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We all know that light travels at a finite speed of nearly $3\times10^8\mathrm{m\,s^{-1}}$. But how did we come to think that light has a finite speed? Why did the scientists try to calculate the speed of light? Or did we just shoot in the dark, thinking that maybe light has a finite speed, let us try finding its value?

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    $\begingroup$ This really belongs on the History of Sciennce SE, but there it would be a duplicate as AFT points out. $\endgroup$ – John Rennie Apr 13 '16 at 18:18
  • $\begingroup$ @JohnRennie Actually that question is about basically how we first calculated the speed of light, but i have to agree that the answer mentions about how we first guessed at the speed of light being finite, although a fleeting one. I guess we could let the question be, with a complete answer? $\endgroup$ – FreezingFire Apr 13 '16 at 18:26
  • $\begingroup$ I think the question is - did people think of light speed as infinite till certain time? And if so, what made them think/realize, it could be finite, so the experiments to prove, would have started and eventually succeded. More like a science history question $\endgroup$ – kpv Apr 13 '16 at 18:51
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    $\begingroup$ Related: physics.stackexchange.com/q/18472/2451, physics.stackexchange.com/q/5194/2451 and links therein. $\endgroup$ – Qmechanic Apr 13 '16 at 19:12
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As far as I know the first time anyone proved that light has a finite speed was when the astronomer Rømer discovered variations in the timings in the transits of Jupiter's moons. He correctly attributed this to the time light took to reach Earth from Jupiter.

His calculated value for $c$ was about 26% too low, but that was pretty good given the state of the art at that time.

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    $\begingroup$ There have been measurements of the speed of light before that, though, but their conclusion was always, that light is either infinitely fast or at least to fast to measure. $\endgroup$ – pfnuesel Apr 13 '16 at 18:41
  • $\begingroup$ Agreed: everyone knew light was really really fast (tho' not as fast as the Infinite Improbability Drive), but until proper observations and equipment could be manufactured, all scientists had was a lower bound on its speed. $\endgroup$ – Carl Witthoft Apr 13 '16 at 19:35
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It was discovered through the Michelson-Morley Experiment. in 1887.

The main goal of that experiment was the discovery of aether, the medium through which light travels, as scientists hypothesized back then.

In their setup they had two light sources moving at perpendicular directions, with the whole experimental apparatus (light source, some mirrors, detector) moving at the same direction with one of the beams. What was realized, was that no matter what the speed of the apparatus was, or its direction, the experimental output was always the same, even when the apparatus wasn't moving at all.

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Measuring light speed is rather trivial now a days, we have laser pulses which have durations of the order of 10^-14 sec, photodiodes which can measure the light pulses as fast as 10^-11 sec, and oscilloscope which can display these events with a resolution of 10^-9 sec. so overall you can detect a nanosecond (10^-9 sec) very easily. now what you can do if you have access to some laser facility you just take a fast laser pulse, split it into two parts steer one pulse by a known distance (lets say 1 meter) then make them fall on photodiode, connect the photodiode with a oscilloscope and you will see pulses on the oscilloscope and hence with a very little effort you can get the speed of light (if you have correct tools).

This question is of historical importance now, maybe in some 1800 AD this would be really plausible question but there is no doubt that light has finite speed as of today. The only question is how accurately we can measure it.

Edit: I am sorry, I have misunderstood your question. It was correctly told by John Rennie that the astronomer Roemer was first to realize that light has finite speed. Roemer observed that the duration of eclipse of one of the moons of Jupiter is shorter when the earth is moving towards the Jupiter and longer when earth is moving away from Jupiter.

This is one of accidental discoveries, One such interesting discovery I remembered was presence of atmosphere on Venus. When Venus pass over the surface of sun (and world was well equipped to see it for first time in 1761) a white ring appeared around the Venus, and an Russian scientist Lomonosov has commented that Venus should have a atmosphere.

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    $\begingroup$ I don't see how this answers the question. $\endgroup$ – John Rennie Apr 13 '16 at 18:31
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    $\begingroup$ Besides, I think you are technically wrong (but not in spirit) when you say "The only question is how accurately we can measure it" since these days the speed of light is defined as 299792458 $m/s$, and this defines the metre. $\endgroup$ – jim Apr 13 '16 at 20:19

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