It was interesting that Ole Romer succeeded where Galileo failed. This is a snippet from the article above in the Washington Post which was written by Mark Weston:
"In the 1630s, Galileo had tried and failed to measure the speed of light as it traveled from one mountain to another. Light took less than 1/20,000th of a second to move between the two peaks that he chose, and no clock was even remotely accurate enough to measure that tiny duration. Galileo was baffled. Light seemed to have no speed at all, but to be instant. "