Archimedes

This book was written by my father, Wayne Thurston. He began the book in the 80s and finished it in the 90s. He gave me permission to put his work on the web.

His first act was to check in his personal locker. The various small emergency belongings he had been allowed to take with him were in place. He rummaged back of these until he felt the hard shape of the object he had been searching for. A smile crossed his lips. Transit, he said to himself. Cissy had been able to smuggle it aboard.

He went back to his seat, and sat idly watching the time machine's chronometer tick off the centuries that separated him from his destination.

"But why is it so important to you?" Cissy had asked. "I mean, so you get to meet Archimedes   so big deal   so what?"

"I just want to meet him, that's all. I just want to see what he looks like   talk to him   you know..."

"In Greek?"

"I know it won't be easy. But I've been studying a lot."

"What makes you think he would agree to see you?"

"I've thought of that. I need something to get his attention..."

"But they won't let you take anything back. You know that."

"Yes, I know that, but you're the one who gives the time machine its final checkout..."

"Are you suggesting that I..."

"It's only a simple instrument. Small. Unobtrusive. You should be able to smuggle it aboard."

"What kind of instrument?"

"A surveyor's transit. Archimedes was interested in measurement of angles. I think it would get his attention."

"But wouldn't there be the risk of changing his future   the present for us?"

"I don't see how a little thing like a transit could do much. The kind I'm talking about would be pretty crude by modern standards. The most it could do would be to show him a little more precision than the instruments of his day."

"Well, I don't know. It's horribly against all the rules of the project."

"Ok, so I do it without the transit. I'll find some way to get him to listen to me."

"Is it really that important to you?"

"Yes."

"I'll think about it..."

She had. The transit was in his locker.

The rate of change of the chronometer began to slow as he approached 200 B.C. He was scheduled to come out of hypertime at the year 210, but he had the ability to modify the target date. He did so now, changing it to 212 B.C. 210 would be too late. Archimedes would be dead, killed during the siege of Syracuse. He set the point of entry for January 1, to allow for historical errors about the date of the fall. Jeff leaned back in his seat, watching the final days tick down. If all went well, he would soon realize his dream.

The time machine functioned flawlessly. He was deposited, just before dawn, on the island of Sicily, outside the city of Syracuse. From the sand he could feel underfoot, he assumed he was on a beach, but he could see nothing. Accustomed to the light pollution of the 21st century, he was astonished to find total blackness. A light rain was falling.

He projected a quick signal and the machine rose swiftly to take up geostationary orbit. He felt a little chill of apprehension. He was now alone in the world of antiquity. He was also radically violating his training for the mission. Being alone bothered him a good deal more than the violation.

The dull light of morning found Jeff beginning to wonder if the whole adventure had been a mistake. He was dripping wet and thoroughly chilled. He was more than tempted to call down the time machine and get some clothing. Somehow, being stark naked struck him as a little ridiculous   although he understood the reasons for his instructions.

Nothing that could distort the past. Even clothing is too dangerous.

He felt a little guilty about the waterproof package containing the transit. He did not have very long to feel guilty.

"Halt!"

How can I halt? I'm sitting on a rock.

There was a clatter of metal as the small group of men ran up to surround him. In the light of torches, their swords had a nasty look, and their spears seemed very appropriate for... spearing.

"Who are you?"

Jeff realized that the question was in Latin. He was suddenly glad he had studied that ancient language too.

He got up slowly, being careful not to make any threatening movements. "I come from far away," he said, searching for the words. The men did not seem to understand him. Again the question: "Who are you?"

"A friend."

There was a grunt. "Friend?"

There was a quick conversation among the men. He recognized the language. They were speaking Greek, but it was much too rapid for him. He could read their looks, however. They were full of suspicion.

"I am a friend of Archimedes," he said, in Greek. He had practiced this sentence many times. He thought he said it fluently.

There was a derisive laugh. Strong arms seized him. He found himself hurrying away into the early morning mist.

***

The cell was small and damp. It was also impossibly dark. There was not a trace of light anywhere. Jeff felt about and found a few rough planks that seemed to be intended for a bed. He sat down on them and considered his situation. He did not like what he considered.

He got up and made a circuit around the walls. They were rough stone, fitted closely together. There was the door he had been thrown through. Nothing else. He felt his way to the bed and sat down again.

Time stretched out interminably, but finally, at a distance, he heard the clang of a gate being opened. There was the sound of footsteps, and the door of his cell creaked back.

Several figures, shrouded in darkness, stood in the doorway. One of them carried a small flickering lamp. That smoky lamp was the best object that Jeff had ever seen. It lifted the oppression he had been feeling.

He got up slowly and faced the light. He had the words ready. He spoke them slowly, carefully. "Take me to Archimedes."

There was a silence. The men facing him studied him carefully, lifting the lamp so it illuminated his face. "Why?"

"I have heard about him. I want to meet him. That is why I came here."

"From Rome?" There was disbelief, but also a question in the voice.

"Give Archimedes the thing I brought. He will understand."

"That has already been done."

"Then take me to him."

"Archimedes is a busy man. He is building engines to destroy the invaders. He has no time for a Roman spy."

Suddenly Jeff understood. His use of Latin. His halting Greek. Syracuse was in a life or death struggle. Of course they suspected him. He was suddenly glad about the transit. It was his only hope.

"Does Archimedes think I am a Roman spy?"

There was a rapid series of quick exchanges among the group.

"Come with us."

Jeff followed them down a long corridor, up a flight of stairs and into a room lighted by a small opening in one wall.

"Sit down."

The men tied his hands behind his back and pulled up a stool for him.

"There is one who wants to see you. Beware! No tricks! We will be waiting outside."

The men went out, leaving Jeff alone. In a few moments a slender figure entered. It was fully robed with a cowl covering the head. One hand held a fold of the cloth across the face, just below the eyes. Jeff was suddenly embarrassed. He was still naked, and the eyes that studied him were a woman's eyes.

"You wish to see Archimedes. Why?"

Jeff sensed that this woman held life or death in her hand. That knowledge did not improve his Greek. The words came out slowly, "I come from far away. I have heard of Archimedes. I wish to meet him."

"And the instrument you brought, where did you get it?"

"They are common in the place I came from."

The woman seemed startled. "Common?"

"We used them in the past to measure land."

"In the past? What do you use now?"

"We have much better instruments now."

The woman considered this carefully. "The instrument makes far away things seem near."

A warning bell rang in Jeff's mind. He had not thought about the transit's telescope. He had thought only of its ability to measure angles. Perhaps it had been a mistake to bring it. Perhaps a telescope might conceivably change the future. But now it was his only hope.

She leaned forward eagerly. "And the curious symbols on the instrument form a strange pattern."

Jeff sensed an opportunity. "I could explain this pattern to Archimedes. He has the power to understand such things."

He sensed laughter in the woman's eyes. "And you think I do not have this power?"

Jeff did not know what to say, so he decided to say nothing. Instead he smiled. "I meant no offense."

"I ask you again: why do you wish to see Archimedes? Answer honestly this time."

"Because I have read about him in... manuscripts."

"You mean you are familiar with his works? Which?"

"On the Sphere and the Cylinder."

"Explain to me his proofs."

Jeff was suddenly embarrassed. He knew that this work was such a favorite of Archimedes that he had asked to have a sphere and the cylinder inscribed on his tomb. But Jeff could not follow the mathematical thinking. He decided to be frank.

"I do not understand them."

"Then you are not a mathematician?"

"No."

"Why then are you interested in Archimedes? Is it perhaps his engines of war?"

"No. I have no interest in war."

The woman fell silent, studying Jeff closely. When she spoke again, it was evident that she was puzzled. "What you say is very strange. You are found, naked, outside the city. You have a remarkable instrument   nothing else. You are not a mathematician. You are not interested in Archimedes' engines. You say that you have no interest in war..."

Again she was silent, pondering. "How did you come here?"

"I came in a ship."

"And how did you pass the Roman blockade?"

Jeff felt himself being drawn deeper and deeper into a suspicious situation. Indeed, the woman turned slightly as if she were about to call the guards.

"No! Wait! I shall answer your questions."

"Honestly?"

"Honestly. The ship that I came in does not travel on water."

The woman studied him closely. "Yes?"

"I cannot explain it to you easily. It travels... in the air."

"And what is the name of the place you came from?"

"You would have no name for it."

"And why is that?"

Jeff hesitated. He had the distinct feeling that this woman would not easily be deceived. Indeed, he did not wish to deceive her. But how could he explain time travel? And did he dare to do it if he could? He began to see the folly of what he was doing. He had only wanted to meet Archimedes-- to talk a little with him   to see this man who was considered to be one of the greatest minds of all history.

"Because it is in a place that you know nothing of."

"Perhaps because it is on the other side of the sphere?"

Jeff seized the chance gratefully. "Yes."

"Sailors say that beyond the Gates of Hercules is a great ocean..."

"The land I come from is on the other side of that ocean."

Jeff sensed that she had made a decision.

"My father will wish to meet you."

She lowered the fold of cloth that had concealed her face. Jeff was first struck by her beauty. Then he was acutely aware again that she was a woman. He squirmed in embarrassment.

She smiled. "The customs of your land must be very different."

"But... but will I be able to meet Archimedes?"

"My father is Archimedes," she said quietly.

***

It was mid afternoon when Jeff was taken from the prison. A brilliant sun looked down from an impossibly blue sky. Clothed now, he was conducted through the streets of the city. When his escort reached an imposing house, they knocked and stood waiting. The door was opened by a huge man who paid no attention to the guard but studied Jeff closely. Then the woman who had visited Jeff's cell appeared.

"You are dismissed," she said curtly to the guard. "Enter, Stranger!"

The Captain of the guard seemed undecided. "But your safety..."

"Xanthus can assure our safety. We will summon you if there is need."

"As you wish."

Jeff followed the huge man into a small courtyard. The roof was open to the sky. There were carefully attended flowers in abundance.

"Be seated," the woman said, motioning to a marble bench.

Jeff did so, having some trouble with the loose fitting garment that he was wearing. It seemed to want to fall open in the wrong places.

"My father will return shortly. He is supervising the installation of a new catapult. In the meantime..." She clapped her hands, and a female attendant appeared, bearing two golden goblets on a salver. "Would you like refreshment?"

Jeff took one of the goblets, and sipped the wine. It was delicious.

"My name is Penelope. Perhaps you would tell me yours."

"Barklay. Jeff Barklay."

"Is it customary for the women in the land you come from to drink with the men?"

"Yes..."

"Very well," she said archly. "Then I shall drink with you." Regarding Jeff closely over the rim of the goblet, she took a sip of the wine.

In the sunlight flooding the garden, she was much more beautiful than in the dungeon. Her flowing white garment revealed a slender form. Her movements were exquisitely graceful. Her hair was a shining black, and her eyes were emerald green. She lowered the goblet and stood smiling at Jeff.

"You find me attractive."

It was a statement, not a question. Jeff nodded.

"Men do," she said. "But it is not my body that I wish them to find attractive. The men of Syracuse who appreciate my mind are old and fat. The young men are athletes, or soldiers, or empty headed nobles. Are you one of those, Jeff Barklay?"

It was Jeff's turn to smile. "No, I am not an athlete, nor a soldier, and I am certainly not a noble."

"And you are not a mathematician?"

"No."

"But, in the place you come from, surely there is mathematics?"

"Yes."

"And surely you have studied some of it?"

"Yes, but..."

Penelope smiled deferentially. "My father, as you know, is interested in mathematics. He is also a very busy man. Perhaps it would help if you showed me a few things while we are waiting. Then I could explain them to him. Something simple, of course. Perhaps the symbols you use for counting..."

Another warning bell rang in Jeff's mind. Would this information be enough to significantly alter the course of history? He did not see how it could. And, after all, this was such a charming young lady. It would not be gracious to refuse her.

"I would need something to write on..."

"Come," she said, taking his hand, "there is a sand table in my father's work room."

Closely followed by Xanthus, they went to an inner part of the house. Jeff looked about in amazement. There were many scrolls, but what surprised him most were the scientific devices. He decided that the Greeks of Archimedes' time were more advanced than he had thought.

"You see the instrument you brought," Penelope said, indicating. "My father has studied it carefully. But he is not familiar with your number symbols. Would you explain those to me?"

Jeff went to the table in the middle of the room. It was a shallow box, perhaps six feet square, filled with the finest white sand. There were diagrams drawn in the sand.

"Go ahead. My father is done with these."

Jeff smoothed the sand and carefully drew the numeral '1'.

"Tell me the name in your language," Penelope suggested. "That will be easier for you."

"This is the symbol we use for 'one'." He held up one finger. "And this is 'two'. And this is 'three'..."

"Let me repeat them," she said, when he had reached 'nine'. Her memory was excellent. Her accent was charming as she said the English words. "But how do you write this many?" She held up the fingers of both hands.

"Oh, we write a 'one' and a 'zero'." He drew it in the sand.

Penelope looked puzzled. "I do not understand. What is this symbol?" She pointed to the zero.

"That is 'zero'. It stands for..." Jeff hesitated, uncertain how to proceed. Finally he smoothed the sand and pointed to it. "'Zero,'" he said, hoping she would understand.

From her look, he could see that she did not. Penelope was having the same difficulty as mathematicians centuries later did when first encountering 'zero'.

"You mean   nothing?"

Jeff nodded. "But when it is written after a 'one' the two together stand for 'ten'." He drew the numeral again, and made ten marks under it for emphasis.

Penelope still looked puzzled. "And after ten?"

"Eleven." Jeff wrote it. "And twelve..." He wrote the rest of the numerals, up to nineteen.

"Wait," Penelope said, "Let me see if I can find the pattern." She studied the numbers in the sand. "Ah! Is this correct?" She drew the numeral '20', and glanced quickly up at Jeff.

He nodded, and she smiled happily. "My father will be pleased with me." She went back to considering the numbers. "This is very impressing!" she said.

Jeff's ego expanded considerably. The look that this charming young woman gave him made him feel ten feet tall. His self-satisfaction did not last long.

"I can show a proof for the theorem of Pythagoras," he said, hoping to intensify the look she was giving him.

"Not so elementary," Penelope laughed. "I knew five proofs of that when I was... how do you say it?... six."

"Oh," Jeff said, rather crestfallen.

"But show me more of your numerals. Can you write one for a very large quantity?"

"If you wish." Jeff drew a 'one' followed by six zeroes. "That is one million."

Penelope looked fixedly at Jeff. "Let me be certain," she said. "After 99 you would write 100?"

"Yes."

"And, after 999, 1000."

"Yes."

"And then 10000, then 100000, then 1000000?"

"Yes."

Penelope clapped her hands with glee. "My father will love this! He admires large numbers! He can write a number for all the grains of sand in the universe! He considers it one of his greater accomplishments."

"We have another way to write very large numbers," Jeff said, eager to regain his reputation. "We call it scientific notation."

Her look encouraged him. He explained the use of exponents.

"That is an... exponent?" she said, pointing.

"Yes, we use them to show how many times a number is multiplied by itself."

"And this number... what do you call it?"

"'Ten'"

She stamped her foot in irritation. "I know that you call it 'ten'. You told me that. I want to know the word that tells what it is."

"Oh." Jeff began to understand her question. "We call it the 'base'."

Jeff was puzzled by Penelope's expression. She seemed lost in thought. He did not know, of course, that he was dealing with a mathematical mind that functioned on the same order as her father's   a mind that had been developed and trained by her father. And her father was Archimedes... Unknown to Jeff, the world he came from trembled.

Finally, Penelope's expression showed that she was back in the room with him. She now looked suspicious.

"And you still say that you are not a mathematician?"

"No, I am not a mathematician."

"We shall see," she said. "Perhaps you do not wish to tell me now..."

Then, changing mercurially, "But, in any case, my father will be pleased. Your symbolism is most economical. Your concepts are beautifully simple. I can see that they facilitate calculation."

She smiled, impish now, "I suspect he will want to reward you handsomely."

She moved closer to Jeff, looking into his face. Closeup, her eyes were even more dramatic than at a distance. He felt himself surrendering to the spell of her beauty. He sensed that she was expecting him to kiss her. He reached out with that in mind, but he was interrupted by the sound of voices in the outer hall.

"Oh," she said, "that is my father returning. You wanted to meet Archimedes. You are about to do just that."

But, before she turned to leave him, she raised on tiptoes and kissed him quickly. "That... for now..." she said.