Directionality
The answer to a question posed by a lecturer is, as a rule, meant to carry some metaphor within it.
That is especially true when the question has no plain answer at all—when it asks after the essence of being, or of a concept, or some other metaphysical thing. The aim is never really to obtain an answer. There is no self-evident reply to the question What is life? And yet a lecturer asks it anyway, for a simple reason. Sometimes a question threaded with analogy draws an audience's attention more surely than any straightforward explanation.
That was what I had assumed. Until now.
A dry laugh escaped me as I looked at the words on the blackboard.
Vector. A thoroughly academic term for a physical quantity possessing both magnitude and direction.
At a glance the definition can seem complicated, but in truth it is remarkably simple. To describe nature, two kinds of physical quantity are required: one that merely expresses an amount, and one that expresses direction and magnitude together. Mass is an example of the first; force, of the second. And that latter quantity—the one that has both direction and magnitude—is what we call a vector.
Setting the finer points of the definition aside.
"……."
Why was that coming up now?
She had asked what magic was, and I had been utterly certain some figurative turn of phrase was about to appear. I was not the only one thrown off. Little by little the lecture hall began to fill with hollow laughter.
Then a student raised his hand. He was the first to have been called on, and the one who had just been dressed down—a boy with a shrewd look about him.
"Professor. I request a correction regarding your remarks."
"What do you mean?"
"We are freshmen of the Federal Magic University. Elementary concepts such as vectors are something we have already mastered."
"Oh?"
"We failed to give a fitting answer because the question was vague. As for the contempt your previous remarks showed us, I would ask that you correct it, but……."
Slick smiled a slick little smile.
Heard in passing, it almost sounded like a fair argument. Most of the class stayed silent, but a few who looked like Slick's friends lifted one corner of their mouths in agreement.
Even so.
Even if he happened to be right, it was no attitude for a student. They had already cut the incoming class down to seven hundred, and someone like this had still made it through.
I was sighing inwardly when Luise stepped in front of Slick, head tilted, wearing a look of pure incomprehension.
"You wouldn't happen to be an auditor, would you? Or a student here on a visiting tour?"
"Ah—no, but—"
"Otherwise I can't imagine why you would say something so foolish."
"……!"
A flat, offhand mutter. Luise's voice carried exactly as a professor's should; all seventy students heard that aside with perfect clarity. Slick's face flushed red, then went pale, but whether it did or not was of no concern to her. She returned to the lectern.
"I will say two things. First. In my lectures, I attach neither empty words nor unnecessary asides. And second. This is rather embarrassing to say with my own mouth, but."
Luise rolled up her sleeves and parted her lips.
"The people who will stand before you from now on are university professors. That means, at the least, holders of a doctoral degree. Do not confuse this place with school."
Silence swept the hall. The excitement that had lingered since the entrance ceremony sank in an instant. Only now did it truly feel as though we had become students of the Federal Magic University.
"Come now. This is a delightful hour of magic studies. Don't sit there so wilted."
Into that bright stillness, Luise laughed, clear and easy. In the next moment she shifted her grip on the chalk, red hair flying, and began to write.
"That magic is a vector is no joke. At least, not in the classical magic you will study in your first year."
"Take an example. I expect you all know the Type 1 spell Disc."
I nodded. It was a spell I could fairly call my specialty, and the foundation of my innate magic, Flowering.
Luise turned back toward us and posed another question.
"What conditions are required to cast Disc?"
Silence returned to the lecture hall, but its character was different this time. Every student wore the look of someone thinking in earnest. So did I.
The conditions of Disc.
Years ago, when I had cast magic by sense and instinct alone, I could not have answered. Now, after Benjamin's teaching, I could arrive at an answer of my own. First, as with all magic, one must set the amount of mana to be invested. Then, through rotation, one forms the shape of the disc. That much is easy. The problem is how to make the disc into a perfect plane. That part I had always managed by instinct, and so I could not reduce it to a condition.
The other students, too, were slow to raise their hands. There is a considerable gap between actually casting a spell and explaining its theory in orderly logic.
After a while, Luise spoke.
"Setting the mana, and the rotation—you will have learned those at school, so you know them well enough. The remaining element is this."
She drew a long vertical line down the blackboard, then sketched an arrowhead at its top.
"A vector. A normal vector, to be precise."
Direction and magnitude. A line that possesses both.
I was puzzled. What did that have to do with making a disc into a plane? Used that way, the vector would make the axis of rotation easier to picture, but it would be no help in forming a plane.
Or so I was thinking.
"There is more than one way to define a plane. And in Type 1 and Type 2 magic, using a vector is the most effective. Think about it."
The equation Luise had written was the equation of a plane in three dimensions. And I knew from experience that it was not something one could actually use when casting.
"It isn't intuitive. You might say it contains the result, not the principle. A vector is different."
In the next moment she wrote a new equation. In boldface—that is, in vector notation.
"You do not need to understand this equation at all right now. What matters is that this concise expression is of a piece with the principle of casting."
"……."
As she said, it was a concept we had not yet learned. Even so, I tried to imagine the principle the equation held. A starting point. A vector n extending from that starting point. A dot product of zero means perpendicular to n. And the equation must name every point perpendicular to n, free in both distance and direction…….
Ah.
I could feel it. This was a way of defining a plane that I had never known—one profoundly friendly to magic. Friendly enough that I wanted to pull out my grimoire and try it at once!
As if speaking for that wish, Luise drew out a grimoire of her own. A professor's grimoire, no less. Unlike students, who for convenience double their lecture texts as grimoires, a professor may give a grimoire any form she likes. It need not even be a book.
The fountain pen in Luise's hand shone blue.
Shwaaa—!
Droplets spun. The disc, unfolding nimbly in the space of about five seconds, formed a plane so keen it drew a gasp of wonder all on its own. Had it stopped there, there would have been little to astonish anyone. The reason my eyes went wide was that its axis of rotation had tilted some forty-five degrees, and the disc had begun to spin at speed.
The whole of it entered precession.
"Wh—what?!"
"That was possible……?"
Cries broke out here and there. Luise parted her lips.
"In the Type 2 Magic I am teaching this term, you may not have many occasions to apply direction. We will deal mainly with optics, heat, waves, and the like. But whenever I take the freshmen's first class, I tend to speak about direction."
She walked slowly, sweeping her gaze across the faces of the seventy students. The disc, of course, was still whirling in the middle of the hall.
"The Federation's educational system is exceedingly well paved. Follow the road, and you can receive an education of a very high standard."
But.
"If the road you have walked until now was a single narrow path, then the road that will open before you at the Federal Magic University is a fork."
The university is no longer kind. Not even the Federation's foremost institution for the study of magic is an exception. Of the freshmen, those who go on to walk the path of a magic scholar amount, at a generous estimate, to about three in ten. No one guarantees anyone's future. Every decision rests with oneself. What matters is effort, and luck, and will.
The instant Luise smiled, the madly spinning disc collapsed into a handful of droplets. Left as it was, gravity would have drawn it down to stain the floor without resistance.
Paaak—!
In an instant it split into thousands, tens of thousands of tiny droplets and scattered into the air.
This was Type 2 magic. A formless miracle that, until a few centuries ago, had not even been called magic, but something along the lines of a divine art.
When Luise declared it so, I was sincerely in awe.
A moment later, I found nearly every student clutching their head and letting out a heavy sigh. Why were they all like this? I was still wondering when the girl in the seat beside me muttered under her breath.
"What the hell kind of bullshit was that…… there goes my life."
Only then did I understand. The equations that had run wild through the middle of it, and the magic demonstrated before our eyes that left common sense behind. In other words, we were going to learn all of it from here on—and it was going to be on the exam.
"Ah."
Deeeng—
Just then the bell rang, marking the end of the lecture. Why did it feel like a bell announcing the start of a new road of penance?
Fortunately, the days ahead at the magic university did not seem destined to be nothing but penance.
Today's second lecture.
A general-education course every freshman was required to take. Its reputation was that it offered some comfort to students crushed beneath the difficulty of magic studies. The lecture was held in the grand hall I had visited once before, as an examination site.
In a somewhat lighter mood, I opened the classroom door.
And.
The moment I stepped through and into the room, my eyes met those of a brown-haired girl.
She was in a wheelchair.