Sylrandis Manor.

Second floor bathroom.

Sylrandis lay in the bathtub, eyes closed comfortably, murmuring dreamily.

"My dearest apprentice, you must remember, imagination is always the foundation of all magic. Concretization and geometry are only aids to imagination, not imagination itself."

By the tub, Rosen's mentality had changed tremendously.

Yesterday, he was full of lustful thoughts, with the sheep in his heart desperately trying to break out of the pen.

Now, just thinking that the woman lying before him might be a cruel murderer, he felt chilled inside.

Forget about lustful thoughts, staying calm was already the limit.

But his pursuit of magic had not changed. Hearing Sylrandis' words now, his thoughts were inevitably led in that direction.

After careful consideration, he said, "Master, do you mean a mage doesn't necessarily have to imagine a hand?"

"Correct."

Sylrandis nodded approvingly, then flipped over in the tub again, her snowy peaks peeking out amidst the rose petals.

"As a foundational cantrip, Mage Hand is easy to grasp precisely because it borrows the concrete image of the hand deeply ingrained in the soul."

"But think carefully, to move an object from one place to another, do you really need to imagine an entire hand and arm?"

At these words, Rosen felt a flash of insight strike his mind, instantly shattering his preconceived notions.

'To move an object, you don't need a hand. The essence is an external force acting on the object!'

'More fundamentally, it's the transfer of energy!'

'Ah~~ the wicked master is truly harmful! Celestius only told me half the story, trapping me in a pit.'

At this moment, Rosen even had the thought that even if Sylrandis was a homicidal maniac, as long as she didn't harm him, she would still be his most respected mentor in the arcane arts.

Feeling the towel stop rubbing, Sylrandis turned her head. Seeing him standing there in a daze, she didn't speak, patiently waiting instead.

After a long while, feeling the towel start moving again on her body, she finally spoke, "My dear apprentice, tell me your thoughts. Let me see if you've gone astray."

Her last sentence reminded Rosen, dispelling his thoughts of concealment.

Of course, he naturally wouldn't directly state the physics concepts from Earth.

After carefully organizing his language, he began to articulate his ideas.

"Master, I was just thinking, there are countless ways to move an object, far more than just using a hand. Gale winds, flowing water, or even just levitating an object halfway, they would all make it move."

"I suspect there must be some commonality among them. And this commonality is likely the most direct and effective way to imagine the Mage Hand."

Hearing this, Sylrandis' languid expression disappeared, her eyes staring straight at Rosen's face.

She gazed at him for a long time before suddenly sighing softly, "I really didn't expect to hear the most outstanding words of wisdom in this world from the mouth of a 15-year-old apprentice of the arcane arts."

'Splash~'

She stood up from the tub, disregarding Rosen's resistance, and drew him into her embrace directly.

"My dearest apprentice, don't struggle. Let your mentor properly feel your existence. After all, as you continue to grow, opportunities like this will become fewer and far between."

There was no lust in her tone, only lament.

Rosen stopped struggling, "Master, I must remind you, not everyone appreciates such aggressively intimate expressions of emotion."

"Hehe~ As long as you're still my apprentice, you'll have to endure this."

Sylrandis released Rosen and lay back in the tub again.

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"What you just called 'commonality' is exactly what all mages have been pursuing for nearly a century now."

"Unfortunately, up until now, no convincing explanation has been found. We can only approach that goal as closely as possible."

Rosen was slightly surprised, and couldn't help thinking inwardly, "Isn't this commonality just the transfer of energy? How is that so hard to discover?"

But on second thought, he didn't feel it was strange.

Even on Earth, it took humankind thousands of years of civilization to evolve modern science.

And in this world with magic, subjective and objective were extremely indistinct, idealism and materialism mixed together.

Especially the observer effect, which was crazily powerful here.

A mage with mana, just by focusing intently on an object, would inevitably greatly interfere with it.

To distill the fundamental laws of the material world from such chaos was surely even more difficult than on Earth, not less.

"Master, what's the current progress then?"

Sylrandis smiled lightly, "Haven't you seen it already?"

Rosen was startled, then immediately realized, "Do you mean, precise geometry?"

"Clever indeed," Sylrandis couldn't help praising him. "From ancient times until now, mages have always been pursuing the most efficient ways to utilize mana."

"This process is also one of gradually stripping away daily experience and entering a purely logical world of thought."

"So, moving an object transforms into visualizing the trajectory of the object's movement."

As she spoke, a rose petal floated up from the water's surface, then drew a perfect circle in the air.

"Do you understand now, my dearest apprentice?"

Rosen slowly nodded. At the same time, he willed another petal to rise up and draw a standard equilateral triangle in the air.

Sylrandis encouraged, "Can it be more complex?"

Rosen thought for a moment, then changed the equilateral triangle to a parabola.

"From lines to curves is a huge step forward, but can it be even more complex?"

Rosen immediately switched again, changing the parabola to a sine wave oscillation using trigonometric functions.

Sylrandis' eyes lit up, "Ah ha~ it's a water wave? Now that's more interesting."

Rosen nodded. With a slight exertion of will, the sine wave's amplitude and phase began continuously shifting, as if it were a forward flowing water wave.

Watching for a while, Sylrandis also became interested and tried mimicking the water wave with a petal.

But very quickly she realized that while she could mimic a water wave, it was far less stable than Rosen's, and the curve transitions much less natural and precise than his.

She stopped and looked at Rosen, confusion flashing through her gaze.

"You...discovered some mathematical tool to describe water waves, didn't you?"

Only numeric description of geometry could truly standardize imagination.

Precisely because of this, Sylrandis could draw perfect circles, yet couldn't do what Rosen did without mathematical methods to describe water waves.

Rosen didn't conceal it, because it wasn't difficult.

While this world may not have it yet, with the current mathematical foundations, sine waves described using trigonometric functions were already obvious, their discovery just a matter of time.

"Master, when I was reading Principles of Precise Geometry, I accidentally discovered that expanding triangular numbers on a grid becomes water waves. By modifying some parameters, you can arbitrarily change the intensity and phase of the waves."

"Um...phase is a term I came up with myself. It means the timing of wave peaks."

Sylrandis didn't speak. She fell silent, gaze blank, clearly lost in thought over Rosen's words.

After a good while, she 'woke up' and looked at Rosen, her gaze now containing a trace of shock.

"Rosen, I don't know what words to use to describe you anymore. Please forgive my earlier frivolity."

"Oh no, this is a major discovery with extremely broad application potential. I must immediately write an arcane thesis on it!"

She no longer cared about her bath, floating directly out of the tub.

The new bath towel swiftly flew to her side, quickly drying the water on her body. At the same time, her arms reached out and she slipped into the new bathrobe. The robe seemed to come alive, 'automatically' contouring to her body, then 'automatically' cinching the belt tight. The quill pen and wood grain paper were already prepared on the desk.

Once she was seated, the quill pen sprang up as if alive, speedily writing on the paper.

Concurrently, another towel flew over, scrubbing her thick, still dripping fiery red hair.

The entire process was smooth and effortless, peaking at over 10 concurrent threads of manipulation.

Moreover, in each thread, her use of Mage Hand reached a seemingly innate level.

It was as if she were the sovereign of this small world.

Seeing this level of skill, the tiny bit of smugness Rosen had just started feeling immediately vanished.

He silently admonished himself, 'Rosen, Rosen, trying to strut with just a tiny accomplishment? You still have a long way to go!'

After writing for a while, she turned and asked, "Rosen, do you want to be first author on the paper?"

Rosen shook his head without hesitation.

"No no no, Master. My only contribution was providing some inspiration, but my shallow arcane knowledge and narrow experience are insufficient to write a proper arcane thesis. So please don't include my name."

He already understood mathematics was just an aid for casting, the truly valuable part was applying it to magic.

And he was still only half-baked in this area.

Sylrandis thought seriously for a moment, then nodded in agreement, "You're right, you're too young and too weak, can't yet shoulder such fame."

After writing a few more sentences, she pointed at the tub, "Clean that up spotless, then leave here. Don't make any sounds that could disrupt my train of thought."

"Bring meals, Master?"

"Yes!"

"Yes, Master."

Rosen silently cleaned up. Once everything was spotless, he was about to go downstairs when a voice sounded from behind him again.

"My dear apprentice, tonight's lesson plan remains unchanged. Don't forget now."

"Yes, Master."

He left the second floor room in high spirits, steps light.

Closing the door, a thought flitted through his mind.

"Could such a powerful master really be a deranged homicidal maniac?"

"Based on my observations so far, absolutely impossible."

"But to protect themselves, murderers are usually extremely adept at pretending, one face in public, another in private, it's common."

"Besides, the master exhibits a split personality. Looks like two for now, but who knows if there's a hidden third!"

"Still, a rabbit doesn't eat around its burrow. I should be safe for now, keep observing."

Back in his little room, he started seriously testing his new knowledge.

In the time that followed, the two, master and apprentice, one furiously writing on the second floor, the other studying magic on the first floor, did not disturb each other.