EQUATION _for_ TIME

                         by R. R. WINTERBOTHAM

           [Transcriber's Note: This etext was produced from
                          Comet December 40.
         Extensive research did not uncover any evidence that
         the U.S. copyright on this publication was renewed.]


There is no one today who has seen a living horse. The creature became
extinct a couple of centuries ago, about the year 2,800. Man, who
betrayed the horse into what he became, hardly regretted the passing.

However, and I speak with all sincerity, there will be men of the
future who will see a horse. Perhaps men of the future may ride
horseback like knights and cowboys of the Middle Ages.

The secret of time travel has been discovered. No one has traveled
through time as yet, although man has explored the universe for more
than twenty light years from the sun. But the day of time travel is not
far distant. It had simple beginnings. All great things began in simple
ways. Newton and the apple were the beginnings of modern understanding
of the laws of the physical world; Watts and the teakettle were the
origins of industry and the machine age. A very beautiful young woman
and an unscrupulous man were responsible for time travel.

I met the man early in the morning of July 2, 3002. I remember the
date because on the day before I had visited in Alexandria, Egypt, and
I had eaten dinner in Shanghai, China. It was nearly midnight when I
reached the rocket port in Chicago and a jam in the pneumatics delayed
my arrival home until nearly one o'clock in the morning.

Blake, fully dressed, met me at the door. There was a worried look in
his eyes.

"There is a gentleman to see you, sir," Blake said. "I explained that
you would not return until quite late and I tried to get him to leave,
but he said it was urgent that he see you the minute you returned."
Blake glanced over his shoulder toward the library and lowered his
voice to a whisper. "I was a little frightened of him, sir. He doesn't
seem quite--ah--quite right, sir, if you know what I mean. Shall I call
the police?"

"No, Blake." I felt confident of licking my weight in madmen and I
entered the library.

A tall, distinguished, dark haired gentleman rose to greet me.

"Ah! Dr. Huckins! I was afraid you would not get here in time!"

As he spoke I noticed a peculiar light in his eyes. It seemed to be a
reflection from the fluorescent lamps of the library, but it showed a
little too much of the whites of his eyes and I thought of what Blake
had said about the man not being "quite right."

I did not feel that I owed him an apology for keeping him waiting,
since I usually received visitors by appointment.

"I am Gustav Keeshwar!" he introduced himself. He seemed to expect some
reaction, but unfortunately the name meant nothing to me, although if
I had paid more attention to the newspapers I would have known who he
was at once.

"I am the president of the Stellar Transport Company," he announced.

As he spoke he glanced secretively about the room, as though he feared
an eavesdropper. Then he picked up a brief case which was lying on the
table. With no explanation he opened it and pulled out package after
package of thousand dollar bills.

"You may count it if you wish," Keeshwar said. "There are 1,000 bills,
each of one thousand dollar denomination. One million dollars in cold
cash."

There are any number of bank presidents who have never seen a million
dollars in one pile. Spread out before me, I could scarcely grasp the
amount of wealth it represented. As I recall now, my clearest mental
reaction was a curiosity about how he managed to tuck it away so neatly
in a brief case. Then I wondered if it was real money. A closer glance
at the bills convinced me that it was.

Suddenly I came to my senses. I closed the library door and locked it.
I glanced nervously at the shades to make sure all were pulled down.

"Great Scott, man, you shouldn't carry all that money around with you
in a brief case!" As I said it, I spoke with the realization that the
man was mad.

"I brought the money to you," Keeshwar said. "It is yours if you will
do one thing for me."

"I must ask you to leave and to take your money with you," I said,
realizing that I was turning down the ransom of a king. "No honest task
ever called for a million dollars compensation--"

"But you have not asked me what I wish you to do!" Keeshwar exploded.
"Look! Do you see how much money a million dollars is?"

I do not wish to pose as a man over-stocked with principles. A million
dollars is more money than I ever hope to see again at one time.
But I had a good income, a nice little fortune tucked away in worth
while investments. I had a good name and my position in the world was
better than average. I did not trust this man. I had a feeling that the
million dollars he offered would not be worth the price.

"I am a surgeon," I said. "If you wish my professional services, I will
charge you a reasonable fee."

"I want your services," Keeshwar said. "I want them for one day."

"You may have them. I will send you a bill after I complete the task."

"I want your services tomorrow," said Keeshwar, persistently.

I shook my head. "I have a delicate operation scheduled tomorrow. It is
an operation I cannot postpone."

"It is an operation on Trella Mayo?"

I started. "How did you know that?"

"It is this operation that I wish you to perform for me," Keeshwar
said. "Would it not be simple to let your knife slip, or to allow
something to happen to her--for one million dollars!"

I do not remember clearly what happened next. I think I knocked the man
down. I do remember stuffing his million dollars into his brief case
and throwing it after him out of the door.

When I closed the door I was excited and unnerved. I found some
sedative tablets and swallowed one. Then I sat down to think. Trella
Mayo, beautiful, young and intelligent, a woman in a billion! Someone
wanted to kill her.

She was only twenty-eight, yet her discoveries in physics had astounded
the world. She might have taken first place in any beauty contest, yet
she preferred working in a laboratory with men too old to notice her
charms.

Her operation was not serious, except that it involved delicate skill.
I resolved that nothing must happen during that operation the following
day.

Two weeks later I visited Trella, now convalescing from her operation.

"I've wanted to talk to you, Fred," she said after I had taken her
temperature, felt her pulse and gone through the usual ritual.

"I must warn you that I'll send you a bill for any medical advice I
give you," I replied, laughing.

She smiled only a little and then puckered her brow seriously.

"I wanted to ask you about that operation. Wasn't it performed under
unusual circumstances?"

I was taken by surprise and I am afraid that the truth forced its
indications through my professional manner. "Why do you ask?"

"I noticed Blake standing near the door. There seemed to be a bulge
in his pocket. It couldn't have been a gun, could it? And you kept
watching, as if you were afraid a tribe of Indians would drop in for a
massacre. I wonder if there couldn't have been a tall, dark gentleman
mixed up in these unusual precautions?"

I did not reply.

"And I've noticed during my convalescence that the internes that
continually hover around my door have a look as if--well, shall I say
that they look more like policemen than internes?"

I laughed nervously. "I think you are a mental case, Miss Mayo," I
said. "I shall have to call in a specialist."

"You do not need to deny it, Fred," she said. "Why do you suppose I
insisted that you perform the operation? Why didn't I let you call in
someone else? It was because you are the only man in the world that I
trust, Fred. How much did Gustav Keeshwar offer you to do me in?"

Before I could stop myself I opened my mouth and blurted the truth.

"One million dollars!"

"Whew!" Trella whistled softly. "I'm worth a lot to you! I must be
getting close if Keeshwar will pay a million to see me out of the way."

"Trella," I pleaded. "What is it all about? What's behind this
mystery?"

"If you turned down a million dollars for my sake, I think I can trust
you," she said. "Supposing I was about to invent a new method of
locomotion? Can you see where Keeshwar might find me obnoxious?"

"A new kind of space ship?"

Trella shook her head. "A new kind of locomotion. Animals either swim
or walk. Man also uses wheels."

"He also can fly. So can birds."

"Flying is simply swimming through the air and crawling, as a worm
or snake, is gliding, like swimming. Space ships swim, too, after a
fashion. Boats swim through the sea and sleds swim on ice. Therefore we
have only three kinds of locomotion: Legs, wheels and sleds. Another
might revolutionize everything."

"But there couldn't be any other way to travel. Even the planets 'sled'
through ether."

"There is another way. It will open exploration to the furthest limits
of the galaxy."

"I can see why Keeshwar was so interested."

"As soon as I'm out of bed, I want you to call on me at my laboratory,
Fred. I'll show you something that will make your eyes pop out of your
head."

I turned to leave, when something on the window pane caught my eye.
It was a small, cherry-red spot, about the size of a twenty-five cent
piece.

The minute I saw it, I knew what it was. I shouted to the
interne--really a detective--outside the door, and lifted Trella into
my arms. I must admit that I handled her a little roughly and she
groaned as I hurried her out of the room. But what I did was necessary.

As I left the room, the glass of the pane melted and a beam flashed
across the room, striking the bed where Trella had been an instant
before. That beam was an Oronic Ray, 5,000 degrees hot, of the type
used in welding the rockets of space ships.

It was evident that Gustav Keeshwar intended to finish Trella Mayo
whether I would help him or not.

       *       *       *       *       *

A few weeks later I visited Trella in her laboratory.

"I'm anxious to see this incomprehensible conveyance," I explained.

"At least, I'm glad you are taking an interest in something besides my
safety and my operation scar," she replied.

She led me through a corridor toward a heavy steel door, which she
unlocked.

"You are the first person besides myself to go into this room in the
past five years," Trella added.

I scarcely know what I had expected to see. What would anyone expect
to see, if he was told he was going to be shown a machine that
neither walked, glided nor rolled? Such a contraption is beyond human
experience.

It was a long, hollow tube, large enough to hold a human body. It was
made of quartz and on each side was a cylindrical, low power atomic
energy machine.

"This," Trella said, "is the translator."

"The what?"

"I call it my space-time translator, which someday will make the rocket
as obsolete for space travel as the horse for surface travel. It will
take an object from one point in space-time to another instantly."

"Instantly?"

"There is a small lapse of time," Trella confessed. "You see the
machine has two motors, one for starting the operation and the other
for completing it. It takes about one second's time to switch the
motive power from one motor to the other."

The machine, except for the motors, was made entirely of quartz and
silver. On the right side of the machine was a long strip of silver
running the full length of the tube. It was about three inches wide and
it was connected with a knife-like blade of silver on the left side
of the tube by a strand of silver wire. Silver was used, of course,
because it was the best known conductor of electricity and other forms
of energy.

"It would be wonderful if it worked," I said.

"It does work," Trella said. "We sent two guinea pigs to the Sirius
system yesterday morning. We got them back in an hour with a copy of
yesterday's issue of _The Sirian Daily Universe_. Here's the paper."

She held out a copy of the beautifully printed daily magazine. On the
cover was the date, August V2, 504 (3002).

It was customary for terrestrials to use terrestrial dates wherever
their outposts were located in the stellar system. But instead of using
the terrestrial year--as shown in parenthesis on _The Sirian Daily
Universe_--the year always was reckoned from the date when the planet
was first visited by an expedition from the solar system. Although days
were not always the same, twenty-four hour periods could be reckoned
quite easily so that on some planets a single day might have more than
one terrestrial date, and on others a single day would be a fraction of
a legal day. The number of actual days usually was indicated by a Roman
numeral preceding the Arabic figure. Thus August V2 indicated that
Sirius had risen and set five times while the sun had done so twice
during the month of August.

"Unbelievable!" I said. "How does it work?"

"It operates through time," Trella explained. "It takes a short cut
between two parallel instants."

She took a guinea pig from a cage in the laboratory. She put the
wriggling animal inside the quartz tube and strapped it firmly in the
center.

"Watch," she said.

She turned a switch on one of the boxes. A low hum arose from the
atomic motor. Trella watched a dial located in the top of the quartz
tube until an arrow pointed to a gold star. Then she pressed a button
in the motor on the right side of the machine.

I noticed that the translator had controls that could be operated from
inside the tube as well as from the outside.

There were two distinct gasps of the motor. Half of the guinea pig
disappeared with the first gasp and the remaining half disappeared with
the second.

Where the tube had been a second before, there was nothing now.

"He's on Proxima Centaur now," Trella said. "I managed to equip a
laboratory there about two years ago. It was through that laboratory
that Keeshwar learned of my experiments in translation. My men on
Proxima will send back the guinea pig in a few minutes."

We sat down and waited. Trella explained the machine, although at the
time the explanation was a little over my head. The actual translation
was accomplished by the pushing of one motor and the pulling of another
across an extra-dimensional space. Half of the object to be translated
was hurled across space by the pushing of the first motor. The second
motor, which operated automatically, began pulling the other half,
including the first motor, after it as soon as it materialized at the
end of the journey.

By means of radio signals the exact location of every explored planet
had been determined. It was therefore only a matter of mathematical
calculation to find the target. There was some risk, of course, if a
mathematical error were made in computing the range but considering the
risks involved in ordinary methods of interstellar flight everything
was in favor of the translator.

"The whole secret of the invention lies in locating the proper _Now_ in
space-time," Trella explained.

"The proper _Now_?" I asked.

"Of course," she said, "the _Now_ we experience on earth is not the
same _Now_ that exists simultaneously on Rihlon, the second planet
of Proxima Centaur. We are dealing with space-time, Fred. Time is a
dimension, it stretches like a line through space. If we connect the
_Now_ of the present with the _Now_ of ten minutes ago, we have a
straight line, just as we would have a straight line if we connected
any two points in the universe. The _Now_ of the present and the _Now_
ten minutes ago on Rihlon also would be a straight line, but it would
not be the same straight line."

"But it would be parallel!" I exclaimed, beginning to see her point.

"Oh, so you _do_ know something about mathematics?"

"Of course! If you connect the _Nows_ of the present on both the earth
and Rihlon, you have a straight line, perpendicular to the parallel
time lines of both the earth and Rihlon. Why couldn't your invention be
used for time travel? Couldn't you connect the present--_Now_ of Rihlon
with any _Now_ in the time line of the earth--any _Now_ of the past or
future?"

"The idea occurred to me, but it won't work," Trella replied. "There's
a serious obstacle we can't overcome. In going backward or forward in
time we do not travel in lines perpendicular to the parallel time lines
of the earth and Rihlon--or for any other planet for that matter. But
we travel like this--" Trella drew a figure on a piece of paper.

[Illustration:

    A      G    E    B
    -------+----+------
            \   |
             \  |
              \ |
    C          \|F   D
    ------------+-----

Figure 1]

"The line AB represents the time line of the earth and the line CD
represents the time line of any other planet X. The two lines are
parallel. E represents the earth--_Now_, and F the _Now_ on planet X. A
line connecting the two is perpendicular to both AB and CD. Supposing
we should travel from F to a point G, a _Now_ in the earth's past. If
we connect F and G we would have a right triangle GEF. The hypothenuse
GF would be the square root of GE squared plus EF squared."

"There is nothing mathematically implausible in that," I said.

"There is nothing implausible, yet to determine the exact distance from
G to F is in most cases impossible. Unless the distances involved are
of the proper ratio, say, 4 and 5, the line GF becomes an irrational
number, of which it is impossible to find the exact value. Supposing
the distance from E to F was one light-year and the distance from G
to E, one year. Then GF would be the square root of one squared plus
one squared, or the square root of two. Because we are dealing with
such immense distances and because even the smallest decimal point of
error might lead to disastrous results, we cannot attempt time travel
unless we know the exact value of the square root of two, or any other
irrational number."

As Trella finished speaking there was a coughing hum and the translator
appeared in the room, containing the unharmed guinea pig and a copy of
the _Rihlon Gazette_ for Aug. 3rd, which was this day.

"Do you believe me?" she cried gleefully, waving the paper over her
head.

It was quite convincing, I admitted.

"Now I am going to make a trip in the translator!"

"You!"

It was the beginning of a long argument. There was danger in the trip,
I told her, and Trella had come to mean a great deal to me. She scoffed
at my fears and told me that if I didn't care to witness the first
translation of man to another planet in another star system she would
do it when I wasn't there.

Of course, no man can win an argument with a woman.

Trella climbed into the translator.

I closed the opening. Her hand rose to the switch that operated the
mechanism from inside the tube. She smiled and her lips moved in a
cheerful good-by. Then she touched the switch.

The indicator on the dial crept upward toward the gold star.

Suddenly the unexpected occurred.

The door of the laboratory opened. Trella had forgotten to lock the
door when we entered the room.

As I heard the noise, I turned and saw Gustav Keeshwar leveling a gun
toward the helpless young woman in the glass tube.

I sprang toward him just as the gun went off.

Apparently he had not expected to find me in the room, for as I lunged
he uttered a cry and threw the gun at my face. Then he turned and ran.

I managed to duck in time to receive only a glancing blow on the head.
I started to pursue, when my eyes fell on the translator.

Something terrible was wrong.

Half of the tube had disappeared and, with it, half of Trella's body.
The other half, containing half of the most beautiful woman on the
earth, remained in the laboratory.

My spring toward Keeshwar had spoiled his aim enough to keep the bullet
from striking Trella, but the bullet had struck the small silver wire
that ran from the atomic motor on one side to the atomic motor on the
other. The translation had been only half completed.

Half of Trella's body was on the earth, while the other was on Rihlon,
four light years away!

Her single eye was open and her half-face was frozen in an expression
of terror. She did not move and she was not breathing. There was no
blood. It was a complete suspension of animation.

Suddenly I realized that I was losing precious seconds. Unless
something was done, Trella would die.

I picked up the bit of wire that had been broken off by Keeshwar's
bullet. I lifted it toward the end dangling from the motor.

Then Trella moved! It was not suspended animation, but something
else--something new!

Her eye swung toward me. Her half-head visibly shook. Her half-lips
moved but no sound of her voice reached me. But I understood. She was
telling me not to replace the wire.

She lifted her hand and drew a right-angled triangle on the side of the
tube.

I understood. Trella was alive and she would continue to live, but it
would be impossible to restore her component halves merely by mending
the broken wire.

Trella was linked in time. She was still whole, but half of her body
was visible in one _Now_ and the other half in a _Now_ on Proxima
Centaur, four light years away.

To join the halves of her body, would mean joining the two _Nows_ and
to do that would form a triangle, at least one side of which would be
an irrational number. Unless the riddle of time travel were solved, it
would be impossible to make Trella whole.

I walked around the half-tube. Her appearance was not what I expected
to see. It was not a case of sawing a woman in half. The cross section
of her body appeared only as an opaque blankness. When I touched
her side I felt something cold and hard. It was as if I had touched
eternity.

The laboratory officials were called in for consultation. It was
decided that the matter should be hushed, at least until we knew what
should be done. There was too much to do now to be bothered with police
and reporters. We would not have a warrant issued for Keeshwar. There
would be time to deal with him later.

We discovered that Trella could eat and she seemed to be in perfect
health. But I knew that she was doomed unless we could restore the
parts of her body. Her muscles would atrophy. Inaction is more deadly
to the human machine than millions of disease germs.

If it would be possible to locate some day in the future when the wires
might be pieced together and the linking of Trella's two halves might
be accomplished without rationalizing irrational numbers, our problem
would be solved. But the nearest date in the future when this could be
done was three years ahead.[1]

[Footnote 1: Three years from the time this accident occurred would
make the sides of the triangle between the past event, the present, and
the present on Rihlon (four light years away) equal to the units 3, 4
and 5. Three squared, plus four squared equals five squared.]

But in three years Trella would be dead. We could not wait for the
coordinates to adjust themselves. We had to make the coordinates
adjustable to our purposes.

A small chronometer located in the atomic energy machine on the quartz
tube gave us the exact time the silver wire had been broken.

Even Blake, my servant, offered a suggestion:

"If you could take the earth half of Miss Trella's body to Rihlon, or
bring the Rihlon half to earth and bring the two _Nows_ together, would
that form a rational triangle?"

I took paper and pencil and tried to figure it out.

[Illustration:

    B      E               H         A
    -------+---------------+----------
           |              /|
           |             / |
           |            /  |
           |           /   |
           |          /    |
    -------+---------+-----+----------
    C      F         G     J         D

Figure 2]

The line BA represented the time line of Rihlon. The line CD was the
time line of the earth. The points E and F were the _Nows_ on Rihlon
and earth, respectively, at which the accident occurred. The point G
represented the _Now_ at which a space ship would leave the earth for
Rihlon carrying Trella's half body. The point H represented the _Now_
of arrival on Rihlon and the point J the parallel point on earth.
We still had a right-angled triangle and we still had to deal with
irrational numbers. But hold on--

I gazed at my drawing. Before my eyes was the answer! The whole thing
was clearly and completely solved. The secret of time travel was
solved. Trella was saved. The invention of the translator had been
perfected so that all danger of becoming lost in time was removed![2]

[Footnote 2: As a mental exercise, I would suggest that the reader look
at Figure 2 for a minute or two and figure out the answer. The answer
is there and high school mathematics should enable a person to discover
how to extract the irrational number.--Dr. Fred Huckins.]

"Blake," I said to the servant, "bring me my automatic pistol."

"Wh-what?" Blake stuttered.

"I said bring me my automatic pistol. I'm going to save Trella, or
murder somebody."

"Perhaps I should call your lawyer."

I threw a book at him and he left hurriedly, to return in a few minutes
with my pistol and holster. I strapped the weapon about my waist and
slammed my straw hat on my head. In a few minutes I stepped from a taxi
in front of the Galaxy building, in which the officers of the Stellar
Transport Company are located.

A clerk with thick glasses interviewed me.

"I want to charter a ship for a trip to Proxima Centaur," I explained.
"I want one of your late model cruisers which can go about ten times
the speed of light. I want to get there quickly."

The clerk nodded. I have often wondered about the composure of clerks
who never seem to be astonished at anything. "We have a ship available
that could get you there in three months, that's sixteen times the
speed of light. But to charter it would cost one million dollars."

He never batted an eye when he named the price. I doubt if the clerk
was receiving more than forty a week.

"I should like to transact the deal directly with Mr. Keeshwar," I said.

"He will be pleased, I'm sure," the clerk replied. "What is your name?"

"Andrew J. Colt," I said, for lack of more originality.

The clerk disappeared into the sanctum. He returned presently with:

"Mr. Keeshwar will see you, Mr. Colt."

I had counted on Keeshwar being--or pretending to be very busy as I
entered. I expected him to pay no attention to my entry, and not even
to glance in my direction, as if a million dollars were a trifling
matter, until we were alone.

I judged Keeshwar right. When at last he glanced at me he was unnerved
by the presence of an automatic pistol which was pointed directly at
his head.

"I must warn you not to touch any of those buttons on your desk," I
said. "It would give me a great deal of pleasure to drill you and I
won't go out of my way for an opportunity."

"Wh-what d-d-do you w-w-ant?" he asked, turning pale.

"One day you offered me a million dollars to take Miss Mayo's life,"
I said. "Now I'm asking you to contribute an equal amount to save it.
However, I'm willing to take it out in trade. I want you to pilot one
of your ships for me to Rihlon."

"Impossible!" Keeshwar said, regaining some of his composure. "I
couldn't leave my business for a period long enough to make the trip."

"If you don't leave your business to make the trip right now you won't
exist any more," I warned casually. I reached into my pocket and
brought out a silencer, which I fitted to the end of the pistol barrel.
I unfastened the safety and aimed deliberately.

       *       *       *       *       *

The space ship containing the terrestrial half of Trella Mayo, in
company with myself, Blake, two other scientists and Gustav Keeshwar,
arrived on Rihlon three months later. Keeshwar, who had had a pistol
trained on him almost every instant since I had called at his office,
was released and permitted to return to earth. He did not know that I
had left the instructions on earth for his arrest for felonious assault
the minute he landed.

We located Trella's Rihlon laboratory. It was the matter of a few
minutes to make the connection of the broken wire and to finish the
translation of her two halves.

Trella stepped out of her quartz prison, swayed unsteadily for a second
on her feet, and then collapsed.

"How on earth did you do it?" she asked. "How did you reconcile the
irrational number?"

I sketched the figure roughly (Figure 2). "The distance from F to G and
the distance from E to H does not enter into the equation," I said.
"The only thing we are interested in is the distances GJ, JH and GH."

"And GH is an irrational number," Trella said.

"Quite right, although like most things that appear absurd on the
surface, it is not as irrational as it seems. The distance G to J
is three months, the time required for the flight from the earth to
Rihlon. We will represent this by the unit 1. The distance JH is
four light years, the distance in space from earth to Rihlon. This,
therefore, would be sixteen units. Using the formula (GJ)^2 plus (JH)^2
equals (GH)^2 we find that GH is the square root of one plus 256, or
257. The square root of 257 is 16.031228, etc., an irrational number.

"It can't be expressed in figures! We do not need figures when we can
draw a picture. The triangle GHJ is a picture of an irrational number.
We had only to go to Rihlon to complete the equation."

"Time can be traveled," Trella said.

"Where would you like to go on our honeymoon?" I asked.

"To the Garden of Eden," she said.