A Discourse on Change
Contrary to his promise that he would teach me magic once I had mastered Frauvian, Benjamin had put off any real instruction for three full years.
It was a strange way to shape the studies of a pupil he had taken in after judging his magical talent.
He must have his reasons. I had pressed the doubt down with no more than that, telling myself the answer would come once the lessons began, and in the meantime I had given myself to laying a proper foundation.
I never thought I would hear that answer today.
Benjamin turned the first page of the calculus text and asked me, "The reason I haven't taught you the study of magic. Do you know it?"
"......I'm not sure. I wondered whether there might be some reason one ought not to learn magic too hastily."
"Magic carries no such outlandish restriction. If it did, I would have taken the magic books from you before anything else."
He was right. Now that I could read Frauvian with some ease, I had spent those three years secretly working my way through the books, and the spells I could actually cast had come to twenty-two in all — every one of them contained in the primer Benjamin himself had written, Type 1 Magic — Foundations.
So it isn't a restriction.
I followed the thought on. What if knowledge of that kind — mathematics, natural philosophy — were indispensable to the working of magic?
"Benjamin. As your pupil, I was taught only what I needed, and taught it intensively."
"That's right."
"Which is to say that if one means to take up the study of magic, a foundation is required... mathematics, natural philosophy, and the like."
"Hm."
"And without mastering that, one must not advance to the next stage of magic."
Benjamin nodded. "Close enough. It isn't that you mustn't learn it rashly, or that you'll be led down some wrong path. Nothing of that sort."
"Then what is it?"
"You stall."
With a thud, he roughly shut the window that had been standing a little open.
"You strike a wall, and no expedient will carry you past the foundations. Strain however you like — you cannot understand it, and neither application nor progress is possible. You have no choice but to stop where you are. And those who reach that pass, more often than not, no longer have the courage to go back and rebuild the foundations of their learning from the start."
Because it feels like going backward.
Benjamin's gaze settled, and he looked at me. "Well. You needn't worry on that account."
"Truly?"
"Yes. Provided you learn this as well."
Thock. He rapped the book, and the gesture drew my eyes to it. It was exceedingly thick, and badly worn — a volume he seemed to have used in his younger days. A book of several decades, at the least.
So he declared.
"A quiz. How is a discipline made?"
Benjamin held up two fingers.
"First. A genius appears and writes a new theory down in a single sweep. Second. Several geniuses put their heads together and bring a new theory into the world."
So a genius has to appear either way.
I answered at once. "Hm. Wouldn't it be the latter?"
"Why do you think so?"
"Because a theory must, at bottom, be rigorous. However great a genius someone may be, working alone they will make mistakes and leave gaps. There is a limit to what one person can do, so creating a discipline by oneself seems impossible to me."
"A decent intuition."
I had listened to Benjamin's private dialect long enough to know that his decent was a word of denial.
"Was I wrong?"
A nod, and another. Meaning yes.
Then the first is the right one? And yet the great monuments of learning did not seem the sort of thing a few geniuses could raise with their own hands.
Even as I thought it, Benjamin lifted a third finger.
"Third. A sheaf of papers a genius scribbled down as he pleased, and a sheaf that several geniuses produced while clutching their heads, lie discarded in some dusty corner — and the people who need them weave the two into one."
"......"
"That is the answer."
"It wasn't among the choices."
"Life is like that."
"Ha."
Whether my eyes went cold or not, Benjamin pressed on, unperturbed.
"Calculus is the representative case. The digression has run long enough. Let us begin."
He picked up my magic book in order to cast. True to a master who was a scholar of magic, he had a habit of using spells themselves as teaching tools.
A soft tap—
The most elementary of elementary spells. A point formed by condensing the water vapor in the air came to a halt in the void.
"There. A point exists in space. I made it a point for the sake of the example, but we could just as well change it into a two-dimensional figure, or a three-dimensional one."
The point shifted its shape — equilateral triangle, square, cylinder — and then returned to a point.
"Geometry."
"Yes. But doesn't it feel a little dull?"
It did. The thing one ordinarily expects of magic was absent from Benjamin's Point.
"There's no movement."
"Then let us move it. Concentrate. Consider how this motion ought to be described."
Sssk— the point flew through the empty air, at a constant speed.
"What do you make of it?"
"Couldn't it be written as a linear function? With time along the horizontal axis and position along the vertical."
"That was natural philosophy, wasn't it. And speed?"
"The quantity given by dividing distance by time."
A formula I remembered clearly. But Benjamin grinned, and that mischievous smile, too, was a form of negation.
"By that definition, the slope of the function is the speed itself. But the motion of the world we live in is not like this."
In that instant the blue point burst into flight in every direction at once. It accelerated and slowed, drove forward and reversed, climbed and dropped, and at last even traced a spiral.
And in the midst of it Benjamin asked again, "How are we to describe the speed of the point now?"
"......"
"The slope, yes. But how do you find the slope of a curve?"
The point's motion fell into slow motion.
You cannot find the slope of a curve. Then what?
"You divide it."
At that moment three graphs appeared in the air, taking time as their horizontal axis, their values renewing as the moment passed. One was a graph of position, locked in step with the point. Another was a graph of velocity. And there was one more.
"'The method of dividing without end.' That revolutionary method went by many names. The method of fluxions, and the rest — even the geniuses who invented it could not agree. It is different now. The proof was completed some eighty years ago."
"The discipline that describes the motion of all things in the world. Its first step."
Benjamin's eyes flew wide, and my mind went light and far away.
Winter was over. The weather had at last become fit for wandering out of doors.
A morning off. At the hour when Benjamin was usually sunk in a nap, I went out into the yard, and Dorothy came with me.
It would have been about two years before. I had seen her swinging a wooden sword with the sweat pouring off her, and since I was then complaining of a severe want of stamina, I had asked her to teach me the exercises of a soldier. The training club that began that way — suspended in winter, on account of the cold — was now entering its third year. In an age when guns and warships ruled the world, I could not quite see why we drilled with wooden swords, but it did improve my stamina, and so I had nothing to say about it.
I steadied my breath and swung. Thwack — a clean, bright sound. Dorothy had turned the wooden sword aside with ease.
Edge against edge. Wood or not, a blow hurt badly, and so I had to meet her with a resolve approaching a fight to the death. That I had not taken so much as a single bout from her of late only spurred my competitive spirit further.
"Here I come."
Dorothy twisted her stance. A sharp overhead cut came whistling in, splitting the air, and I raised my blade on the diagonal to meet it. Unlike me, who tried to overwhelm by strength, hers was an attack with a flow to it. It looked as though I could shed it cleanly.
"Ngh—"
I could not. With a grinding creak the blades locked, and wood-dust scattered. I had thought the block a good one, and still my wrist went numb.
The bout ran on, one exchange and then another. I looked at Dorothy and could only shake my head. A light outfit, nothing but a shirt over her underclothes, and a face entirely calm. It was the same as always — and yet, whenever she showed skill of this order, it was plain that Dorothy, too, had lived no ordinary life.
Only after we had traded blows several times more did the sparring at last come to a pause.
"Ugh... let me rest a moment."
"I'll rest too. You've improved a great deal."
We sprawled on the lawn without ceremony, neither of us standing on pride. For the first few months Dorothy had been perfectly fresh even after a bout, but now that my frame had grown, she actually spent a measure of stamina of her own.
"Haa......"
I stripped off my top, and Dorothy undid the buttons of her shirt as well. After three years under the same roof, there was nothing left between us to be shy about.
At the end of my gaze was Dorothy, her tied hair let down, wiping the sweat away. Across the white skin briefly laid bare ran countless scars — cuts and lacerations, and patches where the skin had discolored. The most striking of them was a scar near the left trapezius, as if she had been seared by fire. It was the reason she favored an officer's dress uniform, which left little of the skin exposed.
Dorothy rarely spoke of herself, and she disliked doing so. There was, then, no way for me to know her past. Only this was certain: the unhurried life in this two-story house was the life she had wanted. As one of the Oslo household, she must have been tasting a happiness of her own making.
I was the same. If there was something I wished for, I knew by now that Benjamin would not stint on his support of it. Even the frugal life of the Oslos was, by my measure, abundance itself. Reckoned purely in practical terms, remaining in this house was the best choice there was.
There was only one problem.
A person is not a creature who reckons by practicality alone.
"......"
All at once I thought of the savings tin I kept with such care in a corner of my room. A tin canister holding eighty pounds.
Still.
I want to leave this land. I want to set my foot in the Federation, the fountainhead of magic.
With no justification, I could only devote myself to receiving their kindness with gratitude. But if a justification were ever to arise....
The thought broke off there.
"Let's go in."
"Ah. Yes."
The trailing edge of spring, when I was leafing through the calculus text somewhere near its middle. It was a day I had been turning over how the usefulness of differentiation might be applied to magic.
"By any chance, is an Imperial named Eugene residing at this house?"
Five soldiers of the Federal Army knocked at the door of the Oslo household, as if to prove that no secret lasts forever.