It was peaceful and agreeable away from the tensions of the city, and I discovered that Denton, in spite of his long silences, was an amiable and intelligent man.

I had lost track of the days we had been away, but it was certainly at least twenty. Denton showed no sign of wanting to return.

We encountered a small settlement, nestling in a shallow valley. We made no attempt to approach it. Denton merely marked it on the map, with a rough estimate of the population.

The countryside was greener and fresher than that to which I had grown accustomed, although the sun was no cooler. It rained more often here, usually during the night, and there were many different sizes of streams and rivers.

All the features of the region, natural or man-made, difficult for the city to pass through or suitable to its peculiar needs, Denton marked without comment on his map. It was not the job of the Future Surveyors to decide which route the city should take; we worked simply to establish the actual nature of future terrain.

The atmosphere was restful and soporific, the natural beauty of our surroundings seductive. I knew the city would travel through this region in the miles to come, and pass it without registering appreciation. For the city’s aesthetics, this verdant and gentle countryside might equally be a windswept desert.

During the hours when I was not actually engaged in any of our routine tasks, I was still lost in speculative thought. I could not get out of my mind that spectacle of the manifest appearance of the world on which we existed. There must have been something, somewhere in those long years of tedious education that would, subconsciously, have prepared me for that sight. We live by our assumptions; if one took for granted that the world we travelled across was like any other, could any education ever prepare one for a total reversal of that assumption?

The preparation for that sight had begun the day Future Denton had taken me outside the city for the first time, to see for myself a sun that revealed itself to be any shape but that of a sphere.

But I still felt there must have been an earlier clue.

I waited for a few more days, still worrying at the problem when I found time, then had an idea. Denton and I had camped one evening in open country beside a broad, shallow river, and as sunset approached I took the video camera and recorder and walked alone up the side of a low hill about half a mile away. At the top there was a clear view towards the north-eastern horizon.

As the sun neared the horizon, the atmospheric haze dimmed its glare and its shape became visible: as ever, a broad disk spiked top and bottom. I switched on the camera, and took a long shot of it. Later I replayed the tape, checking that the picture was clear and steady.

I never tired of the spectacle. The sky was reddening, and after the main disk had passed beneath the horizon, the upright pinnacle of light slid quickly down. For a few minutes after its passing there was an impression of a bright focus of orangewhite at the centre of the red glow… but soon this passed and night came on quickly.

I played the tape again, watching the image of the sun on the recorder’s tiny monitor. I froze the picture, then adjusted the brightness control, dimming the image until only the white shape remained.

There in miniature was an image of the world. My world. I had seen that shape before… long before leaving the confines of the crèche. Those weird symmetrical curves made an overall pattern that someone had once shown me.

I stared at the monitor screen for a long time, then conscience struck me and I switched off to conserve the batteries. I did not return to Denton straight away: I was straining my memory for some key to that faint recollection of the time when someone had drawn four lines on a sheet of card, and held up for all to see the place where Earth city struggled to survive.

The map that Denton and I were compiling was taking on a definite shape.

Drawn on the long roll of stiff paper he had brought, the plan took the form of a long, narrow funnel, with its narrowest point at the patch of woodland a mile or so to the north of where the city had been when we left it. Our travels had all been within the funnel, enabling us to make measurements of large features from all sides, to ensure that we compiled information as nearly accurate as possible.

Soon the work was finished, and Denton said that we would return at once to the city.

I had, in the video recorder, a complete and cross-referenced visual record of all the terrain we had covered. In the city, the Council of Navigators would examine as much as they felt necessary to plan the city’s next route. Denton told me that other Futures would go north soon, draw another funnel map of the terrain. Perhaps it too would start at the patch of woodland, and take a course five or ten degrees to east or west, or, if the Navigators felt that a safe route could be found in the terrain we had surveyed, the new map would start further up the known territory, and push forward again the frontier of the future we had surveyed.

We headed back towards the city. I had expected, in some melodramatic way, that now we had the information we had been despatched to obtain, we would ride through day and night with no regard to safety or comfort; instead, the leisurely ride through the countryside continued.

“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.