But the others were finishing their preparations and securing their loads, and it wasn’t long before they began trundling forward. The old home was growing dim now, but Seth trusted the Southites to keep their bearings; he’d learned to recognize many parts of the bowl himself, regardless of their state of illumination, and to anyone who’d grown up here, the more distant regions, which would look much the same from either chasm, ought to be enough to show them the way.
Ada moved first, then Seth followed her, and they soon settled into a position half a dozen paces to the rear of the Southites, who advanced with a fixed, methodical gait on six of their legs, using the other two to hold the towing ropes. Seth had been afraid that he and Ada might have trouble keeping up, but the Southites were held in check by their own carts; even if they’d had the strength to pull them along faster, a heavily laden cart hitting a bump at speed would be asking for trouble.
«I have a great idea for a scam when we get back home,» Theo said.
«I’m listening.»
«We go from town to town, offering to arm-wrestle the locals, and make sure that the bets are placed while people can only see you from the right.»
Seth said, «I’m not sure if the most optimistic part of that is assuming that I can somehow remain to the north of all our marks until the last moment, or assuming that my left arm will still be good for anything after its millionth turn of the crank.» All this talk of asymmetry wasn’t helping; his neglected right arm, resting uselessly along his side, positively ached to make a contribution, and his left arm really couldn’t understand why it had to shoulder the entire burden.
«Rock,» Theo warned him.
«Sorry.» Seth had let his attention wander; he backed up the cart and steered around the outcrop. «I hope your pingers don’t end up as unbalanced as my body.»
«Test me,» Theo replied. Seth waved his right hand above the side of his head; it appeared in Theo’s view, as crisp as ever.
The line of lights swung slowly around the bowl, retreating into the distance. Seth drew his blanket tighter against the chill. Up on the surface, if the camp at the edge of the steamlands was still there, people would be sitting around listening to the evening rain.
«If this hole is too large to get around . . .» he began.
«Then the armchair surveyors will need to make a choice,» Theo replied. «Or maybe the logistics will make the choice for them.»
«Between the bridge of balloons . . . and what?» Seth had thought Theo had reconciled himself to the futility of trying to trap the sun.
«Between the bridge of balloons, and a journey like our own—only better prepared, and lasting much longer. If the migration’s blocked in every direction, it will need to take a detour to the southern hyperboloid while the habitable zone passes over the chasm.»
Seth couldn’t decide if he was serious. «Is this your idea of finding an alternative that makes building a mountain sound easy? I can’t see how they’d get that many people down the cliff, let alone all the way down the slope. And once they got here, what would they eat? Four guests seems to be about the limit of the locals’ hospitality, and even that’s still touch and go.»
Theo said, «You’re right about all of that, but if there’s no other choice, there’s no other choice. However hard it is, people aren’t going to give up and die quietly: some fraction of them will make it down here, and some fraction of those will find a way for their families to last out the wait.»
Seth still balked at the whole idea, but whatever the prospects were of it becoming a reality, the notion itself seemed dangerous. «Don’t mention any of this to Dahlia. If the Southites suspect that letting us go back could bring on an invasion from the other hyperboloid, then once they stop laughing at the thought of a million people flopping around on their sides, and picture them all equipped with armored carts instead—»
«I’m not stupid,» Theo said. «But nor are they. The fact that no one’s ever questioned us deeply about why we were exploring the chasm, and why it’s so important to us to know its size, has got more to do with Dahlia’s limits as a translator than anything else. These people have been in a position to understand the shape of the world for far longer than we have. It might be a stretch for some of them to imagine life on our hyperboloid, and the nature of the migration there, but there must be a few who can work out most of it, without any help from us.»
«So . . .?»
«So whatever results we measure, if they ask, we tell them that the hole’s small enough for the migration to detour around it. We can go back with the good news, and no one like us will ever bother them again.»
Seth laughed. «And we drop a subtle hint that, in the absence of any such confirmation, our people won’t risk trying to go around the hole and ending up trapped. Keeping us here against our will starts to sound like the more dangerous choice—and anything less than giving us the best boat they can make to take us safely up the slope would be a false economy.»
«Yes.»
«I like that better than your arm-wrestling scam. But do you really think we understand these people well enough to fool them?»
«We won’t need to lie about the measurements,» Theo stressed. «I’m sure they know how big the hole is, even if they don’t know how to tell us. But the real question is, how big a hole does it take to block the migration? And that depends on something they’ve never experienced: the geology of a landscape where the axial direction is horizontal, not vertical. If we claim that we can go around the hole, most Southites would think: of course . . . what could stop you? But even the ones who’ve thought it through carefully, and are capable of carrying out all the purely geometric calculations, still won’t be able to quantify the geological part of the problem. They can choose to doubt us, or they can choose to take our word for it. But there’s no way they could know for sure that we’re not telling the truth.»
when the sun began shining from the chasm behind them, the expedition finally stopped to rest. Seth let his arm hang loosely against the crank and tried to find a position that took the load off his shoulder and elbow.
“This must be what it’s like to try to tear your own arm off,” he said.
“Really?” Ada replied. “I could easily keep going for another half a day.”
Seth ignored her. “How are you feeling, Dahlia?”
“I’m all right. Do you think we’ll find the missing people soon?”
“It’s early days.” Seth didn’t want to raise her hopes too high, but it would be cruel to start speculating about how badly things must have gone for them if they couldn’t send word of their plight, or even signal their presence. He hadn’t been paying much attention to the land ahead, but if anyone had spotted a campfire or rising smoke, Dahlia would have been the first to know what they were rejoicing about.
“Iqbal wants me to meet all his cousins. He gave me a message to pass on to them if I see them first.” Dahlia emitted a long string of squawks; Seth saw some of the Southites turn to stare at her. “He said they’ll be surprised that I know so much about them.”
“That might not be their biggest surprise,” Theo replied.
A Southite approached, and spoke with Dahlia. “He wants to know if we’re comfortable, and if there’s anything we need,” she said.
“Tell him ‘thank you, we have everything.’” Seth didn’t recognize him as anyone they’d named back at the settlement. “Can we agree that he’s called Marco?” He wasn’t sure that Dahlia could tell a Southite’s sex any better than he could himself, either from their appearance or anything they said about each other, but he’d largely stopped caring.
Dahlia passed on the reply, including the adopted name.
“Nkko,” the Southite echoed.
“We haven’t insulted him, have we?” Seth asked anxiously.
“No,” Dahlia assured him. “But see if you can say his real name.” She uttered it, and Seth dutifully attempted to repeat it—leaving her helpless with laughter, and Marco app
arently entertained. Theo tried as well, and did a slightly better job. Ada flatly refused. “I know my limitations,” she said.
“Thith, Tho, Ata, Tatya,” Marco declaimed proudly.
Another two Southites joined them; Ada named them Lana and Niall. Dahlia translated some of their banter, but it came too fast for her to explain everything. To Seth all three of them seemed friendly and curious; they might have been shy about approaching the guests back at the settlement, but now that they were traveling together they were less inhibited.
“They want to know if you and Ada are going to have more children,” Dahlia told Seth.
“Not here and now,” he said. “We’ll need to get back home before we even think about that.”
Dahlia translated his reply; the Southites jabbered and hooted back at her. “Lana said she doesn’t believe you’re my parents. She said I don’t look enough like either of you.” Dahlia sounded puzzled, and a little hurt by the suggestion.
Ada said, “Just tell her that things are different where we come from.”
“Surely not that different,” Dahlia translated.
“Tell her she’s welcome to come and see for herself,” Ada replied.
“She’d rather see the proof closer to home.”
“Don’t say any more about it,” Seth interjected. “They’re just teasing us. You should talk about something else.”
Dahlia continued, offering no further translations. When the Southites finally went back to their group they seemed to be in the same good spirits as ever, but Seth couldn’t help feeling a twinge of anxiety. If they knew that he and Ada had lied to Dahlia, or that Dahlia was lying to them, what would they think? It was impossible to be entirely honest with them, without being entirely honest with Dahlia.
Theo said, «So when will I have a little brother, Daddy?»
«Ask your mother.» Seth watched the Southites chatting among themselves as they lit a small fire. This close to the chasm it wasn’t really cold at this time of day, but they were using the fire to melt some of the ice on the ground into drinking water.
«We need to be more careful,» he said.
«That’s a nice sentiment,» Theo replied, «but I don’t know what it actually entails.»
«Neither do I,» Seth admitted. Keeping more secrets? Telling better lies? «I don’t know how much longer I can do this,» he said. «If I can’t put my feet on the ground soon . . .»
«You’re going to crawl inside a Southite’s skull and start drinking their blood?»
Seth laughed. «Would that help? How do you live like this, and not lose your mind?»
«By getting my own way, enough of the time.»
«Yeah? But how do I manipulate these Walkers, when I can’t even speak their language?»
Theo said, «By finding a way to convince them that their interests coincide with yours.»
twelve days into the journey, Seth could see the mouth of the chasm stretched out across the bowl behind him, a jagged ellipse that blazed and darkened as the line of lights swept by. But it was only on the fifteenth day that he was able to take his first measurements; before then, the angle that the chasm subtended had been larger than the maximum azimuthal span of the elegant Southite instrument that he’d borrowed.
Dahlia had done her best to translate the owner’s instructions, but Seth would have had no trouble making sense of the device even if he’d found it discarded on the ground. The annotations on the calibrated plates for the alidade and plumb line weren’t important in themselves; it was easy enough to guess, and then verify, what scheme of divisions had been used.
Seth chose a dozen points on the chasm’s ellipse and measured, first, the angles between them, and then the angles from each sighting to the gravitational vertical. It was then an easy calculation to determine the angles between a line from the center of the world to his present location, and the corresponding lines from the center to the twelve points on the chasm’s rim. Together, all of these measurements could tell him everything about the size and shape of the chasm’s mouth—with one proviso. All of the results were expressed as fractions of the fundamental scale of the southern hyperboloid: the more-or-less constant distance from any point on the surface to the center of the world.
The equivalent measure of his own hyperboloid—the radius of the midwinter circle—had been established by the ancients and refined over generations. Seth gathered, from his tortured conversation with Siméon—as Theo had magnanimously dubbed the instrument’s owner—that the same was true for the Southites. But whether it was due to some deep cultural fissure, or simply Dahlia’s inexperience and limited vocabulary, Siméon had been unable to communicate this fundamental quantity in terms that Seth could understand. And Siméon’s own measurements of the chasm, taken before he’d left the old home, were mired in even more opaque and bewildering conventions; Seth could have had infinite faith in his goodwill and technical prowess, but he still would not have trusted his own potentially garbled interpretation of the numbers to guide the migration and the fate of millions of people.
He had been on the verge of reconciling himself to returning home with all the crucial results expressed in an unknown scale, leaving it up to a second expedition to determine the missing parameter. But Ada and Theo had argued about the problem for days, and finally reached a solution.
Siméon—the original Siméon—had established a formula linking the period of a swinging pendulum to the strength of gravity. The period also depended on the length of the pendulum—and they’d lost all the calibrated objects they’d brought with them from the surface—but Ada insisted that she knew the length of the bone that ran from her wrist to her elbow with enough precision to use it as a standard. Seth had no reason to doubt her; the size she claimed looked plausible to his eye, and when Iqbal transferred it to a length of rope and used it to measure Seth’s own height, the figure tallied with one he recalled from a surveying class when he’d needed it for an exercise in the geometry of shadows.
The Southites had no mechanical clocks, but the bowl itself made an exquisite timepiece. Once Seth had measured the angle between two distant chasms, he knew what the time between the line of lights illuminating first one and then the other would be, compared to the full circle of a day.
Iqbal had set up the pendulum for them, three Ada-forearms long, then Seth and Ada had taken turns with the observations, one of them counting the swings while the other watched the light come and go from the chasms. They’d repeated the experiment more than fifty times, then averaged all their measurements.
The final verdict had been uncanny: within the range of uncertainty of their method, the strength of gravity here was exactly the same as it had been back on the surface. So if the rock beneath them was of the same density, their distance from the center of the world was also the same. Theo had struggled to find an explanation for the coincidence—some kind of equalizing process that would have reshaped the world, over the eons, had the values not been identical. Seth had listened to all of his wild theories, but in the end he’d decided that it made no difference. If they’d found that gravity had been one-fifth more than usual, or one-twelfth less, he would still have trusted the method—and if Ada turned out to have been mistaken about the length of her forearm, they could easily adjust their results to account for that once they were back home.
As Seth took the measure of the chasm, he scribbled all the numbers onto a slate—another gift from Siméon. When he was finished he carefully packed the instrument away, then he stared at the figures, reluctant to proceed with the calculations.
Theo said, «I think I have the main results, but we should do this independently, then get Ada to check it as well.»
«All right.» Seth wrote out the geometric formulas he needed across the top of the slate, and began substituting the angles he’d measured. «When we were memorizing tables of hyperbolic functions for the surveyors’ exams, did you ever think we’d be applying them to the southern hyperboloid?»
«I never even thought we’d need to remember them,» Theo replied. «How could anyone possibly lose their paper copies?»
Seth’s concentration faltered. «Is that correct? Or have I lost the ability to multiply?» What if none of them could carry out basic arithmetic any more, let alone remember long lists of numbers from a lifetime ago? And Ada hadn’t memorized those tables; to thoroughly check the results, she’d need to recalculate the values from first principles.
Theo said, «We’re meant to be doing this independently, but I don’t see any mistake there.»
Seth pressed on, converting all the angles into distances. When he was finished, he stared at the results. «Am I right?» he asked Theo. «Or have I dropped a digit somewhere and shrunk everything by an order of magnitude?»
«I think you made a small error in the fourth row.»
«How small?»
«Smaller than the precision of any of this, but you really ought to fix it anyway.»
Seth laughed. «I’ll leave it in there, and see if Ada catches it. Or maybe she’ll decide that I’m right and you’re wrong.» Then without warning, he found himself sobbing with relief. What handful of numbers had ever cost so much to measure? And he knew now that if the answer had been different, he could not have borne it.
“What’s wrong, Seth?” Dahlia called to him anxiously.
“Nothing,” Theo replied. “We’re happy.”
“Why?”
“Because we think the hole in the world is small enough for everyone to walk around safely. No one else is going to have to make the hard journey we made.”
every time the search party set up camp, Seth repeated his measurements. At first, Ada was willing to indulge him and check his calculations, but eventually she lost interest, and even Theo decided that any further repetition was pointless. «When you average all these results, you’re reducing the effect of random errors—but there are sure to be a few systematic errors as well, which are the same every time. The instrument won’t be perfectly calibrated. There’ll be optical effects around the rim of the chasm. And the mouth of the chasm up on the surface won’t be exactly the same shape as the mouth down here.»