Chapter 57: The Calculus Conundrum

Inside the High Tower’s study.

Lia’s fingertips traced over the two replicated ancient documents.

Beneath the coarse touch of parchment lay the intricate thought processes of two geniuses, both mad and brilliant.

The Aether Wave Theory.

The Consciousness Field Hypothesis.

One sought to construct a unified model for the external world, while the other turned inward, delving into the very essence of spirit.

Lia read quickly.

More than merely studying, she was reviewing.

She was like an examiner from the future, grading papers written centuries ago.

“The premise of aether, indeed, unifies the manifestations of various magical schools to some extent, making it a rather clever theoretical model.”

Lia murmured to herself.

“However, the problem lies in him mistaking this model for reality. Should observational precision improve, or phenomena inexplicable by aether emerge, the entire theoretical framework would instantly crumble.”

As for Weber’s consciousness field…

“This fellow is quite ruthless.”

Lia’s fingertip tapped on the words ‘collective consciousness resonance’.

“He actually intended to link all human brains in parallel to create a Skynet? What kind of cyber-charlatan is this…”

She read with keen interest, occasionally jotting down a few lines of annotations on another sheet of paper, using simplified symbols and physics terminology from her previous life that Klein would not understand.

Unbeknownst to her, Klein had silently positioned himself behind her.

He observed the illegible scrawl that resembled celestial script, then glanced at Lia’s relaxed expression, which suggested she was perusing cabbages rather than ancient texts, and finally spoke.

“Can you understand it?”

“It’s alright.”

Lia did not lift her head.

“It’s quite interesting, especially this Weber; his approach is very bold. Unfortunately, his technical path was misguided, and his tools were far too rudimentary.

If only he had a foundation in quantum mechanics and information theory…”

Mid-sentence, she abruptly snapped her mouth shut.

‘Damn, I spoke too freely.’

As expected, Klein cast a probing gaze upon her, yet he did not press her on the unfamiliar terminology, choosing instead to change the subject.

“Your theoretical foundation is already quite solid. However, your horizons are still confined within this tower.”

Lia looked up, her eyes reflecting confusion.

“Your understanding of magic derives entirely from my notes and the dusty tomes of the library.”

Klein paused, then articulated his thoughts.

“You are like a swordsman who has studied countless sword manuals but has never once engaged in a real duel.

“Starting tomorrow, you will attend classes at the academy.”

“Classes?” Lia was stunned. “Learn what? To cast fireballs, which I already know?”

“Go and learn what you do not know.”

Klein walked to the window, gazing at the academy buildings bathed in sunlight in the distance.

“Observe what those of comparable strength to you are thinking, and what they are learning.  See how existing theories uphold the functioning of the entire magical world.”

Lia pouted slightly, then assented.

***

The very next day, Lia became a distinguished transfer student, having been directly placed by Klein into the academy’s highest-tier elite class.

Most of the students in the class were prodigies from noble families or mage lineages across various regions; their ages varied, yet each held themselves with immense arrogance, their spiritual energy so abundant it seemed poised to overflow at any moment.

Lia’s appearance was like a stone cast into a placid pond.

“Who is that transfer student? I’ve never seen them before.”

“I heard they’re Master Klein’s new apprentice… Really? That ‘Mediocrity Crusher’ is taking on an apprentice?”

“They look so tiny, are they even an adult?”

“Hmph, probably just a connection.”

Lia paid no mind to these whispers, finding a corner to sit in and opening her textbook.

Principles of Advanced Evocation.

On the first page was the classic model for the Blazing Shield.

A single glance was all it took for Lia to spot the energy circuit she had optimized and presented to Klein.

She closed the book, her face devoid of expression.

‘Alright, it seems this class can be used for meditation.’

However, after only a few days of such routine, Lia discovered an unexpected source of amusement.

Her knowledge base, far surpassing that of her peers, coupled with the occasional ‘ingenious insight’ that slipped through her fingers, quickly dispelled any notion of her being a mere ‘connection’.

Gradually, a few young individuals with genuine curiosity for knowledge began to gather around her.

Among them, the most lively was a red-haired youth named Finn.

Hailing from an old Earl’s family, he possessed exceptional talent and a flamboyant personality, always eager to prove himself an unparalleled genius before others.

Another was a girl named Ella, quiet by nature, always clutching a thick notebook, and prone to silently deducing and contemplating Lia’s various propositions on her own.

Before long, the small group spontaneously formed a sharing circle, convening once a week in a secluded corner of the library.

***

That day, Finn arrived last, clutching a thick stack of manuscripts with an air of mystery.

“You will absolutely never guess what I’ve brought today!”

He slapped the manuscripts heavily onto the table, producing a muffled thud.

His entire face glowed with exhilaration, pride practically oozing from his pores.

The others at the table were drawn to him.

“Finn, what is this? A new assignment from your mentor?” a boy asked curiously.

“No.”

Finn puffed out his chest, lowering his voice to a deep, resonant tone, as if imparting a divine revelation.

“This is… the key to a new era.”

He picked up the topmost sheet of manuscript, on which a title was transcribed in elegant handwriting.

—The Mysteries of Change: An Introduction to Calculus.

Lia’s hand, holding a cup of hot tea, froze.

She gazed at the familiar title, then at Finn’s face, which practically screamed ‘worship me now,’ and nearly spat out her mouthful of hot tea.

“Calculus?”

Ella pronounced the unfamiliar word curiously, “What is that?”

“It is a brand new language, one used to describe ‘change’!”

Finn’s voice was impassioned and fervent, as though he were preaching.

“My mentor, Professor Malcolm, stated that this is the foundational portion he transcribed from a groundbreaking thesis not yet cleared for public release!”

“The original thesis is profoundly esoteric, yet even this introductory segment alone is sufficient to revolutionize all our understanding of motion and change!”

He surveyed the group, highly pleased with the expressions of shock and bewilderment on their faces.

“Consider this: an object is in motion, its speed constantly changing. How do we determine its speed at a specific ‘instant’?”

“Take an extremely short interval of time, calculate the displacement during that interval, and then divide?” Ella immediately replied, recalling the approximate solution found in textbooks.

“No!”

Finn waved a hand, his posture resembling a deity guiding the lost.

“That’s merely average speed! It’s false! Not real! Calculus, however, allows us to grasp that infinitesimally small instant, one that doesn’t exist in reality, yet is truly present!”

“It can calculate the speed at that very instant!”

Lia quietly took a sip of her tea.

‘Good heavens, is he playing philosopher here?’

‘What does ‘an instant that doesn’t exist in reality, yet is truly present’ even mean?’

‘It’s just finding a derivative; he’s making it sound like Zen enlightenment.’

Seeing that he had successfully captivated the audience, Finn grew even more triumphant, picking up a sheet of manuscript.

“My mentor gave me a problem as a test of my learning.”

He cleared his throat and read aloud.

“A stone, enchanted with a ‘Feather Fall’ spell, drops from a high tower. The relationship between its descent distance h and time t can be described by the formula: h = 10t – t².

“Question: What is the instantaneous velocity of the stone at the instant t=3 seconds?”

As soon as this problem was posed, everyone furrowed their brows.

This was a classic example of variable speed motion, for which traditional methods could not yield an exact instantaneous velocity.

“I spent two days, and finally grasped the profound secrets of ‘Calculus’!”

Finn’s chin lifted even higher.

“Watch closely!”

On a blank piece of parchment, he commenced his demonstration.

Instead of directly calculating the derivative, he employed a cumbersome method of infinite approximation.

He first calculated the positions at t=3 and t=3+Δt, then determined the displacement difference, and finally divided it by the time difference Δt.

[ (10(3+Δt) – (3+Δt)²) – (10*3 – 3²) ] / Δt

The entire process filled a whole page of paper, and he even crossed out several lines due to calculation errors.

Lia watched from the side, a serene calm settling within her.

‘It’s merely a basic derivative problem: differentiate h(t) = 10t – t² to get v(t) = 10 – 2t, then substitute t=3, and the answer is 4.’

‘Something that could be solved with mental arithmetic, he had turned into an endeavor akin to proving Goldbach’s Conjecture.’

Finally, amidst everyone’s bated breath, Finn, drenched in sweat, arrived at the final formula:

(4Δt – Δt²) / Δt = 4 – Δt

“Do you see!”

Finn pointed excitedly at the result.

“When this time difference, Δt, approaches ‘nothing’ infinitely, it simply vanishes!”

“What remains is that sacred instant, its true velocity!”

With a trembling hand, he wrote a large ‘4’ on the parchment.

“Oh my goodness…”

“Finn, you are a genius!”

“Is this… the key to a new era?”

Exclamations of awe rose one after another, and Finn’s face flushed crimson with excitement and pride.

Basking in their admiring gazes, he felt as though he stood at the pinnacle of magical history.

“Um…”

A faint voice, ill-timed, broke the silence.

Everyone turned to look.

There, Lia had raised her hand, her face etched with a look of eager, yet perplexed, inquiry.

“Finn, there’s something I don’t quite understand.”

“Speak!” Finn gestured expansively, feeling at that moment as if he knew everything and understood all.

“Here,” Lia pointed to his final equation, “you said that when Δt infinitely approaches ‘nothing,’ it simply vanishes.

“But, in these calculations, Δt is in the denominator.

“If it were to equal ‘nothing,’ that is, zero…

“…wouldn’t this division become meaningless?”