How do astronomers measure distances to stars, galaxies, and other celestial objects? A summary of techniques will be provided at this page at a later date. The references below are excellent resources that begin with the history of astronomy.
It is well known that already in the 3rd century B.C.E. the Greek philosopher Eratosthenes calculated with a high degree of accuracy the radius of the earth. What may not be as well known is the fact that his value was not generally accepted as the true value from the time of Ptolemy (2nd century C.E.) until the 15 century C.E. Ptolemy accepted a lower value arrived at by Poseidonius (2nd-1st centuries B.C.E). The story is told in Webb, pp. 20-21:
"The estimate made by Poseidonius proved to be more acceptable than that made by Eratosthenes. This was mainly due to the influence of the Greek astronomer Claudius Ptolemy (fl.AD 140), who accepted the lower value. For more than 1000 years, scholars all over the world accepted Ptolemy's word as law. It was thus commonly held that the Earth's circumference was 7000 miles smaller than it really is. Ptolemy's mistake was to have important repercussions. When the Italian explorer Christopher Columbus (1451-1506) sailed westward from Spain he did so in the belief that Asia was only three or four thousand miles distant. Had he known that Eratosthenes was correct, and that it was 12000 miles west, it is unlikely that he would even have dreamed of sailing. It is certain that no one would have financed him. Columbus was lucky that the Americas happen to occupy the space where he thought Asia would be.
"It was not until 1522 that mankind had direct proof that the Earth was round..."
(Webb, Stephen, Measuring the Universe, pp. 20-21)
Van Helden, Albert, Measuring the Universe: Cosmic Dimensions from Aristarchus to Halley. Chicago & London: University of Chicago Press, 1985.
Webb, Stephen, Measuring the Universe: The Cosmological Distance Ladder. Springer-Praxis Series in Astronomy and Astrophysics. Chichester, UK: Praxis Publishing Ltd, 1999.