'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
