Mathematical Analysis
Fran, far away in the Material Realm, was completely unaware of the Lord of Radiance's expectations. She was engrossed in writing in a small room specially arranged for her by the Northern Lands Society.
At the same time, she began to visualize those earthly works in her mind for reference.
She is now preparing for her first Paper, in which she intends to introduce a mathematical tool— Calculus —in the first Paper Paper, no matter what.
Since Fran is going to write down all the scientific research results on Earth, she will definitely need to use a tool like Calculus.
Because the theories she learned required Calculus to explain these theorems, and many of them were even derived by working backward from mathematical formulas.
If she were to directly present all those various theorems that followed, she would have to wonder if the reviewers could even understand them.
Therefore, no matter what, she must first promote Calculus as a mathematical tool.
When the time comes, if she publishes something the reviewers can't understand, she can use her existing posts to debate with them.
However, after carefully understanding the publication format of this Paper, she realized that she had misunderstood something.
In this other world, any Paper that does not contain research on magic is simply not accepted for publication.
For example, after Fran's mentor, Harbos, gained fame for his Paper on the use of mechanical aids, many mages started writing Paper using mechanical structures they had scavenged from the common people...
In any case, none of these Paper were approved after they were submitted. Only a few papers that actually studied the underlying mechanisms and wrote a few methods to enhance Wind Element Paper were barely approved.
You asked how Fran knew? Well, of course, it's because she spent a dollar to get it from that mage dealer; it was written in the Snowflake Weekly.
Because Harbos's Paper was so incredibly popular, especially at the Northern Lands Society where it was Paper, almost every featured section was analyzing it.
And Snowflake Weekly explicitly mentioned:
The Northern Lands Society's reviewers would like to explicitly remind everyone: please do not submit Paper that have no effect on any spells. No matter how good your mechanical structure analysis is, we will not approve them.
To reiterate, the Northern Lands Society is a sorcerer's academy; all achievements should be based on magic...
This completely shattered Fran's idea of submitting a Paper based solely on mathematical analysis; she didn't even need to consider it.
When she submits this purely mathematical Paper, the reviewers will definitely criticize it, saying it's from a sorcerer's society...
Therefore, she had to change her approach, such as using Calculus to analyze a spell to improve its effectiveness, so that her Paper could be approved.
This is easier said than done, as it can easily lead to all sorts of unforeseen difficulties.
She ultimately chose the Fireball as the subject of this analysis.
Fireball, as the symbol of the Elemental School, is one of the oldest spells in the world, and is known to almost everyone.
On [Snowflake Selection], the Paper "How to Cast Fireball" was even placed in the [Classic Chapters] section.
Furthermore, this Paper is a rare free chapter, which any mage can learn directly once they arrive in the Northern Territory.
Many mages, upon seeing that such a famous spell could be learned for free at the Northern Lands Society, naturally went straight to search for and learn it.
A spell that an apprentice can cast, a spell that even the guards of the Northern Lands Society can cast.
Fran wanted to research how to create the largest fireball in the current environment with the least amount of Magic Power power.
She initially thought the real challenge lay in formulating a truly useful mathematical expression, but unexpectedly, her biggest difficulty was actually in defining it.
That's right, in a Magic World, there isn't actually a universally accepted standard unit of measurement.
In all these Paper, when describing Magic Power, terms such as the Magic Power possessed by an ordinary apprentice or the Magic Power possessed by First Order Mage are used to describe the total amount of Magic Power required, without a specific unit of Magic Power.
If she really wanted to calculate the Magic Power cost of Fireball, she would first have to define exactly how much a unit of base Magic Power is.
She tried every possible method to squeeze out the smallest unit of Magic Power to make it easier for her to calculate, but after discovering that her Magic Power could be divided into units invisible to the naked eye and mind, she gave up the idea.
She chose a more opportunistic approach, first defining a basic unit of Magic Power using the Fireball.
[In a room devoid of any elemental activation and free of any dissipating Magic Power, a mage without elemental spell specialization fully releases Magic Power into the spellcasting structure of Fireball. The Magic Power consumed by the released fireball is defined as one standard fireball.]
During the experiment, Fran cast Fireball several times without activating her spell feats, confirming that the Magic Power cost of a fully Magic Power- infused Fireball remained unchanged.
Then, through multiple experiments, the volume of the fireball was measured under different amounts of Magic Power, and these data were plotted on a coordinate axis as an inverted U-shaped curve.
By importing the input Mental Power and the size of the fireball into a coordinate system, a curve describing the relationship between them is formed.
From this, we can derive a simple mathematical formula...]
After writing the formula, Fran felt a surge of inspiration and became completely absorbed in the writing process.
She didn't notice how quickly time was passing until she wrote the last letter and breathed a sigh of relief, only to find that the sun had already risen the next day.
Fran patted her face. "I haven't slept all night..."
But she felt no fatigue at all; on the contrary, her mind was in a state of even greater excitement.
After she wrote down the mathematical analysis of the Fireball, all sorts of ideas kept popping into her head, like, what if other spells would work...?
She rubbed her throbbing temples, abandoning the idea of continuing her research. Right now, she needed to submit the article first, and deal with the rest later.
Just then, a Panel appeared in her mind.
[You spent an entire night analyzing the Fireball in detail, drawing many useful conclusions, and summarizing these conclusions in a way that others can understand. Your understanding of the Fireball has deepened; Fireball proficiency +11000.]
Fran looked at the pop-up notification on the Panel with delight. This notification pointed her in a direction for quickly improving her Arts proficiency in the future.