" 'Tis there!" Geoffrey cried.
"Directly where we said 'twould be," Magnus said, with pride.
" Tis a darker gray." Cordelia pursed her lips. "Would it be harder, Fess?"
"Since this stone is harder than the first we found, and since it maintains that hardness comes with age, I should say the prospect is likely, Cordelia. Can you make any other inferences?"
The children were silent, startled by the question. Then Magnus said slowly, "Thou dost mean that an we seek farther in this direction, we shall discover more rocks."
"That does seem likely."
"And then the farther away they are, the harder they will be?" Geoffrey asked.
"I would presume so, though the spy-eye cannot test it."
"Yet it can see if they are there. Send it farther"— Magnus eyed the angle of the sunlight that streamed through gaps in the trees—" farther west, Fess."
The scene on the screen shrank as the spy-eye rose, then shot into a blur as it swept away.
"Hypothesis: that the farther west we go, the more rocks we shall find, at an interval of approximately three hundred meters," Fess postulated, "and that each rock shall be harder, though we cannot test it…"
"And darker!" Cordelia cried.
"And with louder and more driving music!" Gregory added.
"Harder, darker, and with more raucous music," Fess summarized. "Why do we extrapolate so?"
"Why, because the farther west we go, the older the rocks must be!" Magnus said triumphantly.
"A warranted inference, Magnus! Yet that insight should yield one more."
The children were silent, staring at the screen.
"I would I had had such a tutor," Gwen murmured.
"Why, that the first rocks… must have come from the west country," Magnus said slowly.
"Excellent, Magnus! And what does that, in its own turn, tell us?"
"That the crafter who sent out the first rock must be also in the west," Gregory breathed. "I had forgot that there must needs have been a person who did make the first of these rocks."
The scene steadied on the screen, and there it lay, neatly centered, a dark gray rock. Fess turned up the speaker, and twining music with a hard, quick beat boomed out at them. They winced, and the sound dwindled quickly. "Hypothesis validated," Fess said, with a trace of smugness.
"I could use this form of thought to discover an enemy's camp," Geoffrey whispered.
"It is a powerful tool," Fess agreed.
"Yet this is not the only hypothesis involved," Gregory said, his little face puckered in thought.
"Indeed?" There was an undertone of anticipation in Fess's voice.
"We have tracked the trail of this one rock," Gregory said, "yet wherefore should the crafter have made but one?"
His brothers and sister stared at him, startled, and Rod and Gwen shared a proud glance.
Then Cordelia said slowly, "Aye, 'tis unlikely. E'en an he did it solely for the pleasure of it, would he not have made many, to delight in his own prowess?"
"That is possible." Fess carefully said nothing of the plague of songbirds that had struck the area around the Gallowglass house earlier that spring. "But how could we answer that question?"
"An there were other rocks," Magnus said slowly, "they would have split and flown three hundred meters at a time, even as these did."
"That is sensible, if we assume such rocks were identical to the ones we have already found."
Magnus shrugged, irritated. "There is scant reason to think aught else. They should therefore be each at a distance north or south from each of these we've found, but at an equal distance east and west."
"Why, how is that?" Geoffrey demanded.
"Oh, see, brother!" Magnus said, exasperated. He caught up a twig and dropped down to sweep dead leaves aside, exposing bare earth, and scratched with the twig. "An the rocks begin from the crafter, there in the west—let this dot stand for him—then the rocks we've found sprang from him three hundred meters at a time, here… here… here… and so forth." He made a series of dots moving farther and farther east. "Yet if another stone so split, and sent forth offspring, 'twould be either hard by each of these—and we know 'tis not, for we'd have seen them—or at some little distance, here… here… here…" He punched another line of dots, moving farther north as they moved east. Then he froze, staring down at his own diagram.
So did his parents.
Slowly, Gregory reached in with another twig and punched another line of holes south of the original line, moving farther south as they moved east, then another line south of that, and another, and another…
" Tis a set of circles," Cordelia breathed.
"With a common center," Geoffrey agreed.
"The term for such circles is 'concentric,'" Fess explained.
Magnus looked thoughtfully at Fess. "There is no reason why this could not have happed, Fess."
"I agree," the robot said softly. "Let us send the spy-eye searching north and south—though as you have noted, children, it must search in an arc, not a straight north-south line."
"Yet how shall it know how sharp a curve that arc must have?" Geoffrey asked.
"Why, by the distance from the center of the circle, brother!" Magnus crowed. "Dost thou not remember that the circumference is equal to pi times the diameter?"
Geoffrey glared at him.
"But in this case, we do not know exactly where the center is," Fess reminded him.
Magnus looked startled for a second, then had the grace to look abashed.
"Fortunately, the rocks we seek send forth sound," Fess added. "I shall turn amplification up to maximum, children. If the spy-eye comes near a rock, we shall know by its music."
The children waited in breathless silence, trying to ignore the droning of the stone behind them.
A tinny, clattering sound came from the screen.
" 'Tis there!" Geoffrey said.
"We shall proceed in the direction of maximal increase of signal," Fess told them. On the screen, the scene swooped down and around, and steadied on…
"Another rock!" Magnus cried, and Cordelia clapped her hands. Gregory only smiled up at the screen, his eyes glowing.
The rock lay in the center of the frame, medium gray, and heavily thumping under its cascade of metallic notes.
"Seek again," Magnus urged.
"Seeking," Fess answered, and the picture blurred once more. The children held their breath as one sound dwindled and another grew, then…
"'Tis there!" Magnus pointed, and the other children cheered.
The rock lay in the center of the screen, almost identical to the last two, both in appearance and sound.
As the sound slackened, Gregory piped, "Fess—canst thou determine an arc from three points?"
The robot was silent for a beat, then said, "If we assume it is an arc, Geoffrey, yes."
"Then do so, please! And show us it on a map of Gramarye."
Rod stared, amazed, as he realized what the boy was getting at.
"Remember," Fess said slowly, "that this is only an hypothesis."
"Hypothesis! Hypothesis!" Geoffrey protested. "Doth one hypothesis lead ever to another?"
"Yes, Geoffrey. That is how human knowledge progresses."
The screen flickered, and the children found themselves staring at an overhead view of the Isle of Gramarye. Then a circle appeared over it, cutting through the western corner of Romanov, down along the western edge of Tudor and the western corners of Runnymede and Stuart, to intersect the Florin River in the middle of the Forest Gellorn, and on through the western corner of Loguire to cut Borgia in half from north to south.
The children stared at the screen.
Then Magnus asked, in a hushed whisper, "Where is its center, Fess?"
"Where radü meet," the robot answered, and a large red dot appeared at the western edge of Gloucester.