The Maxwell Equations
Anatoly Dnieprov
Действие происходит во времена ламповых компьютеров. Молодой физик, безуспешно решая сложное уравнение, натыкается на объявление в газете о создании вычислительного центра в его городишке. Чрезвычайно удивлённый, он заказывает решение своей задачи, а затем, поражённый результатом, пытается выяснить способ его получения, не подозревая об угрозе своему рассудку и самой жизни.
Anatoly Dnieprov
The Maxwell Equations
Translated from the Russian By Leonid Kolesnikov
1
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?
2
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.
I nearly cried out in delight when at the end of a synthesis of countless theorems, intermediate calculations, formulae and equations the final solution emerged-a mathematical formula taking up three whole lines.
And to add a touch of the exquisite, the unknown mathematician had given himself the trouble of resolving the long formula into a simpler one. He had found a brief and precise form containing only the more elementary algebraic and trigonometric expressions.
At the very end, on a small inset, there was a graphic representation of the solution.
I could wish for nothing better. An equation which I thought could not be solved in the final form had been solved.
When I had recovered a little from my initial surprise and admiration I went through the photo copies again. Now I noticed that he who had solved my problem had been writing in great hurry and very closely as though trying to save on every scrap of paper and every second of time. Altogether he had written twenty-eight pages and I pictured mentally what a titanic work that had been! Try and pen a letter of twenty-eight closely-written pages in one day or just copy, without following the meaning, twenty-eight pages out of a book, and you will surely find it a hellish job.
But what I had in front of me was not a letter to a friend or a chapter copied out of a book. It was the solution of a most intricate mathematical problem-done in twenty-four hours.
For several hours I studied the closely-written pages, my surprise mounting with each hour.
Where had Kraftstudt found such a mathematician? On what terms? Who was he? A man of genius nobody knew? Or perhaps one of those wonders of human nature that sometimes occur on the border line between the normal and the abnormal? A rare specimen Kraftstudt had unearthed in the Wise Men's Home?
Cases have been recorded of brilliant mathematicians ending their days in a lunatic asylum. Maybe my mathematician was one of those?
These questions plagued me for the rest of that day.
But one thing was clear: the problem had been solved not by a machine, but by a man, a mathematical wizard the world knew nothing about.
The next day, a little calmer, I re-read the whole solution for the sheer pleasure of it this time, just as one will listen again and again to a piece of music one loves. It was so precise, so limpid, so beautiful that I decided to repeat the experiment. I decided to give Kraftstudt Co. one more problem to solve.
That was easy, for I was never short of challenging problems, and I chose an equation which I had always thought impossible to break down so that it could be handled by a computer, let alone be finally solved.
This equation, too, dealt with radio-wave propagation, but it was a specific and very complex case. It was an equation of the type that theoretical physicists evolve for the fun of it and soon forget all about because they are much too complex and therefore of no use to anybody.
I was met by the same young man blinking in the daylight. He gave me a reluctant smile.
"I have another problem-" I began.
Nodding briefly he again led me all the way through the dark corridors to the bleak reception hall.
Knowing the drill now, I went up to the window and handed him my equation.
"So it's not computers that do these things here?"
"As you see," he said without looking up from my equation.
"Whoever solved my first problem is quite a gifted mathematician," I said.
The young man did not say a word, deep as he was in my equation.
"Is he the only one in your employ or have you several?" I asked.
"What has that to do with your requirements? The firm guarantees-"
He had no time to finish, for at that moment the deep silence of the place was shattered by an inhuman scream. I started and listened. The sound was coming from behind the wall beyond the partition. It was like somebody being tortured. Crumpling the sheets with my problem, the young man, throwing a side glance and seizing me by the hand, dragged me to the exit.
"What was that?" I asked, panting.
"You'll have the solution the day after tomorrow, at twelve. You'll pay the bearer."
With those words he left me by my taxi.
3
It is hardly necessary to say that after this event my peace of mind was completely gone. Not for one moment could I forget that terrible scream which had seemed to shake the very stone vaults sheltering Kraftstudt and Co. Besides I was still under the shock of finding such a complicated problem solved by one man in one day. And finally I was feverishly waiting for the solution of my second problem. If this one too was solved, then…
It was with shaking hands that two days later I received a package from the Kraftstudt's girl. By its bulk I could tell that it must contain the solution to the monstrously complicated piece of mathematics. With something akin to awe I stared at the thin creature in front of me. Then I had an idea.
"Please come in, I'll get the money for you."
"No, it's all right." She seemed frightened and in a hurry. "I'll wait outside…"
"Come on in, no point in freezing outside," I said and all but dragged her into the hall. "I must have a look first to see whether the work's worth paying for."
The girl backed against the door and watched me with wide-open eyes.
"It is forbidden…" she whispered.
"What is?"
"To enter clients' flats… Those are the instructions, sir…"
"Never mind the instructions. I'm the master of this house and nobody will ever know you've entered."
"Oh, sir, but they will, and then…"
"What then?" I said, coming nearer.
"Oh. it's so horrible…"
Her head drooped suddenly and she sobbed.
I put a hand on her shoulder but she recoiled.
"Give me the seven hundred marks at once and I will go."
I held out the money, she snatched it and was gone.
Opening the package I nearly cried out with astonishment. For several minutes I stood there staring at the sheaf of photo paper unable to believe my own eyes. The calculations were done in a different hand.
Another mathematical genius! And of greater calibre than the first. The equations he had solved in an analytical form on fifty-three pages were incomparably more complicated than the ones I had handed in die first time. As I peered at the integrals, sums, variations and other symbols of the highest realms of mathematics I had a sudden feeling of having been transferred into a strange mathematical world where difficulty had no meaning. It just didn't exist.
That mathematician, it seemed, had no more difficulty in solving my problem than we have in adding or subtracting two-digit numbers.
Several times I tore myself away from the manuscript to look up a thing in a mathematical manual or reference book. I was amazed by his skill in using the most complex theorems and proofs. His mathematical logic and methods were irreproachable. I did not doubt that had the best mathem
aticians of all nations and ages, such as Newton, Leibnitz, Gauss, Euler, Lobachevsky, Weierstrass and Hilbert, seen the way my problem had been solved they would have been no less surprised.
When I finished reading the manuscript I fell to thinking.
Where did Kraftstudt get these mathematicians? I was convinced now he had a whole team of them, not just two or three. Surely he couldn't have founded a computer firm employing only two or three men. How had he managed it? Why was his firm next door to a lunatic asylum? Who had uttered that inhuman scream behind the wall? And why?
"Kraftstudt, Kraftstudt…" hammered in my brain. Where and when had I heard that name? What was behind it? I paced up and down my study, pressing my head with my hands, tasking my memory.
Then I again sat down to that genius-inspired manuscript, delighting in it, re-reading it part by part, losing myself to the world in the complexities of intermediate theorems and formulae. Suddenly I jumped up because I recalled that terrible inhuman scream once more and with it came the name of Kraftstudt.
The association was not fortuitous. No, it was inevitable. The screams of a man tortured and- Kraftstudt! These naturally went together. During the Second World War a Kraftstudt served as investigator in a Nazi concentration camp at Graz. For his part in the murders and inhuman treatment he got a life sentence at the Nuremberg trials.
I remembered the man's photo in all newspapers, in the uniform of an SS Obersturmfuhrer, in a pince-nez, with wide-open, surprised eyes in a plump good-natured face. People wouldn't believe a man with such a face could have been a sadist. Yet detailed evidence and thorough investigation left no room for doubt.
What had happened to him since the trials? Maybe he had been released like many other war criminals?
But what had mathematics to do with it all? What was the connection between a sadistic interrogator and the solutions of differential and integral equations?
At this point the chain of my reasoning snapped, for I was powerless to connect those two links. Obviously there was a link missing somewhere. Some kind of mystery.
The Maxwell Equations Page 1