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Demonstration touchant le mouvement de la lumiere trouvé par M. Römer de l’Academie Royale des Sciences - [anonymous author 7-December-1676]

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[selected passages by Giampiero Barbieri – founding member]
... in some passages the author uses a cryptic kind of language, trying to hide his non complete understanding of the problem ... these passages have been cancelled ... following, a particular calculation has also been worked out by "Pighin", giving back to Römer all his merits and greatness ....

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« Since long, Philosophers were trying to put up some kind of experiment, [to decide] whether the light moved [straight away] from one point to another, whatever the distance was or if such motion, instead, required a certain time.
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Römer, a member of the Royal Academy of Sciences, when carrying out observations on the first satellite of Jupiter noticed an average delay … » (*)

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Cassini discovered this delay well before Römer and ascribed it to the limited velocity of the light ...
... anyway he did not pursue this intuition further as, with the other three Medicean satellites, he was not able to measure this type of delay... the problem was not of logic nature, as he assumed. It was simply a matter of measurements... actually, measurement becomes unforeseeable when satellites are slower... least, but not last, trying to fix the moment when a satellite enters or outcomes from the shaded zone is a problem and computation becomes uncertain ....
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«… Let A be the Sun, B Jupiter, C the first satellite eclipsed by Jupiter and out-coming in D, and let the Earth, positioned at different distances from Jupiter, be EFGHLK.
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Assuming the Earth to be in L, near to the second Quadrature with Jupiter, just when the first satellite emerges from the shade in D; if 42 hours and a half are needed to observe a revolution of the said satellite, the Earth will find itself in K the next time the satellite outcomes from D.
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It is obvious that if the light takes its time to cross the distance from L to K, the satellite shall be seen some time later at D. This would not happen if the Earth could stand stationary in L.
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The revolution of this satellite, observed at its out-coming from the shade, will be delayed that much time that light will take to go from L to K.
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On the contrary, in the other Quadrature FG, where the Earth is becoming nearer, it meets the light and the revolutions of the eclipses will also be shorter, while the emersions will be longer... »
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From theory to facts ...

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«… The theory that light does not require time is not tenable. After examining facts in detail, it has been found out that what is not noticeable during two revolutions, becomes remarkably relevant in the case of numerous revolutions, i.e. 40. If such revolutions are observed from F, they will be considerably shorter than those at the opposite side of the Zodiac. Difference is 22 minutes between H and E (**) double the distance Earth-Sun ... »
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«… The necessity for this new Equation, regarding the delay of the light, was a routine followed up during all the observations carried out at the Observatory of the Royal Academy since the last 8 years. It was confirmed further by the Emersion of the first Satellite observed in Paris on 9 November last at 5h 35’ 45’’ in the evening [ this observation was not recorded correctly... "Io’s" emersion actually took place at 17h 11' 15'' on 9 November 1676 - Universal Time – when the Sun was about to set, but still above the horizon ... ], 10 minutes later than expected, compared to the observation carried out in August, when the Earth was much nearer to Jupiter. An event Römer anticipated to the Academy at the beginning of September ... »
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From facts to a clever analysis of the same ...

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«… “in toto” to make sure these inequalities were caused by a delay of the light, it is necessary to prove this delay did not derive from any type of eccentricity or from other causes, as those generally taken into consideration to explain the anomalies of the Moon and of the other Planets.
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For certain it has been confirmed Jupiter’s first satellite is eccentric and its revolutions are anticipated or delayed in accordance to how much Jupiter moves near or far away from the Sun. The revolutions of the first Satellite are unequal ... [a point of view of the anonymous author]

... without these three last causes of inequality preventing the first [the light’s finite velocity] to become manifest ... »

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This value should correspond to the true value of light’s velocity equal to 227,000 km/sec. ... but this is against what affirmed in "Companion to the Cosmos by John Gribbin – Phoenix Edition"- see: Römer, Ole (1644 – 1710) - at page 417:
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«… The figure he came up with was equivalent, in modern units, to 225,000 km per second. The difference from the true value was given by the value of the mean distance Earth-Sun, accepted at the time. Using the same calculation with the modern value, for the size of the Earth’s orbit, gives 298,000 km/sec; the modern value for the speed of light is 299,792 km/sec ... »
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This last result is in accordance, in order of size, with the error of "Pighin’s" particular computation ...

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selected passages from an article written by an anonymous author... in some passages the author uses a cryptic kind of language, trying to hide his non complete understanding of the problem ... these passages have been cancelled ... following, a particular calculation has been worked out by "Pighin", giving back to Römer all his merits and greatness ....

Let us reckon ... the matter is not exactly trite ...

As suggested by the anonymous author, who wrote of Römer, it is better to be near to a quadrature and to carry out evaluations on 40 satellite periods .... in this instance, the second quadrature has been chosen, where Earth is moving away ... and where the satellite is out-coming from the eclipse... the graph shows the variation of the distance Earth-Jupiter as days pass by... if the orbits were perfect circles the graph would simply show a sine-curve, but as the orbits are elliptical a slight modulation may be seen, the period being similar to the synodic period of Jupiter... only 35 orbits were selected, as orbit no. 40^th was below Leivi’s horizon, working seat of the Association ...
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Description of the used methodology ....

With the use of a spreadsheet let’s now compute:

results are on the left...

on the right is a detailed explanation of the different steps...

"Io’s" out-comings from shade have been measured with the precision of a second, using Perseus ... an extremely difficult operation for both Cassini and Römer, who instead, based their calculations on definite measures...

The problem also showed a certain chaotic aspect in the value of "Io’s" synodic period where significant errors are due to the variations of the fifth decimal number ... trying to exactly measure the synodic period of "Io" is quite difficult, because of its delays and of its advances...
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The above shows the mathematical difficulties and the true greatness of Cassini and of Römer who, both were able to carry out such accurate measurements and to use statistical methods which, at the time, were just at their starting stage. They also used a good deal of "common sense" to reduce the variance ...
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"Io" takes 3.8 minutes to transit from shade to light. Its vision, when it outcomes from the eclipse, is conditioned both by the atmospheric changes and by the eye sight of the observer... an additional risk when taking measures...

We are about the second Quadrature with Jupiter ... the satellite fully emerges on 24 Nov. 1678 at 21h 20m 51s [Orbit no 01] and again it is seen on 25 Jan. 1679 at 20h 9m 29s [Orbit no 35] ...
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Starting data are the synodic period of "Io", the two Julian dates [first three lines in green] and the distance Earth-Jupiter measured at the two dates a.m.[second two lines in green]...
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The Perseus time, equal to the difference between the two dates JD, is computed and divided by the synodic period of "Io". The number of orbits is obtained by rounding to the nearest whole number [first line in blue] ....

The synodic period is multiplied by the number of orbits to obtain the nominal time .... difference between the two times is computed and expressed in days, hours , minutes and seconds ...

The difference between the two distances from Jupiter divided by the difference in seconds gives the light’s velocity in Km/sec. [last line in blue] ...

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Values in Julian Days were computed with the aid of Perseus
Values of the distances Earth-Jupiter were computed with the aid of Solex91 of Prof. Aldo Vitagliano