by Ian Miller
"An enemy army appears on your left flank and you didn't even know it was there," Timothy said harshly. "What do you do? Bleat?"
"That's not . . ." Gaius suddenly paused, then grinned. "Yes, I can think of a change. Sheep. Wild sheep do not have wool, but rather they have hair!"
"Strictly speaking," Timothy said, "that's not quite true, but it's an example. Find out what you can about sheep. Don't tell me, just work it out for yourself. Sooner or later you will have to use that knowledge, but I'm not going to tell you when. Now, the most important contribution from Anaximander was contraries. Go, read up what you can about contraries. I want you to think generally, but also concentrate on locomotion." Gaius nodded, and as he left the room, Timothy mumbled to himself, "That should give him something to do for a while."
* * *
Gaius also had occasional sessions with the Greek Geminus, who was also somewhat intrigued by a Roman who wished to be a soldier and who seemed interested in Aristarchus. At first, Geminus announced that he would only teach if Gaius successfully showed aptitude and interest, and immediately began discussing what a hypothesis, a postulate, and a theorem were, then what a line was, a surface, and so on. Then he gave Gaius some observational data on planetary motion and asked him to formulate an explanation in terms of Aristarchus' hypothesis. Gaius began to find his days were filling.
It took more than a week looking at pages and pages of data before he was ready to make calculations, then he used a surprising amount of papyrus to record the calculations he made on wax tablets, and with string and markings on a large floor, but eventually he was able to return to Geminus and inform him that he had something: the paths of the planets were consistent with the Aristarchus model, and the reason they turned around and went backwards was because the Earth was travelling inside their orbits, and passed them. He showed Geminus how he had worked out how the data showed how long each planet took to go around the sun, and from their observed paths, he made an estimate of their distances relative to the Earth-Sun distance.
"The distance estimates are a bit rough," Gaius admitted. "I took observations from the retrograde motion, predicted the background of the planet from that point on the Earth's orbit, and selected data from where it actually was later. I know how long each planet takes to return to the same position, but I am not sure that the observed angles are very good, because that wasn't why they were measured. The distances are very large, so maybe they're wrong."
"Very good," Geminus nodded, after he checked what Gaius had done. "If you believe Aristarchus, your geometry seems to be correct, even if the data are wrong. Anything else?"
"Yes," Gaius nodded, "at least I think so. If we look at how far Mars is, and how much it dims, then consider how bright Jupiter and Saturn are, they must be huge if they shine by reflected light. And one last thing: if the stars are suns, as many suspect, then they must be very far away, and not only that, because they have different brightness, they can't all be the same distance. Putting them all on one sphere would be wrong."
If nothing else, this pleased Geminus, for he had advocated this same point. What surprised him was that Gaius had heard of it. As it happened, Gaius had not heard of Geminus' propositions, but his statement had had an effect: he would learn more mathematics. While he felt that, to some extent, this was something he simply had to put up with, nevertheless a time would come when this would save his life.
* * *
Timothy had no clear expectation. He had set Gaius off on this intellectual expedition partly to test the young man, but also to get more time to himself. For a week he was quite pleased. Gaius had not come running back, bleating that he could not do anything, and he had enjoyed his spare time. After two weeks he was beginning to think Gaius had given up, and with time this feeling grew. At first Timothy had pleasant dreams of freedom, for he was sure young Scaevola would be too ashamed not to keep his word. Then there were moments approaching despair, for he suddenly began to realize that he had no idea how he could earn a living commensurate with the lifestyle he had adopted. So far he could buy what he wanted because the bills had gone to Tiberius, and Tiberius had seemingly not bothered to question them. Then his dreams and concerns were burst when, after four weeks, Gaius appeared.
"This was harder than I thought," Gaius began. "It appears that these contraries are always cited, but I don't agree with them."
"You don't?" Timothy snorted. "May I enquire why not? Surely you agree there are the contraries of lightness and heaviness, of hot and cold, and . . ."
"Cold could be the absence of heat," Gaius interrupted, "just as, according to the great Aristotle, darkness is the absence of light." It was only as he said that, he paused. This was more or less what Athene had told him in that dream, or whatever it was. Perhaps all he was doing was recalling what she had said without knowing it, but that did not make sense either. How could he dream about something about which he knew nothing, yet would turn out to be true?
"So what else have you thought about contraries?" Timothy asked, in part to gain time. The concept that cold could be the absence of heat had stunned him, but the pest was correct in that Aristotle had pointed out that darkness was the absence of light. But if cold was the absence of heat, then . . ?
"If I go back to Anaximander's forces that you were discussing last time . . ."
"Yes?"
"It appears ridiculous to assert that heat is being generated at the equator, and its contrary cold is generated at the pole."
"It may appear to be ridiculous," Timothy pointed out, "but it happens. If you go south you get hot, while if you go north you get cold."
"What I am saying," he replied, putting some of his newly acquired mathematics to good use, "is that cold is the absence of heat. The equator is hot because the sunlight strikes it square on, while at the poles geometry says it essentially slides past. The poles are colder because they get far less heat on a given amount of surface."
"I see," a bemused Timothy said.
"There are contraries," Gaius said with more conviction, "but Aristotle was wrong to think they are different things. There is just one, and the contrary is its opposite, or lack thereof."
"And what lead you to that conclusion?" Timothy asked.
"Divine inspiration," Gaius shrugged, but the more he thought of this, the more concerned he became. In the dream, or whatever, Athene had stated that this opinion of contraries was important for his future, but it was also important in that it would be an early means of validating the predictions. This analysis of contraries had come to pass, and Gaius was only just beginning to realize that Timothy had never heard any interpretation like that. That meant he had to start his journal with this, which presumably would take him towards his destiny, but how?
"It would be," Timothy shook his head. This was a new problem. Perhaps this boy really did have talent. "Have you any further examples?"
"You asked me to think about locomotion. If I drop something, it has a force driving it towards the centre of the Earth," Gaius said. "Now if it falls further, it falls faster. It's weight presumably stays the same, so force is proportional to acceleration."
"Are you sure?" Timothy asked with a smile. "You have a cart going along the road at a good walking pace. Let the horse go, and what happens?"
"The cart stops," Gaius nodded.
"So the horse is exerting a force," Timothy smiled. "If the horse exerts more force, the cart will go faster, or the same with your stream. Clearly force is proportional to speed, and it is change of force that is proportional to acceleration."
"I see," Gaius frowned. In fact he did not. Something seemed to be wrong, but he could not work out what it was.
"There're plenty of clues, if you stop and think," Timothy smiled. "Which is the heavier, a rock or a feather?"
"Obviously the rock," Gaius replied, in the tone of someone who could see where this was going, and should have an answer for it, but did not.
"So the rock has more force," Timothy smiled,
"and, if you don't believe me, drop a rock and a feather on your toe! The more force, the more speed results. Everything has a force; therefore it accelerates you to the centre of the Earth, no matter where you are, which, I might add, is the critical fact the great Aristotle used to show why the Earth must be a sphere. No other shape arises from a universal acceleration towards a point. Now, think about the bow. The arrow from the stronger bow will receive more force and that arrow will go further. Go try out some bows, and see if I'm right."
As Gaius left, Timothy stared pensively at the retreating back. The young man seemed strangely interested in quite a variety of things and he was undoubtedly intelligent. A problem remained, however: what was he doing? Creating a more educated tyrant? The fact remained that Aristotle had helped create a monster, and if Aristotle could not restrain his charge, how could he? But was this young Roman a monster? And if not, was he doing the opposite: teaching and encouraging a Roman in physics, a skill he might find totally unsuited for later life, for who, after all, had heard of a Roman interested in Greek science. Romans found it too abstract, and in truth, too useless.
Meanwhile, Gaius was anything but happy. Something was wrong with Timothy's argument! In the normal course of events, he would not care, but there were those predictions! Athene, or whoever that had been, had stated that his life depended on understanding, and he was in trouble over something so simple!
* * *
Gaius remained in correspondence with Claudius, who kept him informed of the happenings in Rome, and Gaius became increasingly despondent as he heard of the machinations, and the growing number of executions. As time passed, it became apparent that nobody was safe; being a direct descendent of Augustus offered no protection, nor, it appeared from the fate of Sejanus, did being the right hand man of Tiberius. Then there was the method; the senate had bowed to Sejanus so many times that Tiberius must have ordered his fate. Tiberius seemed to be playing brutal games with those he disliked, and, as Gaius was only too painfully aware, Tiberius appeared to be playing a game with him. He must keep a low profile, which, with the continual delivery of messages with the Imperial seal, was easier said than done.
* * *
Then came a message with a different seal: the personal seal of Tiberius, Princeps. His hands were shaking slightly as he opened it, and the relief was obvious when he read the cryptic message: "So far, so good. Congratulate Timothy for getting something into your head."
"So," Gaius shrugged to Timothy, "the Princeps thinks you are doing a good job."
"That old goat knows squat!" Timothy growled. "What sort of a job I'm doing'll be resolved later rather than sooner. In the meantime, think about why Pythagoras did what he did. He felt his discoveries brought him closer to an understanding of God. Comment."
'The question then was,' Gaius thought to himself, 'If Timothy is not reporting to Tiberius, who is?' Still, he had to answer the rapid change of subject, so he gave a challenging smile and said, "Assume he actually said that, then for him it would be true."
Timothy stared at him, then laughed a little. "Ha! Signs of a philosopher trying to get out! Good! Now Pythagoras also found the rules of harmony, the relations between the lengths of pipes and scales, and he made two further propositions that are quite fundamental. These are that the Universe is essentially constructed from numbers, and governed by symmetry."
"As put," Gaius muttered, "that is sheer nonsense."
"In what way?"
"I have a rock," Gaius said, "and if I threw it at you, it would hurt, so it is more than numbers. And the rock can be any shape."
"True," Timothy nodded, "but what holds the rock together may be governed by numbers. Now, Pythagoras further argued that if something followed from something else, you should be able to prove it by logical argument. I presume through Geminus you have seen some of the proofs Euclid wrote?"
"I can recall them," Gaius said, without any sign of enthusiasm.
"What Euclid did was to show how you could produce conclusions that you could prove to be correct. The ancient Egyptians built huge buildings, and they clearly knew many of the rules of geometry, but they did not prove what they knew. Comment?"
"I don't know anything about that," Gaius admitted.
"And that's a comment?"
"It's an admission that I have no idea what the Egyptians knew or whether they could prove it. For all I know, Euclid could have merely translated a lot of Egyptian scrolls."
"Interesting response," Timothy said. "Continuing the anti-Greek theme, but at least there's some thought there. Anyway, according to Pythagoras, everything has a cause and the effects will be related to the cause by mathematics. Thus if you throw a spear with twice the force it will go twice as far. As Aristotle noted, the motion of the spear is a constrained motion, not a natural motion, so no matter how hard you throw, sooner or later it will stop, and it will fall to the ground because falling towards the centre is a natural motion, and it will not stop until it cannot go further."
"What do you mean, constrained motion?"
"Basically, there're two sorts of motion, natural and constrained. Natural is eternal, like the Sun going around the Earth . . ."
"Or, as the great Aristarchus would have it, the Earth going around the Sun!" Gaius interposed.
Timothy laughed a little. "I shall ignore that particular attempt to rile me up. To continue, constrained motion contains within it its own contrary, so eventually it slows down and stops. Now, a real exercise. Deduce something about constrained motion. Don't argue about labels; that eternal motion is termed natural is simply a convenient definition. Also, don't go to the library. Your job is to think."
Chapter 8
Since once again he needed inspiration, he walked out to his temple but neither thoughts nor goddesses came. He ate some bread and cheese and lay back in the sun. A bird flew overhead, going towards the sea, and since he needed inspiration he walked down to the little cove. In the sandiest spot, a small fishing boat was having its catch unloaded by a small family, while out on the water there were a number of seabirds fishing. They looked so graceful as they swung effortlessly around in the sky, circling, looking for food. It even seemed so effortless when, like a bolt from Jupiter, they would dive into the water, later to emerge, gulping down food.
The speed they entered the water, he thought, must put them in danger of hitting the bottom but they did not. They were too clever for that, which was more than he could say for himself. Another day gone, and no further ahead. With a shake of his head, he turned away, began to eat the last of his bread and cheese, then he threw a piece away.
A stupid question! Think of something new. Maybe there was nothing new. Motion that slows down and stops is constrained, motion that doesn't is natural. He had never seen anything speed up and disappear, other than coins at the tavern, and there were no other options. The one sentence said everything. What else could be said?
Squark!
Gaius turned around to see a gull staring at him. The gull must have got the piece of bread he had thrown away. Perhaps he was saying, "Thank you!"
Squark!
Perhaps he was demanding more. Gaius was about to shoo it away, but then suddenly something struck him. Why the bird did not strike the bottom! The bird would go a lot slower in water. Perhaps that was the answer to his question. The water slowed the bird down! A strangely simple observation, yet when you thought about it, perhaps the secret to constrained motion!
"Thank you," he nodded towards the bird, and threw some more bread, which was gobbled greedily.
Gaius walked down to the beach and picked up a long stick. There was a pool between two large rocks. He walked to the side of the pool and lowered the stick until it reached the bottom, which was waist-deep. He nodded to himself, placed the stick against the rock, then he walked back to the beach. He needed a small piece of driftwood and some pebbles that were as near as he could find to being the same weight. He then heard giggles, and looked up to see two young women stari
ng at him.
He walked back to the pool and carefully placed the piece of wood on the water, and balanced a stone on top. More giggles. Somehow, he felt self-conscious, which made no sense, because these girls were nothing but trouble. He stood up, held the stick in one hand near his piece of driftwood, while he held another pebble at arm's length, the same distance from the water as the water was deep. Then he overturned his little boat and dropped the pebble at the same time. More giggles.
As he expected, the stone splashed well before the pebble reached the floor of the pool. Just to be sure, he did it again, and the same thing happened, then he did it again, but with the stones reversed. Still the same result, and he had his answer. Something to add to his tiny journal, Timothy would not get rid of him that easily, and more to the point, he would not be drawn to Tiberius' attention as a failure.
* * *
"So, you have thought of something?"
"Of course I have," Gaius replied. "I would not have returned had I not."
"I'm beginning to believe that," Timothy muttered to himself.
"Your constrained motion does not carry its own contrary," Gaius said firmly.
"It doesn't?" a surprised Timothy asked, then he added in a more irritated tone, "You can't avoid the obvious just by declaration, you know."
"I didn't say there were no contraries," Gaius wagged a finger of chastisement. "I said the constrained motion does not contain them. Your Aristotle may have been careless here in not using his own procedures."
"What procedures?"
"Logic! Either the contrary comes from within or from without, and all Aristotle did was assume the first. But suppose it comes from without? Air gives a small contrary, water a bigger contrary, honey an even bigger one, while rock provides a contrary so big it stops everything in its tracks."
Timothy stared at him. This was something that had never occurred to him, nor, as far as he was aware, to anyone else. But he could not say that to Gaius. The young Roman's head would swell to an intolerable size. He resorted to that great teaching aid: if in trouble, bluff! Eventually he nodded. "Well done. That is quite logical."