"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 staring 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."
"There's more," Gaius added.
"There is?" Timothy frowned in surprise. This was unwelcome, because he, the teacher, was being taken into increasingly unfamiliar territory.
"Yes. Remember when you said that force equalled velocity?"
"So?"
"It doesn't! You should be able to see that."
"Then perhaps you should enlighten me," Timothy replied, in the tone of a teacher who knew, even though he didn't and was trying to get his pupil to save him from having to admit it.
"The horse and cart example," Gaius chided. "The horse doesn't provide a force to provide the velocity, but rather to overcome the contrary force from the ground."
"That is possible," Timothy agreed, "but it doesn't prove it."
"Ha! But do you remember your own example?"
"I'm afraid I don't know what you're talking about," Timothy admitted.
"You said it was change of force that gave acceleration."
"So?" Timothy asked in a puzzled tone.
"The rock falls faster as it drops from a tower, but it doesn't get any heavier, does it?"