The first evidence
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In 1671 Römer went to Hven, an island community near Copenhagen, to help re-determine the longitude of the observatory located there.
With others, he began observing a series of eclipses of Io, Jupiter's largest moon. In the end they had eight months of observations or, since Io
makes one revolution of Jupiter in 42 hours, timings on about 140 eclipses over 2/3 of the year. The time intervals between these eclipses
were not regular but appeared related to where the Earth was in its orbit. The length of the interval became shorter as the Earth approached
Jupiter and longer as it moved away; the mathematically predicted time of an eclipse was too early if the Earth was near Jupiter and too late
if the Earth was far from Jupiter. This systematic lack of fit allowed Römer to announce in Paris in September 1676 that the eclipse
predicted for November 9 that year would actually occur 10 minutes later. The observation bore him out and Römer argued that the
discrepancy was due to the finite speed of light. The light takes longer to reach us the farther we are from its source.
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(source: www.stats.uwaterloo.ca/~rwoldfor/papers/sci-method/paperrev/node22.html)