The Sand-Reckoner

by Gillian Bradshaw

  There was a sound of footsteps on the stairs, and Straton hurried up, holding a letter. He glanced round the catapult platform, then looked irritably at Marcus. "Where's your master?" he demanded.

  "Gone into the city to see about arranging a demonstration of ideal mechanics," said Marcus bitterly.

  "He should have waited for the authorization for it!" said Straton, flapping the letter. "Where's he off to? The naval docks? Herakles! Does he really think he can move a ship single-handed?"

  "Yes," replied Marcus. "You want to bet he can't?"

  Straton looked at him, tapping the letter uncertainly against his hand.

  "You owe me a stater," said Marcus deliberately. "You want to try to win it back?"

  Straton sucked his teeth. "I don't owe you anything! The bet was that your master would be offered the job of whoever was in charge of whatever he was set to do. Eudaimon still has his job."

  Elymos gaped at them.

  "You're quibbling," said Marcus. "Eudaimon was in charge of catapults. Now Archimedes is in charge of catapults- isn't he?"

  Straton shrugged uneasily. "King Hieron hasn't said."

  "No," agreed Marcus sourly. "King Hieron hasn't even said whether he's going to pay my master the fifty drachmae that are owing to him. But the whole sense of our bet was that my master's war machines would be better than anyone else's. Now you know that's true- so pay up!"

  Straton cast an embarrassed glance at the Welcomer. For all his ignorance of catapults, he was aware that this one was exceptional. He sighed, and fumbled in his purse.

  "Of course," said Marcus, with deceptive casualness, "if you like you can add another stater to your stake, and bet that Archimedes can't move a ship single-handed."

  Straton frowned, hesitating, staring at Marcus. Then he shook his head. "I'm not betting against your master again," he declared. Then suddenly he grinned and flipped Marcus the Egyptian stater. "Here," he said. "Take it and good luck. I know how to get it back! I'm going to lay Philonides odds of three to one that your master shifts that ship, and I don't doubt for a minute he'll take 'em!" He slung his spear over his shoulder and hurried off with the letter, still grinning.

  Marcus scowled as he put the stater in his own purse. He had expected to enjoy winning that bet, but the image of the of the king's bright smile hung in his mind and soured the pleasure. Jobs were one thing: you knew what you were expected to give and what you could expect to receive. What Hieron was offering was undefined, and who knew what he might want in return for it?

  "You bet that soldier that your master would be offered the job of any engineer he was set under?" asked Elymos, into the heavy silence.

  "That's right," said Marcus shortly.

  "Kallippos is good," said Elymos doubtfully.

  Marcus shot him a look of irritation. "As good as Archimedes?"

  Elymos looked at the Welcomer. Then he shook his head. "I suppose not," he said wonderingly.

  For some reason, Marcus was even more irritated, and suddenly eager to get home. He glanced around the catapult platform once more, and noticed Archimedes' cloak lying abandoned in a crumpled heap under the artillery port. He went to pick it up, then paused and gazed out at the road north.

  The king expected a siege. "It was very stupid of him to expect me to sack a catapult engineer," he had said, "when I'm expecting a siege." Soon, perhaps, a Roman army would be encamped there, in that field before him where goats browsed now. Marcus shut his eyes and imagined the camp: the neat squares pitched within an entrenchment, the campfires smoking, the sound of voices speaking in Latin. There was a bitter surge in the back of his throat. He had heard no Latin spoken for thirteen years now. Soon the Romans and their allies would be here: his own people. They had come to Sicily in a bad cause, and they threatened the city which had become some kind of home to him, the people he had come to care about. If they conquered, he would probably die. But they were his people still. He glanced up unhappily at the menacing shape of the catapult beside him, and reflected that if he were really loyal to his own, he would cut Archimedes' throat.

  7

  That evening Delia was informed that her brother wished to speak to her in his library. She was somewhat taken aback at the choice of location. Hieron generally received the leaders of Syracuse's army and city council in his dining hall or study, and talked to members of his household wherever they happened to be. The library was his private retreat. She picked her way through the gardens and along the colonnade with a mixture of curiosity and foreboding.

  The library was a small room- the book collection of a private individual, not of a city- and it faced onto the smallest of the house's three courtyards. Three of its walls were lined from floor to ceiling with book racks, a neat crisscross of lathes from which the parchment title tags of the scrolls hung down, making the whole room flutter; the fourth wall held the door and a window. The only furniture was a couch, a small side table, and a lampstand. When Delia entered she found her brother reclining on the couch, frowning over a book which lay scrolled open in the light of the three lamps burning on the stand.

  "Hieron?" she said, and he looked up with a smile, then sat up, swinging his feet off the couch and gesturing for her to sit. As she did so, she glanced at the open book, then stared at it hard. It was full of geometrical diagrams.

  Hieron grinned and offered the scroll to her. The title tag informed her that it was Euclid's Conics, Book 3. She waved her hand at it in refusal and mock terror.

  "I don't understand it either," said Hieron. "I was just trying to see if something I saw today was in it. It isn't."

  At this Delia guessed the reason for the summons. "You've seen Archimedes son of Phidias?" she asked eagerly. She had told her brother about her discovery as soon as he returned from Messana.

  Hieron nodded. "And you're right about him," he said. He rolled the scroll up carefully. "He is a very, very clever young man, and could undoubtedly be of value to the city." The rollers clicked together; he tapped them straight and slid the book into its parchment case. "The question is," he went on in a low voice, "how valuable is he, and how much am I willing to pay for him?" And he rested the scroll against his chin, eyes fixed thoughtfully on nothing.

  "Did the catapult work?"

  "Oh, the catapult!" said Hieron dismissively. "Yes, it works. As far as your friend is concerned, it's a good medium-sized catapult, and he hopes it will earn him fifty drachmae and a job alongside Eudaimon."

  "Oh," said Delia, disappointed. "Alongside."

  Hieron lifted his eyebrows. "I'm keeping Eudaimon. I can't afford to lose any engineers just now, and his work is acceptable when he has a machine he can copy. Now he can copy Archimedes'. Once he understands what it is he's copying, I expect he'll be downright enthusiastic about it. It will take him a while to work it out, though, and unfortunately he's going to have to be kept on a tight leash while he does. That's in hand." The king tapped the scrolled book against his chin again. "The question is, what am I to do with Archimedes?"

  "Hire him, of course!" exclaimed Delia.

  Hieron shook his head and sighed. "As what?"

  "As an engineer- what else? And if you expect Eudaimon to copy from him, you ought to make him Eudaimon's superior."

  "Yes, but do I gave him a rank and salary equal to Eudaimon- or to Kallippos? Or do I make up my mind that I'm going to keep him in Syracuse whatever he costs, and plan accordingly? I was hoping, sister, that you, who know the man better than I do, could give me a bit of advice."

  Delia stared. "I-" she began; then changed it to, "But you said it was just a good medium-sized catapult!"

  Hieron shook his head. "I said, as far as he's concerned. It's a one-talenter with a range of five hundred feet and an accuracy equal to the best arrow-shooter, and it can be pivoted with one hand. Archimedes is too young and inexperienced to realize how exceptional it is, but Kallippos didn't know whether to go wild with admiration or with jealousy." There was a pause, and then the king added, with a s
mile, "Being Kallippos, of course he did neither. He just scowled at it and hissed. But I'd bet anything he's in the workshop right now trying to replicate the pivot."

  "I don't think I can advise you at all," said Delia, in a small voice. "I didn't expect- I just thought it was a matter of him replacing Eudaimon. Is he really that good?"

  Hieron nodded seriously. "He may be even better. I've asked him to give a demonstration of ideal mechanics. He offered to move a ship single-handed. I'll see how that turns out before I make up my mind what to do about him."

  "I don't understand," said Delia after a moment. "Why do you have to make up your mind about him now? Why not just- well, give him a job and keep promoting him?"

  Hieron shook his head. He hitched himself up on the couch and turned himself to face her squarely. "Imagine I'm him."

  "You don't look a bit like him," she said, smiling.

  "Now, what is that supposed to mean? You think I should lose some weight? No, imagine I'm the son of Phidias, a mathematical engineer raised by a mathematical astronomer, the sort of man who amuses himself during his idle moments by working out theorems too advanced for Euclid. I studied in Alexandria at the Museum. I liked it. I didn't want to come home. But there's a war starting, my father's ill, and my family depends on me. I am a dutiful and affectionate son. I come home, I look for work making war machines, I find it. Right so far?"

  "I think so," agreed Delia, beginning to be intrigued. "You're certainly right that he liked Alexandria. He talked about it even to me."

  "Everyone Agathon spoke to about him mentioned it! He was apparently supposed to come home two years before he did. Don't look so surprised- you're the one who set Agathon onto him. All right, my first catapult has passed its trial and I've happily agreed to work for what Leptines offered me. I make some very large, very advanced catapults; I also produce countermeasures to seige towers and mines. I'm good at that, of course- the key to siege machinery is the accurate calculation of size and distance, and the key to that is geometry, at which I am adept. At first I don't notice that I'm exceptional, because I haven't made war machines before and I don't have any standard of comparison. But before long it dawns on me that none of the other engineers in the city can do the things I'm doing. And eventually the fame of my machines spreads, and other cities and kingdoms try to hire me. Now: am I a loyal citizen?"

  "I think so," said Delia. "After all, you did come home when you heard about the war, and you hurried to place your abilities at the disposal of the city."

  "Ye-es- but on the other hand, making catapults is the easiest way for an engineer to earn money during a war, and with my father ill my family needs money. Still, we'll say I'm a loyal Syracusan as well as a dutiful son. I reject the offers of Carthaginian Akragas and Roman Tarentum; I scorn Cyrene and Epirus and Macedon- but I feel aggrieved. My family's not rich, my younger sister is of an age to marry and needs a dowry, and I know that I'm worth more than I'm getting. Besides, it's mathematics, not war machines, that is my soul's passion: the yoke frets me. When one of my old Alexandrian friends writes to tell me that King Ptolemy would give me a job in Egypt- five times the salary and half the work- I accept it, take my family, and go. Any comments?"

  Delia frowned. "You wouldn't abandon your city in time of war!"

  "Maybe we'll be out of the war by then: gods, let us be! But if we aren't, won't it mean that I'm eager to take my family out of danger? Particularly when it means returning to a place I love and never wanted to leave. Besides, Egypt is an ally: serving her isn't betraying Syracuse."

  "Would Ptolemy really offer that much?"

  "Oh, that's certain!" exclaimed Hieron in surprise. "Ptolemy's spent a fortune on investigating catapult design, and his advisers perpetually scan the horizon for improvements. And Egypt is rich."

  "Well then," said Delia, smiling with satisfaction, "you should offer him more from the start, so that he's got no cause to feel aggrieved and discontented!"

  Hieron took a deep breath. "Perhaps. But start again. My catapult has passed its trial, and I'm made the equal of Kallippos and paid two or three times as much as I expected. On the strength of this, I can arrange for my sister to marry a man of good family, and perhaps marry a woman of good family myself. I become a citizen of some standing. I have wealth, I have respect. I'm grateful to the city. Even when I realize that my reward is merited, I'm still grateful, because the city recognized me before I recognized myself. When the offer from Egypt comes, I reject it…" Hieron paused, then went on softly, "Or do I?" He stood up suddenly and crossed the room to the book rack. He ran one plump finger down the shelves, then put the scroll of Euclid's Conics back in its place. "The thing I don't know," he went on quietly, "is whether he's merely very good, or invaluable. If he's merely good, treating him generously should be enough to keep him. But if he's what I think he might be, he'll be off to Alexandria eventually however much I pay him- unless I take steps to prevent it. Ptolemy can offer him the Museum, and that's something for which I have no substitute. So perhaps I would do better to save my time and money, treat him as nothing out of the ordinary, and profit by what he's willing to do before he leaves. Or perhaps- perhaps I should decide to keep him whatever he costs, and start chaining him to Syracuse now, before he can realize his own value and assert his freedom." Hieron dropped back down on the couch and put one foot on the cushions beside Delia. "So, what do you think, sister? Is he merely a clever young man, or is he inspired by the Muses?"

  "I don't know," said Delia, low-voiced in confusion. She had imagined herself drawing her brother's attention to merit, and watching proudly as the merit was rewarded. Hieron, however, was not talking about reward, but of use, even of exploitation. She remembered Archimedes laughing with excitement at the thought of what his friends in Alexandria were doing, and suddenly regretted that she had mentioned him to her brother at all.

  "What's the matter?" asked the king.

  "You talk about him as though he were a slave," said Delia uncomfortably.

  Hieron shrugged. " 'One man is my master,' " he quoted softly,

  " 'Custom, yours- and he masters a myriad others too.

  Some are slaves to tyrants, tyrants to fear.

  Men are slaves to kings, kings, to the gods, and gods, to Necessity: for Necessity, you see, endows all things with natures great or less and so forever is master of us all.'

  "Although," he added, in a more normal tone, "I didn't feel like a king's slave even before I was a king myself, and tyrant as I may be, I don't think I'm slave to fear. But I'll grant the poet Necessity and the gods." He smiled at his sister. "Don't worry," he added. "I'm not going to hurt your fellow aulist. In fact, I've invited him to dinner."

  Archimedes was late for the dinner party. he had spent the day at the naval docks, preparing his demonstration of ideal mechanics; when he did not come home to change his clothes late in the afternoon, Marcus was sent to fetch him. The slave found his master covered in dirt and soot and smelling strongly of mutton-fat pulley grease, perched on the roof of a ship shed fixing a pulley to the main roof beam.

  Marcus hauled him down and bore him off to the public baths, ignoring the enthusiastic attempts to explain the system of compound pulleys and wheels- "toothed wheels, Marcus, so they won't slip"- by which Archimedes expected to move a ship. He saw to it that his master was washed and barbered, then brought him home, where a frantic Philyra was waiting.

  "You're going to be late!" she told him furiously. "You're going to be late for dinner with the king! Medion, how do you expect him to pay you if you're going to be rude to him?"

  "But he's the one who ordered the demonstration!" protested Archimedes, blinking.

  Philyra gave a shriek of frustration and hurled his good tunic at him. "You never care about anything except your stupid ideas!"

  Arata, calmer by nature and more resigned, ignored her children's quarrel and drew Marcus aside. "You go with him tonight," she ordered quietly. "But be careful."

  Marcus looked
at her with narrow-eyed reserve. He'd guessed that he'd be ordered to accompany Archimedes to the king's house. A guest did not arrive at a dinner party carrying his own flutes, like a hired musician: a slave had to act as porter, and he was the most natural slave for the job. But- be careful? "Is there some special reason for caution, mistress?" he asked.

  Arata sighed and brushed back a wisp of graying hair. "I don't know," she said slowly. "But- there have been these people asking questions about my Archimedion. I suppose it's just because of the catapults, and understandable- but I don't like it, Marcus. Who can tell what's in the mind of a tyrant? Watch what you say to them in the king's house."

  "Yes, mistress," said Marcus grimly.

  She smiled. "I know I can trust you," she said. "You've served us well, Marcus. Don't think I haven't noticed."

  Marcus hefted his shoulders uncomfortably and looked away.

  When they finally reached the king's house, Archimedes was ushered into the dining room, where the king was already reclining, along with his father-in-law, Leptines, and two army officers (one of them Dionysios), three Syracusan noblemen, and Kallippos- with Archimedes, a conventional total of nine diners. Archimedes was shown to the lowest place on the couch to the left of the table, the most junior place for the youngest guest.

  Marcus was led to a workroom which adjoined the kitchen. Most of the other guests had been attended by their own slaves, and the narrow dirt-floored room was packed with a small crowd. Most were men of about Marcus' age, dressed plainly, though one pretty, long-haired boy in a fine tunic had taken the only stool and sat sniffing disdainfully at the others. Marcus returned the boy's look of contempt: there was no doubt why that one was wearing such fancy clothes.

  "Sit down," said the king's doorkeeper genially; he had been the one who showed Marcus to his place. "What's that you're carrying?"