Roemer knew that the true orbital period of Io could have nothing to do with the relative positions of the Earth and Jupiter. In a brilliant insight, he realized that the time difference must be due to the finite speed of light. That is, light from the Jupiter system has to travel farther to reach the Earth when the two planets are on opposite sides of the Sun than when they are closer together. Romer estimated that light required twenty-two minutes to cross the diameter of the Earth’s orbit. The speed of light could then be found by dividing the diameter of the Earth’s orbit by the time difference.
The Dutch scientist Christiaan Huygens, who first did the arithmetic, found a value for the speed of light equivalent to 131,000 miles per second. The correct value is 186,000 miles per second. The difference was due to errors in Roemer’s estimate for the maximum time delay (the correct value is 16.7, not 22 minutes), and also to an imprecise knowledge of the Earth’s orbital diameter. More important than the exact answer, however, was the fact that Roemer’s data provided the first quantitative estimate for the speed of light, and it was in the right ballpark.
What’s interesting was that at the time the true distance from the Earth to the Sun was unknown, making this measurement either incredibly lucky or incredibly insightful.
Excerpt from Cosmic Horizons from the American Museum of Natural History.