Anatoly Dnieprov - The Maxwell Equations

Translated from the Russian By Leonid Kolesnikov

It all began on a Saturday evening when tired from my mathematical pursuits I took up the local evening paper and came across this advertisement on the last page:

Kraftstudt & Company Ltd.

accept orders from organisations and individuals for all manner of calculating, analytical and computing work. High quality guaranteed. Apply:

12 Weltstrasse

That was just what I needed. For several weeks I had been sweating over Maxwell equations concerning the behaviour of electromagnetic waves in the heterogeneous medium of a special structure. In the end I had managed by a series of approximations and simplifications to reduce the equations to a form that could be handled by an electronic computer. I already pictured myself travelling up to the capital and begging the administration of the Computer Centre to do the job for me. For begging it would have to be, with the Centre working full capacity on military problems and nobody there giving a damn for a provincial physicist's dabblings in the theory of radio-wave propagation.

And here was a computer centre springing up in a small town like ours and advertising for custom in the local paper!

I took up the receiver to get in immediate touch with the company. It was only then I realised that apart from the address the advertisement gave no particulars. A computer centre not on the telephone! It just didn't make sense. I rang up the editors.

"Sorry, but that was all we received from Kraftstudt," the secretary told me. "There was no telephone in the ad."

The Kraftstudt and Co. was not in the telephone directory either.

Burning with impatience I waited for the Monday. Whenever I looked up from those neatly penned equations concealing complicated physical processes, my thoughts would turn to Kraftstudt Co. Men of vision, I thought. In our time and age when mankind endeavours to clothe its every idea in mathematical garbs, it would be hard to imagine a more profitable occupation.

Incidentally, who was this Kraftstudt? I had been resident in the town quite a long time but the name rang no bell. As a matter of fact, I did vaguely recollect having heard the name before. But I couldn't remember when or where, no matter how hard I jogged my memory.

Came the Monday. Pocketing the sheet of equations, I started out in search of 12 Weltstrasse. A fine drizzle forced me to take a taxi.

"It's a goodish way off," said the cabby, "beyond the river, next door to the lunatic asylum."

I nodded and off we went.

It took us about forty minutes. We passed through the town gates, went over a bridge, skirted a lake and found ourselves in the country. Early green shoots could be seen here and there in the fields along the unmetalled road, and the car stalled between banks of mud every now and then, its back wheels skidding furiously.

Then roofs appeared, then the red brick walls of the lunatic asylum standing in a little depression and jocularly referred to in town as the Wise Men's Home.

Along the tall brick wall bristling with bits of broken glass ran a clinker lane. After a few turnings the taxi pulled up at an inconspicuous door.

"This is Number Twelve."

I was unpleasantly surprised to find that Kraftstudt Co.'s premises were in the same building as the Wise Men's Home. Surely Herr Kraftstudt hasn't ganged up the loonies to do "all manner of mathematical work" for him, I thought-and smiled.

I pressed the doorbell. I had to wait long, the better part of five minutes. Then the door opened and a pale-faced man with thick tousled hair appeared and blinked in the daylight.

"Yes, sir?" he asked.

"Is this Kraftstudt's mathematical company?" I asked.

"Yes."

"And you advertised in the newspaper?"

"Yes."

"I have some work for you."

"Please come in."

Telling the driver to wait for me, I bent my head and slipped through the door. It closed and I was plunged in complete darkness.

"Follow me, please. Mind the steps. Now to your left. More steps. Now we go up"

Holding me by the arm and talking thus, the man dragged me along dark crooked corridors, up and down flights of stairs.

Then a dim yellowish light gleamed overhead, we climbed a steep stone staircase and emerged into a small hall.

The young man hurried behind a partition, pulled up a window open and said:

"I'm at your service."

I had a feeling of having come to the wrong place. The semi-darkness, the underground labyrinth, this windowless hall lighted by a single naked bulb high at the ceiling, all added up to a thoroughly odd impression.

I looked around in confusion.

"I'm at your service," the young man repeated, leaning out of the window.

"Why, yes. So this is the Kraftstudt and Co. computer centre?"

"Yes, it is," he cut in with a trace of impatience, "I told you that before. What is your problem?"

I produced the sheet of equations from my pocket and handed it through the window.

"This is a linear approximation of those equations in their partial derivatives," I began to explain, a little uncertainly. "I want them solved at least numerically, say, right on the border line between two media This is a dispersion equation, you see, and the velocity of radio-wave propagation here changes from point to point."

Snatching the sheet from my hand the young man said brusquely:

"It's all clear. When do you want the solution?"

"What do you mean-when?" I said, surprised. "You must tell me when you can do it."

"Will tomorrow suit you?" he asked, his deep dark eyes now full on me.

"Tomorrow?"

"Yes. About noon"

"Good Lord! What a computer you've got! Fantastic speed!"

"Tomorrow at twelve you will have your solution, then. The charge will be four hundred marks. Cash."

Without saying another word I handed him the money together with my visiting-card.

On our way back to the entrance the young man asked:

"So you are Professor Rauch?"

"Yes. Why?"

"Well, we always thought you'd come to us sooner or later."

"What made you think so?"

"Who else could place orders with us in this hole?"

His answer sounded fairly convincing.

I barely had time to say good-bye to him before the door was shut on me.

All the way home I thought about that strange computer centre next door to a madhouse. Where and when had I heard the name of Kraftstudt?

The next day I waited for the noon mail with mounting impatience. When the bell rang at half past eleven I jumped up and ran to meet the postman. To my surprise I faced a slim pale girl holding an enormous blue envelope in her hand.

"Are you Professor Rauch, please?" she asked.

"Yes."

"Here's a package for you from Kraftstudt's. Please sign here."

There was only one name-mine-on the first page of the ledger that she held out for me. I signed and offered her a coin.

"Oh, no!" She flushed, murmured good-bye and was gone.

When I glanced at the photo copies of a closely-written manuscript I couldn't believe my own eyes. From an electronic computer I had expected something entirely different: long columns of characters with the values of the argument in the first column and those of the solution in the second.

But what I held in my hand was a strict and precise solution of my equations!

I ran my eye through page after page of calculations that took my breath away with their originality and sheer beauty. Whoever had done it possessed an immense mathematical knowledge to be envied by the world's foremost mathematicians. Almost all the modern armoury of mathematics had been employed: the theory of linear and non-linear differential and integral equations, the theory of the functions of a complex alternating current, and those of groups, and of plurality, and even such apparently irrelevant systems as topology, number theory and mathematical logic.