Pryn tried to take an imaginary strip of vine the length of the diameter and lay it around the edge. ‘Two and a half lengths…?’ she hazarded. ‘Three? It looks to me it would go about three times.’
The earl nodded. ‘Belham’s first estimate, when he was only a year or so older than you. Within days of making it, however, if not hours—because he was that kind of young man—he took a real piece of vine, anchored one end down, drew a real circle, measured out the diameter on a strip of vine, then laid it out around the edge in order to see.’ The earl’s finger went to the top of the circle, moved along the circumference till it reached a small red mark, somewhere below the first quarter. ‘One diameter’s length around the circle, as Belham laid it out.’ The finger moved along the circle, down under the bottom, and started up the other side till it reached a second red mark. ‘Two diameters’ lengths around the circle.’ The finger continued up the far side until it reached a third red mark a hand’s span from the circle’s top where it had begun. ‘This is three diameters’ length around the circle…which still leaves this much left over.’ Here he switched fingers to outline the remaining arc.
The circle on the parchment was perhaps twice as big as the earl’s head, like a full moon low on the horizon—with its palm’s-width anomaly exceeding the three diameters laid about it. ‘Is it three-and-a-third, then…?’ Pryn suggested. As she said it, though, she immediately saw that the remaining arc was much less than a third of the diameter drawn down the center. ‘Three and…a half of a third?’
‘Belham’s next estimate, which, in this northern tongue we southern aristocrats teach our children and our slaves to speak in deference to the High Court, till it has become the language even of our peasants, can only be talked of—clumsily—as three and a half of a third. In Belham’s own notation, that becomes nineteen divided into six equal parts: one could say three-and one-sixth. To a northerner, I suppose, where all fractions are expressed as thirds, halves, quarters, or tenths, though you’d be able to figure out what it meant, it still must sound clumsy.’
It did.
‘I will not reproduce the thinking which led Belham, after much speculation, to revise that estimate to three-and-one-seventh, or, indeed, the later reasoning that led him to the inescapable conclusion that even three-and-one-seventh, while it was closer than three-and-one-sixth, was still not absolutely accurate. Three-and-a-seventh, in Belham’s system, is “twenty-two divided into seven parts.” When Belham returned from a trip to the western desert where he had been called on to supervise the construction of such a circular monument for some reigning desert potentate, my father told me he’d actually taken the time off to experiment. He told my father: “Three-and-a-seventh is certainly close enough for any practical use one might want to put it to in building any real building on the good, solid ground. But just suppose one wanted to build a circular fortress an entire fifteen stades in diameter! If one laid out the diameter across the land and used the figure three-and-one-seventh to calculate, say, a length of a rope to wrap precisely once around the outside wall, one would have—using such a figure—too much rope by the height of a good-sized man.”’ The earl laughed. ‘He’d apparently found this out, he told my father, by laying out the outline for such a fortress on the western earth and measuring it with real vine. Such experiments, of course, can only be carried out in a locale with slaves—as well as potentates obsessed by the desire for such knowledge, or at least potentates who can be convinced to finance the experiments. But then, they’ve always been particularly harsh on slaves in the west.’ The earl laughed again. ‘At any rate, this ultimate accuracy became Belham’s problem, Belham’s challenge, Belham’s obsession. One of Belham’s other early inventions, as you no doubt recall, was the lock and key—till then, slaves’ collars had been permanently welded closed. But frequently he used to say that the existence of this problem was as if his key no longer fit his lock, and he was now its slave forever. This was the problem he presented to anyone for whom a claim of mind was made: find two numbers such that one divided by the other will express exactly the number of times the diameter of the circle wraps its circumference. This was the challenge Belham presented young Venn, when my father introduced them. I must tell you, Belham explained his system of numbers to Venn in this room, just as I explained it to you, but just as I would not be surprised if you had heard it before—’
Pryn hadn’t.
‘—I would not be surprised if Venn knew of it already. For it was, as was the problem by then, famous in the circles that concerned themselves with such things. That explanation took place in this very room. My father stood where you stand now. Belham stood where I stand; Venn stood near the balcony where Jenta is sitting. And I stood—‘He looked about—‘just at the door, hoping not to be sent from the room for coughing too loud or asking an importunate question.’ The earl took an inking stick from a seashell on the shelf below the diagram. ‘With this inking stick Belham drew this very diagram I have just shown you. Using this stick as a pointer, he explained his problem. When he finished his explanation, he took the diagram—’ The earl reached for the circled parchment and slipped it from the several metal clips by which it was held to the yellowed backing board—‘and gave them to young Venn—you must remember that parchment in those days was regarded as even more valuable, since there was more need for it. “Use the back of this for your solution. You may come here at this same hour tomorrow morning to show us what you’ve found.” Venn seized them both, parchment and stick, I remember, and practically fled the room. That evening, at about this hour, she sent a slave to call my father and Belham to come here to the chamber at the seven o’clock bell—she had found her answer! My father was taking an early evening nap at the time; yawning and complaining about these mad commoners who ordered titled lords about like slaves, he arrived here five minutes late. In a fury lest he be presented with another hopelessly garbled non-solution, Belham arrived five minutes early. Because I was a child and could lurk more or less unobserved, I watched Venn wait nervously in a lower hallway just until she saw the slave go to ring the hour bell; then she dashed up the steps with the parchment and the marking stick in her hands so that—as I dashed after her—she walked through the door, there, just as the bell rang. When my father arrived, Venn looked nervously about, then laid the parchment on the floor—’ Rather imperiously (for a nervous young woman from the islands, Pryn thought), the earl tossed the parchment before him, face down. Across its back were inked evenly spaced parallel lines, forming a grid across the whole of it. ‘Clutching the marking stick—’ The earl held the stick up—‘Venn explained in an intense, soft voice: “I have measured out the lines across the parchment so that they are the same distance apart as the length of the writing implement, with which I inked them. They run edge to edge across the whole piece. Now, if I toss the stick down onto the parchment, giving it a little spin, you can see that it will fall—on the parchment—in one of two ways: either it will fall touching—or even crossing—one of the lines; or it will fall so that it lies wholly between the lines, not touching or crossing any line either side of it. Belham,” she said, “you will never find two numbers that express exactly the number you are seeking. But if you throw down the stick repeatedly, and if you keep count of the times it falls touching or crossing a line, as well as the total number of times you toss the stick at all, and if you then divide twice the number of tosses by the number of times the stick touches or crosses a line, the successive numbers you express, as you make more and more tosses, will move nearer and nearer the number you seek. Sometimes the number you express will be more, sometimes it will be less, but it will always, eventually, return; and when it returns, it will return to an even closer approximation than before. Thus you are limited in the accuracy of your estimate only by the number of times you toss.” Then Venn thrust the stick at my father, blinked at Belham, and stepping across the parchment, fled past me down the stairs—and went walking in one of the gardens with my moth
er, where they talked deeply and intently with each other several hours of matters I never heard for myself.’ The earl looked thoughtful a moment. ‘Venn always got along better with my mother than my father…At any rate: My father, surprised, dropped the inking stick, I recall—people did not usually thrust things into his hand that way. Belham snatched it up off the floor, paced back and forth, tossed the stick onto the grid, some ten, fifteen, twenty-five times. He frowned a while. Then he ordered me and my father to go away—he was perhaps the only man who could give my father such orders. Belham stayed here for most of the night, calling for lamps when the sun got too low.’ The earl spun the stick and tossed. It landed on the lined parchment, its upper end a-slant a line. ‘One,’ announced the earl, “to be doubled and divided up into what number of equal parts we do not know…?’ He laughed and let his cloak fall over his hand. ‘Quite late, Belham called my father back up here—I came along too, because I was a curious boy. “She’s right, you know,” Belham said. “What’s worse, I don’t know how I know she’s right! But I’ve already been able to determine that with only five hundred tosses, I’m now at an approximation more accurate than my twenty-two divided by seven! Another five hundred and I shall be a good deal more accurate! Now, the ends of the stick describe a circle as they fall, turning—two circles, actually, one for each end—two interlocked helices, that may be interrupted at any arbitrary point. But then, there’re always two lines on either side they might fall on to compensate…and the lines are the same distance apart as the length of the stick. The sum of all possible angles at which the stick can land so that it crosses a line, divided by the sum of all possible angles it can land so that it doesn’t cross—but what sort of sum is that?”’ The earl laughed again. ‘Of course they called Venn up to talk to them about it in the morning. And she did talk with them, quietly and intently, late into the afternoon. One thing I remember she said before I grew too bored and I went down to my suite: ‘The problem you have put me will remain a problem till the globe of the world and the globe of the sun meet in their common center and the one consumes the other. This answer I have proposed, however, humanity will know and forget, know and forget, know and forget again. And that knowing and forgetting will approximate the peaks and depths of civilization as closely as the quotient of your tosses approximates that number which, rationally, we know is not there.’ And as I turned from the door to go, I thought: “What can be known…What can be forgotten…” And I became a magician—though no doubt I have left out all sorts of details that might elucidate what certainly will strike you as an enigma—’
‘Daddy—’ Lavik got down from the railing—‘what you’ve left out—forgotten, I suppose—is the reason grandfather called Belham and Venn together in the first place!’
Jenta walked from the rail, picked up the lined parchment, the inking stick, and returned them to the counter.
The earl said: ‘But I didn’t think our guest was really interested in that…’
‘Of course she is!’ Lavik said.
Jenta looked back out between the columns. ‘She probably wants to know…’
‘Well.’ The earl shrugged beneath his cloak; the edges swung before his robe. ‘What my father had called them both here for, you see…He wanted them to build an engine. He wanted them to build him an engine that would raise a city from beneath the waters where it had sunk.’
‘Now,’ Lavik said, ‘you know he doesn’t think you’re a spy—anymore. At least he’s decided to treat you as though he doesn’t.’
‘Really, Lavik,’ the earl said, ‘why should I think she was a spy? She’s only a girl, even younger than you are.’ He looked at Pryn. ‘Building the engine, of course, was a job they never completed. Belham gave up on it in a week, after driving my father almost to distraction by doing lots of things that required lots of money and lots of time and had as little to do with his assigned task as laying out miles of vine in the desert when you’re called on to build a three-story circular fortress. “It can’t be done,” he said at last; and besides, he was more interested in other things. But Venn finally did invent a sort of engine—another approximating engine. It’s been working, now, for quite a while. It included a story, and a magic astrolabe…’
Pryn looked out at the waters where the late sun no longer revealed a city. ‘The engine,’ she said, ‘was this astrolabe, and the taleteller’s story, and the old tales of Olin’s wealth and madness, the rumors among the slaves, all the signs around Nevèrÿon that bring heroes to this spot in the Garth…’
‘Heroes and spies, heroes and spies—though it’s sometimes hard to tell the difference.’ The earl’s smile returned to its radiant absolute. ‘One might revise your details. But you have outlined—approximately—Venn’s solution to my father’s problem; although for all its efficacy it was as far from successful as were Belham’s attempts—at least he abandoned his. We should be going down to supper shortly. I’ve told you the history of this chamber—but Venn’s “engine” was put together downstairs where we shall be eating. Perhaps we should discuss it there?’
‘Your father wanted the city raised in order to get the money,’ Pryn said.
‘One assumes.’ The earl took the edge of his cloak which had drifted open and pulled the brilliant blue closed.
‘And heroes come with swords, don’t they,’ Pryn said, ‘all kinds of swords, seeking the same treasure?’
‘All kinds of heroes,’ the earl said. ‘All kinds of spies—’
Jenta said: ‘The spies usually carry small knives—’
‘—which they leave at home tucked under the straw of their sleeping pallets when they’re invited for dinner.’ Lavik laughed.
Pryn looked at the earl again. Was he gazing at the double swords more than at the single?
‘I suppose my father felt, like so many of his breed, that the discovery of the treasure would restore a certain glory to the south that had already begun, even then, to drift north. For once, in the days of Neveryóna, this was a very different land.’
‘And this astrolabe…?’ Pryn looked down. ‘Mad Olin’s magic circle of different stars—it was to guide people here to…the treasure?’
The earl nodded.
Suddenly Pryn bent her head to loose the links from under her hair. ‘Here…’ She lifted the chain from her neck. ‘I’m not a spy. And I’m certainly not a hero. You take it!’ Surprised she’d actually said it, she’d wondered, rehearsing it in her mind, how much her hesitation had been idealism and how much, indeed, simple fear. ‘I think when I want to find a fortune, I’ll go back to Kolhari and see what I can win in the markets there. I have the key to Belham’s lock, I’ve been shown the memory of water, and I’ve flown a dragon—there ought to be something I can do. Go on. You take it.’
The earl raised an eyebrow. ‘You would just…give it to me?’
Pryn extended her two fists. ‘It was just…given to me.’ On its chain, the bronze disk turned and turned back in the late sunlight. ‘I have no use for it, need of it, nor, really, knowledge about it—at least that I knew, when I got it, would turn out to be knowledge.’
‘Oh…’ The earl pondered. ‘Well, the truth is…I have no need of it either!’ He smiled again; the stocky little man’s cloak drifted open. ‘I don’t, believe me! Take it with you back to Kolhari—if you go. Pass it on to someone else. You see…how shall I put it?’ He coughed. ‘It’s as if my voice deserts me when I most need—’ He turned first left, then right, looking among the unknown writings on the various parchments for some prompt to articulation. ‘The astrolabe is a tool to bring people here. But once it, itself, is here, it has finished its job—until it is recirculated abroad. Once it reaches the origin, the center, the heart of the system, however, it is, so to speak, excluded from the system, and the system itself threatens to come to a halt without that vital part.’
‘But perhaps you want the money…?’ Pryn said, tactfully, she hoped—though it just sounded suspicious. ‘Take it. Certainly it wo
uld be an easier job if you had it to help you than if it were off, sliding and slipping all over Nevèrÿon.
‘Jobs, work, tools, engines, hearts—the engines that drive the workers to use their tools and function on their jobs! Production! Ah!’ the earl cried. ‘I have been associating with Old Rorkar and his like so long now I only know how to speak of the world as if it were all a huge brewery! Your astrolabe is a sign in a system of signs. It has a meaning, yes, but that meaning is supplied by the rest of the system, which includes not only the tales of history, madness, and invention, but the similar instruments sailors use to orient themselves at sea, the play of power in the land, and the language by which they can all of them be—systematically—described.’