You don't have to wait until 16th century to estimate the distance to
moon. According to a NASA site:
Aristarchus around 270 BC derived the Moon's distance from the
duration of a lunar eclipse (Hipparchus later improved that method).
His argument, in a nutshell: if the Moon circles the Earth, then in
about a month it completes a full orbit, the length of which (assuming
it is a circle) is about 6 times the distance of the Moon.
For the question of the mass of earth, we have to come to the Newton's
age. Isaac Newton showed that the gravitational acceleration "g"
experienced by an object caused by the gravitational attraction of a
second body, is directly proportional to the mass "M" of the
attracting body, and inversely proportional to the square of the
distance "R" between the two bodies.
g = (G * M) / (R * R)
Here, G denotes the gravitational constant. once G is known the mass
of the Earth can be obtained from the 9.8 m/s2 gravitational
acceleration on the Earth surface.
G was first measured in the laboratory; in 1798 by Cavendish and
co-workers accurate to about 1%. More information can be found at:
Bonus: Using this information, the Sun's mass can also be obtained
from the size and period of the Earth orbit around the sun.
Hope this helps
Google Answers Researcher