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?”