Chapter 33: The Unification of Heaven and Earth

Klein remained silent, his blue eyes devoid of emotion as he regarded Lia.

Lia felt a shiver run down her spine under his gaze, and the smile on her face faltered slightly.

“Alright, there was a small mishap in the middle,” she cleared her throat. “An assassin burst in, named Silent Shadow, according to Horace.”

She pointed to her neck.

“They held a knife to my throat, intending to take me hostage.”

Klein’s gaze followed her finger, lingering on the pale skin of her neck, where no mark remained.

“Then Master Horace arrived. He told the assassin that my life didn’t matter, and that capturing all the ‘rats’ was the priority.” Lia spread her hands. “The assassin seemed to believe him, and in desperation, launched a psychic attack against me.”

She patted the pendant on her chest.

“This thing Master Horace gave me works wonders; it blocked their dying struggle. Then Master Horace dealt with them.”

“Oh, and the assassin’s knife didn’t actually touch me. I could feel the pressure of the blade, but it was as if there was a slippery layer between my skin and the knife. They didn’t even notice it themselves.”

Klein finally reacted. He stepped forward, picked up the silver necklace, and his fingertips brushed the crystal pendant.

The pendant’s surface shimmered with light, but its internal energy fluctuations were noticeably weaker than when he had first given it to her.

“I understand.”

He released his grip and turned to exit the study.

Lia watched his retreating figure, feeling that the temperature of his final words was considerably colder than usual.

***

As Klein descended the stairs, Adèle stood in the ground floor hall, anxiously wringing her hands.

Upon seeing Klein, she seemed to find her anchor, quickly stepping forward to meet him.

“Master.”

“Speak,” Klein uttered a single word.

Adèle was compelled to take half a step back by the invisible pressure emanating from him.

“Three days ago, a powerful magical fluctuation suddenly erupted from Miss Lia’s room. Immediately after, Master Horace appeared directly in the tower via a teleportation array.”

Adèle’s voice trembled uncontrollably.

“Master Horace said that ‘rats’ had infiltrated. He sealed the entire tower, then proceeded to the third floor. After that, I knew nothing; Master didn’t allow me to go up.”

She paused, then added.

“When Master Horace appeared, his expression was grim. He declared that anyone daring to act within your tower was courting death.”

Having heard her account, Klein asked no further questions, heading directly towards his alchemy laboratory.

The magical lights in the hall flickered erratically behind him, and the air temperature plummeted, causing Adèle to shiver involuntarily.

***

Lia paid no heed to the commotion downstairs.

Three days of brainstorming with Horace had filled her mind with new ideas and logic chains awaiting organization.

The assassination’s greatest revelation was that her theories now possessed a power so profound that the old guard would stop at nothing to eradicate it.

This realization filled her with both peril and exhilaration.

She had to accelerate her pace, to present irrefutable evidence that would firmly nail the world’s scientific framework onto the foundation she was constructing.

She settled back at her desk and unfurled a fresh sheet of parchment.

She was about to formally compose the treatise that would irrevocably alter the world’s paradigm.

‘The Principle of the Unification of Celestial and Terrestrial Motion’

She began by writing the title.

The first step was definition.

The world possessed powerful concepts, yet they were exceedingly vague. She needed to provide precise, quantifiable definitions.

She inscribed the title of the first chapter: ‘Definitions of Fundamental Physical Quantities’.

“I. Mass (symbol m): An inherent property of an object. It characterizes an object’s tendency to maintain its original state of motion.

The greater an object’s mass, the more difficult it is to alter its state of motion. This property does not change with the object’s position.

It is important to note that mass and weight are two distinct concepts. Weight is the force exerted on an object due to gravity, and it varies with changes in gravitational field strength.”

“II. Force (symbol F): The cause of a change in an object’s state of motion. In the absence of an external force, an object will remain at rest or continue in uniform linear motion.

The effect of a force depends not only on its magnitude but also on its direction.”

Having defined these two most fundamental concepts, Lia proceeded to introduce the next core idea.

“III. Centripetal Force (symbol F_n): The force that compels an object to move in a circular path. This force always points towards the center of the circle.

Without this force, the object would move tangentially in the direction of its current velocity. For an object of mass m moving at a velocity v in a circle of radius r, the required centripetal force is: F_n = m × v² / r”

She did not merely present the formula; on a scrap piece of paper, she performed a brief derivation using the principles of calculus and geometric figures.

Approximating a small arc of a circle as a straight line, she calculated the change in the velocity vector, ultimately arriving at the centripetal acceleration a = v²/r, which, when multiplied by mass m, yielded the centripetal force formula.

She simplified this derivation process, writing it as an annotation to the formula on the parchment.

The foundational tools were now prepared.

Next came Chapter Two: ‘From the Laws of Celestial Motion to Universal Gravitation’.

“According to the ‘Records of Celestial Trajectory Observation,’ we know that the motion of stars follows these laws.

Among them, the third law states: The square of a planet’s orbital period is directly proportional to the cube of the semi-major axis of its orbit.”

T² ∝ r³

“To simplify the derivation, we will first approximate planetary orbits as circular.

Planets orbit the sun precisely because the sun exerts a gravitational force upon them, which provides the centripetal force required for their circular motion.”

“From this, we can establish the equality:”

F_gravity = F_centripetal

“Substituting the centripetal force formula, we obtain:”

F_gravity = m_planet × v² / r

“The orbital velocity v of a planet is equal to its orbital circumference 2πr divided by its orbital period T. That is, v = 2πr / T. Substituting this into the above equation:”

F_gravity = m_planet × (2πr / T)² / r = (4π² × m_planet × r) / T²

“At this point, we introduce the third law of celestial motion, T² = k × r³ (where k is a constant of proportionality). Substituting this relationship:”

F_gravity = (4π² × m_planet × r) / (k × r³)

“Simplifying, we get:”

F_gravity = (4π² / k) × (m_planet / r²)

“Observing this equation, (4π² / k) is a constant that is the same for all planets orbiting the sun.

This equation clearly demonstrates that the magnitude of the sun’s gravitational pull on a planet is directly proportional to the planet’s own mass and inversely proportional to the square of the distance between them.”

Lia paused at this juncture.

This was the derivation of universal gravitation from Kepler’s Third Law. While logically sound, it was not yet complete. She needed to introduce a more fundamental principle to perfect the formula.

“According to the principle of action and reaction (where if one object exerts a force on a second object, the second object necessarily exerts an equal and opposite force on the first), the force with which a planet attracts the sun must be equal in magnitude to the force with which the sun attracts the planet.

Therefore, the magnitude of the gravitational force must also be directly proportional to the mass of the sun.”

“Synthesizing the above deductions, we can conclude: The gravitational force between two objects is directly proportional to the product of their masses and inversely proportional to the square of the distance between them.”

She picked up her pen and solemnly inscribed the ultimate formula onto the parchment.

F = G × (m₁ × m₂) / r²

“Here, m₁ and m₂ represent the masses of the two objects, and r is the distance between them. G is a universal constant, which I term the gravitational constant. Its numerical value must be determined through precise physical experiments.”

She let out a long breath.

The core of the theory was now complete.

Next, she intended to prove that this formula, derived from celestial motion, was equally applicable to objects on the ground.

Chapter Three: ‘The Free-Fall Motion of Terrestrial Objects’

“Objects on the ground fall because they are subjected to the gravitational force from the entire Earth. According to the formula we derived:”

F_Earth = G × (m_Earth × m_object) / r_Earth²

“Here, m_Earth is the mass of the entire Earth, and r_Earth is the Earth’s radius. Since, for objects near the surface, changes in height are far less significant than the Earth’s radius, we can treat r_Earth as a constant. Likewise, m_Earth and G are also constants.”

“Therefore, we can combine G × m_Earth / r_Earth² into a new constant, denoted as g.”

g = G × m_Earth / r_Earth²

“Consequently, the gravitational force experienced by terrestrial objects can be simplified to:”

F_Earth = m_object × g

“This is the weight of an object. This formula demonstrates that, neglecting air resistance, all objects falling near the Earth’s surface possess the same acceleration, g. This aligns perfectly with experimental observations.”

Heaven and Earth were thus thoroughly unified within a closed logical loop.