In 1676 a Danish astronomer, Olaf Roemer, had assisted in making observations of the innermost satellite of Jupiter.

First, let's try to understand the figure. As the moon moves around Jupiter, there is a place (A to B) when the moon is in Jupiter's shadow - in other words, it is eclipsed as far as observations from the Earth are concerned. The Earth observations might be made from positions 1 and 2 of Earth's orbit about the sun; Jupiter's moon would therefore be seen to enter the shadow at A and emerge at B.

Roemer noticed that the time between eclipses (NOT the eclipse time) was 42 ½ hours when observed from points X or Y; that is, when the Earth was neither moving away from, or toward, Jupiter.

But when the Earth was moving away from Jupiter (position 2) he noticed that the period was 14 seconds longer. He deduced that light had to have a finite speed, because the 14 second lag indicated that the light had to "catch up" with the Earth (go a larger distance). Observations were made from many points on Earth's orbit. The details are more complex, but a summary of his observations is that he estimated that the time needed for light to travel a distance equal to the diameter of Earth's orbit was 22 minutes (the modern value is 17 minutes).

This information was taken from Holton & Roller, Foundations of Modern Physical Science, Addison-Wesley Publishing Company, pages 551-552 (1958). H & R point out that "some of Roemer's contemporaries attacked his result as implying a ridiculously large speed...others continued to believe that the speed of light is infinite."

Although it may be out of print, if you can get a copy, Holton & Roller's book is an excellent and very readable historical Physics textbook.