As Leddravohr tilted his head back to finish a glass of brandy his gaze drifted towards the zenith — and suddenly he had his answer. The great disk of Overland was now almost fully illuminated and its face was just starting to show the prismatic changes which heralded its nightly plunge into the shadow of Land. Deepnight — that period when the world experienced real darkness — was beginning, and it had its counterpart in Leddravohr’s mind.

He was a soldier, professionally immune to fear, and that was why he had been so slow to acknowledge or even identify the emotion which had lurked in his consciousness for most of the day.

He was afraid of the Overland flight!

What he felt was not straightforward apprehension over the undeniable risks involved — it was pure, primitive and unmanning terror at the very idea of ascending thousands of miles into the unforgiving blueness of the sky. The force of his dread was such that when the awful moment for embarkation arrived he might be unable to control himself. He, Prince Leddravohr Neldeever, might break down and cower away like a frightened child, possibly having to be carried bodily on to the skyship in full view of thousands.…

Leddravohr jumped to his feet and hurled his glass away, smashing it on the balcony’s stone floor. There was a hideous irony in the fact that his introduction to fear should have taken place not on the field of battle, but in the quietness of a small room, at the hands of stammering nonentities, with their scribbles and scratchings and their casual visions of the unthinkable.

Breathing deeply and steadily as an aid to regaining mastery of his emotions, Leddravohr watched the blackness of deepnight envelope the world, and when he finally retired to bed his face had regained its sculpted composure.

“It’s getting late,” Toller said. “Perhaps Leddravohr isn’t coming.”

“We’ll just have to wait and see.” Lain smiled briefly and returned his attention to the papers and mathematical instruments on his desk.

“Yes.” Toller studied the ceiling for a moment. “This isn’t a sparkling conversation, is it?”

“It isn’t any kind of conversation,” Lain said. “What’s happening is that I’m trying to work and you keep interrupting.”

“Sorry.” Toller knew he should leave the room, but he was reluctant to do so. It was a long time since he had been in the family home, and some of his clearest boyhood memories were of coming into this familiar room — with its perette wood panels and glowing ceramics — and of seeing Lain at the same desk, going about the incomprehensible business of being a mathematician. Toller’s instincts told him that he and his brother were reaching a watershed in their lives, and he had a longing for them to share an hour of companionship while it was still possible. He had been vaguely embarrassed about his feelings and had not tried putting them into words, with the negative result that Lain was ill at ease and puzzled by his continuing presence.

Resolving to be quiet, Toller went to one of the stacks of ancient manuscripts which had been brought from the Greenmount archives. He picked up a leatherbound folio and glanced at its title. As usual the words appeared as linear trains of letters with elusive content until he used a trick which Lain had once devised for him. He covered the title with his palm and slowly slid his hand to the right so that the letters were revealed to him in sequence. This time the printed symbols yielded up their meaning: Aerostatic Flights to the Far North, by Muel Webrey, 2136.

That was as far as Toller’s interest in a book normally went, but balloon ascents had not been far from his mind since the momentous meeting of the previous day, and his curiosity was further stirred by the realisation that the book was five centuries old. What had it been like to fly across the world in the days before Kolcorron had arisen to unify a dozen warring nations? He sat down and opened the book near the middle, hoping Lain would be impressed, and began to read. Some unfamiliar spellings and grammatical constructions made the text more oblique than he would have liked, but he persevered, sliding his hand across paragraph after paragraph which, disappointingly, had more to do with ancient politics than aviation. He was beginning to lose momentum when his attention was caught by a reference toptertha: “…and far to our left the pink globes of the ptertha were rising”.

Toller frowned and ran his finger across the adjective several times before raising his head. “Lain, it says here that ptertha are pink.”

Lain did not look up. “You must have misread it. The word is ‘purple’.”

Toller studied the adjective again. “No, it says pink.”

“You have to allow a certain amount of leeway in subjective descriptions. Besides, the meanings of words can shift over a long period of time.”

“Yes, but.…” Toller felt dissatisfied. “So you don’t think the ptertha used to be a diff—”

“Toller!” Lain threw down his pen. “Toller, don’t think I’m not glad to see you — but why have you taken up residence in my office?”

“We never talk,” Toller said uncomfortably.

“All right, what do you want to talk about?”

“Anything. There may not be much… time.” Toller sought inspiration. “You could tell me what you’re working on.”

“There wouldn’t be much point. You wouldn’t understand it.”

“Still we’d have been talking,” Toller said, rising to his feet and returning the old book to the stacks. He was walking to the door when his brother spoke.

“I’m sorry, Toller — you’re quite right.” Lain smiled an apology. “You see, I started this essay more than a year ago, and I want to finish it before I get diverted to other matters. But perhaps it isn’t all that important.”

“It must be important if you’ve been working on it all that time. I’ll leave you in peace.”

“Please don’t go,” Lain said quickly. “Would you like to see something truly wonderful? Watch this!” He picked up a small wooden disk, laid it flat on a sheet of paper and traced a circle around it. He slid the disk sideways, drew another circle which kissed the first and then repeated the process, ending with three circles in a line. Placing a finger at each end of the row, he said, “From here to here is exactly three diameters, right?”

“That’s right,” Toller said uneasily, wondering if he had missed something.

“Now we come to the amazing part.” Lain made an ink mark on the edge of the disk and placed it vertically on the paper, carefully ensuring that the mark was at an outermost edge of the three circles. After glancing up at Toller to make sure he was paying proper attention, Lain slowly rolled the disk straight across frie row. The mark on its rim described a lazy curve and came down precisely on the outermost edge of the last circle.

“Demonstration ended,” Lain announced. “And that’s part of what I’m writing about.”

Toller blinked at him. “The circumference of a wheel being equal to three diameters?”

“The fact that it is exactly equal to three diameters. That demonstration was quite crude, but even when we go to the limits of measurement the ratio is exactly three. Does that not strike you as being rather astonishing?”

“Why should it?” Toller said, his puzzlement growing. “If that’s the way it is, that’s the way it is.”

“Yes, but why should it be exactly three? That and things like the fact that we have twelve fingers make whole areas of calculation absurdly easy. It’s almost like an unwarranted gift from nature.”

“But…But that’s the way it is. What else could it be?”

“Now you’re approaching the theme of the essay. There may be some other… place… where the ratio is three-and-a-quarter, or perhaps only two-and-a-half. In fact, there’s no reason why it shouldn’t be some completely irrational number which would give mathematicians headaches.”