At a time when many thought the Speed of Light was infinite, a few brave souls tried to come up with a definite number for it. Most of them failed. There was one guy, however, who wasn't even trying, but came up with a great way to guess. Take a look at how he took his measurement, and what it has to do with the moons of Jupiter.
During the 1600s, people probing into the nature of light were, metaphorically, groping in the dark. They weren't sure whether it arrived instantaneously anywhere it went, or if it took time to travel. They weren't entirely sure that it wasn't an emanation from the lit objects themselves — how else could they show up as different colors in the same light? But a few researchers were convinced that light actually traveled, and were trying to measure its speed.
Not Ole Roemer. Roemer was taking advantage of light, not studying it. He and his new telescope were peering out at the moons of Jupiter. Particularly he was looking at Io. Io speeds around Jupiter in just under two Earth days, and Roemer was hoping that timing Io's eclipses might help provide a tool for ocean navigation.
Obviously, Roemer wanted to be on time, and peering through his telescope, just as each eclipse started, but he noticed a funny thing. As one season set in there was a slight delay in the time the eclipses began. Half a year later, all the eclipses were earlier than expected. Was he calculating wrong? After what had to be some fraught computations, he realized his calculations for the time periods of the eclipses weren't off. Only one major factor was different. At one point of the year the Earth was closer to Jupiter, and at another point it was farther away.
The orbit of the Earth spans a vast distance. Roemer realized that the light from Jupiter takes longer to get to the Earth when it's farther away. He took a look at the time delay — there was about twenty two minutes difference between the early and late eclipses — worked out the distance of Earth's orbit, and calculated that light must move at a rate of 131,000 miles per second. This is not the correct speed of light, which goes at 186,000 miles per second, but it was definite proof that light traveled, and that it took time to do so. In the next few decades, devices that could help measure the speed of light popped up, and our calculation of light speed has been refined ever since.
But this was the first proof that light traveled, and the first informed guess as to how fast, and it was done by a guy who wasn't even trying.