Galileo tried to measure the speed of light by having a servant take a lantern to a distant hill top. Galileo was to open his lantern and as soon as the servant saw the light he was to open his lantern. When Galileo saw his servants lantern he would have a measure of the round-trip-time. But of course Galileo could only conclude the light travel time was small compared to the servants reaction time…i.e. it was very fast.
The first real measurement of the speed of light was made in 1675 by the Danish astronomer Ole Roemer. He kept track of the times that moons of Jupiter were eclipsed.
When he compared these data to calculations assuming uniform orbital motion he saw the the eclipse intervals got a little longer as the Earth moved further from Jupiter and a little shorter then as Earth approached Jupiter in the Earth's annual orbit around the Sun. He realized this was due to the increasing travel time for light. So knowing the size of the Earth's orbit he calculated the speed of light to be 200,000 Km/s.
Roemer's result wasn't very accurate. The first reasonably accurate value was calculated in 1728 by an English physicist, James Bradley.
Because of the Earth's speed in orbit a telescope has to be tilted slightly in the direction of motion toward a star. This is called stellar abberation. By comparing the angles for observations of a star six months apart and knowing the Earth's speed in orbit, Bradley calculated the speed of light to be 301,000Km/s. Quite close to the current value.
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