King and Emperor

“Let's try it again,” he said.

“Very well. Draw your lines.”

Shef drew five vertical lines down the sand, making four columns. He drew two horizontal ones across the top, three more across the bottom.

“We will try to solve your range problem, only with harder numbers than the ones you told me. Let us say that you have thrown a boulder of two hundred and eighty pounds, and it has reached one hundred and twenty yards, six score in your language. But you need to reach one hundred and forty, seven score. Your throwing weight was eleven hundred and forty pounds. You must increase that in the proportion of seven to six. So, first, we must multiply eleven hundred and forty by seven and then divide by six.”

Shef nodded: the words “multiply” and “divide” were strange to him, but he understood the ideas.

“First,” Solomon went on, “write the numbers for eleven hundred and forty in their correct place.”

Shef pondered for a moment. He had soon grasped the idea of a number-system based on position, but it still did not come easily to him.

“Try writing it the Roman way first,” said Solomon gently. “Write it in the sand away from your table. The Romans wrote a thousand by ‘M’.”

Shef wrote a wavering “M” in the sand, followed by a dot. Then a “C” for one hundred. Then the familiar four “X”s for forty. His hand reached out to make the hole sign for al-sifr, nothing, but drew back. The Romans would have left such a number as M.C.XXXX.

“Now enter the Arabic signs at the head of your table,” Solomon prompted. “Eleven hundred and forty at the top, seven at the bottom.”

Shef hesitated again. This was the tricky part. Now. The right-most column was for numbers below ten. In the horizontal line one above the bottom, the line for the multiplier, he wrote the shaky hook of a ‘7’: the multiplier. The topmost horizontal line was for the multiplied, as he could see from his Roman numerals, one thousand, one hundred and forty.

In the left-most column he wrote a ‘1,’ corresponding to the ‘M’. In the next, another one, corresponding to the ‘C.’ In the next, the more complex shape of a ‘4,’ for the four ‘X’s. And finally, in the right-most column, the column for the ten numbers below ten—ten, not nine, as Solomon had proved to him—he drew the al- sifr hole.

Now to do the sum. The first thing, the most vital thing, was that seven noughts were nought. So write in the nought. Deliberately, watched by four pairs of eyes, Shef began to work his way through the calculation on the sand-table, primitive ancestor of the abacus. For long moments he thought of nothing else, lost in a sensation no man of his race had ever before experienced: the fascination of number. In the end, hardly aware of how he had come by it, he stared with surprise and deep satisfaction at the result written in the sand. Seven thousand nine hundred and eighty. He had never seen a number like that before—not because it was too large, not when every part of his realm was counted in hides, thirty thousand for a province, a hundred thousand for what had once been a kingdom. No, because of its exactness. Over a thousand, numbers to him were counted in scores, or tens, or hundreds, or half-hundreds: not sixes and sevens and eights and fours.

“Now you are to divide seven thousand nine hundred and eighty by six,” went on the quiet voice of Solomon. “Make your table for divisions.”

Shef's hand brushed the sand away, began another table. Prompted again and again by Solomon, he worked his way doggedly through.

“And so in the end,” said Solomon finally, “you know that you would need to increase your throwing weight to…”

“Thirteen hundred and thirty pounds,” said Shef.

“So you would add to your original weight…”

“Two hundreds less ten. One hundred and ninety pounds. Exactly.”

“But by the time you'd set up your sand table and done all this,” objected Skaldfinn, “your enemy would have buried you under a hail of boulders. Because he'd have been shooting while you'd been drawing!”

“Not with the new machine,” said Shef flatly. “I could have done all this while they were still taking the weights out to winch the arm back up again. Then I could just tell them to put the same weight back in with a measured one hundred and ninety on top. And besides: I can see this is just like the far-seer.”

“Like the far-seer?” asked Solomon.

“Yes. You remember that fool Mu'atiyah. He could make the far-seer, he had been shown how the glass should be ground, something we might never have realized in a thousand years. But he never thought to go a step further. I swear, by the time we return to our own home my wise men will have learned more of glass and sight than Mu'atiyah, or even his master. Because they will not rest till they have! Now, this calculation you have shown me, this too I might never have learned for myself. But now I have learned it, I see it can be made quicker, much quicker. The sand table—it's not needed, nor the stones in my hand. I use them because I am not practiced. If I were, then I could put my counters on wire and carry it all with me, like a harp. Or I could do it without any columns at all. Ideas are hard to come by, easy to improve. I will improve this one yet. Improve it till—what was his name, Solomon, al-Khwarizmi?—till al-Khwarizmi himself would not recognize it. And I will use it for more than catapults! This is the anvil for a smith-craft on which the Way could beat out the world! Is that not right, Thorvin? Skaldfinn?”

Solomon fell silent, hand going up to his beard as if for comfort. He had forgotten the furious spirit of the Northerners, and of the One King in particular. Was it wise ever to tell them anything? Too late, now, to reconsider.

Chapter Twenty-two

Dimitrios, senior siphonistos to the Greek fleet, listened to the orders of his admiral with doubt tinged with alarm. The orders were, of course, expressed with great care, somewhere between suggestion and persuasion. The siphonistoi were almost a law unto themselves, curators of the ultimate weapon of Byzantium, in their own opinion more important in the last resort even than emperors, let alone admirals. Dimitrios, if he chose, could make a fortune and live like a lord anywhere in the civilized world, Baghdad or Cordova or Rome, simply by selling the information in his head. A wife, a mistress, seven children and two thousand golden hyperpers in the banks of Byzantium stood forfeit to his treachery. But in any case the whole city knew that no siphonistos would ever reach mastery of his trade if there were the slightest doubt about his loyalty to the cause of Church and Empire: loyalty from principle, not compulsion.

That loyalty caused Dimitrios's doubt. One of his greatest compulsions was to preserving the secrecy of the fire. Better to lose a battle by destroying one's equipment, he had been told a thousand times, than to risk its capture for a purely temporary victory—all victories, for the beleaguered Empire of the Byzantines, had long been recognized as temporary. Now the admiral wanted him to risk a ship and a projector close inshore, in doubtful battle against Jews and strange heathens. Dimitrios was a brave man, who would take his chance of being sent to the bottom by the strange mule-stones, as the barbarians called them. Risking the fire and the secret were another thing.

“The small boats will go in first,” repeated Georgios cajolingly. “Only if they make a lodgement will we bring up your galley.”

“To do what?”

“If they can detach the booms you may be able to get into the harbor and burn out all the stone-thrower ships at once.”

“They'd sink us as we came in.”

“It will be black night, a thin moon, barely a hundred yards to row to the nearest. After that you can always keep an enemy ship between you and their catapults till you are close enough to pump the fire at them.”

Dimitrios considered. He had indeed been enraged at the sinking of the galley and the butchery of its men in the inconclusive battle at sea weeks before: an act of defiance which had left him for the first time in his career feeling helpless. It would be a delight to avenge that shame by burning every enemy ship to the waterline, as all enemies of Byzantium deserved. To show the devil-worshipers the power of God and his Patriarch. And close range