Rømer predicted at the Académie in early September that the emersion of 9 November 1676 would come about 10 minutes later than tables based on August observations said. August was when Earth was much nearer Jupiter. The emersion was observed at 5:35:45 pm, about 10 minutes late. Making the forecast publicly, months ahead, is what made this a demonstration so impactful. Fun fact: the correction for light travel time across Earth's orbit in pulsar timing models is called the "Rømer delay." This is the key logical step. In modern terms the effect per orbit is: ΔT ≈ T·(v⊕/c) ≈ 152,900 s × (29.8 / 299,792) ≈ 15 seconds 17th century timing of an eclipse was uncertain by tens of seconds, partly because Io dims gradually as it moves into the shadow. So one orbit can't show the effect. But the delays add up: 40 orbits gives roughly 40 × 15 s ≈ 10 minutes. That's why "forty revolutions observed on the side F might be sensibly shorter than forty others." Rømer considers 3 other causes for the irregularity: 1) Io's orbit is eccentric (not circular). 2) Jupiter's own speed changes as it moves toward or away from the Sun (Jupiter's eccentricity). 3) "Unequal revolutions of the primum mobile." This means irregularities in the reference clock: the daily turning of the sky, or the equation of time. The argument is about what the effect depends on: An effect caused by Io's orbit would follow the position of Io in its orbit. An effect caused by Jupiter's eccentricity would follow Jupiter's position around the Sun, repeating over about 12 years. The observed effect follows the Earth–Jupiter distance, repeating over about 13 months (Jupiter's synodic period). Suppose Io's eclipses actually occur at times $$ t_n = t_0 + nT . $$ We observe each one when its light reaches us: $$ \tau_n = t_n + \frac{d(t_n)}{c}, $$ where $d(t)$ is the Earth--Jupiter distance. The interval between two successive eclipses, as seen from Earth, is therefore $$ T_{\text{obs}} = \tau_{n+1} - \tau_n = T + \frac{d(t_{n+1}) - d(t_n)}{c} \approx T + \frac{\dot d\, T}{c} = T\left(1 + \frac{\dot d}{c}\right), $$ where $\dot d \equiv \mathrm{d}d/\mathrm{d}t$ is the rate at which the distance changes. This is the Doppler effect applied to a clock. When Earth moves away from Jupiter ($\dot d > 0$), the eclipses arrive spaced out; when it moves closer ($\dot d < 0$), they arrive bunched up. This is exactly the paper's claim that the eclipses are \emph{retarded} along $LK$ and \emph{shortened} along $FG$. ### TL;DR The first measurement showing light has a finite speed. In this paper from 1676 Ole Rømer proved that light doesn't travel instantly using Jupiter's moon Io as a clock in the sky. Io is eclipsed by Jupiter's shadow about every 42.5 hours. When Earth is moving away from Jupiter, the eclipses show up late. When Earth is moving toward Jupiter, they show up early. The light carrying the news of each eclipse has a different distance to cover, so it reaches us at a different time. The delay per orbit is only about 15 seconds, too small to measure with the clocks of the time, but it adds up: over about 40 orbits it reaches several minutes. Rømer estimated light takes about 22 minutes to cross Earth's orbit. Rømer predicted that the eclipse of 9 November 1676 would come 10 minutes late, and it did.  Ole Rømer (1644–1710) was Danish and trained in Copenhagen under Rasmus Bartholin, the man who discovered double refraction in Iceland spar. In 1671 the French astronomer Jean Picard came to Denmark to fix the exact position of Tycho Brahe's old observatory. He was impressed with Rømer and brought him back to the new Paris Observatory and the Académie Royale des Sciences. Rømer was 32 when he wrote this paper. The Observatory was run by Giovanni Domenico Cassini, who had built the best tables of Jupiter's moons. This paper is partly a disagreement with Cassini: - Cassini seems to have briefly suggested a finite speed of light himself, in August 1676, and then dropped the idea. - He preferred to explain the irregularity through features of Io's orbit. - He never accepted Rømer's explanation. Christiaan Huygens and Isaac Newton were won over quickly. The Cartesians were not: Descartes had argued that light is transmitted instantly, like pressure through a rigid stick. He even claimed lunar eclipses proved it: if light took time to travel, the Sun, Earth and Moon would look out of line during an eclipse (his argument fails because the delays involved are only seconds). Rømer went back to Denmark in 1681 as royal mathematician. Later he became Copenhagen's police chief, and in that job he installed street lighting :)