Well, there are not only two pieces of evidence from which the ancients did in fact know about the spherical shape of the Earth. The first two of them came from ship navigation. When a ship sail away from the earth the line of shore disappears gradually. The same is when the ship approaches to the shore. The land appears gradually. If the Earth would have been flat the land would simply have the same high all the time of the ship trip. The second evidence that comes also from navigation is similar to this: when a ship disappears at the horizon on the sea, first the body of the ship disappears, then the sails. This shows that the Earth need to have some curvature for the body of the ship to be under this curvature first and the sails above it.
The third piece of evidence is more direct and it is related to the shape of other celestial bodies, like the sun and the moon which are seen to be round by the eye. Why Earth should not be of the same shape? And an even more solid evidence would be the fact that the shape of the Earth shadow on the Moon during a lunar eclipse has a certain curvature.
There have been also some other pieces of indirect evidence from which the ancients knew that the Earth was spherical. First of them was the periodicity of the seasons. Second was the trajectory of the Sun in the sky, which is an arc of circle. Also the motion of the stars which revolve around the Earth during the night could have been another indirect evidence.