The Sand-Reckoner

by Gillian Bradshaw

  Straton gave a sigh of relief, set his spear and shield against the wall beside the door, then dropped onto the free end of the couch and began to unfasten his baldric. Dionysios grinned again, this time with knowing sympathy for long hours standing guard, for sore feet, a stiff back, and boredom.

  Archimedes, feeling much the odd one out, took the least comfortable place, in the middle of the couch between the two soldiers. The obsequious waiter bobbed about taking orders, then retreated.

  "Straton tells me you've just come back from Alexandria and are looking to be of service to the city in the war," said Dionysios.

  Archimedes nodded. "But," he added awkwardly, "I've found that I can't go up to Messana to join the army. When I got home- That is, my father's dying. I can't leave Syracuse until- You understand what I mean. If there's something I can do here in the city…" He trailed off with an uncertainty he did not feel. He had left his father to endure illness alone: he would stay with him now, until the end.

  "Ah," said Dionysios. "I am sorry."

  "Bad thing to come home to," said Straton sympathetically. "That, and the war."

  Archimedes made an inarticulate noise of agreement.

  There was a decent silence, and then the captain asked about Alexandria.

  They talked about the city through the first course of the mealthe Museum, the scholars, the temples; the beauty of the courtesans. Straton was silent at first, nervous in his commanding officer's presence, but Dionysios was cheerful and relaxed, there was plenty of wine, and before long they were all three talking freely. Dionysios swirled the fragrant red in his wide-bowled cup and praised Egypt. "The House of Aphrodite," he said. "That's what they call Alexandria, isn't it? Everything that exists anywhere in the world is there, they say- everything anyone could desire. Money, power, tranquility, fame, learning, philosophy, temples, a good king, and women as beautiful as the goddesses who once came to Priam's son Paris to be judged. I'd love to go there!"

  "It's the House of the Muses," agreed Archimedes warmly. "It draws the finest minds in the world as the Heraclean stone draws iron. I didn't want to leave it."

  "But you've come back to Syracuse, because of the war?"

  He nodded. "And because my father was ill."

  Again there was a moment's silence, and this time Archimedes realized that it was more because of the mention of the war than from tact over his father's illness. The war was a subject that weighed heavy on the minds of the two soldiers, but one they did not want to discuss. Twelve years before, the Roman Republic had defeated an alliance consisting of all the Greek cities of Italy, half a dozen rebellious Italian tribes, and the army of the kingdom of Epirus across the Adriatic. The forces had been commanded by the brilliant and adventurous Epirot king, Pyrrhus, who was said to have been the finest general of the age. How could Syracuse alone succeed where such an alliance had failed? Her only hope of victory lay in the alliance with Carthage- and Carthage had always longed for her destruction. How could anyone discuss this war? What was there to say about a conflict where one's enemies were preferable to one's allies?

  The waiter returned with a dish of broiled eel in beetroot saucethe main course- then filled up the wine cups and departed again. Dionysios helped himself to some fish. "Do you know anything about catapults?" he asked, finally getting down to the business that had brought them there.

  Archimedes' earlier discomfort had melted: the company and talk had been almost easy enough to be Alexandrian, and the food was better. Sicilian cooking had always been famous as the finest in the Greek world. He scraped some fish onto his piece of bread, took a bite, and gave the answer that came most naturally to him. "The really interesting thing about them," he announced around his mouthful, "is how you make them bigger. The critical feature is the diameter of the bore in the peritrete. To increase the power of the throw you have to increase the other dimensions in proportion to the increase in the diameter of the bore. So it's the Delian problem in another form!"

  Captain and guardsman stared confounded, and he realized that the company wasn't very Alexandrian after all. "The problem of how to construct a solid a given amount larger than a similar solid," he explained apologetically. "You, um, have to calculate the mean proportionals."

  "What's Delian about that?" asked Straton.

  "People started trying to do it when the priests of Apollo on Delos wanted to double the size of an altar."

  "Don't you just double all the measurements?"

  Archimedes gave him a look of astonishment. "No, of course not! Say you have a cube measuring two by two by two; that gives a volume of eight. Doubling the measurements to four will give you a volume of sixty-four- eight times as big. What you need-"

  "What I meant," interrupted Dionysios pointedly, "was, do you know how to build catapults?"

  "What's the peritrete, anyway?" asked Straton.

  Archimedes looked from one of them to the other. "Do you know anything about catapults?" he asked.

  "Not me!" declared Straton cheerfully.

  "A little," said Dionysios. "The peritrete's the frame, Straton."

  "The part the arms stick into?"

  Archimedes dipped his finger in his wine and traced on the table the peritrete of a torsion catapult: two parallel boards separated by struts. He added the two pairs of boreholes, one at each end of the frame, with a column of twisted strings running from the top bore to the bottom. Each mass of strings gripped one arm, which extended from the frame so that the catapult looked rather like an immense bow, lying on its side and with a gap in the center to allow passage for the missile. A bowstring ran from the tip of one arm to the tip of the other, and a beam with a slide was fixed beneath the frame's center to hold the missile.

  The two soldiers leaned over the table and examined the sketch. The waiter came back to refill the cups, regarded the blotted table with displeasure, but, at a glance from Dionysios, refrained from wiping it.

  "So which is the critical feature?" asked Dionysios.

  Archimedes tapped the boreholes. "All the force of the catapult comes from the strings," he said. "The twist in them is what makes the arms of the catapult spring forward after being drawn back. The thicker the column of strings, the more force they exert and the heavier the missile you can throw. The greater the diameter of the bore that holds the strings, the more powerful the catapult."

  "And how powerful a catapult could you, personally, build?"

  Archimedes hesitated, blinking. Dionysios' question seemed to have missed the point of his own explanation. "There's no theoretical limit!" he protested. "The most powerful catapult I've ever examined was a one-talenter in Egypt, but-"

  "A one-talenter?" interrupted Dionysios eagerly. "You could build a one-talenter?" Stone-hurling catapults were classified by the weight of the missile they could throw. A talent- about sixty pounds- was officially a man's load, and the one-talenter was generally the most powerful catapult in a city arsenal. A few larger machines had been made from time to time, by exceptional engineers for great kings, but ordinarily even one-talenters were rare. Many cities had nothing heavier than a thirty-pounder.

  "Of course!" agreed Archimedes. "Or one bigger than that- but you'd need special equipment to load and draw it."

  Straton had been looking more and more uncomfortable; now he cleared his throat and said anxiously, "Sir- yesterday he said he'd never actually built any war machine."

  Dionysios looked at Archimedes with surprise and indignation.

  "You don't need to have actually built one to know how it's done!" Archimedes declared, defending himself against the unspoken charge of deception. "You just need to understand the mechanical principles. I do. It will take me a little bit longer than it would take a more experienced engineer, but I can produce a catapult that works."

  Dionysios regarded him a moment longer, unconvinced.

  "Look," said Archimedes, "you don't need to pay me anything until I've produced a working catapult."

  Dionysios' eyebrows
shot up. "A one-talent working catapult?" he asked.

  "If that's what you want. If you have the wood and the strings for it. You know it will be big, don't you?"

  "Obviously," agreed Dionysios. "The king has one at Messana, and it's nineteen feet across." He studied Archimedes a moment longer, very thoughtfully now: he was not sure whether he had found a treasure or a self-deluding fool. But there was no need to decide, if no money needed to change hands until a catapult was completed. He turned back to his food. "When the army went off to besiege Messana," he said, "King Hieron left one of his engineers- Eudaimon son of Kallikleshere in the city with orders to make sure that all the watchtowers in the city wall were equipped with their full complement of catapults. Mostly that's just meant renewing the strings, but there are quite a few new machines to be built as well. Some of the old ones are completely worn out, and some of the watchtowers were never supplied to begin with. Now, Eudaimon has had no trouble building the arrow-shooting catapults, but he's not so good with the stone-hurlers. Unfortunately, stone-hurlers- particularly big stone-hurlers- are what the king wants most. So if you can do some, you've got a job."

  "I can build stone-hurlers," said Archimedes happily. "When do you want me to start?"

  "Come to the King's house in the citadel tomorrow morning," replied Dionysios. "I'll introduce you to Leptines the Regent, and he'll approve your conditions of employment. I warn you, though: I am going to take you up on your offer, and recommend that you not be paid until the first catapult you build has been seen to work."

  Archimedes smiled. "Thank you!" he exclaimed. He glanced down at his sketch on the tabletop, and felt a sudden thrill of excitement. A one-talent stone-hurler would need careful planning if it was not to be unwieldy. This was something new, something interesting. He wiped up the drawing with his napkin, dipped his finger in his wine cup again, and began to calculate.

  The other two watched him a moment. Then Dionysios looked at Straton and raised his eyebrows.

  Straton's answering look was glum.

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

  "I think I may have lost a bet," replied the soldier.

  Dionysios looked at him, looked at the now deeply absorbed Archimedes, guessed the general nature of the bet- and laughed. "Never mind!" he said consolingly. "Your loss will be the city's gain- and they have flute girls in this place who could make you forget far worse griefs than that." He clapped his hands, and the waiter, who had been standing impatiently outside the door, entered to carry out the dishes and usher the flute girls in.

  In the house near the lion fountain, Philyra was waiting up for her brother. Phidias in his sickroom fell early into a restless slumber; Arata settled on a mattress on the floor beside her husband, where she would wake easily if he needed her during the night. The slaves went to the hot upper room they shared at the back of the house. But Philyra went into the courtyard with the wide-necked lute her brother had given her, sat down on the bench beside the door, and began tentatively to pluck the strings.

  Lutes were comparatively new instruments to the Greeks, unknown before the conquests of Alexander the Great. Philyra had seen them before, but never held one: one of her own was the best present she'd ever been given. This one was marvelously beautiful, with a round body of polished rosewood and a neck inlaid with shell. It had a deep, sweet tone, too.

  Philyra plucked each of the eight strings in turn, then, with a breathless thrill, stopped them near the top of the neck and plucked them again. She was an accomplished player on the kithara, and knew how to raise the pitch of a string by stopping it on the crossboard with her finger- but for kitharists, such fingering was a virtuoso exercise and its use was limited. The lute promised a whole new dimension to music.

  The whole family had always been musical. For as long as Philyra could remember, Arata and Phidias had played together almost every evening, he on the kithara, she on the lyre. Archimedes, as he grew older, had usually joined them on the auloi- the soft-voiced woodwind flutes that were played in pairs- and when Philyra in turn learned an instrument, she too had joined in the concerts. Sometimes the family had played for hours, late into the night, one offering a melody which the others would take up, alter, and pass back. It had often seemed to Philyra that music was an ideal world, that all the best things in the real world were there, but clearer, stronger, more poignant. There was her mother's steadiness, maintaining the balance and rhythm of their common life; there were her father's dreamy gentleness and his sudden tumultuous excitements. And there was her brother, not vague, as he so often was when you spoke to him, but fearfully, ruthlessly precise, and so deep and complicated that she had often struggled to follow him- though in the end he had always resolved his musical knots into an affectionate simplicity. When he left for Alexandria she had tried to play the auloi for a bit, because the strings had sounded so bereft without the flutes' voice to wind among them. But in the end she had gone back to her lyre and kithara. There was something disreputable about a girl playing the flute- and anyway, nobody could play like Medion.

  She had missed him. She'd been angry that he hadn't come home when he was supposed to, and bitterly angry when their father fell ill- but now that he was back, the anger was already melting away. She hoped that he would soon return from his drink with the soldier, so that they could play some music together.

  She experimented with the lute for perhaps an hour, then, tired by the intense concentration, put it away in her room and came back with her old kithara instead. Easily she plucked out a slow, soft tune with her left hand, while her right struck an occasional ripple of accompaniment with the plectrum.

  "Remember once," sang Philyra, her low voice blending with the strings,

  "Remember when,

  I told you this holy word?

  'The hour is fair, but fleet is the hour,

  The hour outraces the swiftest bird.'

  Look! It's scattered to earth, your flower."

  She was very good, thought Marcus, standing at his window and listening to her. But that was no surprise. She'd played well before he left, and she'd had three years to get better.

  Behind him, Chrestos was curled up on the pallet they shared, while Sosibia and her daughter shared another bed behind a curtain. But he could not sleep, so he stood there, gazing into the darkness of the courtyard below, and listening to the music.

  When he had first entered the household, he had found the nightly concerts disturbing. In his own home there had not been much music. His mother had sung sometimes while she worked, and he and his brother had sung in the fields, but apart from that, music had been something one paid others to perform. He had bought some whenever he had money, because he loved it; now he could not afford it, and had it all the time, for nothing. At first he had resented his own pleasure in it: surely it degraded him that he enjoyed any aspect of his slavery? But he'd got used to music, accustomed to having it around, sensitive to its patterns and undertones. He'd almost forgotten what life was like without it.

  Philyra sang on, her voice rising clear and sweet into the dark, old songs from the countryside, new songs from the royal courts, love songs and hymns to the gods. Marcus stood silently at the window, listening and watching the stars above the rooftops of Syracuse. After a while she stopped singing and simply played, passing the tune from right hand to left and back again, and he sat down against the bedroom wall, but kept listening, wondering why this ripple of notes should say so many things more than any human tongue.

  At last Philyra stopped, yawned, and sat silent, her kithara upon her lap. Marcus stood up hurriedly, so that he could watch her go- but she did not. He understood then that the music had simply been to amuse herself while she waited for her brother to come home. He hesitated, nervous of approaching her. But what harm would it do for a household slave to advise her to go to bed? He turned from the window, crept out of the room- silently, so as not to disturb Sosibia- and down the stairs.

  "Mistress?" he called, steppin
g out into the courtyard, and even in the dark he saw how she jumped.

  "What do you want?" she called, guilt from her suspicions of him adding sharpness to her tone.

  Marcus stopped a few feet away, faceless in the dark. "Mistress, don't wait up all night," he said gently. "Your brother may not be back for hours yet."

  She made an exasperated noise. "He's bound to be back soon! He's been gone hours already."

  "Probably he's treating this man to an evening's entertainment. That means he may not be back until after midnight. There's no reason for you to wait up. I'll open the door for him when he comes."

  Night hid Philyra's frown, but not the suspicion in her voice when she said, "He never used to go out drinking until after midnight!"

  Innocent! thought Marcus affectionately. To expect that he'd keep the same hours after three years on his own in a famously luxurious city! "He was often out late in Alexandria," he told her. "And tonight he'll have to go along with whatever the other fellow wants, to be sure of his help. It's probably a good sign that he's late: means there's something on offer."

  For a moment, Philyra said nothing. She told herself that Marcus was implying that her brother had picked up expensive habits in Alexandria, and that this was exactly what Marcus would say, to account for the missing money. "What was he doing out late in Alexandria?" she asked at last, in a brittle voice. Truth or lies, she didn't really want to know, but it was unfair to go on suspecting Marcus without knowing what he had to say.

  But Marcus answered at once, and mildly, "Nothing you need to worry about, mistress. He had a pack of friends, and they'd sit about drinking and talking and… making music, half the night. When there wasn't a lecture on next day, they'd see the sun up."

  It still didn't sound like her brother. He'd never been inclined to either drinking or talking, and he'd never had any close friends. She tried to think of a question that would catch Marcus in a lie, but at that moment there was a quick rap on the house door.