Chapter 52: The Dynamic Measurement

On this particular day, a letter arrived uninvited. It wasn’t delivered by a servant; instead, it flew directly into Lia’s window.

The missive was a paper bird, intricately folded from parchment, fluttering its tiny wings before gently alighting upon the draft Lia was currently working on, emitting a crisp, magically enhanced birdsong.

A voice, imbued with a hint of amusement, emanated from within the paper bird—it was Laplace.

“To Master Lia, Opener of the Star Gate, the desserts at Laplace Observatory have piled high, and the star-track observation device feels lonely. Might you be willing to grace us with your presence and share some secrets of the cosmos with a curious old man?”

Lia picked up the paper bird.

In her palm, it transformed back into a flat sheet of parchment, bearing the same message written in elegant, flowing script.

She raised her eyes, gazing at Klein, who was engrossed in reading documents nearby.

Klein did not look up, his gaze still fixed on the pages, his tone unruffled.

“Do you wish to go?”

“He has prepared desserts.” Lia’s response was both succinct and irrefutable.

“Then go.”

Klein turned a page of his book.

“I will accompany you.”

Lia paused, momentarily taken aback; she had expected him to offer some excuse, perhaps citing ‘pressing guild matters’ or the like.

“You don’t have guild matters to attend to?”

“Unimportant.” Klein closed his book, rising to his feet. “Prepare yourself. We depart now.”

***

Laplace’s mage tower was not situated within the bustling districts of the capital; instead, it stood majestically atop a solitary hill just outside the city.

The tower’s structure deviated from traditional stone, fashioned instead from a pristine, bone-china-like material, seamless and monolithic, without a single visible joint.

Its apex was not a spire but a colossal, hemispherical dome crafted from transparent crystal.

Even beneath the midday sun, simulated stars continued their slow, intricate, and precise orbits within the dome.

The light of teleportation dissipated in the tower’s ground-floor hall.

Laplace was already waiting there.

Today, he had forgone his customary purple, gold-trimmed mage robes, opting instead for a loose grey gown that made him resemble an ordinary, amiable grandfather from next door.

“Little one, you’ve finally arrived!”

He chuckled, striding forward to greet them, his gaze bypassing Klein entirely to focus solely on Lia.

Only then did he feign noticing Klein, casting a glance his way with a hint of exaggerated complaint in his voice.

“I invited only Miss Lia, not her chaperone. Klein, are you truly worried this old man might abscond with your precious apprentice?”

“Yes.”

Klein’s response was a single, unadorned word, spoken with absolute candor and directness.

Laplace was thoroughly choked by that solitary word; he huffed, puffing out his beard, and rolled his eyes at Klein with theatrical exasperation.

“Youngsters these days have no respect for their elders! Come along, little one, pay no mind to this block of wood. I’ve prepared some wonderful things for you.”

He naturally took Lia’s hand and led her deeper into the tower.

Lia followed passively, glancing back over her shoulder.

Klein trailed behind them, his expression unreadable, his steps unhurried and measured, each one seemingly calculated, his presence undeniably potent.

Instead of ascending, they entered a room located in the tower’s central core.

The moment the door swung open, Lia’s pupils dilated almost imperceptibly.

Within, there was only an enormous, circular oak table.

The table itself was laden with an array of dazzling, delectable desserts.

There were mille-feuille dusted with powdered sugar, cream cakes adorned with vibrant red berries, translucent fruit puddings, and mountains of cookies in various whimsical shapes.

The air was thick with the sweet aroma of cream and baked wheat, so rich it was almost suffocating.

“Well, what do you think? I had the kitchen prepare all morning,” Laplace announced proudly. “Knowledge requires energy, and sweets are the finest source of it. That is my theory.”

Lia sat down without ceremony, reaching for the closest cream puff, generously drizzled with chocolate sauce.

“I concur with this theory.”

She took a bite, her eyes narrowing in blissful contentment, her cheeks happily working.

Laplace’s smile deepened as he watched her contented expression. He, too, picked up a biscuit and crunched on it audibly.

Klein, meanwhile, sat quietly to the side, a single glass of plain water before him, his aura utterly incongruous with this saccharine sea.

“Klein, your life truly is dull,” Laplace remarked, shaking his head.

Klein offered no response.

Lia swiftly devoured three cream puffs and a small slice of cheesecake, then contentedly sipped half a glass of sparkling fruit juice.

“Alright, little one.”

Laplace wiped his hands with a napkin and set down his empty plate.

His smile remained, but the light in his eyes shifted, transforming from amiable warmth to a gaze of intensely focused sharpness.

“The desserts were merely an appetizer. Now, let us discuss matters of importance.”

He waved a hand.

Instantly, all the food and tableware vanished from the massive oak table, leaving its surface gleaming anew.

Following this, colossal rolls of blueprints and thick stacks of calculation manuscripts materialized from thin air, completely covering the tabletop.

“Your thesis—I’ve read it no less than twenty times,” Laplace’s voice deepened. “Especially that final sentence; it was truly the devil’s whisper, leaving me sleepless and without appetite for the past month.”

He pointed to a corner of one blueprint, where a line of text was annotated in red ink.

“‘Regarding how to determine the value of G through an ingenious experimental setup, I already possess a brilliant conception.’”

Laplace read it aloud, enunciating each word carefully.

“And then came that damned ‘Pity there wasn’t enough blank space to write it all down.’”

His finger tapped against the intricate blueprints, producing a dull thud.

“I followed your line of reasoning, pondering it for a solid month.”

He began to elaborate on his ideas to Lia, his eyes gleaming with the excitement of a seeker who had found their path.

“Your theory predicts the existence of universal gravitation and provides its law. This gravitational constant G is the bridge connecting celestial bodies and terrestrial objects.

To measure it, one must quantify the incredibly faint, almost nonexistent gravitational force between two ordinary objects here on the ground.”

“My concept is as follows.”

He picked up a core design diagram and unrolled it before Lia.

“I propose constructing a lightweight horizontal rod, securing two small spheres of known mass at its ends.

Then, using a sufficiently thin and resilient thread, suspend this rod from its midpoint, allowing it to rotate freely in the horizontal plane.”

“Next, I would prepare two larger, heavier spheres and move them simultaneously near the smaller spheres.

According to your theory, the heavier spheres would exert a gravitational pull on the smaller ones, creating a torque that would cause the entire rod to twist.”

He pointed to an angle marking on the blueprint, his eyes sparkling with intellectual brilliance.

“If I can precisely measure the final angle of the rod’s torsion, and combine that with the torsional properties of the suspension thread, I can deduce the magnitude of the gravitational force.

With the gravitational force F, the masses of the two pairs of objects m₁ and m₂, and the distance r between them, plugging these into your formula will yield the value of G.”

Having finished, Laplace looked up, his gaze intensely fixed on Lia.

His eyes held anticipation, yet also a hint of uncertainty, much like a student awaiting a teacher’s judgment on their homework.

“How similar is my approach to your ‘brilliant conception’?”

Lia did not answer immediately.

She lowered her head, meticulously scrutinizing every blueprint and every line of calculation on the table.

Laplace’s design was remarkably detailed, accounting for numerous specifics, from material selection to external environmental shielding, all meticulously planned.

The room was silent, save for the soft rustle of parchment being turned.

Klein sat quietly, the glass of clear water before him untouched, his deep blue gaze unwavering from Lia.

After a long moment, Lia finally looked up.

“The general direction is correct.”

She began.

“Using torsion to amplify minute torques—that core idea is sound.”

Laplace’s face brightened with a hint of joy, and he unconsciously straightened his posture.

“However,” Lia’s tone shifted, her voice soft, yet it caused Laplace’s joy to freeze on his face, “your design contains two fatal flaws.”

“They will render all your measurement results into nothing more than meaningless random numbers.”

Laplace’s smile lines had not yet faded, but the curve of his lips stiffened.

Lia extended a finger, tapping the spot on the blueprint that specified the suspension thread material.

“The first problem lies here: the suspension thread. You plan to use enchanted spider silk, which is tough enough and sufficiently thin.

But you’ve overlooked its ‘material inertia’ and ‘fatigue deformation.’ This means that after it’s twisted and then recovers, it won’t fully return to its initial position.

Furthermore, under prolonged force, it will undergo irreversible microscopic deformation. Its torsional coefficient isn’t a constant value; it will change with time, temperature, and even air humidity.

Using it as a baseline means your measurements are flawed from the outset.”

Laplace’s expression changed.

He had evidently not considered the microscopic instability of materials, a blind spot in his knowledge.

Lia’s finger moved to another sketch detailing the measurement method.

“The second problem is even more severe. Your measurement method is incorrect.”

“You are attempting to measure a final, static torsion angle. But the gravitational force you’re dealing with is simply too minuscule.

A gentle breeze, distant footsteps, or even a change in air pressure caused by a bird flying outside the tower will produce interferences several orders of magnitude greater than gravity on your setup.

Your rod will never truly be still; it will constantly oscillate randomly within a tiny range. To extract a stable ‘signal’ from all this ‘noise’ is impossible.”

Laplace’s face had begun to pale.

Following Lia’s guidance, he looked at his highly anticipated design blueprint, and fine beads of sweat appeared on his forehead.

His meticulously crafted design, of which he was so proud, seemed utterly fragile and vulnerable in the face of these fundamental issues.

“Then… what should be done?” His voice was dry. “Could it be that this constant, though theoretically predictable, is fundamentally immeasurable in reality?”

If that were the case, Lia’s theory, though logically perfect, would forever lack its most crucial piece of the puzzle, preventing it from truly moving from theory to application.

“No.”

Lia shook her head.

“Your thinking is confined.”

“You keep trying to find answers in stillness, but the answer is precisely hidden within motion.”

She pulled out a blank sheet of parchment and picked up a quill.

“You shouldn’t measure the angle of torsion.”

“You should measure its ‘period.’”

She sketched a diagram of simple harmonic motion on the paper.

“First, without the interference of the large spheres, you give the rod a slight perturbation, allowing it to oscillate torsionally on its own.

Measure the time it takes to complete one full oscillation in this state—we’ll call this the natural period T₁.”

“Then, place the two large, heavy spheres near the smaller ones. At this point, gravity will act as an additional restoring torque, participating in the entire oscillation process. In this new system, measure its oscillation period T₂ again.”

“The effect of gravity will cause an extremely minute change in this period. There will be a difference between T₁ and T₂. Although this difference is tiny, it strictly corresponds to the magnitude of the gravitational force.

Through a simple formula, you can precisely calculate the value of the gravitational constant G from the difference between these two periods.”

Lia paused her writing.

“This method I call the dynamic measurement method. It doesn’t require the apparatus to achieve absolute stillness; instead, it utilizes the apparatus’s oscillatory characteristics.

Most random, irregular interferences will be averaged out over a long period of measurement. What you obtain is a much more stable, pure result, stripped of most noise.”

As her words fell, Laplace stared blankly at the diagram and concise explanation on the parchment.

Stillness… motion…

Angle… period…

In his mind, it was as if a bolt of lightning had rent through thick fog.

Indeed!

Why fight against unavoidable interferences?

Why not embrace them, utilize them, and seek the constant law within dynamic change?

This idea was simple, elegant, yet it overturned all his past experimental logic!

“Ha… haha…”

Laplace’s laughter began as a low chuckle, then grew louder and louder, finally erupting into a hearty, uninhibited roar that filled the entire room.

“So that’s it! That’s it! Dynamic measurement! How could I not have thought of it! What an old fool I am!”

He clutched the parchment bearing the dynamic measurement method as if it were an peerless treasure, his face flushed with excitement.

Suddenly, the excitement on his face morphed into a hint of slyness.

“Horace! That old man! I bet he’s still racking his brain trying to build an absolute vacuum chamber for his equipment! Just wait!

Just wait until I measure the value of G with this method and announce the results at the council… Hahahaha! I truly wish to see the color of his old face then!”

Laplace’s laughter was filled with the exultation of impending triumph.