"Diana? Can I come in?"

"Hm? Are you done already?"

"Already? Diana it's seven!"

Diana looked down at her tablet. 

"Oh . . . sorry guys, I didn't realize."

"Think nothing of it Lady Diana. As confusing as all this may be, I find it infinitely more entertaining than my normal activities."

"While I cannot quite say the same, I appreciate the opportunity to expand my skill set."

Sanna and Wall were the first to get up from their seats. They had been practicing English, the usual entourage of Wall, Sanna, Len, Titanyana, and Rize joined today by Nemo, Zhoie, and Kerefel. 

"Sanna, Wall, English?"

"Ah, sorry. Um. Thank you for the lessin?"

"I thank you for teeching us this day."

Diana nodded. To someone like her, inexperienced in the art of teaching, this was good progress. They weren't quite at the point of pen on paper, but they had committed to memory a few common phrases. Please and thank you, good morning and other greetings, farewells, the proper way to phrase a question (she permitted split to be used at that time so she knew what they were asking for), and the basic numbers (one through ten).

Diana only spoke English during their lessons. Even when answering their questions, only English. If possible she would have Arc provide a graphic to go along with her words so that they could begin to attach word to concept, however the slower pace at which she needed to speak did throw her off a bit. It wasn't her normal cadence of speech, and she often found herself stumbling over words she would normally say much faster.

"Good. You two are dismissed for today." A bit of emphasis on 'today' to remind Sanna of the proper way to say it. 'this day' may be grammatically correct, but he would not understand why.

Diana would not risk an accent, at least not this early on. Wall and Sanna might not require such attention to detail, but Len, Titanyana, and Nemo were going to be people she relied upon to teach in her stead. If they were to assume that role, they needed to know how to speak in the most correct way.

One by one, everyone in the room thanked her for the lesson and went along with their night. All except for Rize, Len, Titanyana, and Nemo. They were to receive 'extra' lessons from Donovan, who was here to tag in on the education front. She needed dinner, and she planned to eat it while conversing with Zhoie. 

"Alright, I'm going to eat with Zhoie. How was dinner?"

"Delicious."

"What is it?"

"Delicious." He didn't know what to call it.

"I can't wait." Donovan's knowledge of dish names was not the most extensive, Diana knew prodding would be an exercise in futility. "Well, I'm off. See you in bed!"

"Love you." Diana reciprocated Donovan's kiss with vigor before leaving him, secretly monitoring Titanyana's reaction as she did so.

A little bit of an ear flick was all she needed to know Titanyana could hear them. Whether or not Titanyana was genuinely reacting in embarrassment or had just happened to have an itchy ear at that moment was technically a mystery, but was Diana's intuition ever wrong? Yes, actually. Yes it was. In fact it had been wrong about Titanyana before, during the party.

She felt certain she was correct now of course, Diana's close observations ever since had pretty much guaranteed her suspicions. Titanyana was absolutely smitten with Donovan, and Diana needed nothing more than the change in her tail's behavior as proof. Right now her tail was slowly swaying back and forth, but when he spoke to her it would almost seem to stiffen out. Diana didn't know what thoughts or instincts triggered that specific response, but it was probably along the lines of nervousness or a desire to hide her feelings.

Diana had judged Titanyana to be a 'good girl', someone who would not unceremoniously make advances on Donovan. Diana wouldn't mind if Titanyana did make a move, even now there was a strong urge to start something between the two of them, but the more she observed the more Diana realized that would be a bad idea. Titanyana needed time, at least for now, to sort out her feelings and how she wanted to go about courting Donovan.

- - - - -

Donovan took a few seconds to get caught up on what Diana had been teaching them. It wasn't particularly important, Donovan was not here to teach them English, but he would like to know their progress. He needed to know how much they would be able to understand from the name of concepts alone.

"So . . . where did we leave off last time? Variables?" Donovan knew where they had stopped last time. He just wanted to see how much they remembered. Rize's hand shot up. "Yes, Rize?"

"I think you called them 'graphs'?"

"Oh?" Donovan feigned a bit of surprise to boost their confidence in the progress they had made. He wasn't a particularly good actor, but this was easy enough. "Ah, yes. Here we are. Graphs. Before we continue that lesson I would like to review what we went over before that, algebraic expressions and variables raised to a power. Do you recall the specific names attached to the first three powers? Nemo, what is a variable raised to the first power called?"

"Linear."

"Good. Do you remember what that word means at it's base?"

"I do not."

"It means 'of or relating to the properties of a straight line' or 'directly proportional', literally spelled line-ar. Technically, and I expect you to remember this, a variable raised to the power of zero is also linear, however that is just a number. A number on it's own in an algebraic equation is called a-?"

""Constant.""

"Great work. Len, what is a variable raised to the power of two called?"

"Uh, um. Quartic?"

"Close, that's when a variable is raised to the fourth power. Titanyana?"

"Squared?"

"That's one of the two, Quadratic is the one I am looking for. You'll pick up on when you should be using quadratic or squared with time, I won't pester you about it until then. Do you know why we called it 'squared'?"

"Is it because it's like the area of a square?"

Donovan clapped his hands. They hadn't covered much geometry yet, that was planned to happen after the graph segment during their first dive into calculus. It was possible Diana had spoiled something about it to her, but Donovan felt it more likely that she had just looked at the word.

"Yes, that is correct! It is to do with the area of a square! We'll get to this later, but the area of a square is defined by it's height times its base, both of which are the same. If we assign a variable 'x' as the length of the side, the area must be?"

Titanyana furrowed her brow just a little while the rest of them scratched something down on their chalkboards.

"x squared?"

"And if we double x?"

"Four times x squared?"

"Great work Titanyana, but I do want you to practice writing that down instead of doing it in your head. Believe me, it's going to get complicated and you won't be able to keep track."

"Okay." Titanyana began to jot it down, the tip of her tail twitching happily from the praise.

"Good. Now back to your Rize. What is a polynomial equation with a variable raised to the third power called?"

"A cube-ic?"

"Cubic, that is correct. What is the correlation to the cube?"

"The area?" Rize gave a wrong answer, the rest of the class not having confident looks.

"No. It's the volume, or the space the cube takes up." Some 'oh'-s and 'ah'-s indicated they at least understood the reasoning behind it. "Alright, now that we have gone over the relationships between the power of a variable and their real life counterparts, I want to introduce you to a rather abstract concept known as a 'unit'."

Donovan turned his back to them and got to work drawing a straight line on the board, eyeballing distances to put tick marks.

"This line is is roughly one meter long. A 'meter' is the base unit designed to measure length. It has larger and smaller denominations that are all related by ten to the power of a multiple of three, though there are a few notable exceptions. Tell me, what do you think the unit designed to measure the area of something is called?" Donovan looked out on their clueless faces. With the exception of Nemo, someone who had some knowledge of the Holifanian Theocracy, everyone here probably had an inconsistent unit for measuring distance. It could be an arm, a leg, a length of time it takes to walk, or something along those lines "Anybody?"

"I-is it square meters?"

"Correct! Good job, Len. Yes, it is possible to treat these units as variables. Length is a base unit, something that cannot be decomposed any further. Area is length squared, and Volume is length cubed. What makes units so special is that they can be combined with other values that have assigned units, the final combination of them being the description of what it is you have."

". . . What does that mean?" Len did not quite seem to get it.

"It means that I can calculate a value of some property in relation to the rest of the universe. Let me give you an example. The unit for mass is known as a 'gram', however for the purposes of calculation the kilogram, which is one thousand grams is used. The density of a substance is the mass of the object divided by its volume, mass over length cubed. Now let's play around with that.

Let's assume that I have somehow determined the density of some material and I need to know its mass in order to be sure I have enough of it. If I can find the volume, I can determine the mass by multiplying those two values. The length cubed in the denominator of the density will cancel out the length cubed in the numerator and the only unit I am left with is mass. Therefore, the density of some object multiplied by its volume is equal to its mass.

Alternatively, I can find the volume of an object so long as I know its mass and density. Same for the density when I know mass and volume."

Confused stares and tilted heads greeted his overly enthusiastic explanation, the only exception being Nemo who was staring blankly at her tablet. Did none of them understand what density was?

"Okay, how about if I put it this way. Titanyana, your sword is made of steel, right?" Titanyana nodded. "Depending on the type of steel used, the density can usually range anywhere from 7500 to 8000 kilograms per cubic meter. Lets just say that it's 7500 for now. From when I grabbed it, I'd say your sword is about a kilogram. The volume of your sword, the amount of space it takes up, is equivalent to the mass over the density, see? Which means that your sword is about . . . that's a small number in cubic meters so I'll use centimeters . . . about 130 cubic centimeters, give or take."

Donovan put his pinky up on the board and traced the width of his nail. That was about a centimeter wasn't it?

"This tiny cube, sorry if it doesn't pop out of the board, is one cubic centimeter. Your sword is about 130 of these."

The four of them scribbled down a cube of their own. Nothing on there was coherent English but they knew numbers and letters. Whatever they couldn't write out for themselves as a reminder could be written as a symbol of some sort.

"So . . . so units are a measure of 'things'?" Rize wanted to clarify for himself.

"Yes."

"Does that mean Split has a unit?"

Donovan froze. Did split have units? He hadn't heard Arc talk about it before, but he repeated the question to Arc. The answer made him want to bang his head on the table.

"Not yet. We don't exactly have enough experience with it to start calculating anything, so for now we have decided against trying to label anything. Once we have experimented enough with it to find some sort of relationship it shares with the physical plane, we will set up the base unit."

This was a lie, they did actually have a unit for split, it was just so ridiculously small that it was irrelevant now. It was defined as the amount of split contained within one cubic meter of atmosphere under standard atmospheric conditions, a value so low that it effectively registered as zero. Even in the vacuum of space, Arc noted that even the weakest regions seemed to have concentrations millions of time greater, and that the concentration varied wildly over comparatively short distances. To Donovan this was incredibly embarrassing. That wasn't a unit, it was wrong. They would need to find some relationship between Split and another array of units, probably something to do with the properties of water or carbon.

"Anyways, we can get back to units later. I just wanted to plant the idea into your head so that you aren't quite as confused when we talk about them again. Now onto drawing graphs. You all remember the other names for 'x' and 'y' right?"

cakeonfrosting

My hatred for thermodynamics (as a class not a concept) now exceeds my hatred of having to wake up in the morning. The only reason for that is because I no longer have to wake up for thermodynamics, at least not for a few months. Finals are over, and I am getting ready to head back home. You might not have a chapter for a few days while I get settled, you might get more than usual as I desperately try to destress. Who knows? Certainly not me.

Anyways, units. I know this seems like a little bit of a useless chapter but I promise it is one of the more important ones long term (at least for me). This concepts involved in the chapter combined with the previous chapter which demonstrated Donovan's gradually increasing capability to use split will serve as the base from which Donovan and Arc learn to use (and abuse) the qualities and effects inherent in split.

This is a warning for you rule-set lovers, some of the ways split will be used are contradictory at times. THIS IS INTENTIONAL. Split is not magic, but it isn't science either. Science can be used to measure it to some degree and certain processes will always yield a single consistent result, but there are also places where split cannot be counted on to repeat a phenomenon. In many ways it is the ultimate agent of randomness and unpredictability, in others it follows a strict set of rules that cannot be broken.

We already have one example of this, gods. The conditions for a god's creation and destruction are fairly concrete (though they do shift around a little bit with regards to some factors) however the limits imposed on the gods vary drastically. So far there have been three directly introduced and one more mentioned in passing.

The Navigator, The Arboreal Maiden and a mystery god who has not yet been unveiled but has been directly introduced to Donovan and Diana all have pretty vague limits put on them. You don't know what two of them are the god of yet, but the Navigator is just good at navigating, except he isn't only good at navigating. There are other things like his ability with split sorcery that has not yet been expanded upon but are pretty vague. 'Is he good with split sorcery because of his godhood, or is that the result of his own efforts?' 'How can the god of navigation be destroyed if being the god of navigation makes you the best navigator?'

Then of course is the Hunter. I won't get into too many details, but the Hunter doesn't so much control his godhood as his godhood controls him. He is not bound by rules to the same degree as other gods, as what defines 'Hunting' is genuinely a matter of perspective. The quarry of a hunt doesn't have to be an animal, it doesn't even have to be alive, which makes defining who the best hunter is a really vague process. It follows that the abilities bestowed upon him are similarly vague and variable depending on the target of his hunt.