Periodically the Galilean satellites of Jupiter, i.e., Io, Europa, Ganymede and Calisto, pass into the shadow of Jupiter. The interval between such eclipses, as viewed from the Earth, is not quite regular, as you can see by looking at the column entitled Jupiter's Satellites in Sky and Telescope magazine.
If one views such eclipses when the Earth is at greatest elongation (as viewed from Jupiter), the Earth will either be moving directly toward or directly away from Jupiter. If the speed of light is finite, the length of time between successive eclipses (as viewed from Earth) will either be shortened or lengthened. Of course, the actual size of the AU was not yet known in Rømer's time, so the distance travelled by Earth between successive eclipses was also not known. Therefore, all that could be determined was the ratio v/c of the Earth's orbital speed v and the speed of light c.
In this simulation, the distance of Ganymede from Jupiter is greatly exaggerated, and no attempt has been made to depict accurately the relative sizes of the Sun, Earth, Jupiter and Ganymede. Moreover, the eccentricities of the orbits and their differences in inclination have been ignored.
The third Galilean moon of Jupiter, Ganymede, has a period slightly over one week. A table of eclipse times is constructed in the form day hour:min:sec and the differences of successive times are also displayed. Note that the time between eclipses, as viewed from the Earth, is greatest when the Earth is at greatest elongation and moving away from Jupiter and least when the Earth is at greatest elongation and moving toward Jupiter. From the difference, 1 minute 52 seconds, you should be able to calculate the ratio v/c of the Earth's orbital speed v to the speed of light c.