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Estimating the radius of the Moon - The medieval way

TL;DR I myself learnt the method from this page here.

Having estimated the radius of Earth, the next step, celestially speaking, would be to estimate the radius of the Moon, our satellite and closest neighbor. And before we proceed any further, let me just point out the fact that the distances or sizes in astronomy are measured in angles and not in meters. From Earth, it looks to us as though the astronomical objects are moving on a celestial sphere and therefore, their motion can be quantified in terms of angular motion and sizes in angles. How to measure these angles you say? Well using a telescope of course! A telescope can be calibrated such that one can move it along or perpendicular to the horizon with milli-degree accuracy, helping us therefore to estimate the angular size of objects in the sky.

Now that we've cleared that, let's look at quantities in the Earth-Moon-Sun system that can actually be measured. We already know the radius of Earth. We can estimate the angular diameter of the Moon, and therefore of the Sun, using telescopes. It is measured to be 31 arcminutes or approximately half a degree. Why is the angular size of the Moon equal to the angular size of the Sun you ask? Hint : Eclipses!

Coming to point, using the small angle approximation, we can say that the ratio of angular size of the Moon and the angular size of the Earth's shadow at a distance equal to the distance between the Earth and the Moon is equal to the ratio of the actual diameter of the moon to the actual diameter of the Earth's shadow at that distance!






The next step towards measuring the radius of the Moon is to look for an equality among the angles mentioned above. We can clearly see that the angles A, B and C add up to 180 and so do the angles C, D and E. We can therefore say that the sum of angles A and B are equal to the sum of angles C and D. In the small angle approximation, we can roughly say that the angles D and E are equal to the ratio of the radius of Earth to the distance between the Earth and the Moon and the distance between the Earth and the Sun respectively. Angles A and B on the other hand are equal to the ratio of the radius of the Sun and half the diameter of the Earth's shadow to the distance between the Earth and the Sun and the Earth and the Moon respectively. After making use of the equality that the angular size of the Sun is equal to the angular size of the moon i.e the ratio of the radius of the Sun to the distance between the Earth and the Sun is equal to the ratio of the radius of the Moon to the distance between Earth and the Moon, we can finally get an equation for the radius of the Moon in terms of the radius of the Earth and the ratio of the angular size of the Earth's shadow to the angular size of the moon; all of which are measured! You can follow the math here and see that the radius of the Earth is 3.7 times the radius of the Moon.

And to put a cherry on top of everything, now that we know the radius of the Moon and it's angular size, we can measure the distance to the Moon! Voila, now we know the radius of the Earth, the radius of the Moon and the distance between the Earth and the Moon!

You can also refer to this page, which gives a halfhearted explanation as to how to measure the same.

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