Ocular Observatory

While walking Lorna one morning this week, I noticed the Sun and Moon in the sky at the same time, and I started wondering what sort of information I could get about the Solar System based only on some simple measurements of their positions. The picture I had in my head was this:

Earth-Sun-Moon System (Not to scale)

As long as they're both visible, we can measure the angle between the Sun and the Moon with a compass. What occurred to me though is that we can also measure the angle between the Sun and the Earth by looking at the phase of the Moon:

Based on Wulffnet from Wikipedia

We can translate that length into an angle with

With two angles, we know the third, and we can relate them to the sides using the Law of Sines (and some other trig identities):

It also occurred to me that, because the Sun and Moon have similar angular sizes, which allows us to get solar eclipses, we can relate these distances to the relative sizes.

where r is the radius of the Sun/Moon.

Unfortunately, once we try to put some numbers in here, the my idea of "primitive astronomy by eye" breaks down: The actual ratio of distances for the Moon and Sun is about 0.0026, which means you'd need to get the angles within about a tenth of a degree. Thinking about this type of analysis makes me respect people like Galileo and Copernicus all the more.