WSP 032
Approaching the Limit
It was a beautiful day in Willow Hollow. The sun was high in the sky, birds were flying through the air singing beautiful songs, and the people of the town were all gathered together shouting numbers at a stone platform.
“Ten! Nine! Eight! Seven! Six! Five! Four!”
Jeh gave them all a thumbs up from her position in the center of the launchpad, inside a brand new never-before-flown Skyseed IV. The second and third vessels were still at Axiom being used for rapid messaging and advanced mapmaking in Kroan, but the fourth had been shipped here the moment it was ready. It was a somewhat different design to the others—it was wholly spherical, for one, and with four spherical metal knobs poking out of the sides, reminiscent of the design for the Moonshot. The Skyseed IV was not symmetrical in every direction, though, there was still a top and a bottom; the top was the standard sealing lid design the first Skyseed had, but the bottom had a little cylindrical receptacle that stuck a short distance into the ship. This was the functional prototype for Vaughan’s airlock, a simple receptacle with two doors that were sealed by pressing into a rubbery kind of plast. Inside the receptacle was a small potted plant that they were going to expose to space.
“Three! Two! One!”
Jeh put her hands on the drive’s controls and grinned. “Here we go!” She actually did the proper ascent procedure for once, starting slowly and ramping up speed over time. She only needed to use one hand, as the air restorer was wrapped around her wrist for easy contact, so with her free hand she waved at everyone as she ascended.
There was a small party taking place around her. Seskii had set up several tables of drink and food with assistance from Mary, and the entire town was in on it. Since it always took several hours to complete even the shortest of missions, the plan was to see if they could keep the party going until Jeh got back in the evening. The entire Wizard Space Program was present, even Alexandrite, as was Envila and most everyone Jeh knew from WIllow Hollow. Jeh made sure to give a special wave to Ripashi as thanks to him for the bear furs she was currently wearing. There was just something right about having fuzzy mitts and a hoodie with cute little ears. It was also really comfortable, perhaps more comfortable than the last set of bear furs she’d had.
All the better to go into space with.
Soon, she was so far away she couldn’t see individual people and so she focused more on keeping herself balanced. The Skyseed IV was much easier to do this with than the first had been—it had no fins so they didn’t get in the way, and as it was largely spherical the wind didn’t have too much to push on to make it spin, and what little spin there was could easily be rectified by using Orange to grab one of the protrusions and set it back. All she needed to do was make sure the drive pulled her upward. Even though she was a little rusty at this, she found it almost easy.
Once she was far enough that wind was no longer a problem, she allowed herself to relax, kicked the drive to its maximum setting, and just waited until she was “high enough.” She had no reason to try to orbit anything, so “high enough” was wherever she decided to stop.
There was no real trigger when she decided to stop, she just shrugged and turned off the drive at some point, allowing the weightlessness to consume her. She still mentally considered the direction of Ikyu “down,” though, so even when she rotated the ship to get a better look at the world below, she didn’t lose her orientation. A big grin crossed her face as she looked down.
She had really, really missed this.
Margaret is going to love it up here.
She used her imaging device to snap a picture of Ikyu. It had not changed much from the last time she was up here, but it had changed. She could see Benefactor on the border of Kroan and the Wild Kingdoms, visibly smaller than before but still visible from this height. She could also spot the kilometers-wide circle in the Wild Kingdoms where the battle had taken place. One thing she couldn’t see was the square-shaped hole Benefactor had left in Shimvale—the land up there was too snowy to show that feature to her.
However, all these features, visible though they were, were small. Barely scratches upon the sheer girth of Ikyu itself.
Even if we go to the stars themselves, we could barely leave a lasting mark.
Chuckling to herself, Jeh turned to the airlock and applied Orange to it, trying to open the outer door. She found significant resistance to sliding it open, and at first, she was afraid it had jammed, but the moment she opened it a little it slid the rest of the way open.
The plant flew out like it had been shot out of a cannon.
Quickly, Jeh grabbed it with the power of her Orange. Could have seen that coming, the air rushed out of the gondola when Slashy broke it… shaking her head, she returned to observing the plant. Which… currently looked perfectly fine and like nothing had changed.
This was somewhat like what they had been expecting since the “airless” jars they had made down on the ground took quite some time for the plants to wilt. They had just wondered if even less air did anything. Evidently, not really, and she didn’t intend to stay out here several hours and watch the plant die. With a shrug, she brought it back in, sealed the outer door of the airlock and opened the inner one—which offered quite a bit of resistance, but once she got it open, it hissed and let air into the chamber, pushing the plant to the back door of the airlock.
Air is so weird…
She lifted up the plant and examined it, finding that it was limp, though if that was from being exposed to vacuum or due to being thrown around by air, it wasn’t really possible to tell. She set it aside and briefly wondered if she had anything else to test.
A few minutes later she had determined that the vacuum had no effect whatsoever on Colored crystals or paper, at least not in the short term. Lastly, she decided to put water from her flask in the chamber. She already knew the fun property water had of making spheres in midair while in weightless conditions, she wondered if it held outside.
To her shock, the moment she got some water outside it started boiling, turning the blob of clear liquid into a roiling bubbling mess that appeared almost like an explosion in slow motion.
“Bizarre…” Curious, she brought the water back in before it finished boiling and touched it. The temperature hadn’t changed much—and Jeh was sure it was slightly colder.
Fascinating… She realized that her mouth had a lot of water in it, as did her eyes. That would probably be extremely uncomfortable to feel. She chuckled. Guess we’ll just have to wait until we get a larger airlock to see what happens to me! Maybe it’ll feel like having my eyes lit on fire…
She recorded her observations briefly and decided it was time to head down. She prepared for “down” to suddenly become “up” and started her descent, aiming right for Willow Hollow. The trip down was decidedly uneventful, and she came to the launchpad without problem, landing squarely to the applause of a crowd that was only slightly smaller than when she’d left.
“Thank you, thank you!” Jeh called as soon as Blue unscrewed the main lid. “Look forward to a lot more of that in the coming weeks! And months! Until we get the Moonshot ready and go… to the moon!” She thrust her fists into the air prompting another cheer from the crowd.
“Now, there better be some good food left, I’m starving!”
~~~
“Y’know, you could stick around,” Jeh said, munching on some kind of unidentified roasted meat on a skewer. “I’m sure I can get them to let me take you up to space once the Moonshot is ready.”
“I appreciate the offer,” Envila said. “But I am already on my journey. Once I circle the globe, though, I do intend to return—by that point I expect your little Space Program to have advanced considerably, and that you’ll have many more things to show me.”
“Heh, yeah, I sure hope so!” Jeh beamed.
“I do want to see the stars, although… part of my journey’s purpose is to learn more about the world and the people who inhabit it. There are no people up there, but… there is the ever-present beauty of creation.”
“Well, you’ll live forever, wait a few thousand years, then there’ll be people up there. Blue’s already talking about the possible untapped crystal mines on the moon, and you know what that means.”
“That would imply Crystalline Ones, though.”
“Which is really weird, huh? Nobody’s seen any evidence of anything like that on the moon!”
“I look forward to hearing your answers on the subject when I return.”
Jeh nodded, sighing. “…I hope your journey is quick.”
“I myself am torn. Rushing across the Wild Kingdoms as we have, I did not get to truly appreciate them and know the people within.” She looked at her hand with a wistful expression. “Though I agree such action was necessary. I think… I shall go at whatever speed feels natural at any given time. Who knows how long I’ll journey across Kroan before going to the sea, and then the Tempest?”
“Oh yeah, you should talk to the others before you go, they can give you tips about the Tempest.”
“I have already spoken to Vaughan…”
“Suro and Lila have also been there. And…”
“Jeh!” the Sourdough twins shouted, jumping on top of a nearby table.
“Hey!” Mary called. “You two get down!”
“This table…”
“…has our food on it!”
“We can do…”
“…whatever we want with it!”
Mary pointed an accusatory finger but couldn’t think of anything to say.
The two Sourdough twins returned to striking a pose and holding out their hands to Jeh. “Come with us!” they said in unison. “We’ve got something cool!”
“…What is it?” Jeh asked.
“Bread designed to be stacked into a giant tower,” one said.
“We want to see who can stack it the tallest,” the other continued.
“You can be our first test pilot!”
“…Stacking bread…?” Jeh grinned. “I’m in! See ya, Envila!” She waved and went off with her friends.
Envila smiled brightly as the children went… to play.
“A welcome sight?” Lila asked, jumping up on a table next to Envila.
“Yes… children should play.” Envila paused. “Adults should play too, but it’s not unusual when they don’t. Children… need the simple joy of being children.”
“Agreed on all counts.”
“I have heard much of you, Mayor, Keeper, and apparently once an adventurer?”
“I would not describe myself as an adventurer…” Lila chuckled softly. “I was simply born in a land where what people consider ‘adventure’ was the norm.”
“The Tempest?”
Lila nodded. “You will be heading there soon?”
“Soon is a relative term, but yes. Any recommendations?”
“It may seem wild, and like the rules of the world you know do not apply, but in truth, it is a place just like any other. The only difficult part is getting in and out, and the only obstacle to that is the weather. Charter a good ship and you’ll be good.”
“Ah, so applications of common sense are all that are required?”
“Well, that and an expectation that the people who live in the Tempest may seem like they don’t have any common sense at all. Which… I do hate to lump all sorts of people into one category, but that does seem to generally be the case. I was one of them.”
“What changed you?”
Lila smiled, turning her gaze to a little black cat who was currently talking it up with Vaughan. “A meek cat who was so, so out of his depth.”
Envila chuckled. “Ah, love, such a powerful thing. You are fortunate its force was positive for you.”
“A fact I thank Dia for whenever it comes to mind.”
“Wise.”
“I… have a question for you,” Lila said, face becoming serious.
“Oh?”
“You have lived long, extremely long, and have been on the walk far more than I or any of my compatriots. Is… is it true that She always has more to teach you?”
“Absolutely,” Envila said without hesitation. “I have no doubt there is a truly infinite depth to this life, and that we can always grow greater than what we already are… of course, to be actually great is to become no one, something which I am most definitely not doing.” She let out a short laugh that had a hint of sadness to it. “Sometimes I wonder if that means I’m doing it wrong…”
“For what it’s worth, you have a good heart. Though I cannot begin to fathom what sorts of things your life is meant to show.”
“You give yourself too little credit. The struggles of the old mirror the struggles of the young, it is merely the flavor that changes.”
“Curious…”
“I wish you luck leading this town in its spirit, its government… and its exploration of the stars.”
“When you get back I know Jeh will want to take you up there.”
“I look forward to it.”
~~~
And so “normal” life returned to Willow Hollow. Time marched on, as it always did, turning days to weeks, and everyone slowly went back into their routines. Naturally, not everything could go back to the way it was, for the Wizard Space Program was trying to build a ship that could go to the moon, and all the orders for the parts kept having complications. Which resulted in a lot of letters being sent with questions that needed quick replies.
Alexandrite dropped a bag full of letters on the table in front of Vaughan.
Vaughan stared at it blankly. “That’s even bigger than the last one…”
Seskii whistled. “Looks like I’ve got my work cut out for me!”
“Seskii, I’ve said it before and I’ll say it again, but we are so lucky to have you.”
“Behold, the almighty secretary!” Seskii immediately set to rapidly sorting the letters into various piles, ranking them by urgency. “Let’s see… complaining engineers, complaining wizards, complaining jewelers… oooh, and one from all three at once! And…” She ripped open two of the letters and compared them. “Would you look at that, they entirely disagree with what needs to be done, fun.”
Vaughan grumbled. “I’m going to have to deal with that myself, aren’t I?”
“Yeah, probably, letting your secretary dictate engineering decisions probably won’t be seen as proper. But I’ll sort through them for you and organize it.” She continued rifling through the various letters, sorting through them.
“I also have word from Gronge,” Alexandrite said. “He has progressed much with his versions of the Skyseed and has already progressed to what he calls Seaskimmers. They work rather well in the deep sea and he’s fairly sure they should work anywhere, but, again, leviathans.” He frowned. “Gronge is probably going to try something to get past them, soon. I am unsure exactly how. Perhaps he intends to be invisible.”
“Doesn’t work on Leviathans,” Vaughan said. “They have senses we don’t understand, Purple magic doesn’t seem to do anything to them. And killing one just summons more alarmingly quickly…” He shivered. “I remember reading about them, it felt like a horror story. The massive monster takes everything to be killed, only for three more like it to show up…”
“Maybe you just need to move fast enough,” Seskii said, shrugging.
“Water makes that rather difficult.”
“But not impossible!” Seskii pulled out another letter. “Oh, this one isn’t a complaint, it’s a report from the pilots of the other Skyseeds. Lemme see…” She opened it up and blinked a few times. “Huh. The pilots have started complaining about sunburning really easily when up in space.”
Vaughan scratched his beard. “Huh… now that I think about it, that makes sense, the atmosphere no doubt protects us from some of the sun’s heat, like Blue theorized. That’s easy enough to solve though, just add curtains to the windows of the Moonshot.”
“Making a note…” Seskii said, scribbling something down on a notebook. “And there, ‘get curtains.’ “
“Though that won’t help anyone going up in ships made mostly of glass…” Vaughan frowned. “I guess everyone but Jeh will just bundle up or deal with sunburns.”
“I suspect there are going to be a lot of little problems like this,” Alexandrite said.
“Oh yes,” Seskii said. “The Moonshot is not getting built quickly by any means.”
“Gives us time to relax a bit, though,” Vaughan said.
Seskii gestured at the pile of letters she was only halfway through sorting.
“…Well, physically speaking, anyway…”
~~~
Blue drew a squiggle on a sheet of paper.
She drew a lot of straight lines on the edge of the squiggle.
Then she drew another squiggle over it, related to the lines she had just drawn.
“Hmm…”
“Mind if I come in?” Suro asked, standing in the doorway of her “lab.”
“Well… maybe a little, but I’m not really getting anywhere with this, so come on in.” She turned around and smiled at him. “How’s your end of things going?”
“Pretty well, actually. I’ve been getting the air restorers smaller and smaller, and the feedback on my drive designs has been extremely valuable for my work. Amazing what you can learn when people who are absolute masters at the craft honestly critique your designs.”
“At least you get honest critiques, people pull my stuff apart because I’m ‘not a proper wizard’ and all that.” Blue scrunched up her muzzle at the thought. “It’s hard to tell what is legitimate and what’s just there to try and put me in my place.”
“I can see why that would be difficult, but surely even contentions made from petty motives can be useful?”
“Depends, sometimes they just want to find something wrong so they find something different than how they would do it and deny it, despite supposedly encouraging new ideas. It’s really annoying.” She sighed. “Or maybe I’m just overexposed to Rigelia’s particular brand of hypocrisy.”
“Oh, I’ve heard stories about her. When he was younger Vaughan had… quite a few things to say.”
“Not surprised.” Blue looked down at her squiggles on the paper. “Anyway, I’m not actually working on orbits right now. Not directly, anyway. I’ve started to notice that orbit shapes can make nice, smooth shapes that can somewhat easily be described by single equations. I’m trying to figure out what exactly the pattern that relates the initial conditions to the result is, and somehow I ended up here. At waves relating to other waves.”
“Looks… complicated,” Suro observed.
“Well… not really, I’m just making tangent lines on waves and then using those lines to define new waves. Turns out, it cycles… sometimes. I need a stricter definition of how to make a tangent line on any curve, I’m currently just eyeballing it, but I’m sure I can relate it back to the force of the drive… or the velocity… but I’m missing something and I’m not sure what.”
“At the very least you seem involved in your work, just like the last time I saw you.”
“Why wouldn’t I be?”
“Blue, you’ve been through a lot. I saw a little of it when you first came back, but I wasn’t there, so I… couldn’t really be sure.”
“I told you all what happened that day, right?” Blue said.
“I… well yes, strictly speaking, you did. I just…” Suro flicked his tail. “I’m not sure if I’m legitimately concerned for you or if there’s some part of me that’s jealous I wasn’t able to be there and support you.”
Blue looked at him for a moment and then broke out into a large, but somewhat sad smile. “I see you don’t understand your mind either.”
“Eh?”
“Am I just working as I normally am or deep down am I lying to myself to ignore what happened?” Blue shrugged. “I haven’t the foggiest clue. I feel fine, though I occasionally have nightmares. So evidence points to things being back on track. But I also remember ful- well feeling like everything was fine while I was being really, really stupid. So, given that… how am I supposed to know if my judgment of myself is any good?” A slightly haunted expression crossed her face. “That’s what bothers me these days. If I’m lying to myself and don’t know it.”
“Ah…” Suro paused. “I’m afraid even Lila struggles with that. I believe… pride is where most of it comes from. Uh, er, not the lying to the self part, but the being unable to see it. We all just naturally assume we are in the right. Or if we have a different disposition than you or me, that we are always wrong and that we should lament over it every moment of the day, but…” Suro paused. “I’m sorry, I’m not very… eloquent, I think I’m tripping over my words here.”
“It’s fine,” Blue said. “If tripping over some words gets us to the truth of the matter, that’s exactly what we should do.” She paused for a moment, thinking his words over. “Pride… trying to think about that twists my stomach into a knot.”
“You can see why, I presume?”
“Oh, yes, I’m full of myself.” Blue let out a short but legitimate laugh. “Have you seen me give talks at the meetings? Got me in some real trouble up there in Axiom. …Just because I’m right doesn’t mean I should shove it in everyone’s faces. And yet…”
“And yet you’ll probably keep doing exactly that.”
“Yeah…”
“Such is the nature of life. Takes a while for things to move from the mind to the heart, if they even can.” Suro jumped up on the table. “It’s really infuriating how we can be fully aware of our flaws and agree things need to change and then just… not do that.”
Blue tapped her hoof. “Seems like things shouldn’t be that way.”
“How so?”
“Like… we should be able to set our minds and bodies to the problem and solve it if we gave it our all.”
“Now wouldn’t that just make you proud that you did it?”
Blue blinked.
“Pride cannot be slain like most beasts. I think it has to be worked out of you by experience. Such as what you just went through.”
“So what, I’m supposed to be thankful I was an idiot?”
Suro grinned. “Maybe!”
Blue groaned and rammed her face into the table. “That’s so backwards,” she muttered.
“Yeah… I agree.” Suro looked off into the distance. “I’ve had a similar feeling lately. Had I come in earlier I would not have said the criticism was useful. I was… quite insulted that anyone would dare to suggest that my life’s work not only had flaws, but major flaws in technique that wasted tons of time and effort. Didn’t like it at all. But I couldn’t exactly not listen to them, and… everything’s going smoother now.” He looked Blue right in the eyes. “Every time we think we have slain our pride, it comes back to bite us.”
“If that’s the case, then what’s the point?”
“To be a better person than we were yesterday, I’d say. Do you think you are better or worse off for learning the things you learned during your ordeal?”
Blue took a moment to think about that. “…Overall… yeah, I’m glad I know these things about myself now.”
Suro grinned. “And now I’m confident that you’ll be fine. You are right, you’re probably a bad judge of yourself, and I am a bad judge of myself. But that’s why we have people around us to tell us when we’re doing that. And, right now… I think you’re on the right path.”
Blue gave him a soft smile. “Thanks, Suro.”
“Don’t mention it. Now, I’ll let you get back to your math…”
“Actually… stay.” Blue tossed her mane back. “You might be right, you might have missed something by not being there. And I… probably ignored the feelings involved. You… should know.”
“You sure?”
“Think of it as me returning the favor you just gave me. So… I think it all really started getting to me when I realized that I had made basically the entire Academy hate me for being so arrogant…”
~~~
“WHAT!?” Wyett shouted at the top of his lungs.
Via whimpered. “I told you all this wouldn’t go well…”
The royal family was gathered in the room of the Memory, secluded deep beneath the palace where there would be no prying eyes. Hyrii had accompanied them and was currently trying to calm Wyett down and failing miserably.
“Why aren’t we going to war with Shimvale right this very moment!?” Wyett demanded, laying his hands on the table. “They have infected our minds with a curse!”
“They have infected your mind,” Tenrayce said, closing her book and looking her brother right in the eye. “Furthermore, from your and Hyrii’s reports, it is clear that Shimvale as a whole is not on board with this. I am not even sure the Shimvale council is wholly aware of this ‘ancestry’ of Miss Ziggurat.”
“So what, because they’re divided in an idiotic form of government, they’re absolved from all blame?”
“That is the reason you gave in your report that they were not to be held accountable for Benefactor’s actions, despite a major arm of their trade being involved in supporting her. The only difference in this situation is that it now affects you personally.”
“It is a direct threat to the royal family!”
Unauthorized duplication: this tale has been taken without consent. Report sightings.
“Only the kankathi had that quest, the song merely makes you obey Miss Ziggurat otherwise.”
Wyett turned to King Redmind. “Dad, surely you see the threat?”
“I do,” Redmind said. “But war with Shimvale is out of the question, especially since we are theoretically at war with the rigid plague from the Shinelands as we speak. After that, we should be far more concerned with C-R’s people.”
“Agreed,” the Memory said. “The fact that such an enemy has gone unnoticed for so long is deeply concerning.”
“C-R has no intent to destroy us, though…”
“That was before,” Tenrayce said. “If Benefactor’s judgment of their mission is correct, we have crossed a line. They will want to orchestrate our downfall as a kingdom.”
Wyett slammed his fist into the table. “And we have no idea who their people are, where they are stationed, or how they even plan to do that! Meanwhile, we know exactly where the song comes from and we can go eliminate it. Furthermore, once word gets back to Kaykayzee about us allying with Benefactor, she will be livid. I predict she will support direct confrontation.”
“That her government will oppose,” King Redmind said.
Hyrii nodded. “I… yes, Wyett, the rest of the Shimvale Council is not ready to throw everything away.”
“And we have an inside woman,” Tenrayce added. “Crimkle of the Shimvale Council serves Benefactor.”
“…The rigid?” Wyett tilted his head.
“Yes. She’s ours now, and has been doing things under the Council’s nose for decades.”
“Perhaps the best asset Benefactor has given us aside from herself,” the Memory added. “She will look for opportunities to make a move on Miss Ziggurat on our behalf.”
“In short, we are taking the subtle approach,” Redmind said.
“All this deception…” Via said, frowning.
“There will be far less bloodshed this way, Via,” Redmind said, giving his daughter a warm smile.
Via looked like she was going to cheer up at this, but then Queen Riikaz suddenly stood up. “Oh, so now you want to minimize bloodshed?”
Redmind looked to his wife with sad eyes. “Dear…”
“What if my people had been there, huh? What then?”
“I would not have decimated your people without cause,” Tenrayce said. “Had we been approaching them I would have found a way to covertly protect them.”
“You were not the one who made the call to go out there, Tenryace.”
“I pushed Dad, Mom. I was the origin of the plan, I was the one who carried it out.”
“It went through me, Tenrayce,” Redmind said. “I take full responsibility for the order.”
“You never would have come up with such a plan yourself and you know it. For all that occurred, I am responsible.”
“Tenrayce…”
“S-stop it!” Via stammered. “You two, stop trying to take the blame for each other! You’re just making things worse…”
Tenrayce and Redmind kept staring at each other and at the same time they both relented, allowing their bodies to relax slightly.
Riikaz bristled. “In my mind, you’re both responsible.”
“We made a judgment, Mom,” Tenrayce said, voice quiet. “I… I think it was the wrong one, considering how it ended, but…” Her arms started shaking. “I… I’m sorry, I forgot for a moment that we were a family and not a bunch of politicians…”
Riikaz’s expression softened immediately. “Oh… my little Tenii…” She got up and put her hand on Tenrayce’s shoulder. “I… am furious. But you are and always will be my daughter. That will never change.”
Tenrayce grabbed her mother’s hand tightly. “I broke out there, Mom. I broke. It was only for a moment, but… I did. I saw how pointless it all had been, how all my efforts were for nothing, how I’d sacrificed… too much. I…” She looked up to her father. “I… thank you, Dad, for being so strong.”
The King took in a sharp breath and released it. “I am not as strong as I make it seem. But… I try. Thank you for letting me know it’s working.” He paused. “I will not always be here. One day, I will pass the throne to Wyett, and I will expect all three of you to rule in some capacity. These decisions always need to be made. No matter where the blame lies, or who exactly is in the wrong… this experience was good for you, Tenrayce. Yours as well, Wyett.”
Slowly, the King stood up and held his arms behind his back. “We are the Kroan Family, the chosen rulers of this kingdom, and it is our responsibility to rule. All of you need to learn how to do that, and there is only so much education can give you. You need to experience real situations and real crises. This has been one of them. Learn from it. Make your own thoughts. You do not have to agree with me, and I actually hope you do not, as times change so too do the sorts of leadership the kingdom needs. I learned that… from, well…” He held out his hand to the truly ancient gari who was sitting in the corner of the room, snoring; the old Queen Ursulii. “One day, that will be me sleeping over there, and you all will have a newer kingdom, one that faces its own crises. If there’s one thing this day has brought me, it is confidence that you three will do your best when that time comes.”
“E-even me?” Via asked.
“Yes, even you, little one. The other two need you to keep them in check.”
“I-I will, Dad! You can count on it!”
Tenrayce let out a sigh but smiled. “That… appears to be a correct assessment.”
Wyett took in a sharp breath. “I still disagree with most of this, but I shall accept your decision.”
“Your day will come, Wyett,” Redmind said. “I will keep no secret from you, I fully intend to step down as my mother did one day and pass it to you. When that day comes, my only requests are that you put the fate of the kingdom above your own and that you keep your sisters as your closest advisors.”
Wyett stood tall, lifting up his chin. “Of course, Dad.”
“Good. Now… we have a plan for infiltrating Shimvale, and the army is theoretically already at the Western Ch’eni’tho, ready to engage the rigid plague. Which leaves us with a problem we do not know how to solve—C-R’s mysterious people.”
“Are we absolutely certain they don’t have a name?” the Memory asked.
“None at all,” Tenryace said. “Benefactor thinks it’s because they don’t want anyone to know they exist, so they have no way to refer to themselves, no name, no symbol, no trace. The symbol on C-R’s balloon whale was hers and hers alone. She will likely not return to Kroan, and any agents of her people who come will likely be entirely unrelated to her in any traceable way.”
“So we have a threat and no way to know who it is…” Riikaz growled. “That makes it hard to punch them.”
“Benefactor will have some assistance in that regard,” Redmind said. “She does have an information network that puts ours to shame.”
“But she admitted she has difficulty finding them,” Tenrayce said. “They are not a group with large numbers, merely great influence. Everything about them is designed to influence on a grand scale while remaining entirely invisible.”
“To what end?” Wyett asked.
Tenrayce blinked. “Did you miss that part? They may be a mystery, but Benefactor knows their exact goal. They wish to keep the world from uniting and attaining power which ‘should never be attained.’ ”
Wyett stared at his sister blankly. Then, slowly, he turned to look up at the Memory.
“Yes, Benefactor does believe I hold one of those great secrets. I… am unsure if that is what the memory I hold truly is, but it is certainly possible.”
Wyett put his hands to his head. “What are we going to do…?
“That is, in fact, what we are here to discuss. How to fight an enemy we cannot see. We believe they are currently unaware of my existence, the secret has been kept so well. Benefactor only found out by specifically installing one of her listening devices into your great grandfather.”
Hyrii started rubbing Wyet’s back to try and relieve the intense buildup of tension that was running through him.
“I… I can’t think of anything…”
“You are emotionally compromised,” Tenrayce said. “I am too, to some extent, but I do have some ideas. We should expand Benefactor’s information network…”
~~~
Jeremiah grumbled as he woke up. The bed wasn’t his. The room wasn’t familiar. The house was too simple. The garden outside hadn’t started growing yet.
It was all wrong.
“Why have I been reduced to this?” Jeremiah asked, sitting up. “Have I done something to offend you, Eyda?”
With a frown, he pulled out the hexagonal sigil and called up Kirkkok.
“Glad to see you’re still up and kicking,” Kirkkok said with a squawk. “What can I do for you today?”
“Have I angered Eyda?”
“Oh, rest easy, you’re not being punished.”
“Then what is the point of all this?”
“You’re being taken somewhere new!”
“But… my home…”
“You can get a new one, and the place you are now is going to be quite the hub of activity. Think of it—so many followers of Dia so close, many of which have never stopped to consider anything else. They made the mistake of trusting you, and so we have an in-road here.”
Jeremiah nodded. “Yes… Eyda’s glory can be spread, though it will have to be done covertly.”
“Naturally. Of course, this means I shouldn’t get anyone down here to rebuild your home, that would just be suspicious, wouldn’t it? So just… hang with the less-than-optimal living conditions for now.”
“Of course… Eyda will restore to me what is mine…”
“In time, in time…” With that, Kirkkok vanished.
Jeremiah, still grumbling, got out of bed and went to the kitchen. Margaret was cooking up breakfast, a small smile on her face. She set out rice and fried eggs. “You know, Dad, it’s really nice of them to just give us such a nice house.”
“Eh?” Jeremiah said, looking up from his eggs, somewhat dumbfounded.
“I mean, it’s larger and has more things in it than some of the other houses in town, Vaughan even got us an arcane stove, and our neighbors dropped off eggs this morning free of charge!” She dusted off her hands. “Plus, with time, we can get that garden outside sparkling spectacularly, and I don’t have to hunt anymore, they have plenty of people to do that already.” She sat down in her chair.
“Don’t you… miss our home?”
“I do… it was a nice place. But…” She looked up at a painting of the stars and Ikyu she had done based off one of the pictures Jeh had brought back. “I think this place can be home too.”
Jeremiah grumbled. “Don’t forget that we’re in enemy territory here, Margaret.”
“Enemy?” Margaret frowned. “Dad… you never said they were our enemies before.”
“Eyda…”
“Gives us the power to seek for ourselves.” Margaret frowned. “I’m getting worried about you, Dad. I thought all the excitement was just getting to you, but… it’s been weeks, and you’re still grumbling and… angry. What happened to that smile? I…” She bit her lip. “I miss it.”
Jeremiah saw his daughter’s pain and forcibly gave her a smile.
A tear rolled down Margaret’s cheek. “That’s… that’s not it, but thank you for trying.”
“Margaret…”
“Maybe I’m just too stuck on the past as well… just in a different way than you are.”
Jeremiah frowned. “I’m not stuck on the past.”
“You clearly hate this house, even though it’s better than we could have expected and…” Margaret stopped herself. “We… we don’t need to argue about this over breakfast, do we? Let’s just… have this moment. Can we? Please? Enjoy the morning?”
Jeremiah looked out the window, noting that the sun was rising. He felt his stomach twist with a mild sensation of disgust.
“Yes, of course, we can enjoy the morning.”
~~~
Mary leaned on her hoe, stopping in the midst of her work to watch the bustle that was happening a short distance away. Nearby trees in the forest were being marked with red paint, and various people who weren’t from Willow Hollow were running around between the trees, drawing things on their clipboards and making markings in the ground.
They were from the government, surveying the area for the “proper laboratory” Tenrayce had said she was going to build for the Wizard Space Program. Almost undeniably a good thing, and they didn’t even have to pay for it. Better equipment, a safer space, and it was going to be quite a bit closer to the town itself than Vaughan’s cabin was. Plus, there was even going to be a lounge area.
Mary should have been delighted that it was happening.
But she wasn’t. All she could see on the surveyors were official symbols of Kroan. They were from the government. A government that she had gotten to see firsthand and had not been impressed by.
“Mary.”
Mary didn’t look away from the government workers. “Big G.”
Big G folded his arms and stood next to her. “This is going to bring a lot more traffic into Willow Hollow.”
“Annoyingly.”
Big G turned to her, eyes narrowing. “…Scowls don’t look good on you.”
Mary’s expression faltered. “C-come again?”
“I said scowls don’t look good on you. Your usual smile is better, as is your concerned face, and your curious one. Not the scowl. It’s so unlike you.”
Mary sagged. “I should be happy about this, but I’m not. Look at us, getting all these things, proper funding, and all that. It just… it comes from the Crown.”
“Mary…”
“All our fear about the Crown punishing us for criticizing them is entirely unfounded, I saw it firsthand.” Mary scrunched her nose. “They’re just people like us, and people can do some terrible things.”
Big G nodded, simply taking her word that their conversation was safe. “Why does that surprise you?”
“I don’t think it really does, but the things I saw out there…” Mary couldn’t help but scowl again. “Those people shouldn’t be in charge, wantonly throwing away life like that.”
“Would you do any better?”
Mary’s left eye twitched. “That’s exactly what she said. Didn’t get to answer her at the time. The answer is, of course not, but I know people who would. Lila would blow them out of the water. Suro. Vaughan. You. Heck, even the Sourdough twins are probably better!” She threw her hands in the air. “You don’t just… kill hundreds of people for a chance at something you don’t know anything about! That’s acting out of fear! That’s…”
“I’ve had to order my men to their deaths a few times.”
Mary’s heart stopped in her chest. “Wh-what?”
Big G’s expression became deeply solemn. “We found some kind of explosive gas a few years back. The vein was so large it likely could have caved the entire mines in. Someone had to go into the gas itself and seal it. The risk was immense. I knew it when I ordered Mikan to do it. Sure, he was willing… but I still told him to do it. He sealed it up but got caught in the rockfall in the process. He knew it would happen. I knew it would happen.” He paused. “I could have done it myself, but then who would have led the men? That… was what I thought at the time. I’ve heard the story of what you encountered out there. Tenrayce saw the threat and went to handle it herself. I did not put myself on the line. She did. I’m not sure what that means.”
Mary stared blankly at Big G. “I… I remember Mikan’s funeral, I thought it was just… an accident.”
“Nobody wanted to blame me.”
“…It eats at you.”
“Yes.”
Mary looked back at the workers, frowning. “But… you knew there was a danger. In the end, it turned out that Benefactor could be turned into a friend.”
“I’m not one for intense politics, but I do pick up things from time to time. Would things have led to the alliance had there not been an attack?”
“I…” Mary scrunched her face.
“We don’t know.” Big G closed his eyes and shook his head. “I’m not defending the Princess, Mary. She was willing to throw people’s lives away. But so was I.”
Mary had no response. She simply stared at the government workers, face contemplative.
“…How can we trust our leaders?” Mary asked, finally. “You, Tenrayce… even Lila makes mistakes.”
Big G shrugged. “I don’t have answers for you. I do know that if my boys didn’t trust me, I would not be able to lead them. I think the same is true for all leaders. Trust is required… or fear, I suppose.”
Mary shook her head. “…I think I would have preferred it if this town remained unimportant and unconnected.”
“Nothing remains the same forever.”
“Yeah…” Mary paused. “Thank you, Big G.”
“Thank you, Mary. It has been… some time since I spoke of Mikan.”
She gave him a sad smile.
“I do have to go, sadly. I was only passing through, I have a message for Lila. Will you be alright?”
Mary looked back up at the government workers. “Yeah… I think I will.” She turned away and drove her hoe back into the ground. “The Princess was right about something—I am just a farmer. But that’s a good thing. I provide… for everyone.” She continued her work with a strange, newfound fervor. A simple joy came to her through the simple act of working the land.
~~~
“So, how are your plans for world domination going?” Jeh asked as she walked along a log over the river, the Sourdough twins just behind her.
“Going great actually!” one said.
“Been introducing ourselves to those government surveyors,” the other added.
“Free food is an easy way into the hearts of people.”
“And we sure have plenty of that!”
“You two, giving away food for free?” Jeh gasped in mock horror as she jumped off the log to the riverbank. “What is the world coming to?”
“Oh, they only think it’s free. However—”
“—we are really charging them for a positive impression!”
“Two adorable but also dutiful workers—"
“—always there to give them a smile and a snack!”
Jeh jumped up into a tree, nodding. “So, what, they go back and tell the Crown about you?”
“Well, that is a benefit, yes—”
“—but making good impressions on people is just good practice in general.”
“You girls should go exploring,” Jeh said. “I made quite a few friends out there just walking around.”
“While that’s really good for getting new acquaintances—”
“—it is not good for setting up a seat of power. Location needs to be stable—”
“—or else nobody would ask you to lead a land you’re never in!”
Jeh blinked. “Huh, hadn’t thought of that…”
“Good thing you don’t need to!” one said.
The other beamed. “Because we can do it for you!”
Jeh raised an eyebrow. “I don’t want to be in charge.”
“Oh, but you will be! You—”
“—the first pilot of the Wizard Space Program—”
“—now have an apprentice!”
Jeh stopped short. This was a problem as she was currently trying to jump from tree to tree, so she landed face-first in the dirt, upsetting a family of blue rodents. “…You’re right…”
“You complained about how Vaughan taught you—”
“—now you have a chance to do it differently!”
Jeh jumped up and looked at the Sourdough twins, both of whom were still in the tree, smiling down at her. “So. Uh. I think I’m gonna need some help with the ‘leading’ part of that.” She grinned. “Can I count on you two?”
The two of them bowed extravagantly and spoke in unison. “Of course, what are friends for?”
~~~
Blue drew a line and wrote down the math that described it.
Then she drew a single point on the line and turned back to the equation she had written down before.
“So if this works… then…” She did a series of complicated and somewhat obtuse mathematical operations on her equation, things that no mathematician in their right mind would ever do. Make it more complicated? Jump around using a concept that didn’t really exist in reality? Pretend like all the patterns she had recognized just worked without proof? Absurd. Ridiculous. Mad.
And yet, in the end, she had another equation.
To check, she placed two points really close to each other on her original line and drew a straight line through the points. She looked at this, nodded, and wrote down a single number. It wasn’t exactly the same, but it was very close.
The test wasn’t done yet, however. Now she took her second equation and drew a line for it. Then she drew a bunch of rectangles under the curve, assigning a specific number to each. She added them together in sequence.
It wasn’t exactly the same as the original function, but it was very close. And if she made the rectangles smaller, it was even closer.
Blue started to giggle.
This giggle turned into a series of snorts and whinnies, followed quickly by ordinary laughter that evolved into full-on cackling that could be heard throughout the entire cabin.
There was a knock on the door. “You okay in there?” Vaughan asked.
“Vaughan! Get in here, this is amazing!” She flung the door open with her telekinesis and dragged him in.
He was floored by the sheer amount of mathematical drawings everywhere. It was far more than usual, even for Blue, and unlike usual, most of the pages weren’t just lists upon lists of numbers, but rather abstract mathematical notation that covered entire pages in what was essentially an alien language, interspersed by regular graphs.
She shoved the single page she had just done in his face, the one with the two equation graphs. “Look at it! Isn’t it beautiful?”
Vaughan adjusted his hat. “This just looks like two arbitrary equations…”
“Oh, but they’re not arbitrary! They are very closely related!” She cackled. “See, this one here? It shows the slope of the other one at every point!”
Vaughan tilted his head. “You can’t have a slope on a curved line, they’re not…”
“Forget everything you were taught in math class, it’s wrong! You totally can so long as you have this guy!” She pointed eagerly at a symbol she had used all over the mathematics in her room, a single, tiny dot.
“…What is that?”
“That, Vaughan, is the smallest thing! Whatever number you think of, it’s smaller than it. Unless it’s zero, it’s larger than that!”
Vaughan raised an eyebrow. “Blue, that’s…”
“Absolutely insane, doesn’t actually exist, made of imaginary fairy dust from childhood dreams, oh yes absolutely. ‘Things like that don’t belong in math’ you say, ‘there’s no way that would do anything’ you say. Well, to that I say, it works! I can take an equation and find another equation that describes its slope at that moment! If I draw a line through two points really close together, that line’s slope gets closer and closer to the value of this function at that point the closer I make the points!”
“That’s… certainly interesting.” Vaughan scratched his beard. “The mathematicians would probably hate you for it… but what exactly is its use for us?”
“Primarily? I am so, so, sooooo tired of taking hundreds of thousands of calculations just to figure out the shape of my orbits. I want to be able to do all the steps at once, treating each “step” as the smallest possible step so I can figure out exactly where we’ll be at all times—barring unforeseen variables like, I don’t know, space wind or something.” She riffled through her papers until she found one that showed lots of circles and ellipses and spirals around Ikyu. “See, all these lines? They can be made smooth if you do things right. Smooth. Regular. Perfect circles and ellipses. So I thought, surely, there has to be an easier way to do this. And while I’m not done yet, this… smallest thing is certainly going to help me do that! Oh, this is going to make riding on the Moonshot so much less stressful, I’ll be able to give an orbit prediction in minutes rather than hours…”
“Wow.” Vaughan blinked. “Making new math just to go faster…” He chuckled. “Neat!”
“Oh, you ain’t seen nothing yet. See, when I tried to do this in reverse, I found something much more interesting. Vaughan, what do you think the area under this squiggly line is?”
Vaughan sighed. “You would have to estimate it by drawing a lot of rectangles…”
“It’s five.”
“Eh?”
“It’s exactly five.”
“How in…”
“I told you already, just do it in reverse!” Blue started hopping around in delight. “Now I don’t know how to do it in reverse, exactly, there’s been some trip-ups there, but once I have two functions related to each other in this way, one is the slope of the other, and the reverse is the area of the other!” She giggled. “All those Magenta wizards who complain all day about the really awkwardly precise shapes they have to cut their crystals into are gonna lose their minds.”
Vaughan’s eyes slowly began to widen. He stood up tall and looked around the room, taking in the drawings all around the room. Now that he knew what they meant, he could see it. Most of the drawings came in pairs; one for area, one for slope. There were a few sets of drawings that continued the sequence even further. The most interesting one was a bunch of waves that eventually entered a cycle and became each other…
“Blue… this… I can’t even.”
“I got so upset at calculating orbits I invented new math! Take that, space! Your complexity can’t stop me! HAH!” She threw her hat into the air and caught it back on her head. “Nothing can stop me now. Errors will vanish. Calculations will be speedy. And I will get us to the moon!” She grinned at Vaughan. “So, now that you understand… cool, right?”
“Blue, this may be the single most important discovery you’ve made.”
“Well, if that’s true, it’s going to make the bigwigs back in Axiom even more upset.” Blue chuckled. “Ah, probably shouldn’t feel so good about that, but man it’s satisfying to keep doing things everyone says I shouldn’t be able to do. Anyway.” She sat back down and looked at her pages. “I actually haven’t applied this to orbits yet. I know how it’s just going to take some time to get all the variables and directions right. Might have to limit myself to constant forces, but that won’t be a big deal… well, compared to coming up with this in the first place.”
“I think you should try to write up some formal proofs, send it to the mathematicians at the Academy.”
“I have a better idea. Send them my work, show them that it works, and ask them to prove it for me. Because I have absolutely no justification for using that smallest thing. I threw it in there out of frustration a while back. Best decision I ever made.”
“They won’t like that…”
“But they’re a lot more trained in formal proofs than I am. It should be their job to prove things like this.” She snickered. “Let this be called ‘Blue’s Slope-Area Conjecture.’ Maybe it’s not true for every function, but I’ve yet to find one it doesn’t work for yet! …Even if I can’t describe the line sometimes. But I can draw it!” She cackled. “Ah, feels good to have a breakthrough…”
~~~
Rina and Rona were drinking out of the same very large pitcher of fruit juice while they watched Jeh instructing Margaret on how to fly. Jeh had strapped the gari to the same haphazard chair setup she had been trained on.
“Okay, so, nice and easy…” Jeh said. “We’re just going to take it slow and simple first, we get you to levitate in the air.”
Margaret gripped an Orange crystal in her hand. A soft plast rim had been installed around the sharp edges so it wouldn’t cut into her. “Ready.”
“Slow is boring!” Krays called from her position across Vaughan’s backyard, where she was currently testing out various curtain materials. “Shoot her up there, let her flail in the sky!”
“No, we’re not going to do that,” Jeh said. “We start simple and work our way up to the Skyseed.”
Margaret nodded. “I understand. Today we just need to… float.” She focused her will on the Orange crystal and released the spell, pushing up on the brass disc above her. She had to push quite a bit harder than she’d been expecting to make anything happen, so with a grunt, she pushed more will into it.
The chair device hopped into the air. In surprise, Margaret cut the power and fell back to the ground, breaking one of the chair’s legs.
“You’ve got a frog over there, Jeh!” Krays called.
Jeh ignored her, using Green to fix the chair. “Once you get better control you can jump like that, but you should start extremely gradually, ease into it.”
Margaret took in a breath and released it slowly to calm herself. “Right. Slow… very slow…”
She started pushing again, and of course there wasn’t enough force to lift her. She adjusted it as slowly as she possibly could. When she felt a slight shift in the chair, she made sure to keep pushing. One leg lifted off the ground. Then two. Then all four…
…but almost immediately after this she tipped over and landed on the ground.
“One by one the dominos* fall!” Krays called. “I wonder how big of one you can knock over?”
*She’s merely describing the shape of dominos, the actual game does not exist on Ikyu. However, the art of stacking rectangular slabs up and knocking them down is a somewhat common pastime, and there are even several high-profile artists in Kroan who make such “kinetic sculptures” and then invite people to watch them get knocked down as part of a show.
“You’re focusing your force on one point,” Jeh told Margaret as she stood the chair back up. “Try to distribute the force evenly across the disc. That should at least get you in the air.”
“Distribute…” Margaret frowned. “How, exactly?”
“I… hmm.” Jeh scratched her head. “I’m… not sure, exactly? Once I figured out I needed to do it I kind of just did. Uh…”
“Looks like you klutzes need my help after all!” Krays put down her curtain tests and marched over. “So, some people, like Jeh here, just have a natural talent at magic. Jeh’s talent is a bit of a brute-force learn-by-running-your-head-into-the-wall-a-million-times-as-hard-as-you-can one, but it allows her to just kind of force the crystals into submission, but more complicated things elude her.”
“Hey, I can make pictures!”
“Purple is very suited to generating images. But have you ever managed to release even a small lightning bolt?”
“I… well, uh…” Jeh shrugged awkwardly. “Yeah, no.”
“That’s because the technique for creating lightning doesn’t just need a lot of will, it is also highly complex and derivative. Orange naturally pushes objects around a small area about the size of your fist, to spread the force out requires more complexity. I, personally, learned how to do it by first trying to push objects at multiple places at once until it just melted together like slime carcasses in a pool.” She winked at Margaret. “fortunately, unlike lightning, distributing force doesn’t actually require much extra will!”
“How… do you make lightning?” Margaret asked.
“Haven’t the foggiest, read about it in a book once,” Krays waved a hand. “Very complicated and hard to do, just use Red instead, my suggestion.”
One of the twins elbowed Jeh and whispered in her ear. “Krays is taking over here, making Margaret feel inadequate. Her idea may be good, but she’s changing the mood of the situation. You need to take control back.”
Jeh nodded. “Thank you Krays, we’ll be sure to do that. I think we can take it from here?”
“Doubtful,” Krays said. She chuckled and walked back to her experiments. Five seconds later she accidentally lit a curtain on fire. “…Ah! I’ve found a new worst-of-the-worst material! Legendary! Just… bursts into flame! Wow! …This could be useful elsewhere…”
Jeh giggled. “All right, Margaret, think you can try that?”
“Yeah. I can.” Margaret set her jaw. “Just… apply lots of forces all over.”
“Fair warning, you’ll probably get destabilized by something, it took a lot of time for me to figure out how to apply different levels of force across the area to balance. So… once again, take it slow.”
Margaret did, focusing and applying a few dozen separate “force” spells on the disc. She was surprised to jump into the air almost immediately, but she quickly realized that casting more spells meant there was more total force on the disc. She managed to stay in control, however, but instead of increasing the force she just kept it the same.
She lifted into the air.
“Yeah!” Jeh shouted, her and the Sourdough twins bursting into applause. “You got it!”
“I… I’m floating!” Margaret said, beaming. The moment she said this, though, she shifted her weight, and the chair tilted to the side. She didn’t immediately fall, though, but went in a small arc before landing on the ground. Sideways, yes, but it wasn’t violent enough to be a crash. “…That was weird.”
“It was progress!” Jeh said, clapping her hands together. “We just need to train you until you have enough control to correct that. So… for now, let’s not worry about that and just practice getting into the air, all right?”
Miranda grinned. “Got it.”
“Look at you, celebrating!” Krays called. “All you did was not crash!”
“She has become the noncrasher!” Jeh called back. “Soon-to-be conqueror of the sky!”
“Conqueror of the sky…” Margaret looked up at the clouds and the sun, smiling. “I like the sound of that.”
~~~
SCIENCE SEGMENT
Blue has “invented new math.” To us, this math is known as calculus.
Except she hasn’t really invented it, per se. She has constructed mathematical relations based on observations. She has found a trick that allows her to calculate things, specifically things that will let her relate velocity, acceleration, and position together easier without doing a million billion calculations to find the next inch of the orbital trajectory. It’s just a bonus that it also happens to show her the slopes of lines and areas under curves!
She is correct to call this all a Conjecture, she has not tried to prove anything, nor has she looked for edge cases, or even been very thorough. It just works sometimes in her mind.
Of course, in our day and age, we know that calculus is extremely general and works in basically every case, even ones that look like, at first, there may be no answer. You have to use really unusual mathematical constructs, such as the function that is 1 on the real numbers and 0 on the irrational numbers, to get something where calculus truly can’t do anything. (That function is a pain to work with, let me tell you.)
But once the mathematicians Blue sends her notes to get around to proving and justifying everything… what exactly is calculus?
Ultimately, calculus is the study of the infinite, both the extremely large and the extremely tiny. With it, you can add together an infinite number of things in one step, you can find how steep a curvy line is at any point and, you can find the area of any shape you can describe mathematically. All of this emerges from the idea of “the smallest thing”—the infinitesimal. The infinitesimal is defined as a number that is greater than zero, but smaller than any other number. This is a bit of an odd idea—after all, if you think of a number, you can always think of a smaller number by just dividing it by two, right? Well, the concept of the infinitesimal is that, no matter how many times you divide by two, this thing is smaller than it. But it isn’t zero. One might think infinitesimals don’t exist, but according to our current understanding, the location of a point particle is described exactly by an infinitesimal. It has a location, but no width whatsoever. Granted, maybe our understanding is flawed and this isn’t truly an infinitesimal, but it sure looks like it.
With the infinitesimal and a bunch of somewhat annoyingly complicated math, we can arrive at the primary two things calculus does: find the slopes of lines and the area under curves. Without calculus, the best you can do to find the slope—that is, how steep a line is at any given point—is draw two points very close together, draw a line through them, and say that’s the slope. However, this only gives the slope to a certain accuracy, if you draw the points closer together, you can get a more accurate result. With the power of the infinitesimal, you can draw those points closer and closer and closer together until, essentially, they aren’t separated by anything but the infinitesimal, which is effectively not even a separation. With the infinitesimal, you never actually evaluate the points when they are on top of each other, rather you see what the value approaches as the two points get closer—in essence, the math is identifying a pattern and takes it to its natural conclusion to find what the answer really is, not an approximation. The act of finding a pattern as something “approaches” the infinitely small or large is called taking the limit, hence the title of this chapter. Using the limit and infinitesimal to find slope is a process known as taking the derivative, and once it’s properly formulated it can be used to find the slope of an entire function, creating a new function in the process. You can then do this again to find the derivative of the derivative. This is extremely useful since position, velocity, and acceleration are related in this way. Taking the derivative of position (that is, the function of an object that describes its location at any given time) produces the velocity (directional speed at any given time), and then taking the derivative of the velocity produces the acceleration. Very useful for orbits. Without calculus, you have to do this very slowly and with incremental steps, while with the proper formulation, you can just find the shape of an orbit all at once. (This does make it sound easier than it it, admittedly, as in orbital mechanics we regularly get annoying equations that don’t have simple solutions, but solving one of those is better than solving one every time you want to move an inch.)
The other half of calculus is finding areas under curves. This is done without calculus by drawing a bunch of rectangles under the curve and adding their area together. Naturally, rectangles will either miss some area or poke out from under the curve, thereby definitely being incorrect. However, we are free to decide how big these rectangles are, and smaller rectangles will get more accurate results, but there will be a lot more of them. It turns out that you can set the width of a rectangle to the infinitesimal and essentially do the exact same thing as before--identify the pattern that occurs as rectangles get smaller and smaller with the power of the limit, and then tie that all together into a single process called taking the integral.
Properly formulated, calculus does not require thousands of steps to be taken to find the answer. In fact, if a function is neat and smooth and forms like it have had derivatives and integrals solved before, you can often do the entire process in one step. Saves massive amounts of time.
Normally, one might wonder why integrals are useful for space travel, where are we adding areas together? Well, one thing that it can help with is finding the area swept out by planets in their orbits, which is directly related to how fast an object is going according to Kepler’s Second Law. Furthermore, the same process can be used to add together values along a path, such as calculating the distance crossed along an unusually-shaped line.
The most powerful tool, though, is that the integral is the opposite of the derivative. If you take an integral of a function, you can take the derivative of the integral to get back to the original function. Unfortunately, it’s not quite as simple as 1 + 1 - 1 = 1, there are exceptions and subtleties in this relation, so it can’t be applied blindly. But it is very powerful when used correctly—allowing the position to be found from velocity, or velocity from acceleration, with ease.
Calculus is an amazingly powerful tool. Once Blue and the mathematicians refine it, they will wonder how they ever managed to live with doing things the slow, imprecise way. Granted, calculus isn’t always easy, but even in situations where it can’t give you the answer to a complex relation in one step, the relations between various quantities in a system can be described by calculus, which makes things so much easier even if you do have to resort to adding up a nearly infinite list of things by hand.
Personally, I find calculus to be perhaps the most useful thing to come out of math. You can use it to calculate interest nearly instantly, for instance. I liked school as a kid but sometimes I often wondered why we were learning so many seemingly unrelated things in math. Well, when I got to calculus, all of it made sense—we were learning all those weird things because most of them are used commonly in calculus to solve involved problems.
Some people may (rightfully) object to the way I’ve painted calculus here. If you’re actually going into mathematics or the sciences, it’s worthwhile to actually dig into what it means and the motivations behind creating it. But I’m just trying to quantitatively describe what it actually is, not all the complex nuances behind it. Hopefully, I’ve piqued your interest enough to get you to look more into it! If you don’t already know calculus, I recommend starting with the concept of the limit, as it is generally where all the actual interesting work starts. You can’t figure out what the value is at infinity… so let’s just see what it is as we get closer and closer and then evaluate the pattern!
Though if you see a “delta-epsilon” proof in the first chapter/lecture/article, run. Try to understand qualitatively what’s being done before you dig into the specific proofs of the concepts.
…Perhaps I should have called this the math segment today…