I started this off by asking how the ancients could prove the Earth goes round the Sun, so I had better come up with an answer. (Rather interestingly, no reader has. Perhaps I have no readers!)
 
My answer actually follows in part one of the arguments that Galileo used, although he did not quite get it right. First, it has to be possible, so it is necessary to demonstrate the Equivalence Principle, namely that all things fall at the same rate. However, before doing that, you have to get rid of Aristotle's constrained motion. In my novel Athene's Prophecy, I had my Roman protagonist project a small arrow through water and through air, thus demonstrating the presence of frictional dissipative forces. Once you have got that far, you can drop different weights, but ones that are not going to suffer unduly from air friction. Once you accept that all things fall at the same rate in a constant field, then orbital motion becomes possible, even if you do not know the field is inverse square in nature. However, there is a difference between "possible" and "is".
 
In my view, you can do it with tides. In the second book of this trilogy, shortly to be available, my Roman goes to France for the invasion of Britain and sees the big tides in France. The correlation with the position of the Moon is sufficient to realize that the Moon is the cause, but how? Even now, most people would argue the Moon pulls the water towards it, but this is only partially correct. It is obvious there are no tides in lakes, and indeed if you attribute a gravitational force to the Moon, it is nowhere nearly as strong as Earth's at the surface. Further, if everything is falling at the same rate, then the water should be falling at the same rate as the rock, and everything should stay in the same place. So, what is wrong with that argument?
 
When devising a theory, when you run into something like that, the first thing to do is not to abandon your thoughts but rather ask, what is being missed? In this particular case, two things should strike you. First, there are two tides a day. If tides were simply due to attraction, there should be only one, because a single force cannot push and pull both at the same time. The second thing that should occur to you is that just maybe the size of the planet should be included. Now the cause of the tides becomes obvious. The orbital velocity of the planet determines the velocity at the centre of the planet, and as the body get further from the centre, the force is weaker, and consequently the orbital velocity at that point is slower. The tides arise because the changes due to the size of the earth do not correspond to what is required by the orbits at those points. Note that to make that work, the ancients would have to conclude that the force towards the centre attenuates with distance. It does not need the inverse square relationship, but it does require attenuation.
 
Thus the point nearest the Moon is moving too slowly for its distance while the moon's force is stronger, and there is an accelerating component towards the Moon. Of course, the force towards the centre of the earth is still much stronger, so there is no net motion. However, there are points not directly below. Now there is a vertical and a horizontal component, and while the vertical component is overwhelmed by the Earth's gravity, there is no opposing force to the horizontal component, and the water flows sideways to form a wave crest that follows the Moon. The second tide arises because the point farthest from the Moon is moving too fast for the attenuated force there, and the water sustains an accelerating force away from the centre, and this too has components when the water is not on the moon-earth line. The two tides prove the Earth must be moving. If it is moving, and the Sun stays the same size, it must be moving in a circle around the Sun, and by the same argument, there will be a further pair of waves due to the sun tide. That proves the heliocentric theory, with reservations.
 
You now face a problem: you appear to have shown that the Earth goes around the Moon, and not vice versa. In such a case, it is helpful to create a fictitious situation and test the limits. Thus you could ask, what would happen if the Earth and the Moon were the same mass? Which goes around which? The answer is obviously that since each receives equivalent forces, each behaves exactly the same way, and hence each moves around a common "centre of mass". (The ancients may not have put it like that, but they could make that qualitative argument.)
 
The next point in devising a theory is, when you get something you feel should be correct, you should use it to make a prediction. The prediction would ideally predict something that you did not know, but because there are so many observations it may be necessary to simply unify some points that are known. In this case, you should argue that if the Moon causes two tides, so should the Sun, but because it is further away, its effects are weaker. You therefore predict two waves, each with two crests that travel around the Earth, one in phase with the Moon, the other with the Sun. A little bit of geometry and the knowledge of how a wave behaves and you can start to predict relative tidal heights at different times during a lunar period. When you start doing that sort of thing, you begin to know that you understand something.
 
The purpose of the above has been to show one way how theories can be formed, but I have also hoped to show something of classical science, and how difficult it is to understand something for the first time. It is also interesting to consider how science is taught at schools. I wonder how many pupils are merely told that the tides arise because the moon pulls on the water?