to ask, in front of his owners and still stinking of latrines.

  "Fifty drachmae," said her brother, with satisfaction.

  "Fifty!" cried Philyra, her eyes lighting. "Medion, fifty in a month would be good pay; fifty in a week…!"

  Archimedes nodded, grinning. He had not thought fifty a month good pay, but he supposed he'd been spoiled by the water-snails.

  "You don't have to pay for the supplies out of that?" asked Arata anxiously.

  Her son nodded. "I don't need to pay for the supplies unless the machine doesn't work. And you don't need to worry about that, Mama: I know what I'm doing."

  Marcus frowned, suddenly anxious again, and Philyra caught some restless movement he made and glanced over. Their eyes met, and each recognized in the other the same anxiety: how much did the supplies for a one-talenter cost? This worry, however, was almost immediately eclipsed. "What happened to your eye?" asked Arata, and Archimedes told them about Eudaimon, then, obedient to their urging, took off the cheek strap.

  The area around the eye had by this time turned blue-purple and swollen up, and, even worse, the white of the eye itself had turned red, and a veil of blood hung across the light brown iris. "Medion!" cried Philyra in horror. "You ought to sue him for assault!"

  Archimedes just shrugged. "I'm going to stay away from him as much as I can," he said.

  "Absolutely right," said his mother approvingly. "He's senior to you, and you don't want trouble." She frowned, sniffed, and glanced around at Marcus. "Oh, it's you," she said. "Go and wash."

  Marcus bobbed his head and retreated back into the courtyard. He was engaged in cleaning himself when Philyra came out of the old workroom, still frowning. She paused when she noticed him, then came over resolutely. Marcus at once pulled his dripping tunic back on, embarrassed to be naked in front of the young mistress.

  "How much do the supplies for a one-talent catapult cost?" asked Philyra.

  "I don't know," Marcus admitted. "The strings would be the worst of it. They sell prepared hair by the drachma weight, and for a one-talenter you must have to buy it by the pound."

  Philyra was silent for a moment. "He can build it- can't he?" she asked at last.

  "He's good," said Marcus flatly. "He can."

  Philyra studied him a moment, then let out a long unsteady breath. "I don't know anyone else who makes machines."

  He nodded; naturally she had no way of judging her brother's skill. "In Alexandria," he informed her, "the best engineer in the city offered him a partnership. He wouldn't take it, of course- it wasn't geometry- but he could have had it if he'd wanted. He's exceptional. This fellow Eudaimon's absolutely right to be worried. Mistress, the only worries I have are about what happens if something outside your brother's control goes wrong."

  She let out another long breath, looking at him closely, trying to determine how far she would trust his word. Then she smiled, relaxing. "Medion left his cloak at the workshop."

  "At least we know where he left it," said Marcus. "In Alexandria I was always having to run all over the Museum looking for it."

  She giggled. The sweet soft sound seemed to bubble a moment in his heart. "Fifty drachmae a week!" she repeated reverently, smiling over it. "We could buy back the vineyard! And I…"

  She stopped herself. The vineyard sold to pay for her brother's Alexandrian education should have been her dowry, but she had always tried very hard not to acknowledge that painful fact. Her father, she knew, had hoped to save a new dowry for her from his earnings, but his savings had all been eaten away during his illness. She was of an age to marry, she had school friends who were married already, but with no dowry she was unlikely to find a bridegroom. That was a humiliation she tried not to think about, and not the sort of thing a young lady should confide to a household slave. She scowled at Marcus, who was waiting, his face open and alive, for her to finish her sentence.

  Marcus abruptly understood how that sentence would have ended, and busied himself by bending down for his bucket of dirty water. Of course. He had silently disapproved of the vineyard sale precisely because it had seemed to him to cheat the daughter of the house of an essential in order to pay for a luxury for the son. But now he found that he was in no hurry to see Philyra dowried and married off. He would miss her. No need to worry yet, though. It would take some time to amass a dowry for her, even at the rate of fifty drachmae a week. And with the war…

  He was determined not to think about the war. "If you'll excuse me, mistress," he muttered, and went to tip the water over the scraggle of pot herbs by the door. Philyra watched him a moment in surprise, taken aback by the way he had pulled off a sore subject unprompted. She had not thought he had either the sensitivity or the wit.

  The following morning Archimedes set off early for the catapult workshop. Philyra, setting out to do the shopping about midmorning, found only Marcus in the courtyard: Agatha, who normally accompanied her, was helping her mother in the kitchen, and the boy Chrestos had exercised his talent at making himself scarce when he was needed. She looked at Marcus a moment, thoughtfully, then clapped her hands to summon him over and handed him the basket.

  Walking behind her along the narrow street in the morning sun, looking at her straight back, respectably swathed in a white woolen cloak, Marcus found his steps light with an unaccustomed happiness. Philyra was starting to trust him a little. He prayed silently that the gods would offer him the opportunity to prove his honesty. He kept his eyes firmly shut to the reason he wanted her good opinion: there was nothing to be had there, except pain. Getting that good opinion, winning her trust and liking- that was a pleasure no one could deny him.

  They went first to the baker's, and then to the greengrocer's shop around the corner. The groceress, a thin shrewish woman named Praxinoa, looked at them warily. Philyra bought some leeks and some olives, and paid for the goods with one of her brother's Egyptian silver pieces. The groceress studied the money a moment before putting it in her box and taking out the change. "How's your brother settling in?" she asked Philyra, with an eagerness that surprised the girl.

  "Very well," Philyra told her; then, eager for the neighbors to appreciate the family's improved status, went on, "He's found work already. He's building catapults for the king."

  "Catapults, is it?" asked the groceress. "Huh." She glanced around, then leaned closer to her customer and said in a low voice, "Perhaps that explains it, then. I had a fellow in here just before you came, asking about your brother."

  "What?" asked Philyra, startled and alarmed. "Who?"

  "I don't know who," said Praxinoa, with relish. "Never seen him before. He wasn't anybody from the neighborhood. He was smartly dressed, though. Official, I thought. Must be because of those catapults. They're strategic, aren't they?" Her eyes glittered, hungry for scandal.

  "Yes," said Philyra, trying to sound resolute, though her heart had speeded up. In Syracuse, official interest could be very, very dangerous. "They probably ask about everyone who works in the catapult workshop."

  "They do in Alexandria," put in Marcus dismissively. "Saw it there, too."

  Praxinoa subsided, disappointed. "Learned about catapults in Alexandria, did he?"

  Outside the shop again, Philyra looked at Marcus angrily. "You think it really was somebody from the king, because of the catapults?"

  "I can't think of anything else it would be," Marcus told her.

  Anger gave way to anxiety- and embarrassment at asking advice from a househo
ld slave. "Did people come and ask about him in Alexandria too?"

  Marcus shrugged. "No. But in Alexandria he wasn't allowed in the royal workshops. King Ptolemy thinks a lot of his catapults, and never allows foreigners anywhere near 'em. Archimedes looked at some machines on the wall with his engineer friend, that was all. But catapults are strategic. I don't think this is anything to worry about."

  Philyra nodded, but she was still frowning as they walked on. Phidias had never attracted any disquieting official interest. Of course, Phidias had never earned fifty drachmae in a week, either. Things were changing. She wished she felt more confident that all the changes would be for the good.

  Archimedes was obliviously enjoying the workshop. In the past he had always made his machines himself, assisted frequently by Marcus and occasionally by an unskilled slave lent for a particular task: there had always been a great deal of sawing, hammering, and blistered hands in between the interesting parts of machine-making. Now he only needed to say "I want a beam this big to be joined to that beam with tenons," or "I need an iron heel plate this shape to fit that aperture," and within an hour, there it would be. It removed the drudgery from machine-making and left only the agreeable inventive side.

  He wore a linen patch over his eye for his first few days in the workshop, tying it on with Delia's cheekstrap. He had already resolved to give the king's sister her new cheek strap when he went to the house to announce the catapult's completion; in the meantime, it gave him a secret thrill each time he tied on the old one. He did not tell his family where he'd acquired the little leather strap, however. He thought they would disapprove.

  He followed his own advice and tried to stay out of Eudaimon's way. It was impossible to do this entirely, of course. They were sharing the same workshop and the services of the same carpenters. But Eudaimon seemed as happy to avoid speaking to Archimedes as Archimedes was to avoid speaking to him, and for some days all proceeded peacefully. Archimedes made a trip to the nearer forts on the city wall, looking for a catapult whose dimensions he could copy. He eventually fixed on a fifteen-pounder with a particularly vigorous and accurate throw, and corrected the estimated dimensions of his own machine accordingly. The fact that his original was much smaller than his copy raised a few problems, which he enjoyed solving. The one-talenter would have an eighteen-foot arm span and be nearly thirty feet long; it was too heavy and too powerful to aim or draw by conventional methods, and he had to devise systems of pulleys and winches for it. That was fun.