Unfortunately, the exact calculations he used were lost in the Copenhagen Fire of 1728, but we have a pretty good account of things from news stories covering his discovery and from other scientists around that time who used Römer’s numbers in their own work. The gist of it was that using a bunch of clever calculations involving the diameter of the Earth’s and Jupiter’s orbits, Römer was able to conclude that it took around 22 minutes for light to cross the diameter of Earth’s orbit around the Sun. Christiaan Huygens later converted this to more commonplace numbers, showing that by Römer’s estimation, light traveled at about 220,000 kilometres per second. This figure is a little off (about 27% off) from the figure noted in the first paragraph, but we’ll get to that in a moment.
When Römer’s colleagues almost universally expressed doubt in his theory about Io, Römer responded by calmly telling them that Io’s 9th of November eclipse in 1676 was going to be 10 minutes late. When the time came, the doubters stood flabbergasted as the movement of an entire celestial body lent credence to his conclusion.
(collected)
