User talk:Egil/Sandbox/Gabriel Mouton

Comments about the references

The references used for this article have certain discrepancies which should be mentioned:

Connor and Robertson claims that the milliare is based on the degree, but this is contrary to all other sources, and does also not match the pendulum definition or the similarity to the toise.

Bigourdan says that the virgula pendulum would oscillate 3959.2 times in half an hour (une demi-heure). If one interprets oscillation as full cycle, which is the current understanding, this would make the length of the pendulum ~5.1 cm, which is clearly incorrect. The likely explanation is that oscillation should be interpreted as change of direction, i.e. twice the number of cycles, just like the one second pendulum which has a period of 2 seconds. In this case, the resulting pendulum length becomes ~20.5 cm, which is very reasonable.

The NASA document with John Quincy Adams on the history of measures claims Mouton used a one second pendulum, but the other sources, including Bigourdan, who writes about Mouton in much more detail, says otherwise. Using a one second pendulum would highly complicate matters, since the length of a one second pendulum would correspond to 4.846 virgulas, which would make construction of a virgula standard beam hard to do with a high level of accuracy. Using the pendulum described by Bigourdan, one would arrive at the desired length directly. (I have sent a letter to Nasa on this matter, and their document is now updated).

There are other claims that a virgula is 18.5 cm. That would of course have been the case if Moutons definition was made at a time when the Earths circumference was known to a higher accuracy. But as Bigourdan points out, the figure Mouton used was from Riccioli, and the pendulum used as the practical embodiment clearly establishes the length of Moutons virgula as ~20.5 cm. As far as I know, the virgula lived only as a proposal, and there has been no attempt to redefine the virgula after Mouton. -- Egil 12:12, 29 August 2005 (UTC)

Ferdinand Hoefer: "Historie de l'astronomie", Paris 1873, mentions that the error of Riccoli was more than 10000 meters per degree. [1]. Assuming a Bologna foot is 38 cm, we get 122.290 km per degree, which matches this claim well. -- Egil 00:12, 14 October 2005 (UTC)

Comment about the pendulum calculation

The G used for verification of the length of the pendulum is 9.80665, which is the standard for 45 degrees of latitude. The city of Lyon is sufficiently close to this latitude (at 45°46′1″N 4°50′3″E), and at suffiently low altitude to make this a valid approximation for this purpose. -- Egil 12:31, 29 August 2005 (UTC)

That's lowercase, italic g, and g_n corresponds to approximately 45.5° geodetic latitude (the one we always use for this purpose;it isn't exact for any other type of latitude either). The g at Lyon is likely 9.806 m/s² at its latitude and elevation, for more precise figures you'd need a specific latitude and elevation and local gravity anomaly.

But you claimed on my talk page that there was some explanation of the "minute of arc" discrepancies here. I don't find it, maybe you have something that could be added. Gene Nygaard 16:54, 31 August 2005 (UTC). Here is the two relevant sentences in your writing; for once Rktect has a valid point in this regard: "His suggestion was a unit, milliare, that was defined as a minute of arc along a meridian. The base unit would be the virga, 1/1000 of this, corresponding to 64.4 Bologna inches, or ~2.04 m." Gene Nygaard 17:03, 31 August 2005 (UTC)

Thanks for the g, mea culpa. The g I used gives 20.5376... cm, your g gives 20.5363... cm. We are in the 5th digit here, and probably approaching the error budget of our dear abbot anyway, so I don't think we need to go to extremes like finding the coordinates and height above sea level of the Saint-Paul cathedral in Lyon.

The only "minute of arc" discrepancy I know of is the difference between the pendulum length (which we can perhaps accept as ~20.54 cm), and the length I got from converting 6.44 Bologna inches into metric, at ~20.4 cm. I assume is due to an inaccurate value for the 17th century Bologna foot (I used a value 38 cm), so any increase in the precision of that value would be highly appreciated.

Any talk of a value of 18.5 cm for the virgula is circular reasoning based on todays knowledge - but I take it that this was not what you meant. -- Egil 18:46, 31 August 2005 (UTC)

With regards to the length of the Bologna foot, the interesting value is the length that Mouton reckogned for the Bologna foot. Presumably he could not afford to send an expedition to Bologna to find out, so he would have to rely on something else, like a value based on the toise, or similar. I have not seen any information as to those particulars, but it would of course be highly enlightening.

And just to make it perfectly clear: A virgula of 20.4 cm, say, implies that Moutons assumption of the Earths circumference would be 360×60×10000×20.4 cm i.e. 44064 km. That did not make him a fool, it is simply an indication of the uncertainty connected to this measurement at the time, and perhaps units of measurement in general. Of course, Jean Picard would improve the acccuracy significantly only a few years later. But even the definition of the meter wasn't spot on -- they ended up with a definition that gave 40009.152 km, and not 40000 km exactly, as was intended.

The Galileo project at Rice says this about Moutons suggestion [2]:

A fraction of the terrestrial meridian would be adopted as the universal unit of length. The measuring procedures at the time were too unsatisfactory. The topic wouldn't be taken up again until 1790.

Another thing: Mouton did also publish a Lyon-based value for a one second pendulum, which was found to differ from the values obtained in Paris (located at another latitude). This fact is inconclusive wrt which pendulum was used to define the virgula, since it could also be used as a reason to explain the appearant confusion.

Final nit-pick (not that it matters): The value 9.80665 should be pretty standard, but now defined at 45.542, not 45 degrees [3]. (So even closer to Lyon. The altitude of Lyon airport is 188 m).

-- Egil 05:33, 1 September 2005 (UTC)