Chapter 20: The Birth of Calculus: Lia’s Dilemma

Lia scurried back to her room, shutting the door with a decisive thud. Она leaned against the door, her heart still hammered wildly, refusing to settle.

The three laws had been laid bare: inertia, force, and interaction. These three cornerstones were potent enough to fundamentally reshape this world’s understanding of force.

But what then? The ultimate question remained: the heavens and the earth. Why did the fields in the heavens decay so sharply, while those on earth remained almost constant?

Lia, of course, knew the answer. Universal gravitation. That formula, so elegant it was a work of art: F = G * (m1*m2) / r^2.

The magnitude of the force was directly proportional to the product of the two objects’ masses and inversely proportional to the square of the distance between them.

This formula could perfectly unify everything. For planets, the distance ‘r’ was immense, meaning even a minuscule change in distance would lead to a drastic decay in gravitational pull, thereby explaining the “laws of stellar motion.”

For objects on earth, such as a stone falling from a hundred-foot tower, its change in distance relative to the entire world’s radius was negligible, allowing ‘r’ to be treated as a constant.

Consequently, gravity approximated a constant value, which in turn yielded that constant acceleration ‘a’. Heaven and earth were, at this moment, perfectly unified.

The question was, how did this formula come to be? It wasn’t simply conjured from thin air. It was derived through rigorous mathematical deduction from the three laws and the laws of stellar motion.

That derivation, however, required a crucial mathematical tool. Calculus.

Without the concept of limits, without derivatives and integrals, it would be utterly impossible to handle instantaneous rates of change like velocity and acceleration, let alone deduce the inverse square relationship between gravity and distance from the elliptical orbits of planets.

And this world had no calculus. Lia ran a frustrated hand through her hair. ‘Damn it, Newton was Newton not merely because he conceived the three laws and universal gravitation, but because he, alongside others, forged a mathematical tool called calculus to prove it all!’

And she? She was just a research grunt standing on the shoulders of giants, using ready-made tools. Now, the shoulders were gone.

She would have to become the giant herself.

‘Create one?’

The thought had barely surfaced when Lia herself was startled by it. For her to establish calculus from scratch in this world?

This was an order of magnitude more difficult than writing a few papers on the laws of motion. But did she have any other choice?

‘She couldn’t very well tell Klein that all this knowledge came from her previous life, could she?’

Lia straightened, walking to her desk. There was no choice in the matter; she simply had to brace herself and proceed. To do so, she would need some paper.

Lia opened her door, just in time to see Adèle ascending the stairs, a tray of pastries in her hands.

“Lia? Are you alright?” Adèle asked, her voice laced with concern upon seeing her. “You ran so fast just now, I thought you’d argued with the mentor.”

“No, Senior Sister Adèle,” Lia managed a strained smile. “I merely… had some sudden new ideas that I needed to jot down.”

“Is it about that paper?” Adèle’s eyes sparkled. “I heard Martin say that Master Valerius of the Sorcery School… his mental model collapsed because of your paper.”

Lia’s smile stiffened.

“Huh?”

“It’s true! They said Master Valerius’s head just went ‘bang,’ and then it was gone.” Adèle elaborated, gesturing with her hand, her face a mix of horror and a hint of inexplicable excitement.

A chill snaked down Lia’s spine. ‘So her paper could genuinely kill people.’

“Um… Senior Sister, could I ask you for some paper? The practice parchment, I need a lot of it.” Lia quickly changed the subject.

“Of course, how much do you need?”

“A hundred sheets to start with.”

Adèle paused, her movements halting slightly. ‘A hundred sheets?’ ‘That’s enough for an apprentice to copy half a potion grimoire.’

“Alright, wait a moment.” Adèle didn’t press further. After setting down the pastries in Klein’s room, she went downstairs and soon returned, cradling a large stack of cut parchment.

“Thank you, Senior Sister.”

“You’re welcome,” Adèle replied, placing the paper on Lia’s desk. “The mentor said that any research needs you have should be prioritized. If it’s not enough, just come find me anytime.”

With the door shut, she was once again alone in the room.

Lia spread out the parchment, picked up a quill, but did not immediately dip it in ink. Closing her eyes, she began a frantic search through her memories of her previous life.

‘How could she explain calculus to someone completely devoid of the concept of limits?’

‘She couldn’t just jump straight into definitions – delta, epsilon, that kind of language would deter many immediately.’

‘She had to start with the most intuitive, most fundamental problems.’

Opening her eyes, she penned a question at the top of the first parchment sheet.

“How do we describe change?”

She thought of an excellent example.

“An object falls from a tall tower, its speed constantly changing. We can determine its average speed between the first and second seconds, but how do we ascertain its instantaneous speed at the very end of the first second?”

This was an unavoidable question. Lia drew a curve on the paper, representing the object’s trajectory as its distance changed over time. She marked two points on the line, Point A and Point B, corresponding to the first and second seconds, respectively. She connected A and B with a straight line.

“The slope of this line represents the average speed from the first to the second second,” Lia annotated beside the diagram.

“Now, we want to know the speed at the exact instant of Point A.”

“We can make Point B approach Point A along the curve. For instance, by taking its position at 1.5 seconds.”

She drew a new Point B on the diagram, along with a new connecting line.

“The slope of this new line represents the average speed from the first to 1.5 seconds. This value is closer to the true speed at Point A than the previous one.”

“What if we continue to make Point B approach Point A? 1.1 seconds? 1.01 seconds? 1.001 seconds?”

Lia continuously drew new connecting lines on the diagram; these lines rotated around Point A, their slopes constantly changing.

“When Point B approaches Point A infinitely, infinitely close, yet without coinciding with Point A, this connecting line will ultimately become a tangent line, touching the curve at only one point: Point A.”

She drew that tangent line with a decisive stroke of her pen.

“The slope of this tangent line represents the object’s instantaneous speed at Point A, precisely at the end of the first second.”

“I shall call this process, this method of infinitely approaching a target to observe its ultimate trend, the ‘Method of Extremes’.”

Lia wrote down the three characters for ‘Method of Extremes’. She decided against using the word ‘limit’ for now, opting instead for a term that sounded more like a method.

Next came integration. If the Method of Extremes involved infinitely dividing things, then integration was about reassembling those infinitely small parts. Lia took a fresh sheet of parchment. She drew an irregular shape.

“How do we calculate the area of this shape?”

Employing the method of ancient Greek mathematicians, she drew numerous small rectangles within the shape.

“We can fill this shape with many rectangles. Summing the areas of all these rectangles will give us an approximate area.”

“The more rectangles we use, and the narrower they are, the more precise this approximation becomes.”

“Now, we employ the Method of Extremes once more. If we allow the width of each rectangle to shrink infinitely, approaching zero, while simultaneously increasing the number of rectangles infinitely…”

“Then, the sum of the areas of all these infinitely thin rectangles will infinitely approach a definite value.”

“This value is the exact area of this irregular shape.”

“I shall call this method, of adding an infinite number of infinitely small quantities, the ‘Method of Summation’.”

The Method of Extremes, the Method of Summation. One for division, one for accumulation. These were the two most fundamental cornerstones of calculus.