The Sand-Reckoner

15

The Romans did not assault Syracuse again that summer. After the exchange of prisoners, they returned to Messana, where the troops spent the winter while Appius Claudius went home to Rome.

He was not reelected. Reports of the army's many causes of dissatisfaction with him had been circulating widely throughout Rome by the time he returned, and he was received coldly, without honors and without thanks. Neither of the new consuls elected in January belonged to the Claudian faction.

The two legions in Sicily, however, were considered to be insufficient for the gravity of the situation there, and another six specially strengthened legions were enrolled. In the spring both consuls set out for Sicily with their huge armies, and when they landed at Messana, they proclaimed favorable terms for any Sicilian city that would abandon Syracuse. At this all of the dependencies of Syracuse, all her friends and allies on the island, fell away.

In early summer the forty-thousand-strong Roman army arrived at Syracuse itself and laid siege to the city, ringing it about by land with a bank, ditch, and wall of earth and timber. Greek engineers from the subject cities of Tarentum and Croton constructed siege machineswheeled towers with ladders, grapples, and catapults mounted in them, and the thick-roofed, open-floored wheeled carts called tortoises, which protected massive battering rams. In the middle of the summer the besiegers tried to take Syracuse by storm.

They failed utterly. Over the previous summer, Hieron had been calling upon Syracusan allies for supplies- grain to feed the citizens during a siege, wood and iron and hair to shape weapons for her defenses- and Egypt and Rhodes, Corinth and Cyrene had responded. The new season had found the city more impregnable than ever. Extra ditches had been dug about the walls, inside the range of the defending catapults, so that the attackers had to wheel their cumbersome siege machines down steep slopes, then up, then down again, all the while suffering the bombardment of the Syracusan catapults. And that bombardment was of a strength such as the Romans' Italiot engineers had never imagined. Immense stones smashed the tortoises and toppled the siege towers. Men who tried to right them fell under a rain of bolts, and incendiaries smashed into the damaged machines and set them ablaze. The rams never got anywhere near the walls, but were crushed like beetles on the slopes of the Epipolae, and abandoned when the attackers fled. Hundreds of Romans injured or trapped in the wreck of the machines were taken prisoner by the Syracusans; hundreds more were killed.

Manius Valerius Maximus, the senior Roman consul, conferring with his colleague and his senior advisers after the assault, had one of the Syracusan catapult stones rolled into his tent for their consideration. It weighed over two hundred pounds. The Romans regarded it with horrified wonder.

"I'd heard," said the Tarentine chief engineer in awe, "that Archimedes of Syracuse, King Hieron's engineer, could build three-talenters. I thought the stories were exaggerating."

"It seems they fell short of the mark," said Valerius Maximus. "Like our own assault."

The Tarentine had no ideas for countering Syracusan artillery, and was, in fact, fearful that a man who could build three-talenters might have worse things in store for any siege engine that did succeed in getting close to the walls. The Romans considered the possibility of blockading the city and concluded that it was pointless even to try: they had no fleet apart from the few Italiot vessels and the transports which had brought them across the straits, while the Syracusans possessed eighty decked warships to defend their shipping. The number was certain: the Syracusans had proudly displayed the fleet to their Roman prisoners the summer before.

Even more worrying, from the Roman point of view, was the news that General Hanno, the Carthaginian commander in Sicily, had been recalled to Africa, tried by the Carthaginian Senate, and sentenced to death by crucifixion for his inaction. There were rumors that Carthage was now recruiting mercenaries and intended to press the war in earnest.

"We must have peace with Hieron of Syracuse," concluded Maximus. "Carthage is the main enemy, but we cannot fight the Carthaginians with Syracuse hostile at our backs. And it seems that we can't subdue Syracuse by force. Carthage has given Syracuse no support since the war began. Perhaps Hieron will be willing to abandon his alliance."

No one had any objection to this change of policy. The rumors of Syracusan atrocities no longer found wide acceptance: the Roman prisoners released the previous year had had nothing but praise for King Hieron.

The following morning, Maximus sent a herald to Syracuse to ask King Hieron for a parley. The king at once agreed, and Roman consul and Greek monarch met on the plain below the fort of the Euryalus. Maximus was surprised to find Hieron so pleasant and reasonable a man; Appius Claudius had led him to expect a cunning and belligerent monster.

The negotiations went on for three days. Once she had entered a struggle, Rome was not in the habit of accepting anything short of her enemy's total surrender, and however generous she might otherwise be to the defeated, she always required that her new "ally" supply troops to fight for Rome. This was precisely the condition Hieron rejected most emphatically. If Syracusans were to fight and die, they would do it on behalf of their own city, not for foreigners. Syracuse would remain sovereign and independent, or she would remain at war. She could not hope to win, but, on the other hand, the Romans couldn't hope to reduce her and couldn't afford to ignore her. Rome at last, reluctantly, yielded and concluded a treaty such as she had never made before.

Rome not only recognized the independence of Syracuse but also granted the city the right to govern eastern Sicily from Tauromenium, just south of Messana, as far as Helorus on the southern point of the island- in fact, to keep all the territory she had held before the war began, including all the cities which had recently yielded to Rome. All this land was guaranteed exempt from war- which included immunity from attacks by Rome's deplorable allies, the Mamertini. Syracuse, for her part, agreed to provide the Romans with supplies for a campaign against the Carthaginians in Sicily, and to pay a war indemnity of a hundred talents of silver, the payments to be spread over twenty-five years. The latest batch of Roman prisoners were returned without ransom.

The treaty was formally concluded with an exchange of oaths and sacrifices offered to the gods. And its conclusion was celebrated on both sides, with feasting and heartfelt relief. Rome could now concentrate on Carthage, and Syracuse had steered her way through the channel to peace.

When the Romans were dismantling their siege works in preparation for their return to Messana, two men of the second legion went to their tribune and asked permission to go into the city to pay a debt. Since one of them was a centurion of the legion and the other his second-in-command, permission was granted. So Quintus Fabius and Gaius Valerius walked slowly up the long road to the city they had left by night the year before.

It was an August morning, and around them the land baked in the summer sun. The open fields were loud with cicadas, the road white with dust. Fabius tapped his centurion's vine-stem stick unhappily against his thigh as he walked: he had not wanted to come, but Gaius needed an interpreter. He was obliged to Gaius, in an ill-defined, guilty sort of way. He had caused Gaius grief. Fabius' promotions had come rapidly over the past year, and he had taken advantage of them to haul Gaius up through the ranks after him, out of the same obscure sense of obligation.

They reached the Epipolae gate of the fort of the Euryalus, where the Syracusan guardsmen eyed them with suspicion. Fabius explained their errand in his clumsy Greek, and they were allowed to pass, though they were required to leave their arms at the gate. One of the guards escorted them into the city: the peace was still very new, and they were not trusted, least of all at the house to which they were bound.

They crossed the limestone scrubland of the plateau, passed the huts of the Tyche quarter, and descended from the heights into the marble grace of the New Town. They both glanced at the cliff which towered over the theater to their left, the edge of the plateau where the quarries were situated. But their escort led them through the New Town and into the citadel of the Ortygia.

The house they were looking for lay on the north side of the Ortygia, not far from the sea wall. It was a large house, and had been repainted not long before: the front was a crisp pattern of red and white, unfaded by sun and unmarked by dust. The guardsman from the Euryalus knocked upon the pristine door.

Gaius Valerius stood on the sunny doorstep, listening to the guardsman explaining and a boy doorkeeper answering doubtfully, all in the rapid musical language which he could not understand. He had been eager for this meeting, but now that it was almost upon him, he wondered why he bothered. Because of Marcus. What good would it do Marcus? What good would it do anyone? Still, he clutched the small package he had brought with him and asked Fabius, "What's the delay?"

"The slave says his master's working, and doesn't like to be disturbed when he's working," replied Fabius. He interpolated a comment into the flow of conversation between the slave boy and the guardsman, and both turned to look at him. The slave blinked, then shrugged, and stood back, opening the door for all three of them.

"What did you say?" asked Gaius, stepping into a cool marble hall.

"That we only wanted to return his master's property," said Fabius.

The boy walked ahead of them, along a colonnade which enclosed a garden, green and cool after the hot streets, then through a narrow passage, past another, kitchen, garden, and into a workroom which might have been part of another house entirely. The floor was packed clay, and the walls were stacked high with timbers. In the center of the room stood a sinister-looking box more than half the height of a man, formed of wood and lined with lead; sitting on a corner of it was a basin with two large neat holes in it, and scattered about it were oddments of leather, wood, and bone and a blacksmith's bellows. Whatever the device was, however, it had been abandoned, and the only person in the room was a young man who crouched on a low stool not far from it, gazing intently into a box of pale sand and sucking on the hinge of a set of compasses. Gaius had never actually seen his face before, though he had once heard him play the flute, but he knew at once who it was. The magician who could number the sands and make water run uphill, Syracuse's extra army, his brother's onetime master.

"Sir," said the slave boy, with great respect. He had been purchased only the winter before, and he was in awe of his new master.

Archimedes lifted one hand in a wait-a-minute gesture and did not take his gaze off the pattern in the sand.

The boy looked at the visitors and shrugged helplessly.

Gaius cleared his throat, then called out, "Archimedes?"

Archimedes made an indistinct reply around the compasses- then suddenly stiffened. His head jerked up, a smile of delight on his face. For a moment Gaius found himself meeting a pair of bright brown eyes that sought his own eagerly- and then the delight faded, and the eyes became puzzled.

"Oh," said Archimedes. He got to his feet, glanced down at his interrupted calculation, then back at the visitors, questioningly now.

"Excuse us," said Fabius stiffly. "I am Quintus Fabius, a centurion of the second legion; this is Gaius Valerius. We are come to speak to Archimedes son of Phidias."

"You're Marcus' brother!" Archimedes exclaimed, looking at the second man. He could see the family resemblance now, in the wide shoulders and the stubborn line of the jaw, though Gaius Valerius was slighter and fairer than his brother. "You are welcome to my house, and good health to you! When you called my name I thought for a moment it was Marcus. You sound just like him."

Gaius just stared. Fabius turned to his companion and translated, which took Archimedes aback: he had somehow expected Marcus' brother to know Greek.

Gaius nodded, then stepped forward and held out a long slim box wrapped in black cloth. "I came to return this," he said quietly. "I think it was yours."

Archimedes stared at the box, recognizing the shape, and knowing with cold sick grief that something he had hoped would not happen had happened, and happened long before. He did not take the box, even when the translation was finished and Gaius took another step toward him, offering it again.

"Marcus is dead," he said flatly, looking up from the shrouded flute case to meet the eyes of Marcus' brother.

There was no need for translation. Gaius nodded.

Archimedes took the flute case and sat down on his stool. He pulled at the knots that secured the wrapping, then bit the cords and broke them. He unwrapped the case, opened it, and took out his tenor aulos. The wood was dry to the touch, and the slide, when he moved it, squeaked stiffly. A cracked reed was still fixed to the mouthpiece, and the tarnished clamp had left a green stain upon its dry gray side. He unfastened the clamp and pulled the reed out, then began rubbing the mouthpiece clean on the cloth the case had been shrouded in. His hands knew what they were doing; his heart was bemused and numb.

"I don't play," said Gaius. "And I did not want it to stay silent forever."

Archimedes nodded. He spat on the mouthpiece and rubbed it again, then set the instrument down in his lap. He wiped his face with a bare arm, and realized at that that he was crying. He looked back at Gaius. "Your brother was an extraordinary man," he said. "A man of great integrity. I had hoped that he was still alive."

Gaius' face convulsed with pain. "He died last year, the day after your people returned him. Appius Claudius had him sentenced to the fustuarium."

Fabius hesitated over the last word, unable to translate it. "To the beating to death," he supplied eventually.

"Hieron told me that Marcus had offended the consul," said Archimedes wretchedly. "He said that he spoke to Marcus before he sent him back, and urged him to tell whatever lies would save his life. But Marcus was never any good at lying."

"He was a true Roman," agreed Gaius proudly.

The brown eyes fixed him, uncomprehending and angry. "Oh? The people who killed him called themselves true Romans. If they were, he wasn't."

"Appius Claudius isn't a man, let alone a Roman!" exclaimed Gaius hotly.

"You can't disown him that easily!" replied Archimedes. "The Roman people elected him and followed him. His successors are now obliging my city to pay for the war he and his friends started and forced upon us, the war that isn't over yet. Rome hasn't disowned him, and nor can you! Your people murdered Marcus. Gods! Barbarians!"

Gaius flinched, though Fabius, adding the last phrases to his translation, merely looked contemptuous. Behind them, the guardsman from the Euryalus, who had stood watching the two Romans with his spear balanced on guard, grinned. Archimedes looked back down at the flute, trying to calm himself. He fingered the dry wood, and remembered Marcus stroking it. Marcus had never had time to learn it properly. Waste, waste, stupid waste!

"I loved my brother," said Gaius slowly. "And I wanted…"

He hesitated. He did not know how to speak to this man. He wished Archimedes had indeed been the white-bearded sage of his imaginings; he would have known then how to conduct himself. This young man, this foreigner who angrily condemned his own people, confused Gaius, muddling his reactions. He remembered the two voices in the dark courtyard of the house in the Achradina: this one, quick, drink-slurred, questioning and commanding; and the other voice, now silenced. He had been unable to tell then, and could not tell now, what connection bound them, what history of emotions they assumed. He took another step forward and squatted in front of the figure on the stool, trying to meet the eyes, silently raging against the need to pause, and wait at each phrase for Fabius to alter his words and render them comprehensible; longing for direct communication.

"I did not have much time to talk to Marcus last year," he said. "A little while when we escaped; a little while more before and after the trial. But he did talk a little about Egypt, and about you and your household, and about… about Greek things. Mechanics, mathematics- things I have no knowledge of. I do not know very much about what he was like, the last years of his life. I want to know. I lost him when he was sixteen, and almost half his life is missing for me. Please, tell me what you can. I ask it as a favor, as the brother of a man who was your slave, and for whom, it seems, you had some affection."

Archimedes sighed, still fingering the flute. "What can I say? He was, as you say, my slave, and for most of the time I knew him I took him entirely for granted. One does not ask a slave what he's thinking or feeling: one just expects him to get on with his work. My father bought him when I was about nine. We paid a hundred and eighty drachmae for him- it was during the Pyrrhic War, and slaves were cheap. We had a vineyard then, and we needed a worker to help get in the vintage, and a farm- the tenants mostly managed it on their own, but we usually tried to help with the harvest, as one does. So your brother did that, and he did the heavy work at the house, and occasionally helped the neighbors. Marcus hated being a slave- I think I always did know that- but I think apart from that he wasn't unhappy. He lived in the house with me and my parents and my sister and our other slaves. My father was a gentle man and a good master. Your brother didn't seem to dislike his work, and he enjoyed other things. We always played a lot of music, and when we went out to concerts and the theater we usually chose Marcus to carry things, because we knew he enjoyed music. Machines, too- yes, he did like them. I was always building them, and he was always interested in them. He'd help hammer and saw, and he'd make suggestions- tell me that what the next hoist needed was a way to fix it halfway up and so forth, and then when I'd worked out how to make it do that, he'd grin about it. So we liked each other."

"When I was nineteen, my father gave Marcus to me, and we went off to Alexandria for three years. I was not a good master. Marcus would say, 'Sir, we're out of money,' and I'd say, 'Oh well,' and forget about it, leaving him to sort out how we were to live without it. He was very resourceful and always amazingly honest. When he took money out of my purse- he had to, I never remembered to give it to him- he'd always tell me how much and for what, even though I never paid the least attention, and he always used to remind me who I owed money to. And he used to mend clothes and make sandals himself, and he'd do odd jobs for the tradesmen in exchange for this and that we needed. He never complained. But he never liked Alexandria- at least, that was the impression I had. He was always telling me we ought to go home. But the last year we were in Egypt I designed a machine to lift water, and he told me once that he'd enjoyed building that more than any other work he'd ever done."

"The water-snail," said Gaius.

Archimedes smiled at the word, which, being Greek, needed no translation. "I'm not surprised that he told you about that: he loved that machine. We didn't make them for long. I got tired of them. He was furious with me about it; he kept telling me we could make a fortune with the god-hated things. He never saw the point of geometrynot that he admitted to me, anyway."

"He seems to have…" Gaius hesitated: told you what to do a lot hovered on his lips, but he was afraid of offending, and changed it to "… spoken his mind to you freely."

Archimedes snorted. "He always spoke his mind freely. That's what he died for, isn't it?" He looked back at the flute, and continued, "When the war started, we came home. He was… unhappy about the war. We didn't know he was Roman- if anyone asked, he'd say he was Sabine or Marsian or Samnite or whatever- but we knew he had some loyalties to Rome. He kept swearing, though, that he would never do anything to harm the city or our house." Archimedes paused, then added, "Of course, he would have been even more unwilling to harm Rome. And you know how quickly he decided to help you. But afterward he kept saying that he was sorry he had abused my trust. And he was very sorry over the man you killed escaping- a fine man, and a friend." He raised his head and looked directly at Fabius. "If you are the Fabius who was with him that night, he said he was wrong to have given you a knife. And he said that he thought you would have killed me if you'd known who I was."

Fabius looked back a moment in silence; he did not translate the addition. "It was our duty to escape if we could," he said at last. "As for the other, yes, I would have killed you. We had heard of your catapult-making, and I thought you were likely to cost Rome dearly. As you have. Many men are dead, and the peace we achieved has gained us little, because of you and your catapults. I do not say you were wrong to defend your city, but I would have been right to defend mine."

"No one was attacking Rome," Archimedes pointed out coldly. "Your reasoning puts the bully on the same level as the victim who fights back. I find it fallacious. Nor do I understand how your consul could justify putting a brave and loyal man to death merely for speaking his mind."

Gaius had been listening to this incomprehensible exchange anxiously, and now cleared his throat nervously. Fabius resumed his translation with the complaint against the consul. Gaius Valerius looked away with an uneasy hunch of the shoulders that reminded Archimedes suddenly and painfully of his brother.

"The consul was a weak man angry," said Gaius. "As soon as he found out who Marcus was, he had him arrested and tried. He was the judge and the principal accuser. Nobody would have put Marcus to death for what happened at Asculum. Not even at the time- he was sixteen when it happened, and he'd been in the legion only three weeks! But our father had taught us to expect harsh punishments, and Marcus was always hard on himself: he'd convinced himself that he deserved to die, and he'd expected to. But even Claudius couldn't rely on Asculum after so many years. The big charge he had was that Marcus had dishonored the Roman name. Through accepting slavery, you see, and through saying that the Romans were wrong to attack Syracuse."

"And he wouldn't lie and say he thought they were right?" asked Archimedes, with resignation.

Gaius nodded wearily. "I think he meant to. But when it came to the point, he was angry, and he didn't. The consul had accused him of other things as well. Foul things."

Archimedes looked at him, frowning, and Gaius went on reluctantly, "Of prostituting himself to Greeks. To King Hieron, and to you, among others." Archimedes flushed angrily, and Gaius went on hastily, "Stupid accusations, but he had no way to refute them, except to get angry. So he got angry, and did not tell any lies, and the consul sentenced him to death."

Gaius reached over to the flute case and took something else from it: a fat black flask about the size of a child's fist, empty. "I was very glad that he had this," he went on, low-voiced. "The legions knew that Marcus was innocent- but since the beating had to take place, the fact that no one wanted to strike only meant that it would last longer. But in the morning, when they went to take him out of the tent where they were keeping him, he was already dead. He had this with him, this and the flute. Your gifts, weren't they?"

Archimedes shook his head. "Only the flute," he said soberly. "That was from Hieron. He told me that he'd given it to Marcus, just in case."

Gaius looked at him in surprise and doubt, then ran a finger around the top of the flask. "A gift from the king of Syracuse? I am indebted to the king. But I don't understand how King Hieron knew Marcus, or why he bothered."

"He knew your brother through me," replied Archimedes. "And he wanted Marcus to come back to Syracuse after the war and be his Latin interpreter. It would have been a good position, and Marcus would have filled it very well. Hieron told me about it. Your news will grieve him too." Archimedes got to his feet, holding the flute carefully in both hands. "It is waste, and nothing but waste. I don't know what your people will do to the world."

Gaius rose too, and bowed his head in a gesture that was neither denial nor acceptance. "Marcus was a Roman," he said. "I would ask you, sir, to remember that of us as well. But I don't want to quarrel with you. I am grateful for your kindness in speaking with me, and grateful also for your kindness to my brother while he lived. He admired you greatly."

Archimedes shook his head angrily. "I did not realize how exceptional your brother was until too late," he said. "I am much to blame. I hope it is some consolation to you to know that even as a slave he earned the respect of those around him." He hesitated, trying to think if there was anything else he should say, then realized that the visitors must have had a long walk from their camp, and asked them if they would like some wine.

They thanked him and agreed that they would, indeed, like something cold to drink. Archimedes started back toward the main part of the house; as he did, Fabius gestured at the lead-lined box in the center of the workroom and asked uneasily, "What is that machine? A new kind of catapult?"

"May all the gods and heroes forbid it!" Archimedes exclaimed vehemently.

He had never been so sick of anything in his life as he was of catapults. He'd lost track of how many he'd built: one-talenters, two-talenters, three-talenters, three-and-a-half- and four-talenters. And arrow-firing catapults, with particularly long ranges and particularly large bolts. Outworks to the walls had come as something of a relief. The nasty little surprises he and Kallippos had invented for any siege machine that did get close to the wall had seemed like the comedy tacked onto the end of a tragic cycle at the theater. There was a long list of other things that could be done, with time and supplies, if the war went on, and he was infinitely glad to have escaped them- for the time being, anyway. His relief at the peace had been as great as that of anyone in the city. "That is a water-aulos," he told Fabius happily. "Or it will be, when I finish it."

"A what?" asked Fabius confusedly.

Archimedes' eyes lit. "A water-aulos. See, you fill the tank with water, and you put this hemisphere in it." He hooked the holed basin off the tank's corner and held it, upside down, in the empty cistern. "And then you have one pipe coming down to this opening here, and another coming out from here, which is shut off unless you press the keys to open the valves- the valves are the cleverest part- and you pump the air in with the bellows. The water pressurizes the air, so that when you release it through the pipes, it produces a good volume of sound." He put the basin back over the corner of the tank. "I'm waiting for the pipes from the bronzesmith."

"But what is it for?" asked Fabius.

"It's a musical instrument!" said Archimedes in surprise. "I said it was a water-aulos, didn't I? My wife wants one."

"A musical instrument!" exclaimed Fabius, and shook his head wonderingly. "So peace has already reduced the greatest of engineers from the making of catapults to the making of flutes as an amusement for women!"

Archimedes stared at him in complete bewilderment for a moment, then went red. "Reduced?" he repeated furiously. "Catapults are stupid, god-hated lumps of wood which throw stones to kill people! I hope I never touch another of the filthy things in my life! This will sing with a voice like gold to the glory of Apollo and the Muses. This is as much superior to a catapult as… as…" He fumbled for a comparison, then gestured impatiently at the abacus, "… as that is to a pig!"

"But I don't know what that is, either!" said Fabius, amused.

"A calculation of the ratio between the volumes of a cylinder and an enclosed sphere," replied Archimedes with cold precision. He edged back toward it, and looked down, frowning. "Or an attempt to calculate it, anyway." The way into the problem, and the answer to it, had eluded him.

"But what use is that?" asked Fabius, coming over to look down at the drawings scratched in the sand: spheres and cylinders labeled endlessly with letters; letters repeated down the sides in inexhaustible calculations, in curves, in straight lines, in figures balanced and not balanced. So much intelligence, he thought, to be squandered upon air!

"It doesn't need a use," declared Archimedes, still looking down at his diagram. In his mind, a circle slipped up the line of the cylinder, its height the diameter, then revolved at the mid-point to form the sphere: perfect, more perfect than anything on earth. "It simply is." He studied his calculations and saw that they would get him nowhere. He picked up a flattened piece of wood and carefully smoothed the blind alley out.

"What are you saying?" asked Gaius in Latin. Fabius had translated nothing about the water-aulos: he had no idea what a "valve" was, or "pressurize," and he suspected that the words simply didn't exist in Latin.

"The box in the middle of the room is part of a musical instrument," said Fabius. "I said it was a sad descent from catapults to that, and he took offense. He said that music was nobler than war, and that this"- he gestured at the box of sand- "is nobler than anything."

As the dead end vanished into the sand, Archimedes saw suddenly the path along the spinning of the circle to the truth. Breathlessly he hooked the stool over with his foot and picked up his compasses. "Just a minute," he said to the visitors. "I've just seen something. Just go into the house and have a drink; I'll join you… in a minute."

The others looked at him in surprise, but he was already oblivious to them. The compass marked out its precise reckonings in the fine sand, and his face following it was rapt, intense, joyful. For the first time in his life, Fabius felt the foundations of his own certainties tremble. This mind was not bent upon air. The suddenly quiet room was filled with something that made the hair stand up along his arms, something that existed for no human use. Perspective altered dizzyingly, and he wondered what his own use was to a universe. Unaccountably afraid, he ducked his head and backed away.

When Delia came into the workroom a couple of hours later, she found Archimedes sitting on the ground, resting his head on the stool and gazing fondly at the abacus. "Dearest?" she said gently.

He raised his head and beamed at her. "It's three halves!" he told her.

She came over and knelt beside him, putting an arm about his shoulders. They had been married since January, and she was beginning to feel that she was going to be very good at estate management but would never understand geometry at all. "The ratio is?" she asked, trying to take an interest.

He nodded, and swept a hand toward the thicket of calculations. "It all comes out so perfectly," he wondered. "A rational number, after all that. So exact, so… perfect!"

He was so happy that she hardly liked to disturb him. But after a moment she said, "I heard that there were two Romans here earlier. What did they want?"

The happiness vanished. He looked around in alarm. "By the god! I said I'd join them in just a minute. Are they…?"

"They left some time ago," said Delia sourly. "Melias said they talked to you, and then you sank out of sight into the abacus, so he gave them a drink, and they went. What did they want?"

He told her, sadly, and showed her the battered flute. "Though all Gaius Valerius really wanted was to hear about his brother," he finished. "I liked him. He was like Marcus, very straightforward and honorable. The other one, Fabius, was a true Roman. He thought it was a reduction to go from catapults to music!" He rubbed angrily at another spot of corrosion on the reed clamp. "Marcus told me once that the Romans don't think music a fit subject for serious study at all. He said that his father would have beaten him if he'd asked to learn the flute. He wanted to learn it anyway. But they didn't give him the chance."

She put her arm around him again, remembering the slave who had sat in the dark garden, listening to the music. She could not remember his face, but she was sorry he was dead. Sorry mostly for Archimedes' sake, but a little for the slave's sake as well. "I pray the earth is light upon him," she said.

He turned toward her, put both arms around her and kissed her, then held her, feeling the shape and warmth of her against his chest, a comfort for every grief. When he asked Hieron for her, he had not known that it was even possible to feel for a woman what he felt now. From the first day of their marriage, she had astonished him. It now seemed to him that she was best at everything he was worst at, that, like the second leg of a compass or the second flute of a pair, she completed him.

Even with war, even with siege, even with catapults they had been so happy.

He thought, painfully, of Marcus dead; Marcus burned, and the smoke rising from the funeral pyre high into the sky above Syracuse. Perhaps he had seen it, and not known what it was. He had noticed Marcus little enough in life.

Marcus had done his best to fulfill all his obligations honorably, and had died in their contradictions without complaint. He himself, no better a man, had everything to make himself joyful. By what calculation could those shapes be made to balance? Archimedes sighed and glanced down at the small riddle he had solved, the perfect ratio, already dwindled in his estimate.

And yet, that ratio was perfect still. Perfect, and known. It rested whole in his mind, needing no use, sufficient in its existence. Like the soul. But unlike the soul, comprehended.