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This text is published in accordance with the Arcane Schools of Magic curriculum for both the Arcane classes and the Mechanical classes as a foundational textbook for reference and education.
Introduction to Math in the Realm of Mathemagics
For a young mage ready to enter the field of Mathemagic, all of the careful planning, complex symbols and specialized equations probably seem very intimidating. Perhaps you’ve watched the older mages in your community doing grand calculations for spells that you think you cannot possibly follow. Or maybe you aspire to join the mecha-magicians, who engineer and build with such close precision to create their glorious automatons. Well have no fear, young mage, for in this book you will learn everything you need to know to get started in Mathemagic.
Consider the humble bumblebee. According to all known laws of aviation, there is no way a bee should be able to fly. Its wings are too small to get its fat little body off the ground. The bee, of course, flies anyway because bees do not care what humans think is impossible. What the bee does not know, is that its fat body and small wings are perfectly designed in cubes and rectangles of divine structure, a key component of Mathemagical engineering. These structures are imbued with magic and allow us and other creatures of nature to harness and use it, even if they do not understand that they are doing it. Magic not only allows bees to fly, it also allows plants to flourish and living creatures to live and think. Arithmetic, and by extension, magic, is woven into every facet of our world.
The story of the Arcane is more than a catalog of finished Mathemagical achievements. It is a story of passion and collaboration of mages and non-mages attempting to understand the functions of our world and apply them to create a beautiful, prosperous future for their people. This textbook will cover all the foundations that every Mathemagician needs, from essential equations, to concepts, history even to philosophy. In addition to computational proficiency, this book aims to give students the tools to understand unfamiliar theorems and create their own. Blending history, concepts of larger Mathemagics and the equations that go with them will aid in memory and understanding, providing the student with the confidence to excel in a mage's career.
1.1 Theorems, Equations and Mathemagicians
Lesson 1: "Yenergy"
The most famous equation in all of Mathemagics, Mage Issac Newton's Theory of Special Relativity describes the relationship between math, energy and magic, therefore known as "Mathemagic". The formula he devised, Y=MC², is used to not only explain the process through which energy can be turned into "Yenergy", but also calculate the final Yenergy output.
Newton considered the possibility that all the world's radiant energy, usually passive, is not in fact stagnant, as had been previously believed. He observed the perfect unfoldings of nature, discovering patterns in what held energy, and what did not. The structures that create life in plants and animals and people compared to the ragged, element worn coldness of rock and dirt. The perfect balance of a circle, even in all its measurements; the intuit engineering of a triangle, holding itself together with a mere three sides; the possibility for infinite shapes through precisely matching lines, ultimately discovering the channeling possibility in the continuum between the two. Newton theorized that that passive--but not stagnant--energy could be drawn into physical form through relativistic prompting, using the math as the equivalent key.
It works like this. Radiant Energy (C) is multiplied by channeling it through Perfect Measurement (M) which compounds it into physical, visible and usable form, (C²), thus creating magic. He coined this energy of magic, "Yenergy".
But how do we get these hypothetical variables for the equation? In Newton's time, Yenergy and the Theory of Relativity were mostly constrained to being mere theories, but they paved the way for the Mathemagics we know today, inspiring others to discover the secrets to Mathemagics and what exactly is Perfect Measurement. It started with circles and triangles, the simplest forms of Mathemagic to produce and therefore the foundations for all young mages. It is no act of chance that defensive spells are made with these two elements combined--they are, at their core, a wall of pure, condensed Yenergy. When you look, you will see that every shape of Mathemagic is a combination of the triangle and the circle, the continuum that blossoms in its complexity and capability, held together by these clasps of foundation.
Lesson 2: Pythagoras' Theorem (Pythagorean Theorem), Y=mx+C
In the earliest days of the old country, (5th century BM (Before Magic)), the philosopher and polymath Pythagoras discovered the perfect ability for shapes to be combined. He realized that the two sides of a triangle equal the length of the hypotenuse, which was later used to represent how the combination of mathemagically perfect shapes can add each individual's base capability to the end output. It works like this:
Y (Yenergy) is equal to Mathemagics (m) multiplied by the total number of lines in the shapes used (counting overlaid lines as one and intersecting lines as individuals) (x), plus the base amount of Radiant Energy used (C).
Lesson 3: Einstein's Theorem of Radiance
Perhaps the most important element for any of these calculations is the amount of Radiant Energy that exists passively throughout the world. This is represented as the standard variable C, meaning that any capital C in an equation represents the sum of this equation, representing the amount of Radiant Energy in any given space. Similarly, the capital Y always stands for "Yenergy".
Radiant Energy exists as a force similar to magnetism. d and v are the direction and the velocity of the energy field, respectively. As magic is cast, the velocity and direction both change to follow the casting path of a spell. When passive, the direction is 0, making the velocity also 0. What this means is that the direction and velocity do not need to be calculated when casting from rest. C is perpendicular to the plane that contains both v and d. Think of it as energy moving outward from this plane. At rest, this plane does not have any direction. Technically, the force would not exist without this plane to be perpendicular to, however energy contains its own force, as Phlogiston particles vibrate constantly, but for the purposes of standard of Mathemagic, this is not applicable, as it occurs on too small of a scale to be viable. This does mean that when magic is cast, it starts creating something like a magnetic field, where the Phlogiston concentration is low and dispersed in one area and highly concentrated in others. This pushes the unused particles towards each other and away from the highly condensed area where magic was cast, simultaneously drawing the Radiant Energy out of living materials in the area to compensate. You likely already learned about diffusion and how molecules naturally move from areas of high concentration to low concentration to retain equilibrium. Phlogiston works similarly. This explains the phenomena of magic burnout, and in extreme cases, magic withering. Thankfully, a very large amount of magic would have to be cast in a space in a short amount of time to have a noticeable effect. More experienced mages can increase their natural capacity for Phlogiston over time, allowing them more stamina from the raised capacity and the henceforth lessened tendency for Phlogiston to diffuse. Remember, practice increases your stamina and is good for your long term health!
To find the amount of Radiant Energy (C), we use C = qvd sinθ, where v and d calculate the velocity and direction of energy, within the radius a mage can and plans on drawing from (q), and sinθ, representing the general amount of expected Radiant Energy in the atmosphere. Remember: just because v and d may be 0, the radius must still be multiplied by sinθ. Helpfully, Einstein calculated the standard sinθ for us, which can be understood as 3.14. That decimal is extended when performing various levels of Mathemagic, but for most purposes, 3.14 works fine.
You may recognize this number as π, the mathematical constant approximately equal to 3.14159, that is the ratio of a circle's circumference to its diameter. Talk about Mathemagical perfection!
Another side effect of casting magic, as discussed above, is the tendency for Phlogiston to be repelled from areas of casting. Therefore, when casting, the mage must be aware of the direction and velocity of the sum of previously cast spells, as that will determine whether your v variable is a positive or negative value. When it is negative, Radiant Energy is moving away from you, and will be more difficult to utilize. If more powerful spells have been cast away from you than towards you, it may not matter who cast last, that value may still be negative. On the other hand, if a spell has recently been cast in your direction and the Phlogiston particles dispelled, you will likely have a wealth of energy available to collect if you act quickly.
1.2 Concepts of Mathemagic
Phlogiston
Phlogiston is the name of the particles magic is made of. They are incredibly high energy and too small to see with the mortal eye. Some theorists say that the manifestation of magic is due to enough Phlogiston coming together to be perceived visually, while others suggest that the the powerful vibrations of these charged molecules in such close spaces produce the light and heat of magic, and that we are not seeing he Phlogiston itself. On top of those most popular theories come many others, but the truth is, there is very little we know about Phlogiston as the root of magic. Some magicians say that energy doesn't exist as molecules of any kind.
We do know what Phlogiston--or energy, as it is referred to on a non-molecular level--comes from. Energy is continuously emanated from sources of life, and curiously, combustion as well. That means that all humans have an internal well (so to speak) of Phlogiston, or energy, that can be drawn from if taught how. Some people are able to intuit this aility, and some creatures and people have more or less Phlogiston than others.
Casting and enchanting using one's own Phlogiston is not recommended. In small doses, if necessary, it can be done, and when casting you will always be utilizing your own energy. It is much more efficient to draw energy from the passive Phlogiston around us, or from items and automatons that have been made to contain it in larger quantities. Once upon a time, Phlogiston was drawn from other living creatures, including plants, but as knowledge of the Arcane has progressed, the need to do that has become nearly obsolete.
So what is an ideal Phlogiston source?
An ideal Phlogiston source is defined as the two-terminal element in which the charge across these terminals is specified at every instant of time. Also, charge is independent of the current through the source, that is the current can flow in any direction through the source. This definition is expanded on further in mecha-magic, but essentially, energy or Yenergy can be deposited into an object and retrieved from it, while remaining contained inside otherwise. There is no distinct and specified flow path for magic energy. A source may be able to open, close or partially open its "terminals", such as how living creatures take in and radiate Phlogiston constantly, just like breathing. Mages have created items, such as some automatons, that can wholly contain Phlogiston imbued in it, or that can passively diffuse it on a larger, more concentrated scale then most objects. On the other hand, fire is an open Phlogiston source. It has no terminals and it cannot change the rate at which Phlogiston is expelled, although it does expel a significant amount for as long as it is burning.
Does this mean we can create Phlogiston?
No. As with all matter, Phlogiston cannot be created or destroyed. Consider what happens when a spell goes wrong and explodes or disperses. Is that energy gone completely? No, you can still sense it around you, even more so than usual because it is still closely packed together even as it disperses. When absorbing and releasing Phlogiston, living organisms don't destroy and create new Phlogiston, it cycles through them in currents that connect all living things.