Erling Poulsen

While Rømer lived in Paris, he was timing the eclipse of Io, the innermost big moon of Jupiter, by the shadow of the planet.
He discovered that the period between eclipses was changing, it was a Doppler shift.

For wave functions, which propagates with a velocity c, we have the following equations :

λ * ν=c and ν=1/T
where λ is the wavelength, ν the frequency and T the period

If we integrate T from one conjunction between the Earth and Jupiter to the following opposition, we get an accumulated Doppler shift in the period of 16.7 minutes, near the value found by Rømer in 1676 (published in Journal des Sçavans 1676)

COMPUTATION OF THE SPEED OF LIGHT

Rømer never did compute the speed of light, maybe because he felt that the distance between the Earth and the Sun was not known too well. However, there is a value in "Adversaria" (his notebook, Royal library, Copenhagen, fol. α^b, he writes: 1091 earthdiameter per minute).

1676 (september). He announces his explanation of the Io eclipses in a lecture in Academie des Sciences. He was first supported but later criticized by Casini, who found difficulties in observing the same effect with the other 3 moons of Jupiter.
He announces that a eclipse nov. 9^th will be delayed 10 minutes^*).

1676 nov. 9. The eclipse is delayed.

1676 nov. 21. He reads a paper for the Academie des Sciences about the confirmation of the theory.

1676 (december). Rømer's explanation is published in Journal des Sçavans in an article not written by himself and with some misunderstandings (See "Ole Rømer og den bevægede Jord - en dansk førsteplads" by Jan Teuber in the book "Ole Rømer - videnskabsmand og samfundstjener", Gads Forlag 2004, Copenhagen, p. 213).

1677 july 25. London, in Philosophical Transactions, vol XII,no. 136, Rømer's discovery is mentioned. In the 7 months between Sçavans and Philosophical, there was nearly nothing about the discovery.

1677 sept. 16. Huygens in Holland reads Rømer's paper in Philosophical, and he immediately sends a letter to Rømer asking for more information. From the article he understands that light travells one earthorbitdiameter in 22 minutes.

1678. Huygens presents his "Traite de lumiere" for Academie des Scienses. In that, he uses the diameter of earth's orbit and what he thought was Rømer's value for the light's delay in travelling over the orbits diameter to calculate the speed of light (he was the first to present the velosity with terrestrial units, 16²/₃ earthdiam./sec.).

1679. Rømer mentions a eclipse of the moon in the letter above.

1680. In London, Hooke criticizes Rømer, as he maintains that light travels instantaneously. His argument goes as follows:

'tis so exceeding swift that 'tis beyond Imagination; for so far he thinks indubitable, that it moves a Space equal to the Diameter of the Earth, or near 8000 miles, in less than one single Second of the time, which is in as short time as one can well pronounce 1, 2, 3, 4: And if so, why not be as well instantaneous I know no reason....

1686. In the first edition of Newtons "Principia", he uses the 22 min.  

1690. "Traite de Lumiere" by Huygens is printed in Paris.

1692. In his papers, "Adversaria" fol. 17, Rømer makes use of the light to determine the distance to the stars (founded on Tychos starpositions, he found the parallax to be less than 3,5'). He sets the distance to more than eight lightdays.

1704. "Optics" by Newton is published. Here he says that it takes only 8 minutes for the light to pass from the Sun to the Earth. Maybe Newton himself has computed the 8 minutes from "Sçavans" instead of using the 22 minutes he used earlier.

1706. Rømer writes in a remark in "Adversaria" (his notebook, Royal library, Copenhagen, fol. y^a) that Newton uses 8 min. for the Earth-Sun distance, as if he had not seen it before.

1713. In the second edition of "Principia", Newton again uses the 8 minutes.

1842. Doppler explains the Doppler effect.

1849. The French scientists Fizeau og Foucault measure the velocity of light with an instrument placed on the Earth. Fizeau informs the French Academy July 23'th, 1849.

 ^*) From observations in august 1676 he calculates that a mooneclipse november 9th must be delayed 10 minutes. From august we know of three observations the 7^th, 14^th and 23^rd. It was possible to calculate the relative distances in the solar system by help of the Kepler Laws. If we use the software WinStars 2 to find the Earth-Jupiter distances on the relevant dates we get:
nov. 9^th   - 5,526 AU
aug. 7^th   - 4,254 AU, difference to 9/11 1,272 AU
aug. 14^th - 4,316 AU, difference to 9/11 1,210 AU
aug. 23^rd - 4,408 AU, difference to 9/11 1,118 AU
and if these differences should give a 10 minutes delay he must have used a value of the time for light to travel one AU of:
7,9 min/AU if the calculation was founded on the observation of aug. 7^th
8,3 min/AU if the calculation was founded on the observation of aug. 14^th
8,9 min/AU if the calculation was founded on the observation of aug. 23^rd
values much closer to what is accepted to day compared with the 11 minutes (22 min. for 2 AU) which should appear in the atticle in "Sçavans", but does not.
The part of the article where you find the 22 could be interpreted in the following way:
He tells that for a couple of revolutions of Io the effect is very small, but if you compare 40 revolutions from one side with 40 from the other side the effect is considerable and that in proportion to 22.
This could be understood as the Earth moves just as long in the time of the 80 revolutions of Io as light moves in 22 minutes. The time for the 80 revolutions are 42,5 hours x 80 = 141,67 days, in that period earth moves 2*¶*141,67/365.25 AU = 2,436 AU, and if it takes
light 22 minutes to travel that distance the velosity must be 9 min./AU, close to the value above.