I saw this on the blog of a British Astrophysicist:
If you haven’t seen this problem before, why not have a go?
Using only the information given in this Question, estimate the ratio of the mean densities of the Earth and Sun:
i) the angular diameter of the Sun as seen from Earth is half a degree
ii) the length of 1° of latitude on the Earth’s surface is 100km
iii) the length of a year is 3×107 seconds
iv) the acceleration due to gravity at the Earth’s surface is 10 m s-2.
HINT: You do not need to look up anything else, not even G!
The answer you should get is that the mean density of the Earth is something like 3.5 times that of the Sun, although the information given in the question isn’t all that accurate.
I did “have a go” and came out with 3.3 — just like one of the commenters. A very satisfying exercise. No calculus needed to solve the problem.
Looking at this from a historical point of view, all the information needed to solve this problem was known to European scientists by the year 1700.