The Sand-Reckoner

by Gillian Bradshaw

  He stopped, breathing hard. The other two looked back at him in bewilderment; Marcus was rubbing his arm. Archimedes met their gaze for a moment, then looked down at the calculations at his feet, perfect and unsolved. His rage began to trickle away, and he shuddered. What he'd said was true- but they would never, could never, understand it. For a moment he felt fully the pain of his isolation, as he had not felt it for years- not since he was a little boy, and had first understood that all the things he found most wonderful were to the rest of the world mere confusion. He longed for his father, and then, wistfully, remembered Alexandria, house of Aphrodite where existed all things that anyone could desire, magnet of the mind.

  "Even if that's true," said Marcus at last, "you can't calculate in the dark."

  Archimedes gave a small groan of despair, dropped the end of his broken stick, and walked silently away.

  Straton swallowed as he watched the tall black figure slouch off, shoulders hunched and head hanging. "Is he often like that?" he asked Marcus.

  The slave shook his head. "No," he said dazedly. "I've never seen him like that before. I suppose it's the war, and his father dying."

  The soldier nodded, relieved. "Enough to upset anyone. You'd better go look after him. We need his catapults, whether he thinks they're worthless or not."

  They walked in silence as far as the door of the house in the Achradina. There Archimedes stopped, staring blankly at the worn wood. He didn't want to go in. Everything that had happened since he returned from Alexandria seemed to be falling into a kind of a shape inside him- his father's death, the king's favor, Delia- everything. He realized that he needed to see the king, now, while the force of what he felt still armored him against fear and respect.

  "Sir?" said Marcus, and he shook his head.

  "Tell them I'm going to speak to King Hieron," he commanded, and turned on his heel. Marcus called again, "Sir!" but he paid no attention and strode angrily on.

  It was night, and when he reached the citadel the streets were quiet, with no sound but the crickets calling and, far off, the sound of the sea. He made his way rapidly to the king's house, knocked determinedly, and told the surprised doorkeeper, "I would like to speak to King Hieron."

  Lamplight deepened the severe shadows of Agathon's face as he gave the visitor a look to crush stone. "It's late," he said.

  "I know," replied Archimedes, "but see if he'll speak to me anyway."

  The doorkeeper snorted angrily, but nodded. He shut the door; only the sound of his sandals clacking away across the marble floor provided any assurance that he was indeed going to check whether his master would speak to the visitor. Archimedes leaned wearily against a column in the porch and waited. Presently the door opened, and the doorkeeper looked out, more disapproving even than before. "He will see you," he admitted reluctantly, and beckoned Archimedes in.

  Archimedes followed him into the mansion, past the marble antechamber and directly into the dining room. Two lampstands provided a strong soft light, and the remains of a late supper were spread over the table. Hieron was reclining on his couch, while his wife and sister sat either side of him in chairs, as was the custom for a private family meal. Archimedes stopped just inside the door, nodded to the king and his family in greeting, then crossed his arms and rubbed an elbow uncomfortably. He became aware that he was dressed only in the plain black tunic, and that it was covered with dust and oil and not fit at all for a king's house; that he was tired and overwrought and was probably going to say something stupid. Delia's eyes were wide with surprise. He tried not to think of her as he'd last seen her, flushed from kisses and flute-playing, laughing as she untied her cheek strap. She had warned him, then tried to retract her warning: who knew how far she could be trusted? Next to her the queen looked almost as disapproving as the doorkeeper.

  "Good health!" said the king, smiling. "Won't you sit down and have a cup of wine?"

  Archimedes sidled to the nearest couch and sat down on it; one of the slaves at once filled a cup with watered wine and set it before him.

  "What was it you wanted to see me about?" asked Hieron.

  Archimedes cleared his throat, his eyes on the king's. "What is it you want of me?" he asked quietly.

  Hieron's bright pleasantness faltered. He sat up, swinging his legs off the couch, and regarded Archimedes assessingly. Then he said evenly, "You know that you are exceptional."

  Just what Delia had said. Archimedes nodded once, quickly.

  "What do you think a king wants of an exceptional engineer?" asked Hieron, lifting his eyebrows quizzically.

  Archimedes gazed at him for a moment longer, baffled again. Then his eyes dropped to the table in front of him. "I have a… method of analysis," he said. "A way of thinking about geometrical problems mechanically. It doesn't provide proofs, but it helps me understand the properties of things. I think of plane figures as consisting of a set of lines, and then I see if they balance. This is a bit like that. The way a king treats an exceptional engineer- if I think of that as a triangle, then the way you've treated me is more like a parabola of equal base and height. The two don't balance."

  "Don't they?" asked Hieron.

  "No," said Archimedes. He dipped his finger in the cup of wine and carefully traced a parabola on the tabletop: a tall humped curve. Then he traced a triangle inside it, point touching the curve's peak, corners at its edges. It was instantly clear that the two would indeed not balance. Archimedes looked up, meeting the king's eyes again. "The area of the parabola is four-thirds that of the triangle," he said. "I worked it out myself."

  Hieron craned his neck to see, and his quizzical look reappeared. "You don't like getting a third more than you expected?"

  Archimedes made a small dismissive gesture with his hands. "I simply want to understand what I am dealing with. The properties of parabolae are different are from the properties of triangles."

  "Are you accusing my husband of deceit?" interrupted the queen angrily. "After all his kindness to you? What…"

  Hieron raised his hand, and she stopped. Husband and wife looked at each other a moment. Then Philistis sighed. She got to her feet, went to her husband, and brushed back his hair gently. "Don't let him upset you," she ordered.

  Hieron smiled affectionately and nodded, and she kissed him and swished out of the room.

  Delia scrunched herself deeper into her chair, telling herself fiercely that she had an interest here. Hieron didn't know how much of one, but she had a legitimate interest, too. Hieron showed her that he had noticed with an ironic glance, but made no comment. He looked back at Archimedes in silence and made a go-ahead gesture with one hand.

  "You asked me to do that demonstration," said Archimedes. "And it was you who had it posted in the marketplace, wasn't it?"

  Hieron nodded fractionally.

  "They all cheered when it worked," Archimedes went on slowly, "and since then things have been different. I didn't notice at first, but they have been. I was warned"- he did not glance toward Delia- "that I should be more cautious if my demonstration went well than if it went badly, but I didn't understand. I thought it meant to watch the contract- only I haven't been given one. What has happened is that now people know who I am. If I start to do something, they run about to help. People I don't know call me by a nickname which you gave me. Everyone has heard what you said at my father's wake, and how you paid for his funeral- out of respect for me. Everyone has heard, too, that you thought that first catapult I made was worth a thousand drachmae, even though your man only said as much to me in private. You've arranged for me to be famous, haven't you? As an engineer, as an… archimechanic."

  "You would have been anyway," said Hieron, "in time."

  "You arranged for it to happen at once," replied Archimedes. "And you arranged it so that Eudaimon does what I say and Kallippos follows my advice. Even though they have titles and contracts with the city and I don't, still somehow or other my standing is higher than theirs. You tried to give me money the same way,
too- something extra for something undefined. Something that doesn't come from the city, but belongs to me- because I am a great engineer. But I never chose to be a great engineer. That status, like the fame, is something you arranged."

  "Very well," said Hieron, in an absolutely neutral voice, "you've noticed all this. What do you think I want of you?"

  Archimedes blinked at him for a long minute, then said slowly, "I think you do want only what a king wants of an exceptional engineer. But for some reason you don't think I'll give it to you, so you're trying to… to maneuver me into a room to which only you have the key. And if I go in, you'll lock the door behind me, and I won't be able to get out again."

  Hieron looked at him for another moment- then shook his head and gave a long sigh of acknowledgment and disgust. "Oh, Zeus!" he exclaimed. "I've botched it, haven't I? I should have remembered that you're more intelligent than I am." He hitched himself forward in his seat and slapped the table. "But look, I can't lock you into anything, because- unfortunately! — there is no room to which only I have the key. Your parabola has the same base and height as your nice straightforward triangle. I want only what a king wants of an engineer- that you should build things for me- and in return I can offer only what kings have to give- money and status."

  Archimedes' cheeks had flushed with anger. "You were fixing that 'Archimechanic' name onto me as though you were title-tagging a book! In a year or so, if I tried to claim that I'm really a mathematician, everyone would laugh at me and tell me to get on with my real work. My own family would start hiding the abacus. I swore to my father on his deathbed that I would never give up mathematics, and you-"

  "No!" cried Hieron urgently. "May the gods destroy me if that's what I intended! I know you only build machines to get the money to do mathematics, and the main reason I haven't offered you a contract is to leave you free to do just that."

  "Then what is the point of all your arrangements?" demanded Archimedes.

  "To keep you in Syracuse! When Ptolemy of Egypt offers you a position at the Museum, I wanted everyone you know- from your own household through to the man who sells you vegetables- to tell you fervently that you must not accept, that for you to leave Syracuse would be treachery to the city that gave you birth. If I'd really succeeded, you wouldn't even have found a Syracusan ship willing to carry you to Alexandria, and you would have had to stay for very shame. I swear by all the gods, though, that beyond that I intended nothing for you but wealth and honor. Right now you're upset because you've seen what your catapults can do to people, and I understand that- I do, I hate killing, too! But if you think about it when you're calmer, you'll see that nothing I have done is going to oblige you to abandon mathematics. Nothing! With the enemy at our gates, no one can think of anything but war, but I pray to all the gods that we will have peace again, and then there will be space for better things."

  Archimedes blinked at him stupidly for a long time. "Why are you so certain that Ptolemy will offer me a job?" he asked at last. "He has some very clever people in Alexandria already."

  "He'll want you for exactly the same reasons I want you!" said Hieron impatiently. "I don't think you appreciate yet how exceptional you are. You think that compound pulleys and screw elevators are just things anybody else would have used to solve the engineering problems you were faced with. And they are- now. Now they seem obvious to everyone. But last month they weren't, because they hadn't been invented."

  "But- pulleys are used all the time!" protested Archimedes. "And screws have been used to hold things down for ages."

  "So it's perfectly natural to use one pulley to turn another, and a screw to lift things up? Certainly. Only nobody ever did. Only somebody who's happier with the theory of screws and pulleys than with the objects themselves could have adapted them like that. You approach engineering through mathematics- and mathematics is probably the most powerful tool ever employed by the human mind. I knew that before I met you, and when I heard about you I suspected at once that you were going to prove exceptional. Ptolemy had Euclid for a tutor, and he knows the value of geometry even better than I do. Probably the only reason he hasn't offered you a job already is that the problems you were working on in Egypt were so extremely advanced that only about half a dozen men in the world were capable of understanding them, and Ptolemy's head of Museum didn't happen to be one of that half-dozen. But even so, you would probably have been offered a post this summer, if you hadn't come here instead. You have planted your fame in Egypt now, though it's taken a little while to grow. A ship's captain I spoke to recently told me about an irrigation device invented by one Archimedes of Syracuse which obliges water to flow uphill."

  "It doesn't, exactly," muttered Archimedes. "You have to turn it."

  He sat for a moment, contemplating what Hieron had just told him, stunned by it. The unbreachable walls he had sensed closing about him had turned out to be low enough to vault over. The power he possessed could bring him not merely wealth and the favor of kings, but freedom as well. The sea lay before him, and it was his sole choice what course to set on it!

  He looked back at Hieron and managed an unsteady smile. "Thank you for telling me this," he said.

  "I wouldn't have," replied the king sourly, "if you weren't on the point of working it out for yourself. I still want to keep you. I can't offer the Museum, but anything else you might expect to find in Egypt is yours for the asking."

  Archimedes grinned. He picked up his cup of wine and drank it off thirstily, then stood up. "I'll bear that in mind."

  "Do!" said Hieron sharply. "And bear in mind, too, that when Alexandria takes the best minds from all over the world, the rest of the world is impoverished. Syracuse is your own city. She is a great and beautiful city, and fully deserves the love of all her children."

  Archimedes hesitated, looking at the king with curiosity, then replied impulsively, "That calculation about the areas of a parabola and a triangle- it was the parabola I was interested in when I did it. Not the triangle."

  For the first time Hieron was thoroughly taken aback. He stared at Archimedes in honest and straightforward astonishment.

  Archimedes grinned again, and for the first time since he'd come into the room his eyes flicked over to meet Delia's, with a look as though he were sharing a joke with her. "I wish you joy," he said to them both, and departed the room with a swagger.

  The following morning Archimedes set off for the catapult workshop at the usual time, looking tired but determined. Marcus watched him go, then silently let himself out of the house and set off in the opposite direction, toward the Athenian quarry.

  The quarries of Syracuse lay within the city wall. The plateau of Epipolae was composed largely of limestone, a great dry island lying upon the coastal shelf. Its southern, city side broke off in steep cliffs, and into these the Syracusans had cut a series of quarries for building stone. The Athenian quarry was the most famous of these. It took its name from its use nearly a hundred and fifty years before as a prison for the seven thousand Athenian prisoners of war taken at the conclusion of their city's disastrous attempt to subdue Sicily. Within its limestone walls the Athenians had suffered horrors, the living crowded in a narrow pit together with the dead. Many had died, and their bones lay still beneath the quarry floor.

  There was nothing in the appearance of the place now to speak of its grim history. The morning sun was just rising above the overhanging cliffs, casting deep cool shadows, and at the quarry sides a thick tangle of cistus and juniper covered the rock spoil with a canopy of sweet-scented green. There was a stone wall across the quarry entrance, however, and the only gate was guarded. Marcus walked boldly up to the gate and wished the guards good health.

  The guards- there were six of them- looked back at him suspiciously. "What do you want, fellow?" asked their leader.

  "I'm the slave of Archimedes son of Phidias," replied Marcus- and noticed the sharpening of interest as the name was recognized. "He wanted me to check the quarries to find which has th
e best stone for catapult shot."

  At this, suspicion was completely swept away. "Is he building a three-talenter?" asked the youngest man eagerly.

  "He starts it this morning," said Marcus. "It'll probably be ready in six or seven days."

  "Zeus! A three-talenter!" exclaimed the young guard happily. "More than a man's weight! Imagine that hitting you!"

  Marcus forced himself to grin back. "They're going to call it 'Wish You Joy,' " he said.

  All the guards laughed. They reminded one another of the names of the other new catapults at the Hexapylon, and punched the air as they recalled how well they had performed.

  "But why does the archimechanic want you to check the quarries?" asked the man in charge- not suspiciously, but with genuine puzzlement.

  "Think about it," said Marcus. "You can get stone for thirty-pound shot anywhere, but a three-talent boulder is a big piece of rock. If it's flawed or uneven it may not fly straight. So Archimedes told me to go out to all the quarries and check which one would be best for the size of ammunition he needs." He dug into the leather sack he carried and produced a hammer and chisel. "He told me to bring him back a couple of samples, too."

  The guards' leader took the hammer and chisel and examined them thoughtfully. Marcus waited, trying to keep his face blank, trying not to think about what he was doing or what he was about to do. He was already in trouble if news of this visit got back to Archimedesthough not in as much trouble as he would be in if it continued.