"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.