# [Physics] Proof/explanation that Sun is bigger and more distant from Earth than Moon is, Middle Age technology allowed only

astronomyeclipsegeometryhistorysolar system

If I was living in the Middle Ages, how can someone prove or at least explain to me in a simple way that the Sun is bigger and more far from Earth than the Moon?

Can a parallax be used for it?

$$\text{ratio}_1=\frac{\text{diameter}_\text{sun}}{\text{diameter}_\text{moon}}=400.8$$

$$\text{ratio}_2=\frac{\text{distance}_\text{earth-sun}}{\text{distance}_\text{earth-moon}}=385.2$$

$$\text{ratio}_2$$ is only $$3.9$$ % lower than $$\text{ratio}_1$$.

I know that there were four astronomers, that were important in developing the right opinions about the mutual positions and movements: Nicolas Copernicus, Tycho Brahe, Johannes Kepler, Galileo Galilei.

However, this does not tell you how much larger and further away the Sun is. The most basic way I have heard of, but which is hard to measure with ancient technology, is to realise that the Moon is lit by the Sun, and that seeing a half-moon means we have a right triangle, with the right angle between the axis Moon-Earth and Moon-Sun. You can then try to estimate the angle between the axis Moon-Earth and Earth-Sun, which basically appears to be close to $$90^\circ$$, which would render the Sun infinitely far way. Aristarchus, an ancient Greek, measured an angle of $$87^\circ$$, rendering the ratio of distance to be around $$1/\cos(87^\circ)\approx20$$, see here.