Chp.2: Math in magic

Maldor led Haku away from the marching army, far ahead of where it was at the moment; she took her brother to the top of a large rock, where Isaac was waiting for them. Haku was somewhat confused by the presence of the mage: he and his sister could run extremely fast, but he doubted that Isaac could do the same. Even using a magic to fly he shouldn't have been that far along the way yet. "How did you get here?" he asked him.

"I brought him" Maldor answered in his behalf. "We needed a place to work in peace, so I took him here"

"Did you bring him!?" Haku frowned at those words. From a dragon's point of view, carrying another creature on its back was an extremely offensive act. Dragons, after all, were proud creatures and didn't accept being treated like pack animals. The only times a dragon of Haku's family had agreed to be ridden had been when they'd used an avatar, and even then the people who rode them were very close acquaintances of theirs and they didn't even really ride them, but more they clung to their backs and prayed not to fall. Haku couldn't believe that Maldor had allowed Isaac to desecrate her perfect body by allowing him to not only touch her but even sit on her back and be carried away by her.

Isaac evidently noticed his disappointment, and with his knowledge of dragons it wasn't hard to see why. "Yes, that's right! He grabbed me in his jaws and dragged me here! He tossed me around a bit, and I almost vomited, but he saved me a long way!"

The anger in Haku's eyes subsided. "Did she transport you using his mouth?"

Isaac laughed. "Of course! You didn't believe that your sister had allowed me to ride her as if she were any mule, did you?"

After four months of living there together, the man had learned that when you disagreed with dragons, it was best to nip the argument in the bud with a quick, pithy joke or sentence. And technically he hadn't lied: Maldor had actually carried him in her jaws. And indeed he had nearly vomited.

Maldor looked at Haku rather offended. "Really, brother? Did you think I'd let a newcomer ride me?"

"Well... from the way you said it, it seemed that..." Haku tried to justify himself, but then, noticing his sister's increasingly frown, he preferred to let it go. "Sorry, Maldor. I shouldn't have thought so badly of you"

"Mph! Asshole! You're lucky I'm in a good mood today, or I would have punched you already!" Maldor snapped, even though he continued to glare at Haku rather evilly.

Isaac was somewhat surprised by this interaction. He knew that being ridden for a dragon was terribly offensive, but he didn't imagine it was so offensive that even the placid and gentle Maldor would get angry. He was sure that Haku's mistake would have been rather laughed at by his sister, but instead his reaction was much more irritated. Probably, he reflected, the books didn't really record how offensive dragons actually considered carrying someone on their backs. In fact, from their reactions, it looked like Haku had just called his sister a whore.

Haku looked down and didn't retort, which was pretty rare of him, another sign of how offensive what he just thought was from a dragons point of view. "Um! So, what did you wish to show me, sister?" he asked trying to change the subject.

Maldor glared at him again, but she still agreed to talk about the important things (although a small hint of anger remained in his eyes). "Brother, Isaac and I did it! We found a mathematical formula capable of calculating the energy needed for spells!"

Haku's eyes widened. Maldor had told him about this project of hers, about her attempt to 'decipher the universal language', and consequently he knew that she was trying to combine mathematics and magic, but since his sister had never told him anything again he had assumed that this search was destined to be still long. Instead, it seemed, he was far ahead of what he expected! "Really?"

"Yes, really!" Isaac exclaimed almost with tears in his eyes. "We tested it with hundreds of spells and they all gave the same result! If this thing were known to the rest of the world, Maldor's name would become legendary! She would be known throughout the inhabited lands as a pioneer of magic, and surely kings and emperors would fight to have her in their court! I wouldn't be surprised if they would even marry a prince!"

Haku knew Isaac wasn't exaggerating. He himself saw very well the potential of combining magic and mathematics: if this result had been achieved, then using any type of spell would have been much simpler and above all efficient. Knowing in advance how much energy to use, how much mana was needed, how fast this spell was, and so on would give all categories of mages a huge advantage. Even if these concepts were already known today, they were based on the personal experience of each mage; therefore a system that had to be learned individually and from scratch from generation to generation. But if it had been possible to calculate the magic, then everything would have changed: the new mages would have known a priori the necessary energy, the speed of the spell, all the elements that were needed in practice, and they would have been able not only to use the magic much more easily but also study again much faster. Haku was pretty sure that if Maldor were born as a newcomer and achieved the same discovery, she would become one of the most well-known figures in all of the history of magic and all the kingdoms and empires of the world would fight to have her in their court. Surely she wouldn't marry a prince destined to inherit his nation's throne, but it wasn't so unlikely that she would be offered the hand of princes who were third or fourth in the line of succession.

"You tested it on hundreds of spells? So you had discovered the formula a while ago?" Haku looked at Maldor. "Why didn't you tell me about it? I could have helped you"

Maldor was clearly still angry at first, but nevertheless she blushed a little in shame. "You already had too many problems on your mind... and I wanted to be sure that my intuition was right before telling you about it. I didn't want to give you a wrong formula. In short, I understood how it worked by pure chance... it could very well be just a very big mistake"

"It is said that all great discoveries come from chance" Isaac said trying to comfort her. Maldor smiled at him slightly.

Haku snorted. "He's right. Next time don't worry that I'm too busy or that your intuition might be wrong. I'll always be there to help you sister"

Maldor seemed undecided about continuing to offend, but in the end she decided it wasn't worth it. So she finally smiled, no longer holding back. "All right, brother. So, here's what I called the 'magical energy formula'!" she said moving a small rock and revealing a mathematical formula drawn with chalk, which contained the following words:

Em = Cm × Qm × (v1 × v2)

Haku observed the formula with an inquisitive gaze. His calculating mind was screaming with excitement at a formula that seemed to answer his long unanswered question about exactly how magic worked. "Can you explain it to me?" he asked.

"Sure" Maldor said. "So, as you can see, this formula is used to calculate the amount of energy needed to cast a spell, here marked as 'Em', precisely 'Energy for magic'. After several tests on various spells I understood that there are three elements needed to determine the amount of energy required: the velocity at which mana is gathered, the velocity at which that mana is transformed into a spell, and the amount of mana that must be moved to create the spell. First you need to multiply together the two velocities, which I marked here as 'v1' and 'v2'. Strangely I noticed that they are always identical, so it could also be simplified with 'velocity squared'... but since I couldn't test it on each single enchantment existing in the world, I can't be sure, so I preferred to leave them separate. Then we need to multiply the result by the quantity of mana that needs to be collected and moved, which I've marked here as 'Qm' for simplicity"

Haku was extremely interested. His eyes weren't leaving the formula. "And this one?" he asked pointing to the 'Cm' sign.

"Uh... that's the only detail I couldn't figure out where it came from" Maldor admitted in embarassment. "After trying the various formulas over and over, I realized that to get an exact result you always had to multiply everything by 1/2. I didn't understand why, but this was true for all the spells I tested, so I just called that value 'constant of magic', precisely 'Cm' to abbreviate"

Haku wasn't too concerned about that detail. While studying mathematics, he had noticed that in many formulas there were constants that didn't seem to make logical sense. It wasn't so important to find out where they came from as long as they proved to be valid for any calculation. While Haku would have liked to know what the 'constant of magic' was due to, he wasn't concerned that it would cause the spell to fail.

"We've used this formula before to enhance spells we already knew" Isaac said excitedly. "Since we already know how to calculate all the necessary factors, it was easy to see where we were going wrong previously. For example, and I'm sure you'll be happy with this, we've boosted some of the spells needed for plant growth! And Maldor planned to try and enhance runes as well by applying this formula to the system you discovered years ago!"

Haku smiled proudly. In fact, now that they had that formula, getting better at magic would be much easier. Before, both Maldor and Isaac and everyone else had to try and try their spells dozens, hundreds, sometimes even thousands of times before getting a satisfactory result. But now, already being able to calculate the exact amount of all the ingredients needed as well as the energy needed to use them, enhancing the existing spells was very easy. That was the power of mathematics: to make complicated things simple. Just as a builder of houses would have taken years to finish even one, as he would have had to try and try again how to stack the stones to keep everything stable, relying on a project would have taken a few months, if not weeks. And the projects, obviously, were obtained thanks to mathematical formulas, which made it possible to understand how much weight was to be placed in each corner of the building.

It was the same with magic. With a formula that made it possible to understand the correct functioning of any spell, strengthening himself would be extremely simple. It would not be necessary to create any new spells: it would be enough to apply the formula to the ones they already knew, to understand where they were wrong and how to correct any mistakes. "It's perfect" Haku murmured with deep admiration in his voice. "Maldor, you are amazing"

His sister blushed visibly. It was just a few dark spots on the face, but they were a sign of extreme redness in dragons. If Maldor had been a newcomer, the color of her entire face would have been no different from that of a ripe tomato. "It's not all my fault. It was Isaac who gave me the idea..."

"Come on, Maldor, don't be silly" Isaac scolded. "I just gave you a magical jewel. You're the one who had the intuition. This is thanks to you, only and exclusively yours. Your brother is right, you're amazing. Take your credit"

Maldor seemed about to faint from the cascade of compliments. She even started fanning himself with a paw in an attempt to dissipate the heat on her face. "A-Anyway, we're not done yet!" she exclaimed shrilly. "Obviously, I also derived the inverse formulas from this main formula. I'm going to show them to you now, so listen closely..."