“Shouldn’t we hurry?” I said in the end, thinking that perhaps Denton was idling for some reason connected with me; I wished to show that I was willing to move with speed.
“There’s never any hurry up future,” he said.
I didn’t argue with him, but it had occurred to me that we had been away from the city for at least thirty days. In that time, the movement of the ground would have taken the city another three miles away from the optimum, and consequently the city would have had to travel at least that distance to stay within safety limits.
I knew that the unsurveyed territory began only a mile or so beyond the city’s last position.
In short, the city would need the information we had.
The return journey took three days. On the third day, as we loaded the horses and continued on south, the memory I had been seeking came to me. It came unbidden, as is often the case when trying for something buried in the subconscious.
I felt I had exhausted all my conscious memories of the lessons I’d had, and sorting through the memory of the long academic courses had been as fruitless as the sessions had been tedious at the time.
Then, from a subject I had not even considered, the answer came.
I remembered a period in my last few miles inside the crèche, when our teacher had taken us into the realms of calculus. All aspects of mathematics had induced the same response in me — I showed neither interest nor success — and this further development of abstract concepts had seemed no different.
The teaching had covered a kind of calculus known as functions, and we were taught how to draw graphs representing these functions. It was the graphs that had provided the memory key: I had always had a moderate talent for drawing, and for a few days my interest had flickered into life. It died almost immediately, for I discovered that the graphs were not an end in themselves but were drawn to provide a means of finding out more about the function… and I didn’t know what a function was.
One graph in particular had been discussed in great and onerous detail.
It showed the curve of an equation where one value was represented as a reciprocal — or an inverse — of the other. The graph for this was a hyperbola. One part of the graph was drawn in the positive quadrant, one in the negative. Each end of the curve had an infinite value, both positive and negative.
The teacher had discussed what would happen if that graph were to be rotated about one of its axes. I had neither understood why graphs should be drawn, nor that one might rotate them, and I’d suffered another attack of daydreaming. But I did notice that the teacher had drawn on a piece of large card what the solid body would look like should this rotation be performed.
The product was an impossible object: a solid with a disk of infinite radius, and two hyperbolic spires above and below the disk, each of which narrowed towards an infinitely distant point.
It was a mathematical abstraction, and held for me then as much interest as such an item should.
But that mathematical impossibility was not taught to us for no reason, and the teacher had not without reason attempted to draw it for us. In the indirect manner of all our education, that day I had seen the shape of the world on which I lived.
5
Denton and I rode through the woodland at the bottom of the range of hills… and there ahead of us was the pass.
Involuntarily, I drew back on the reins and halted the horse.
“The city!” I said. “Where is it?”
“Still by the river I should imagine.”
“Then it must have been destroyed!”
There could be no other explanation. Had the city not moved in all those thirty days, only another attack could have delayed it. By now the city should at least be in its new position in the pass.
Denton was watching me, an amused expression on his face.
“Is this the first time you’ve been so far north of optimum?” he said.
“Yes it is.”
“But you’ve been down past. What happened when you came back to the city?”
“There was an attack on,” I said.
“Yes… but how much time had elapsed?”
“More than seventy miles.”
“Was that more than you expected?”
“Yes. I thought… I’d been gone only a few days, a mile or two in time.”
“O.K.” Denton moved forward again, and I followed. “The opposite is true if you go north of optimum.”
“What do you mean?”
“Hasn’t anyone told you about the subjective time values?” My blank expression gave him the answer. “If you go anywhere south of optimum, subjective time is slowed. The further south you go, the more that occurs. In the city, the time scale is more or less normal while it is near the optimum, so that when you return from down past, it seems that the city has moved far further than possible.”
“But we’ve been north.”
“Yes, and the effect is opposite. While we ride north, our subjective time scale is speeded, so that the city appears not to have moved at all. From experience, I think you’ll find that about four days have elapsed in the city while we’ve been gone. It’s more difficult to estimate at the moment, as the city itself is further south of optimum than normal.”
I said nothing for a few minutes, trying to understand the idea.
Then: “So if the city itself could move north of optimum, it wouldn’t have so many miles to travel. It could stop.”
“No. It always has to move.”
“But if where we’ve been slows down time, the city would benefit from being there.”
“No,” he said again. “The differential in subjective time is relative.”
“I don’t understand,” I said honestly.
We were now riding up the valley towards the pass. In a few minutes we would be able to see the city, if it was indeed where Denton had predicted.
“There are two factors. One is the movement of the ground, the other is how one’s values of time are changed subjectively. Both are absolute, but not necessarily connected as far as we know.”
“Then why — ?”
“Listen. The ground moves, physically. In the north it moves slowly — and the further north one travels the slower it moves — in the south it moves faster. If it was possible to reach the most northern point we believe the ground would not move at all. On the other hand, we believe that in the south the movement of the ground accelerates to an infinite speed at the furthest extremity of the world.”
I said: “I’ve been there… to the furthest extremity.”
“You went… what? Forty miles? Perhaps more by accident? That was far enough for you to feel the effects… but only the beginning. We’re talking in terms of millions of miles. Literally… millions. Much more, some would say. The city’s founder, Destaine, thought the world was of infinite size.”
I said: “But the city has only to travel a few miles further, and it would be north of optimum.”
“That’s right… and it would make life a lot easier. We would still have to move the city, but not so often and not so far. But the problem is that it’s as much as we can do to stay abreast of optimum.”
“What is special about the optimum?”
“It’s where conditions on this world are nearest to those on Earth planet. At the optimum point our subjective values for time are normal. In addition, a day lasts for twenty-four hours. Anywhere else on this world one’s subjective time produces slightly longer or shorter days. The velocity of the ground at optimum is about one mile in every ten days. The optimum is important because in a world like this, where there are so many variables, we need a standard. Don’t confuse miles-distance with miles-time. We say the city has moved so many miles when we really mean that ten times that number of twenty-four hour days have elapsed. So we would gain nothing in real terms by being north of optimum.”