Chapter 32: The Unification of Heaven and Earth

Silence descended once more upon the study.

The silvery protective barrier slowly dissipated.

Lia remained frozen in the posture of one who had just been held captive, a lingering sting on her neck, and a faint breeze where the assailant had vanished behind her.

She stiffly turned her head, only to find nothing there.

Her gaze then shifted to Horace.

The old man was unhurriedly retracting his finger, as if he had merely flicked away a bothersome fly.

Lia touched her neck.

Her mind finally began to process what had happened.

What had just transpired?

An assassin had burst in, holding a blade to her throat, and then this old man, Horace, had appeared.

Subsequently, the old man had declared his willingness to sacrifice her to capture all the ‘rats’.

And then, the cornered assassin had attacked her, only to be blocked by her necklace, before finally being obliterated by a single poke from the old man’s finger.

Not even a speck of dust remained.

Lia’s legs felt weak.

“You… you just said you were willing to bear the cost of losing me?” Her voice trembled, not entirely from fear, but from a surge of indignation.

Horace adjusted his monocle, casting a glance her way.

“An effective deterrent requires demonstrating a complete disregard for the cost.

This is the foundation of game theory.

Do you not even grasp that much?”

Lia almost choked on her breath.

‘Game theory? To a kidnapper threatening to kill their hostage?’

“If he had actually killed me?”

“He wouldn’t have.” Horace walked over to the desk, his gaze drawn to the parchment inscribed with the formula for universal gravitation.

“With you alive, he had a sliver of hope.

With you dead, his demise would have been certain.

Unless he was an idiot, he would never have made that choice.”

“But he did, in the end!” Lia’s voice rose.

“Which is why he died,” Horace stated, without lifting his head, his tone utterly matter-of-fact.

“A fool who cannot even calculate the optimal solution does not deserve to live.”

Lia was utterly speechless.

Watching the old man’s retreating back, she felt her definition of ‘madman’ needed a complete overhaul.

Horace picked up the parchment, his monocled eye scrutinizing it closely.

“F=G×(m1×m2)/r²… The magnitude of gravity is directly proportional to the product of the masses and inversely proportional to the square of the distance…”

He murmured to himself, completely engrossed, as if the recently deceased assassin had been nothing more than an irrelevant interlude.

“The unification of heaven and earth… described by a single formula…” His breathing grew somewhat ragged.

“That bastard Klein, to have hidden something like this.”

Lia stood rooted to the spot, watching the old man rave over her manuscript, unsure for a moment what to do.

‘Should I flee? Should I continue to argue with him?’

“This G…” Horace finally lifted his head, turning to Lia, his eyes no longer cold but burning with a fervent curiosity.

“The gravitational constant.

You said it needs to be determined through experimentation.

How does one measure it?”

Lia blinked, the abrupt shift in topic leaving her slightly bewildered.

“It requires… a torsion balance experiment,” she stammered in response.

“One must construct an extremely precise apparatus to measure the faint gravitational force between two objects of known mass, and then derive the value of G from that.”

“Torsion balance?” Horace frowned.

“Never heard of it.”

“It’s a device that utilizes the torsional torque of a metal wire to amplify and measure minute forces,” Lia explained.

Horace did not press for details on the apparatus; his mind was clearly operating on a higher plane.

“How do you prove that this formula is equivalent to the third law of planetary motion?”

“It requires calculus,” Lia replied, feeling a slight easing of her nerves.

Discussing academics was, after all, far safer than discussing life and death.

She walked to the other side of the desk, picking up a fresh piece of parchment and a quill.

“Planetary motion can be approximated as uniform circular motion.

Its ability to maintain its orbit is due to the sun’s gravity providing a centripetal force.”

“Centripetal force?” That was another new term.

“A force directed towards the center of a circle,” Lia quickly sketched a diagram on the paper.

“Any object undergoing circular motion requires such a force to maintain its trajectory.

For planets, this force is universal gravitation.”

She wrote on the paper: F_grav = F_centripetal.

Then, she began to write down the formulas on both sides of the equation.

“According to your law of universal gravitation, F_grav = GmM/r²,” Lia explained as she wrote, now fully immersed.

“And the magnitude of centripetal force, as can be derived, equals mv²/r.”

Horace’s eyes lit up.

He understood.

This was an entirely new way of thinking, breaking down motion and establishing equations through the balance of forces.

“v is the planet’s tangential velocity, equal to 2πr/T, where T is the orbital period,” Lia’s quill moved swiftly.

“Substituting this in…”

GmM/r² = m(2πr/T)²/r

“Canceling out the planet’s own mass, m, and simplifying…”

GM/r² = 4π²r/T²

“Rearranging, separating the terms with T from the terms with r.”

T² = (4π²/GM) × r³

Lia completed the final stroke and paused.

In the entire study, only Horace’s heavy breathing could be heard.

He stared intently at the final formula on the parchment.

T² = K × r³.

The square of the orbital period is directly proportional to the cube of the orbital radius.

This was precisely the mathematical expression of the third law of planetary motion.

Horace felt his scalp tingle.

A conjecture describing the fall of terrestrial objects, through a series of rigorous mathematical derivations, had ultimately provided a perfect explanation for the laws governing the stars.

Heaven and earth, in this single moment, were utterly unified by this formula, so concise it sent shivers down one’s spine.

“God… this is the language of the gods…” Horace’s voice trembled as he reached out, wanting to touch the parchment, yet paused mid-air, as if it were some sacrosanct relic.

Lia watched his reaction, feeling her lingering fear and anger inexplicably dissipate considerably.

‘Perhaps… being with a madman who understood her theories wasn’t so unbearable after all.’

***

Over the next three days, the study on the third floor of the mage tower transformed into a bizarre academic symposium.

Horace had completely forgotten his role as a bodyguard.

Like a child who had discovered a new toy, he clung to Lia all day, questioning her on everything from universal gravitation to calculus, from the definition of limits to the application of derivatives.

Lia was initially somewhat reserved, but she soon discovered that the old man was an utterly unrestrained madman when it came to academics.

He would argue with her until his face was red over the definition of a single symbol, and then pound the table in delight when she proposed an ingenious solution.

Together, they derived that tidal phenomena were caused by lunar gravity, and discussed how to calculate the gravitational field of irregular celestial bodies using integration.

Lia even brought up some of her ideas on optics, such as the corpuscular theory of light, to discuss with him.

Horace was utterly astounded.

He felt as if he wasn’t conversing with an eighteen-year-old apprentice, but rather with a prophet from a higher-dimensional civilization.

Every one of Lia’s ideas opened a door to a new world for him.

Lia, too, felt a thrill.

This feeling of meeting her intellectual match, of having a brilliant mind capable of following her every thought, was something she had never experienced in either of her two lives.

When Klein, having dealt with the monster rampage at the border, returned to the tower via a teleportation array, this was the scene that greeted him.

In his study, Grand Archmage Horace and his young apprentice, Lia, were sprawled amidst a chaotic pile of parchments, locked in a fierce argument over the proper limits of integration.

“It must be a closed interval here! Otherwise, the function won’t be continuous at the endpoints!”

“But there are no absolute endpoints in the physical model! An open interval is more realistic!”

Klein stood at the doorway, observing this tumultuous scene, then glanced at the scattered snack wrappers on the floor, and fell into a prolonged silence.

The atmosphere had certainly not been like this when he left.

Hearing the movement at the doorway, both disputants simultaneously turned their heads.

“Klein! You’ve returned just in time!” Horace exclaimed, like one who had spotted a rescuer, grabbing a parchment and rushing over.

“You be the judge! This little monster actually claims instantaneous velocity is the average value of a process—it’s an utter blasphemy against the definition of a derivative!”

Lia also stood up, hands on her hips.

“Then how do you explain my missile immobility paradox?

At any given instant, a missile is stationary, so how does it travel from point A to point B?”

Klein gazed at the two individuals before him, faces flushed, squabbling like fighting cocks, and felt his worldview once again profoundly shaken.

‘He had only been gone for a few days; what on earth had transpired?’

Horace shoved the parchment he held into Klein’s arms, then, without preamble, seized his shoulders and shook him vigorously.

“Where did you find this treasure?

No, she’s not a treasure; she’s an unmined gold mine!

I must go now; I need to rush back and jot down these ideas!

Inspiration, inspiration is overflowing!”

With that, without waiting for Klein’s reaction, Horace conjured a teleportation array on the spot and vanished instantly.

In the study, only the mess on the floor remained, along with Klein and Lia, exchanging bewildered glances.

Klein looked down at the parchment in his arms, covered in strange symbols, then up at Lia.

“He… didn’t do anything to you, did he?”

Lia blinked, then flashed a radiant smile.

“He treated me to the most famous Florin cakes in the capital for three days, and promised that all the Council’s archives would be freely accessible to me from now on.”