Lucien’s voice wasn’t loud, but his words were beyond shocking to the board members present.

As the authorities in math who had won the Arcana Scepter Award, they had to admit that they were more or less proud. However, in fact, they only regarded math only as a tool for studying arcana and analyzing magic spells. Very few arcanists were willing to devote their entire life to the beauty of math, most of them would never explore further in maths past what is sufficient for their purposes.

The Tower’s arcanists who were fond of math more or less shared the same belief before, but it was either a occassional remark of pride or just a flash in the pan. No one had ever, like what Lucien just did, managed to prove by using such shocking reasoning and imposing manner the great value of math: Math went beyond human beings’ intuitive understanding and experience. Thus, maths’ importance and significance had been raised to a degree no one had ever dreamed of before.

Although they understood that Lucien was helping Levski, the board members, who were also mathematicians, were the same excited and encouraged.

Levski’s body was trembling out of great excitement. He knew that math would not lie, or New Principles of Geometry with Complete Theory of Parallels would have never been born. In the furious ocean of arcana, Levski clenched tight to the vessel named Math. Because of his belief, he could travel across the ocean without being overwhelmed by the waves and tides, or by despair!

Fernando did not say anything but merely nodded. He knew that Lucien’s words were indicating something even deeper.

Lucien put back his right hand and removed the projection from the walls. Then he put another thick pile of paper on the magic circle and smiled gently. "Now let’s see if we can find more from another perspective."

historical

Despite all the proof, so far the board members were still reluctant to admit the errors they made, reluctant to admit that they had been wrong for more than ten years. Although, in their mind, they had already begun to accept the new geometry system — They had to, or otherwise Tower Geometry, which was compatible with it, also had to be overthrown.

Besides, they had to admit that Lucien’s words were persuasive. Therefore they were now quite calm and objective towards the new paper.

"Another Perspective on the Parallel Postulate..." Milina read the title in a low voice, and she had some clues in her mind...

"Another perspective..." Levski felt that the mist in front of his eyes was slowly dispersed. He saw something that he had ignored in the past decade.

Like teaching in the magic school, Lucien pointed at the lines on the wall one by one. "So Mr. Levski postulated that ’For any given line R and point P not on R, in the plane containing both line R and point P there are at least two distinct lines through P that do not intersect R’. Here we can make another postulate to complement the entire geometry system: For any given line R and point P not on R, in the plane containing both line R and point P there is no line through P that does not intersect R.

"Now, let’s do the deductive reasoning and see what will we get?"

Levski took a deeper breath and became excited once again. The rest of the board members stared at the paper’s projection and looked very serious.

Lucien went through the paper very fast as he was facing the authorities in math. However, the board members looked more and more grave, more and more confused. Finally, Mabel burst out, "This is another crazy conclusion... different from Levski’s geometry system!"

"Why..." Salgueiro’s murmur was drowned out by Mabel’s voice.

Milina was very perplexed. "Is this a new geometry system different from Tower Geometry?"

Just a few hours earlier, she firmly believed that Tower Geometry was the only geometry system in this world. However, in less than one day, her belief was mercilessly shattered, for twice.

She was glad that knowledge in maths did not affect the cognitive world, or her head would have exploded already. Lucien definitely deserved the titles that people secretly gave him, namely, Headcrusher and Brain Eater...

Levski was not confused, instead, his mind was seized by many thoughts: Why could this happen? Why completely different yet equally correct geometry systems could be deduced from the fifth postulate? Did this have anything to do with the real world?

Under the influence of president Douglas, many arcanists tended to ask "why" very frequently, but few were like Lucien.

"Interesting..." Fernando nodded slightly.

Time flew, and Lucien’s paper had come to the end. Using a spherical model, Lucien proved the system’s compatibility.

Neeshka sighed, "This is another new geometry system... Evans Geometry...

"Math won’t lie... indeed..."

The board members missed the chance of naming a new geometry system after themselves because of their own prejudice and arrogance, as if one accepted Levski’s hypothesis, then the postulate that Lucien Evans just put forward was in fact not difficult to come up with.

Lucien had not finished yet, and he showed the last few pages to the audience. "In my paper, I’ve defined the term ’curvature’. Based on calculation, we can know that when the curvature is zero, we get Tower Geometry; when the curvature is below zero, we have Levski Geometry; when the curvature is above zero, we see the geometry system that I just introduced. They have commonalities in their essence, and they are compatible with each other."

Milina released a soft sigh. Now the two new geometry systems seemed more acceptable to her.

Enlightenment came to Levski. When he was about to applaud out of great excitement, he saw Lucien’s gesture asking them to remain quiet.

What else did he want to say?

This was the question shared by all the members of the board as well as Levski.

Lucien was not roaring anymore, instead, he said in a very soft voice,

"As we can see, under some certain situations, the math questions we’re trying to solve exceed our experience and knowledge. We are therefore restrained, so is the development of math...

"So, maybe we can try to separate our understanding of math from the reality, and temporarily put aside their physical models and meanings; Maybe we can start from the purest axioms and concepts, and, through strict deductive reasoning, derive new consistent math systems logically. As for how to apply those systems, we can think about it later when there are needs...

"This is my personal belief."

Lucien mentioned the notion of axiomatic systems briefly, but did not dig deeper into it. He preferred to wait until most arcanists had accepted the concept. Everything took time.

Levski started applauding. His applause was lonely in the meeting room. He shared so many commonalities with Mr. Evans, and what Mr. Evans just said was exactly those things that he had been searching for — He had the impulse, and he could feel there were things he wanted to tell the public, but he failed to put them together into words. Levski felt that Mr. Evans just saved his geometry system, as well as his life.

Hearing the applauding, the look on Neeshka’s face changed several times. Finally, he slowly lifted his hands and also started applauding.

Following him, the rest of the board members all joined in. After an initial hesitation, their applauding became more and more sincere.

"So, is there anyone who still rejects Mr. Evans’ comment on Levski’s paper?" Asked Fernando after the applauding stopped.

Neeshka rubbed his white brow and sprang up from the chair. Looking at Levski, he said, "Please accept my apology. This is a paper that deserves Arcana Scepter, yet my arrogance and prejudice had deprived me of judgment. I completely agree with Mr. Evans’s comment."

"I would also like to apologize to you, for the past years of criticisms and attacks I imposed on you. Levski, you are a talented, outstanding mathematician. My stubbornness and shallowness had blinded me. You deserve Mr. Evans’s comment and Arcana Scepter." Using the manner of Arcana Above, Milina put her hand on her forehead and bowed deeply to Levski.

Levski knew that the board members would no longer object his theory, but he never expected that they would apologize to him in public. Instantly, his head buzzed, his nose felt sore, and his eyes became blurry. Seeing that the board members all stood up to apologize one by one, he was dumbstruck.

Was there sorrow? There was!

Was there pain? Of course!

But now all the pain and effort had paid off!

Levski’s eyes were red. His lips quivering, but not a single word could come out from them. Neeshka looked at Levski and sighed understandingly. Then he turned to Lucien, "Mr. Evans, I also owe you an apology. You are, without a doubt, a genius and authority in math. Your geometry system and Mr. Levski’s will for sure win Arcana Scepter!"

"Thank you." Lucien replied in a very low voice.

Milina noticed Lucien’s unusual behavior, and she asked, "are you alright, Mr. Evans?"

Lucien smiled wryly. "My throat hurts..."

It was never easy to mimic how the Lord of Storm talked!