Galileo's telescope showed him that venus has a large angular diameter (61 seconds of arc) when it is a crescent and a small angular diameter (10 seconds of arc) when it is nearly full. when venus is "half full," its angular size is about 59 seconds of arc. this sequence of phases and angular sizes indicates that venus:

Indicates that venus orbits closer to the sun than the earth. There are several assumptions that Galileo had to make. They are 1. The smaller the visible arc, the further away the object is. 2. When Venus is nearly full, he sees almost all of the surface of Venus that's exposed to the sun. 3. When Venus is half full, he sees half of the surface of Venus that's exposed to the sun. 4. When Venus is just a crescent, he sees almost none of Venus that's exposed to the sun. So Crescent, 61 arc seconds Large, so it's close to Earth Crescent, so can not see most of the surface that's exposed to the sun. Implies that Venus is between Earth and Sun Nearly full, 10 arc seconds. Small, so it's far from Earth Nearly full, so can see most of surface exposed to sun Implies that Venus is nearly opposite from sun as Earth (e.g. Sun is almost between Venus and the Earth) Half full, 59 arc seconds Almost as large as full, so it's a little further away, but not a whole lot Half full, so from the point of view of venus, there's a 90 degree angle between the Sun and Earth. Those observations indicate that Venus is closer to the sun than the earth is. To show why, assume Venus is farther away from the sun than the earth. If that were the case, then you would never see Venus as a Crescent. The least you would see would be half.