"And beyond power, Wili, there is knowledge -which some say is power." He had slipped into his native English, and Wili didn't bother to pretend ignorance. "Whether it is power or not depends on the will and the wisdom of its user. As my appren-tice, Wili, I can offer you knowledge, for a surety; power, perhaps; wealth, only what you have already seen."
The crescent moon had cleared the pines now. It was one more thing that would never be the same for Wili.
Naismith looked at the boy and held out his hand. Wili of-fered his knife hilt first. The other accepted it with no show of surprise. They stood and walked back to the house.
SIX
Many things were the same after that night. They were the outward things: Wili worked in the gardens almost as much as before. Even with the gifts of food the visitors had brought, they still needed to work to feed themselves. (Wili's appetite was greater than the others'. It didn't seem to help; he remained as undernourished and stunted as ever.) But in the afternoons and evenings he worked with Naismith's machines.
It turned out the ghost was one of those machines. Jill, the old man called her, was actually an interface program run on a special processor system. She was good, almost like a per-son. With the projection equipment Naismith had built into the walls of the veranda, she could even appear in open space. Jill was the perfect tutor, infinitely patient but with enough "humanity" to make Wili want to please her. Hour after hour, she flashed language questions at him. It was like some verbal Celest. In a matter of weeks, Wili progressed from being barely literate to having a fair command of tech-nical written English.
At the same time, Naismith began teaching him math. At first Wili was contemptuous of these problems. He could do arithmetic as fast as Naismith. But he discovered that there was more to math than the four basic arithmetic operations. There were roots and transcendental functions; there were the relationships that drove both Celest and the planets.
Naismith's machines showed him functions as graphs and related function operations to those pictures. As the days passed, the functions became very specialized and interest-ing. One night, Naismith sat at the controls and caused a string of rectangles of varying width to appear on the screen. They looked like irregular crenellations on some battlement. Below the first plot, the old man produced a second and then a third, each somewhat like the first but with more and nar-rower rectangles. The heights bounced back and forth between 1 and -1.
"Well," he said, turning from the display, "what is the pat-tern? Can you show me the next three plots in this series?" It was a game they had been playing for several days now. Of course, it was all a matter of opinion what really constituted a pattern, and sometimes there was more than one answer that would satisfy a person's taste, but it was amazing how often Wili felt a certain rightness in some answers and an unes-thetic blankness in others. He looked at the screen for several seconds. This was harder than Celest, where he merely cranked on deterministic relationships. Hmmm. The squares got smaller, the heights stayed the same, the mini-mum rectangle width decreased by a factor of two on every new line. He reached out and slid his finger across the screen, sketching the three graphs of his answer.
"Good," said Naismith. "And I think you see how you could make more plots, until the rectangles became so nar-row that you couldn't finger-sketch or even display them properly.
"Now look at this." He drew another row of crenellations, one clearly not in the sequence: The heights were not restricted to 1 and -1. "Write me that as the sum and dif-ferences of the functions we've already plotted. Decompose it into the other functions." Wili scowled at the display; worse than "guess the pattern," this was. Then he saw it: three of the first graph minus four copies of the third graph plus...
His answer was right, but Wili's pride was short-lived, since the old man followed this problem with similar decom-position questions that took Wili many minutes to solve... until Naismith showed him a little trick - some-thing called orthogonal decomposition - that used a peculiar and wonderful property of these graphs, these "walsh waves" he called them. The insight brought a feeling of awe just a little like learning about the moving stars, to know that hidden away in the patterns were realities that might take him days to discover by himself.
Wili spent a week dreaming up other orthogonal families and was disappointed to discover that most of them were al-ready famous - haar waves, trig waves - and that others were special cases of general families known for more than two hundred years. He was ready for Naismith's books now. He dived into them, rushed past the preliminary chapters, pushed himself toward the frontier where any new insights would be beyond the farthest reach of previous explorers.
In the outside world, in the fields and the forest that now were such a small part of his consciousness, summer moved into fall. They worked longer hours, to get what crops remained into storage before the frosts. Even Naismith did his best to help, though the others tried to prevent this. The old man was not weak, but there was an air of physical fragility about him.
From the high end of the bean patch, Wili could see over the pines. The leafy forests had changed color and were a band of orange-red beyond the evergreen. The land along the coast was clouded over, but Wili suspected that the jungle there was still wet and green. Vandenberg Dome seemed to hang in the clouds, as awesome as ever. Wili knew more about it now, and someday he would discover all its secrets. It was simply a matter of asking the right questions - of him-self and of Paul Naismith.
Indoors, in his greater universe, Wili had completed his first pass through functional analysis and now undertook a three-pronged expedition that Naismith had set for him: into finite galois theory, stochastics, and electromagnetics. There was a goal in sight, though (Wili was pleased to see) there was no ultimate end to what could be learned. Nais-mith had a project, and it would be Wili's if he was clever enough.
Wili saw why Naismith was valued and saw the peculiar service he provided to people all over the continent. Nais-mith solved problems. Almost every day the old man was on the phone, sometimes talking to people locally - like Miguel Rosas down in Santa Ynez - but just as often to people in Fremont, or in places so far away that it was night on the screen while still day here in Middle California. He talked to people in English and in Spanish, and in languages that Wili had never heard. He talked to people who were neither Jonques nor Anglos nor blacks.
Wili had learned enough now to see that these were not nearly as simple as making local calls. Communication be-tween towns along the coast was trivial over the fiber, where almost any bandwidth could be accommodated. For longer distances, such as from Naismith's palace to the coast, it was still relatively easy to have video communication: The coherent radiators on the roof could put out microwave and infrared beams in any direction. On a clear day, when the IR radiator could be used, it was almost as good as a fiber (even with all the tricks Naismith used to disguise their location). But for talking around the curve of the Earth, across forests and rivers where no fiber had been strung and no line of sight existed, it was a different story: Naismith used what he called "short-waves" (which were really in the one- to ten- meter range). These were quite unsuitable for high-fidelity communication. To transmit video-even the wavery black-and-white flat pictures Naismith used in his transcontinental calls - took incredibly clever coding schemes and some real-time adaptation to changing conditions in the upper atmosphere.
The people at the other end brought Naismith problems, and he came back with answers. Not immediately, of course; it often took him weeks, but he eventually thought of some-thing. At least the people at the other end seemed happy. Though it was still unclear to Wili how gratitude on the other side of the continent could help Naismith, he was beginning to understand what had paid for the palace and how Naismith could afford full-scale holo projectors. It was one of these problems that Naismith turned over to his ap-prentice. If he succeeded, they might actually be able to steal pictures off the Authority's snooper satellites.