Bathing and even bathing together with his mentor had become commonplace after many times.

In the warm, bright, independent bathroom of the new home, Rosen felt no ripples in his heart as he scrubbed Slander's back, and could even focus on discussing issues of magic.

As they talked, he brought up the second purpose for his rushed return.

"Sweetheart, the Earl explained to me in detail the differences between the natural and logical systems of magic."

Slander lay draped lazily over the edge of the bathtub, eyes closed in feline pleasure under Rosen's skillful ministrations.

She murmured, "To have received personal instruction from the Earl so quickly, it seems your father has rendered no small service."

Rosen smiled slightly. "External things matter not. What interests me most is still the knowledge of the logical system." In my free time, I pondered carefully and did get some inspiration."

With war imminent, he naturally wanted to increase his strength by all means. So he brought up calculus without hesitation, in case it could be useful, even extremely useful, so his power could increase substantially.

"Tell me about it. Oh, apply a little more pressure, don't use the washcloth, use your hands directly. This area around my shoulder blades, knead firmly, I think I got a crick in my neck last night."

So Rosen massaged and loosened her muscles and bones while beginning to explain in detail.

"Last time, I talked about generalizing triangular numbers to the two-axis Cartesian plane, right?"

"Mm, yes. A little higher~ Ah~~~~ Ah~~~~ So comfortable!"

"So I wondered, could I also generalize the circle partitioning technique of the mathematical master Archimedes to the two-axis Cartesian plane?"

No sooner had he finished speaking than the lazy look on Slander's face instantly disappeared.

She jerked her head sharply to look at Rosen, but the sudden movement strained her injured neck, forcing a pained shriek. She quickly turned her neck back.

"Quick, massage your daughter's neck~ Ah, feels like my neck will snap!"

She added right away, "Never mind your daughter, Father, please continue!"

Rosen went on explaining.

"Master Archimedes used triangles to partition circles, in order to obtain a precise value of pi. But obviously, no matter how many triangles are used to divide the circle, an exact value of pi cannot be obtained."

"Or rather, only by using an infinite number of triangles to cut the circle, can the exact value of pi be reached."

"So I borrowed the concept of 'infinity', and changed the triangles to rectangles which are easier to compute. See, like this."

With a thought, the mage's hand scooped up water, sketching a Cartesian plane and a parabolic curve on the water's surface.

Because of the soreness in her neck, Slander simply turned around to watch. As she watched, the crimson glow in her eyes shone brightly.

"Ah~~ What exquisite thoughts exist in this world!"

Immersed in his own thinking, Rosen didn't notice Slander's actions, and continued massaging her body absentmindedly. Suddenly feeling a soft, yielding sensation under his hands, he glanced down and pretended nothing had happened, discreetly withdrawing his hands.

Next, starting from the most fundamental concepts, he slowly derived the knowledge of calculus bit by bit.

Slander listened with utmost concentration, her eyes shining brighter and brighter, with one constant thought echoing in her mind.

"By the golden gods, what kind of apprentice have I taken?!"

Rosen didn't notice her change. He focused on controlling the clear water, drawing one beautiful mathematical formula after another in the air, with Slander's own gorgeous snowy figure behind them.

The solid oak tub, flawless beauty, vivid flower petals, clear warm water, all penetrated by a kind of unprecedented, exquisitely refined logical intellect.

In a daze, time seemed to coagulate into eternity.

Slander unconsciously bit her tender lower lip, her misty gaze peering through the dazzling mathematical formulas at the young face behind them, clearly sensing the radiance of wisdom in those azure eyes.

Before long, Rosen finished introducing the core concepts of calculus, namely 'infinity' or 'limits'.

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"Mentor, this is my initial rough idea, very immature, with many flaws that cannot withstand scrutiny."

"But fortunately I have nothing to do these three days, so we can study it in depth together. What do you think?"

"Father, it is my greatest honor!"

She could no longer remain in the tub. After a quick rinse, she put on her clothes and sat in an armchair, pen flying swiftly over the parchment.

After Rosen cleaned up the bathtub, he also stoked up the fireplace to warm the whole room like spring.

Then the two sat at the desk and earnestly researched the application of this brand new mathematical thinking in magic.

In this process, Rosen had advanced modern mathematics ahead of this world. He had an overarching grasp, easily keeping track of the mathematical details.

Even if his memory was hazy, he could open his adventure log and return to his university lectures in his past life for clearer memories.

And Slander was learned and astute, with an unyielding tenacity for immersing herself in scholarship. Her understanding of magical knowledge far exceeded Rosen's.

In short, she was a pure, extremely outstanding scholar-mage.

Under their combined efforts, they advanced swiftly, much faster than Rosen had imagined.

In just three days, they obtained an extremely important and unprecedented application of logical magic!

Late into the night on the third day, the desk was covered in a stack of parchment, the first page titled "Agile Spellcasting".

This was not a spell, but a technique for casting spells.

Slander looked at the parchment, reluctance evident in her eyes.

"Such lovely times are so fleeting. It's a pity you have to leave tomorrow morning."

Rosen was still carefully analyzing the structure of the new spells with a pen, while his mind busily recorded this brand new magical knowledge into his spell analyzer.

"Mentor, it's just a brief parting. I'll return very soon."

"Is the outside world really that fascinating?"

"There are two kinds of books in this world. The first kind is written on paper, the second kind is written between mountains and rivers. And the contents of the latter outnumbers the former infinitely."

Slander sighed lightly. "You always say such thoughtful words."

Just then, Rosen finished calculating the new spells. He passed the calculation notes to Slander.

"Mentor, I'm done. Please check if I missed anything."

Slander took the notes and inspected them carefully.

After some time, she gently put down the notes.

"I don't see any issues either. But after all, this is a completely new mathematical thinking, and we had little time, so I cannot confirm if there are inherent flaws in this thinking itself."

Rosen gave a faint smile. "Whether there are flaws or not, it is certainly very useful. Let's give this new thinking a name, Mentor."

Slander tilted her head, pondering for a bit, then shook her head. "I can't think of one, why don't you suggest something?"

Rosen already had it in mind. "It emerged to depict smooth, continuous complex curves. How about combining 'flow' from 'smooth' and 'num' from mathematics, into 'calculus', what do you think?"

Slander's eyes lit up. "Calculus. What a great name, let's call it that."

Rosen went on. "As for whether this new thinking has flaws, that's easy to verify."

"I'll just try out the new spells. If there's no problem, I should definitely succeed in casting them."

Slander gave Rosen a look. "If you're wrong, the spellcasting will fail. With your fiery magic power, you'll burn yourself into a big candlestick."

Such spellcasting accidents originated internally, so defensive spells were largely useless.

But unexpectedly, Rosen stood up directly and retreated to the corner.

"Let's find out."

In fact, he had already simulated the spellcasting successfully in his spell virtual machine, so he already knew the spells were fine.

Slander was startled. "Don't!"

'Shhh~~'

Rosen smiled slightly as his magic power surged. Following the imagined spell structure, he quickly constructed it successfully.

A tiny, piercingly bright speck slowly materialized in the air, then grew larger and larger, while its brightness remained unchanged.

This meant the spell intensity was unchanged, only the scale had increased.

Anna watched with bated breath. Anxiously, her hands were clenched into tight fists, knuckles white. And Alice, who should have been sleeping, also manifested through magic as an illusion, watching tensely as well.

When the speck reached the size of a jujube pit, its volume temporarily stabilized.

Rosen spoke. "Mentor, this is a twenty percent scale miniature magic missile."

Right after, the missile continued expanding.

"Now it's twenty, thirty, ninety, one hundred, one hundred and fifty, two hundred and twenty percent of max scale!"

"Reached the limit!"

The missile in the air was gently dispelled. After the light faded away, Rosen breathed a slight sigh of relief, and Slander also deeply relaxed.

When everything returned to calm, the two Slanders spoke in unison.

"Success."

Rosen nodded slightly. "Yes, success."

"Mentor, based on the latest Agile Spellcasting technique, the scale of a standard Magic Missile can be adjusted between ten and two hundred and twenty percent, to suit different needs."

Although due to the unique properties of magic power, truly unrestrained spellcasting was still unattainable, such a wide range of flexibility was undoubtedly a huge leap for logical magic!

And with this powerful spellcasting technique, Rosen felt his combat power could increase by at least half.

On the other side, seeing Rosen's successful experiment, Slander was itching to try it herself. She concentrated her mind and began casting.

Compared to Rosen's fiery magic power, hers was lunar, so the speck of light was silver, like stars in the sky.

However, due to the short exposure, her understanding of the infinite concept in calculus was still somewhat lacking. Thus, transforming the form of the spell was not as smooth and natural as Rosen's.

But that was no issue. With ample practice, she would surely become adept.

After completing the spellcasting, her cheeks were flushed and eyes bright.

"Rosen, I can feel this completely new mathematical thinking has immense untapped potential. If we can study it in depth, perhaps we can create advanced, even master-level pure logical magic!"

If that happened, their names would be etched forever in the annals of history. No, just the creation of calculus alone would be enough to shine brightly in history!

Rosen nodded in agreement. "I think so too."

"Ah~ Unfortunately you're leaving tomorrow."

Slander gave her second sigh.

Rosen smiled. "I'll be back very soon."

"Oh, that's right. Physical health is also extremely important. Don't forget to eat and rest properly when I'm away."

Knowing Slander's personality, she would surely devote herself completely to calculus, hence his reminder.

"Rosen, you really worry like an old father."

Slander tried to resist.

"Mentor~~" Rosen raised his tone.

Slander immediately conceded. "Alright, I'll be careful."

(End of Chapter)