So, his ideas were still possible, and he understood why the Moon affected tides. No, that was not quite correct. There was still the question of why there were two tides a day.
* * *
He returned the next day. The problem had gnawed away at him overnight, but he felt no closer to the answer. He simply had no idea what he was missing. As he sat on the rock and watched a seagull, he noticed the tide was now not quite in. It was a little later each day, apparently, which was how Pytheas had determined that the moon was responsible. But two tides a day meant that the Moon couldn't be responsible, unless it was pushing as well as pulling, which was ridiculous. The more he thought about it, the more certain he became that he could not work out what he did not know. Then he remembered Libo's advice: stop worrying about what you can't do, and concentrate on what you can.
What did he know with absolute certainty? The Moon was responsible for the tides somehow. There were two tides per day, one on the side of the Earth closest to the Moon, one on the far side. The Earth and Moon were going around the Sun.
Wait! Did he really know that? He thought for a moment, and decided yes, at least for the time being. He might have to revisit this, but he had to fix as much as possible in place. He knew that all things fell to the centre at the same rate, because he had measured it. He had dropped quite different objects of that bridge, carefully, and he was certain that they did. He was happy to accept that the sea was falling towards the Moon, even if the effect came from sideways motion. He had also been told that the Sun had an effect. When the Moon was full or new, the tides were higher than when the Moon was halfway between. So, what was missing?
What Aristotle would recommend, Gaius thought miserably, was to deduce the answer by logic. But where to start? Make what deductions you could on the simplest system possible that included all you knew.
What did he know about circular motion? Circular motion was two motions, one falling towards the centre, one throwing him away. These were equal, which was why he stayed the same distance from the centre.
Suppose he was in a box, circling the Sun. If so, perforce he would fall at the same rate as the box. What would happen? He sat up with a start. The answer was so strange! If he started in the centre of the box, he would stay there, floating around. There would appear to be no force at all. He would appear to be stationary. As far as he could tell, he would float around without weight! Suppose he was outside the box? Essentially the same thing, except he could eventually work out that he was going around the Sun by looking at the stars. But apart from that, he and the box would fall at the same rate, therefore he would stay on the side of the box, and again he would feel nothing. If the box was replaced by something huge, say the Earth, as far as the Sun was concerned, he was still on the outside of a box, feeling nothing, irrespective of how big the Sun was, because he and the Earth were falling at the same rate. He would fall towards the centre of the Earth, of course, because that was an extra force.
He paused. That was just too weird. It must be wrong! But if it were wrong he could feel the force inside the box, and he would move along the line of the force. But that was impossible if everything fell at the same rate. He had to fall at the same rate as the box, so he could not move towards it. So it must be right. So, for the moment, at least, assume it was right.
At least that explanation also solved an important problem: if the Sun was so far away, the Earth had to be travelling at an incredible speed in a circle, so why couldn't you feel the centrifugal force? The reason was that you, the sand, the air, everything, was falling at the same rate, so there was no force trying to make you move towards or away from anything else.
So, movement was involved in the reason. But that meant that the Earth was going around the Moon, did it not? That was not quite what he was expecting. Could that be right?
Wait a minute! Suppose the Moon was as big as the Earth? What would fall around which? They would each experience force of the same magnitude, but opposite in direction, so both would move. But around what? Maybe both would fall around each other, or around the point halfway between. Now if one was a little bigger, that point would move slightly closer to it! That was it! The Earth was falling around a point somewhere between it and the Moon, and that point was falling around the Sun. Although the Moon was a lot smaller than the Sun, it was more important for the tides, because the Earth was a lot closer to it. The closer you were, the stronger the force! That made sense.
It was then that he saw it. When he thought about it, it was so simple! It was the size of the Earth. The force tending to throw the body away depended on the velocity of the body, while the force of falling depended on the size of the body pulling on it and on the distance between them. For the Earth, these forces were equal and opposite at the centre of the Earth, but the Earth was big, and the forces became out of balance at the edges. Compared with the centre, on the side closest to the Moon the force pulling water towards the Moon was strengthened, and because its distance from the centre of rotation was smaller, its orbital velocity was slower because it did not have the same distance to travel during an orbit. The side furthest away had less force from the Moon on it, but from geometry it was going faster, which meant there was a net force throwing it away from the Earth. Two opposite forces, two tides! And if the Earth was spinning, as Aristarchus needed it to be, the direction was such as to reinforce these speed differences.
He stared across the water. That had to be it. It was so simple! At least it was when you saw it. Then another thought struck him. Not only could he say, if the Earth was moving there would be two tides, but he could say only if the Earth was moving would there be two tides. If the Earth were the centre of the Universe, perforce it was stationary with respect to the Universe. If it were stationary, there would be no force throwing away the water on the far side, because that was due to motion. Therefore the observation of the second tide was clear proof that Aristotle was wrong, and the Earth was moving around a common point between it and the Moon. He had proved that the Earth was moving! You could not directly feel the movement because everything fell at the same rate, but the tides could tell that there was motion because the Earth was big. He felt strangely calm. He understood something.
Suddenly, another thought struck him. If the tides were due to the Earth moving around a common centre between the Earth and the Moon, and the Earth moving around the Sun, there would be two sets of tides, each one like a wave that would add to the other. When they were working together, at full or new moon, the tides would be at their highest, while when the Moon was at right angles to the line of the Sun, at half moon, the two tides would be at cross-purposes, with the low Sun tide adding to the high Moon tide. They would be the smallest tides. The Moon would win out, because the Moon was closest, but these tides would still be the smallest. And a local fisherman had told him that the tides with a full or new moon were always bigger than those with a half moon. That followed from the phase relationship. The bigger tides arose when the sun and moon were working together, the smaller ones when they were at cross-purposes.