The carriage rattled along as it made its way down the road.
Like Lorenz's wagon, this carriage was equipped with leaf spring suspension, which just barely kept Jin’s motion sickness at bay.
The road was composed of earth mixed with gravel. Despite that, no deep ruts had formed, and the carriage made steady progress.
Since only Jin and Kachea were inside, Jin decided to take the opportunity to ask her a few questions.
"I might ask some strange things, but..."
Jin started with a disclaimer, and Kachea nodded, though she looked a bit cautious.
"The moon is round, right?"
"...Huh?"
Kachea’s eyes went wide as she stared at him.
"Have you ever seen a square moon...?"
"Eh? Ah, no, that's not what I meant... If you haven't, then it's fine."
It seemed the moon here did not wax or wane after all, Jin realized. He hadn't been observing it every single night, but for the past half-month, it had remained perfectly round. Still, he had asked just in case there was a lunar cycle that lasted an entire year.
However, Kachea’s reaction made it clear she had never seen a "waning" moon.
"Jin-san...?"
For Kachea, it was a question that seemed to doubt common sense, and the look in her eyes turned slightly cold.
"Hmm... is the moon actually called 'Uni'?"
"...Hah...? The moon is the moon, isn't it? Is there some other name for it?"
By now, her casual demeanor had vanished, replaced by the sharp, focused tone she used when she was concentrating.
"There must be. Like a 'satellite'..."
"What is a 'satellite'?"
"...So we're starting from there, huh..."
Jin had begun to suspect that while the world’s technology was decent, their theoretical knowledge was surprisingly shallow.
"That's why I said I might ask some strange things."
"...Certainly, you did....Is your memory confused?"
Now Kachea was looking at him with worried eyes. He felt somewhat pathetic for it, but he decided to press on with his questions.
"Changing the subject—how do people here learn reading, writing, and arithmetic?"
"In my case... my parents taught me to read and write. I learned arithmetic naturally by helping out at my aunt's shop."
"By arithmetic, do you mean addition, subtraction, multiplication, and division?"
Again, she shot back a question of her own.
"What else is there?"
"Like squaring numbers..."
"What is... that?"
Since Kachea showed a flicker of interest, and since Jin was bored anyway, he decided to explain.
"It's a term used when you multiply a number by itself. For example, two times two is 'two squared.'"
"Let's see... so then, three times three is three squared, and ten times ten is ten squared—is that right?"
"Exactly."
"...What's the point of that? Is saying 'three times three' or 'ten times ten' not good enough?"
"Hmm..."
Jin found himself at a loss. As he wasn't a mathematics teacher, he didn't really know the best way to explain the concept. It was even more difficult because he wasn't teaching her in progressive steps.
"...This is a bit of a tangent, but do you know what area is?"
"The size of a piece of land, right?"
"Well, it's not limited to land....Anyway, how do you find the area?"
"For a rectangle, it's the vertical length times the horizontal length."
"Right. And in cases where the vertical and horizontal lengths are the same, you can use squaring."
"I sort of get it, but I’m not entirely satisfied."
"This is tricky..."
When the topic turned to numbers, Kachea became quite stubborn. Jin decided to try a different angle.
"Then, what would the length of one side be for a square with an area of two square meters?"
"Eh... let's see."
Kachea was taken aback.
"If one side is 1, the area is 1. If it’s 2, the area is 4. If it's 1.5, then it's 2.25... If it's 1.4, then it's 1.96... If it's 1.41, then... let's see... 1.9881. If it's 1.42... then it's 2.0164..."
Jin was amazed by her mental calculation skills, but he cut in.
"Stop....You know how addition and subtraction are pairs, right?"
"Eh, yes. Multiplication and division are also pairs....Based on the way you're saying that, is there something that pairs with squaring?"
Kachea showed sharp insight.
"Exactly. That's called the 'square root.'"
"A square root?"
"Yeah. I think it's because another name for squaring is 'to square' something."
"I see... The square root is what's useful for finding the length of a side from the area. And the thing that pairs with it is squaring."
"That's it."
"This is interesting! Jin-san, you know some amazing things."
"I don't think... it's that much."
"Don't be so modest....Is there anything else?"
As he thought, Kachea had a personality that was naturally drawn to mathematics. Jin wondered what would happen if she were taught the subject properly.
However, there was nothing he could do about that now. Instead, he tried another math-related question that came to mind.
"Do you know how to find the area of a circle?"
"A circle...? You're asking something difficult. Radius times radius... ah, if you multiply the square of the radius by 3, you can find the approximate area... Do you know a more accurate way to find it, Jin-san?"
Jin gave a wry smile.
"Well, I don't know if it's 'accurate,' but using 3.14 or 3.14159 instead of 3 makes it more precise."
"3.14159...? Just how many digits are there really?"
"It's said to be infinite."
"Infinite!?...How can anyone know it's infinite? If you're finding it by calculating... even if it continues for ten thousand digits, it might end at the ten thousand and first digit, right?"
Jin was inwardly amazed. Kachea truly had a mind for mathematics.
"I'm not an expert, so I can't explain it that far. By the way, that value is called 'pi.'"
"Pi..."
Kachea seemed deeply moved.
"Jin-san, you really are mysterious. You don't know things that everyone knows, yet you know things that nobody knows."
"Maybe it really is memory confusion?"
"I wonder. I wouldn't know."
Staring at Jin with sparkling eyes, she leaned in.
"Tell me more!"
The roles had been reversed. While inwardly smiling at the situation, Jin thought for a moment about what to talk about next. Given the flow of the conversation, he decided on the "Theorem of Three Squares."
It was also called the "Pythagorean Theorem," but since he didn't think the name "Pythagoras" would be known in this world, he decided to stick with the "Theorem of Three Squares."
"There's something called the 'Theorem of Three Squares.'"
"The Theorem of Three Squares?"
"Yeah. Let's see, do you know what a right angle is?"
"Yes. That's 90 degrees, isn't it?"
Jin knew that in his original world, Ars, the metric system and other units had come into use because the magus Shuuki Tsetse had spread Earth's mathematics in the past.
Had something similar happened in this world as well? The thought flickered in the back of his mind.
However, while they were having this conversation, one of the guards was eavesdropping from outside the carriage.