Chapter 24: The Calculus of Change and Horace’s Revelation

Klein stepped into the teleportation array, clutching the manuscript.

A flash of silver light enveloped him, and his figure vanished from the spot.

The next moment, Klein materialized within the grand hall of a magnificent white edifice, the very headquarters of the Royal Magic Association and the nexus of the nation’s magical power.

The hall teemed with people, mages in robes of various hues hurrying to and fro. As they discerned the man emerging from the teleportation array, a momentary hush fell over the bustling space.

“It’s Master Klein.”

“Why is he here? Wasn’t it just a short while ago he visited?”

“Judging by his direction, he’s heading for the Theoretical Council.”

Whispers rippled softly behind Klein, yet no one dared approach him. Within the association, Klein’s name signified not only absolute power but also an aura of reclusiveness and severity.

Klein disregarded the myriad gazes, walking directly towards the service counter nestled in the deepest part of the hall.

Behind the counter sat a mage adorned with a three-ring emblem. Upon seeing Klein approach, he immediately rose, standing ramrod straight.

“Master Klein, greetings. How may I be of service?”

Klein placed the thick stack of parchment he held onto the obsidian counter.

“I wish to submit a thesis.”

The receptionist mage’s movements paused for a beat.

“Very well, Master. To which journal would you like to submit it? *Force Field Studies* or *Applications of Mana*?”

“The *Arcanum*,” Klein replied.

The receptionist’s hand trembled slightly. The Arcanum was the Magic Association’s most prestigious journal, publishing only theories profound enough to instigate paradigm shifts across the entire academic world.

The last paper to grace its pages had been five years prior, when a Seven-Ring Master proposed a completely new structural paradigm for the mental force model.

“I understand,” the receptionist said, not daring to be remiss. He produced a form.

“Please fill in the title and abstract of your thesis. According to protocol, we first need to complete preliminary registration, and then the Council will arrange for at least three Masters in the same field to review it…”

“There’s no need for such trouble,” Klein interrupted him. “Simply inform Master Horace.”

The receptionist’s pen tip gouged a long ink mark across the parchment.

Horace.

He was the Vice-President of the Theoretical Council, a Nine-Ring Archmage, and the undisputed paramount authority in mathematics throughout the entire kingdom.

“Master Klein, this… this is somewhat against regulations,” the receptionist stammered, steeling himself.

Klein fixed his gaze upon him.

Those blue eyes, devoid of any discernible emotion, nonetheless exerted immense pressure upon the receptionist. He found his breathing growing labored.

“I… I’ll go inform him immediately.” The receptionist practically fled, spinning around and rushing towards a small communication array behind the service counter.

Klein remained standing in place, waiting in silence.

A few minutes later, the receptionist returned, his face pale.

“Master Klein, Master Horace… requests your presence in his study.”

Under the watchful, astonished gazes of countless association members, Klein was led by another staff member down a long corridor, arriving before an unassuming wooden door.

The staff member knocked on the door.

“Enter,” an aged voice echoed from within.

Klein pushed the door open and stepped inside.

The room was sparsely decorated, with rows of bookshelves reaching to the ceiling, and the air hung heavy with the scent of old paper and ink. A white-haired, gaunt old man sat behind a massive desk, lifting his head to look at Klein.

“Klein, what brings you here?” Horace adjusted the monocle on his face. “I thought you’d never set foot in my dusty old room.”

“I’ve brought something,” Klein stated, approaching the desk and placing the meticulously organized manuscript upon it.

Horace glanced at the stack of parchment.

“A new study? What new contraption has your Force Field school cooked up this time? I’m warning you, if you’re planning to cram those combat-oriented formulas into my mathematics journal again, I suggest you save your breath.”

Klein paid no heed to his taunt.

Horace’s curiosity was piqued. He picked up the manuscript and looked at the title.

“*Principles of Calculus: A Universal Computational Method for Describing Change and Accumulation, and Its Applications in Magical Model Construction*.”

Horace’s brows furrowed.

“Calculus? What newfangled term is this?” he muttered, a hint of curiosity in his tone. “A universal computational method? Quite the bold claim.”

He turned to the first page.

“Introduction: The world is in motion, and all things are subject to change. How do we precisely describe change itself…”

Horace, flipping directly to the main body of the text, commented,

“I read your last published paper. While its format seemed a bit verbose to me, it’s quite suitable for younger mages. After reading the abstract, any competent mage would immediately grasp the paper’s subject matter, saving a considerable amount of time.”

A line of text caught Horace’s eye.

“Chapter One: Limits. We define that when a variable approaches a fixed value infinitely, and its corresponding function result also approaches a fixed value infinitely, then the latter is called the limit of the former…”

Horace’s expression shifted.

He set the manuscript down, adjusted his posture, then picked it up again, beginning to read meticulously from the very first word.

A profound silence descended upon the study.

Klein remained standing before the desk, utterly still.

Horace’s reading grew progressively slower, his fingers tracing the unfamiliar symbols as he silently mouthed words.

“‘Method of Limits’… infinitely approaching, yet never coinciding…”

“Derivative… the instantaneous rate of change of a curve at a certain point… the slope of a tangent line?”

The old man’s breathing became somewhat ragged.

He slowly turned to the later pages, where he found the applied examples Klein had included.

“Using the method of derivatives to calculate the peak energy release of an ‘Elemental Fireball’ spell at the moment of explosion?”

Horace let out a scoff.

“Nonsense! Klein, this is utter nonsense! This is mathematics! It’s the pure logic that studies the fundamental essence of the world! How can it be used to calculate a mere fireball?”

Though he cursed aloud, his hands did not cease their work. He took out a pen and, on a blank sheet of parchment, began to perform calculations according to the example’s steps.

His pen tip glided swiftly across the paper at first, but soon, his movements slowed.

A perfectly self-consistent logical system slowly unfolded beneath his pen.

This system possessed the ability to address problems—specifically, change—that all his previous mathematical tools had been incapable of handling.

Only the soft scratching of the pen against parchment filled the room.

After a long while, Horace stopped writing.

He gazed at the result he had calculated, then at the answer in the manuscript, finding them identical to the last digit.

He lifted his head, his eyes meeting Klein’s impassive face.

“This thing…” The old man’s voice was dry and trembling. “Who wrote this? Don’t tell me it was you, young man; your mathematical intuition wouldn’t allow you to discover something like this.”