The Indefatigable Frog

 

"Zeno was the first great scientist," Professor Hardy stated, looking sternly around his classroom. "For example, take his paradox of the frog and the well. As Zeno showed, the frog will never reach the top of the well. Each jump is half the previous jump; a small but very real margin always remains for him to travel."

There was silence, as the afternoon Physics 3-A Class considered Hardy's oracular utterance. Then, in the back of the room, a hand slowly went up.

Hardy stared at the hand in disbelief. "Well?" he said. "What is it, Pitner?"

"But in Logic we were told the frog would reach the top of the well. Pro­fessor Grote said --"

"The frog will not!"

"Professor Grote says he will."

Hardy folded his arms. "In this class the frog will never reach the top of the well. I have examined the evidence myself. I am satisfied that he will always be a small distance away. For example, if he jumps --"

The bell rang.

All the students rose to their feet and began to move towards the door. Professor Hardy stared after them, his sentence half finished. He rubbed his jaw with displeasure, frowning at the horde of young men and women with their bright, vacant faces.

When the last of them had gone, Hardy picked up his pipe and went out of the room into the hall. He looked up and down. Sure enough, not far off was Grote, standing by the drinking fountain, wiping his chin.

"Grote!" Hardy said. "Come here!"

Professor Grote looked up, blinking, "What?"

"Come here," Hardy strode up to him. "How dare you try to teach Zeno? He was a scientist, and as such he's my property to teach, not yours. Leave Zeno to me!"

"Zeno was a philosopher." Grote stared up indignantly at Hardy. "I know what's on your mind. It's that paradox about the frog and the well. For your information, Hardy, the frog will easily get out. You've been misleading your students. Logic is on my side."

"Logic, bah!" Hardy snorted, his eyes blazing. "Old dusty maxims. It's obvious that the frog is trapped forever, in an eternal prison and can never get away!"

"He will escape."

"He will not."

"Are you gentlemen quite through?" a calm voice said. They turned quickly around. The Dean was standing quietly behind them, smiling gently. "If you are through, I wonder if you'd mind coming into my office for a moment." He nodded towards his door. "It won't take too long."

Grote and Hardy looked at each other. "See what you've done?" Hardy whispered, as they filed into the Dean's office. "You've got us into trouble again."

"You started it -- you and your frog!"

"Sit down, gentlemen." The Dean indicated two stiff-backed chairs. "Make yourselves comfortable. I'm sorry to trouble you when you're so busy, but I do wish to speak to you for a moment." He studied them moodily. "May I ask what is the nature of your discussion this time?"

"It's about Zeno," Grote murmured.

"Zeno?"

"The paradox about the frog and the well."

"I see." The Dean nodded. "I see. The frog and the well. A two thou­sand-year-old saw. An ancient puzzle. And you two grown men stand in the hall arguing like a --"

"The difficulty," Hardy said, after a time, "is that no one has ever per­formed the experiment. The paradox is a pure abstraction."

"Then you two are going to be the first to lower the frog into his well and actually see what happens."

"But the frog won't jump in conformity to the conditions of the paradox."

"Then you'll have to make him, that's all. I'll give you two weeks to set up control conditions and determine the truth of this miserable puzzle. I want no more wrangling, month after month. I want this settled, once and for all."

Hardy and Grote were silent.

"Well, Grote," Hardy said at last, "let's get it started."

"We'll need a net," Grote said.

"A net and a jar." Hardy sighed. "We might as well be at it as soon as possible."

The "Frog Chamber," as it got to be called, was quite a project. The University donated most of the basement to them, and Grote and Hardy set to work at once, carrying parts and materials downstairs. There wasn't a soul who didn't know about it before long. Most of the science majors were on Hardy's side; they formed a Failure Club and denounced the frog's efforts. In the philosophy and art departments there was some agitation for a Success Club, but nothing ever came of it.

Grote and Hardy worked feverishly on the project. They were absent from their classes more and more of the time, as the two weeks wore on. The Chamber itself grew and developed, resembling more and more a long section of sewer pipe running the length of the basement. One end of it disappeared into a maze of wires and tubes: at the other there was a door.

One day when Grote went downstairs there was Hardy already, peering into the tube.

"See here," Grote said, "we agreed to keep hands off unless both of us were present."

"I'm just looking inside. It's dark in there." Hardy grinned. "I hope the frog will be able to see."

"Well, there's only one way to go."

Hardy lit his pipe. "What do you think of trying out a sample frog? I'm itching to see what happens."

"It's too soon." Grote watched nervously as Hardy searched about for his jar. "Shouldn't we wait a bit?"

"Can't face reality, eh? Here, give me a hand."

There was a sudden sound, a scraping at the door. They looked up. Pitner was standing there, looking curiously into the room, at the elongated Frog Chamber.

"What do you want?" Hardy said. "We're very busy."

"Are you going to try it out?" Pitner came into the room. "What are all the coils and relays for?"

"It's very simple," Grote said, beaming. "Something I worked out myself. This end here --"

"I'll show him," Hardy said. "You'll only confuse him. Yes, we were about to run the first trial frog. You can stay, boy, if you want." He opened the jar and took a damp frog from it. "As you can see, the big tube has an entrance and an exit. The frog goes in the entrance. Look inside the tube, boy. Go on."

Pitner peered into the open end of the tube. He saw a long black tunnel. "What are the lines?"

"Measuring lines. Grote, turn it on."

The machinery came on, humming softly. Hardy took the frog and dropped him into the tube. He swung the metal door shut and snapped it tight. "That's so the frog won't get out again, at this end."

"How big a frog were you expecting?" Pitner said. "A full-grown man could get into that."

"Now watch." Hardy turned the gas cock up. "This end of the tube is warmed. The heat drives the frog up the tube. We'll watch through the win­dow."

They looked into the tube. The frog was sitting quietly in a little heap, staring sadly ahead.

"Jump, you stupid frog," Hardy said. He turned the gas up.

"Not so high, you maniac!" Grote shouted. "Do you want to stew him?"

"Look!" Pitner cried. "There he goes."

The frog jumped. "Conduction carries the heat along the tube bottom," Hardy explained. "He has to keep on jumping to get away from it. Watch him go."

Suddenly Pitner gave a frightened rattle. "My God, Hardy. The frog has shrunk. He's only half as big as he was."

Hardy beamed. "That is the miracle. You see, at the far end of the tube there is a force field. The frog is compelled to jump towards it by the heat. The effect of the field is to reduce animal tissue to its proximity. The frog is made smaller the farther he goes."

"Why?"

"It's the only way the jumping span of the frog can be reduced. As the frog leaps he diminishes in size, and hence each leap is proportionally reduced. We have arranged it so that the diminution is the same as in Zeno's paradox."

"But where does it all end?"

"That," Hardy said, "is the question to which we are devoted. At the far end of the tube there is a photon beam which the frog would pass through, if he ever got that far. If he could reach it, he would cut off the field."

"He'll reach it," Grote muttered.

"No. He'll get smaller and smaller, and jump shorter and shorter. To him, the tube will lengthen more and more, endlessly. He will never get there."

They glared at each other. "Don't be so sure," Grote said.

They peered through the window into the tube. The frog had gone quite a distance up. He was almost invisible, now, a tiny speck no larger than a fly, moving imperceptibly along the tube. He became smaller. He was a pin point. He disappeared.

"Gosh," Pitner said.

"Pitner, go away," Hardy said. He rubbed his hands together. "Grote and I have things to discuss."

He locked the door after the boy.

"All right," Grote said. "You designed this tube. What became of the frog?"

"Why, he's still hopping, somewhere in a sub-atomic world."

"You're a swindler. Some place along that tube the frog met with misfor­tune."

"Well," Hardy said. "If you think that, perhaps you should inspect the tube personally."

"I believe I will. I may find a trap door."

"Suit yourself," Hardy said, grinning. He turned off the gas and opened the big metal door.

"Give me the flashlight," Grote said. Hardy handed him the flashlight and he crawled into the tube, grunting. His voice echoed hollowly. "No tricks, now."

Hardy watched him disappear. He bent down and looked into the end of the tube. Grote was half-way down, wheezing and struggling. "What's the matter?" Hardy said.

"Too tight...."

"Oh?" Hardy's grin broadened. He took his pipe from his mouth and set it on the table. "Well, maybe we can do something about that."

He slammed the metal door shut. He hurried to the other end of the tube and snapped the switches. Tubes lit up, relays clicked into place.

Hardy folded his arms. "Start hopping, my dear frog," he said. "Hop for all you're worth."

He went to the gas cock and turned it on.

 

It was very dark. Grote lay for a long time without moving. His mind was filled with drifting thoughts. What was the matter with Hardy? What was he up to? At last he pulled himself on to his elbows. His head cracked against the roof of the tube.

It began to get warm. "Hardy!" His voice thundered around him, loud and panicky. "Open the door. What's going on?"

He tried to turn around in the tube, to reach the door, but he couldn't budge. There was nothing to do but go forward. He began to crawl, muttering under his breath. "Just wait, Hardy. You and your jokes. I don't see what you expect to --"

Suddenly the tube leaped. He fell, his chin banging against the metal. He blinked. The tube had grown; now there was more than enough room. And his clothing! His shirt and pants were like a tent around him.

"Oh, heavens," Grote said in a tiny voice. He rose to his knees. Labori­ously he turned around. He pulled himself back through the tube the way he had come, towards the metal door. He pushed against it, but nothing hap­pened. It was now too large for him to force.

He sat for a long time. When the metal floor under him became too warm he crawled reluctantly along the tube to a cooler place. He curled himself up and stared dismally into the darkness. "What am I going to do," he asked himself.

After a time a measure of courage returned to him. "I must think logically. I've already entered the force field once, therefore I'm reduced in size by one-half. I must be about three feet high. That makes the tube twice as long."

He got out the flashlight and some paper from his immense pocket and did some figuring. The flashlight was almost unmanageable.

Underneath him the floor became warm. Automatically he shifted, a little up the tube to avoid the heat. "If I stay here long enough," he murmured, "I might be --"

The tube leaped again, rushing off in all directions. He found himself floundering in a sea of rough fabric, choking and gasping. At last he struggled free.

"One and a half feet," Grote said, staring around him. "I don't dare move any more, not at all."

But when the floor heated under him he moved some more. "Three-quarters of a foot." Sweat broke out on his face. "Three-quarters of one foot." He looked down the tube. Far, far down at the end was a spot of light, the photon beam crossing the tube. If he could reach it, if only he could reach it, if only he could reach it!

He meditated over his figures for a time. "Well," he said at last, "I hope I'm correct. According to my calculations I should reach the beam of light in about nine hours and thirty minutes, if I keep walking steadily." He took a deep breath and lifted the flashlight to his shoulder.

"However," he murmured, "I may be rather small by that time. . . ." He started walking, his chin up.

 

Professor Hardy turned to Pitner. "Tell the class what you saw this morn­ing."

Everyone turned to look. Pitner swallowed nervously. "Well, I was down­stairs in the basement. I was asked in to see the Frog Chamber. By Professor Grote. They were going to start the experiment."

"What experiment do you refer to?"

"The Zeno one," he explained nervously. "The frog. He put the frog in tube and closed the door. And then Professor Grote turned on the power."

"What occurred?"

"The frog started to hop. He got smaller."

"He got smaller, you say. And then what?"

"He disappeared."

Professor Hardy sat back in his chair. "The frog did not reach the end of the tube, then?"

"No."

"That's all." There was a murmuring from the class. "So you see, the frog did not reach the end of the tube, as expected by my colleague, Professor Grote. He will never reach the end. Alas, we shall not see the unfortunate frog again."

There was a general stir. Hardy tapped with his pencil. He lit his pipe and puffed calmly, leaning back in his chair. "This experiment was quite an awakener to poor Grote, I'm afraid. He has had a blow of some unusual proportion. As you may have noticed, he hasn't appeared for his afternoon classes. Pro­fessor Grote, I understand, has decided to go on a long vacation to the moun­tains. Perhaps after he has had time to rest and enjoy himself, and to forget --"

 

Grote winced. But he kept on walking. "Don't get frightened," he said to himself. "Keep on."

The tube jumped again. He staggered. The flashlight crashed to the floor and went out. He was alone in the enormous cave, an immense void that seemed to have no end, no end at all.

He kept walking.

After a time he began to get tired again. It was not the first time. "A rest wouldn't do any harm." He sat down. The floor was rough under him, rough and uneven. "According to my figures it will be more like two days, or so. Perhaps a little longer. . . ."

He rested, dozing a little. Later on he began to walk again. The sudden jumping of the tube had ceased to frighten him; he had grown accustomed to it. Sooner or later he would reach the photon beam and cut through it. The force field would go off and he would resume his normal size. Grote smiled a little to himself. Wouldn't Hardy be surprised to --

He stubbed his toe and fell, headlong into the blackness around him. A deep fear ran through him and he began to tremble. He stood up, staring around him.

Which way?

"My God," he said. He bent down and touched the floor under him. Which way? Time passed. He began to walk slowly, first one way, then another. He could make out nothing, nothing at all.

Then he was running, hurrying through the darkness, this way and that, slipping and falling. All at once he staggered. The familiar sensation: he breathed a sobbing sigh of relief. He was moving in the right direction! He began to run again, calmly, taking deep breaths, his mouth open. Then once more the staggering shudder as he shrank down another notch; but he was going the right way. He ran on and on.

And as he ran the floor became rougher and rougher. Soon he was forced to stop, falling over boulders and rocks. Hadn't they smoothed the pipe down? What had gone wrong with the sanding, the steel wool --

"Of course," he murmured. "Even the surface of a razor blade. . . if one is small. . . ."

He walked ahead, feeling his way along. There was a dim light over every­thing, rising up from the great stones around him, even from his own body. What was it? He looked at his hands. They glittered in the darkness.

"Heat," he said. "Of course. Thanks, Hardy." In the half light he leaped from stone to stone. He was running across an endless plain of rocks and boulders, jumping like a goat, from crag to crag. "Or like a frog," he said. He jumped on, stopping once in a while for breath. How long would it be? He looked at the size of the great blocks of ore piled up around him. Suddenly a terror rushed through him.

"Maybe I shouldn't figure it out," he said. He climbed up the side of one towering cliff and leaped across to the other side. The next gulf was even wider. He barely made it, gasping and struggling to catch hold.

He jumped endlessly, again and again. He forgot how many times.

He stood on the edge of a rock and leaped.

Then he was falling, down, down, into the cleft, into the dim light. There was no bottom. On and on he fell.

Professor Grote closed his eyes. Peace came over him, his tired body relaxed.

"No more jumping," he said, drifting down, down. "A certain law regard­ing falling bodies. . . the smaller the body the less the effect of gravity. No wonder bugs fall so lightly. . . certain characteristics. . . ."

He closed his eyes and allowed the darkness to take him over, at last.

 

"And so," Professor Hardy said, "we can expect to find that this experi­ment will go down in science as --"

He stopped, frowning. The class was staring towards the door. Some of the students were smiling, and one began to laugh. Hardy turned to see what it; was.

"Shades of Charles Fort," he said.

A frog came hopping into the room.

Pitner stood up. "Professor," he said excitedly. "This confirms a theory I've worked out. The frog became so reduced in size that he passed through the spaces --"

"What?" Hardy said. "This is another frog."

"--through the spaces between the molecules which form the floor of the Frog Chamber. The frog would then drift slowly to the floor, since he would be proportionally less affected by the law of acceleration. And leaving the force field, he would regain his original size."

Pitner beamed down at the frog as the frog slowly made his way across the room.

"Really," Professor Hardy began. He sat down at his desk weakly. At that moment the bell rang, and the students began to gather their books and papers together. Presently Hardy found himself alone, staring down at the frog. He shook his head. "It can't be," he murmured. "The world is full of frogs. It can't be the same frog."

A student came up to the desk. "Professor Hardy --"

Hardy looked up.

"Yes? What is it?"

"There's a man outside in the hall wants to see you. He's upset. He has a blanket on."

All right," Hardy said. He sighed and got to his feet. At the door he paused, taking a deep breath. Then he set his jaw and went out into the hall.

Grote was standing there, wrapped in a red-wool blanket, his face flushed with excitement. Hardy glanced at him apologetically.

"We still don't know!" Grote cried.

"What?" Hardy murmured. "Say, er, Grote --"

"We still don't know whether the frog would have reached the end of the tube. He and I fell out between the molecules. We'll have to find some other way to test the paradox. The Chamber's no good."

"Yes, true," Hardy said. "Say, Grote --"

"Let's discuss it later," Grote said. "I have to get to my classes. I'll look you up this evening."

And he hurried off down the hall clutching his blanket.