Subject: Re: [HM] Gauss experiment on three mountain peaks
From: Herbert Prinz (heprinz@attglobal.net)
Date: Sat Feb 19 2000 - 01:05:22 EST
Abe Shenitzer:
"Why an astronomical, rather than a terrestrial, triangle?
Let c be the constant curvature of a surface. Let A be the area of a
triangle on the surface. The angular excess of a triangle is the sum
of its angles minus \pi. We have:
angular excess = c . A
"
My guess would be, because it's bigger. Angular excess grows
proportionally with area. Curvature of space on terrestrial triangles
is beyond measurability, even with todays instruments.
All I had, when I originally asked my question was Buehler's footnote.
I have not yet had a chance of retrieving and studying the two
articles that Jeremy Gray was kind enough to refer me to. I am
therefore reluctant to speculate about Lobachevski's reasoning, but it
is clear that Gauss' measurement of 1927, while achieving what it was
supposed to, could not possibly allow any conclusion about curvature
of space.
The excess of the GEODETIC triangle Hohenhagen-Brocken-Inselsberg was
15”. This was in agreement with Gauss' own formula that you quote; the
sides of the triangle are 69km, 85km, 107km, the curvature of the
Earth is ca. Sqr(1/6370km), so the excess should be ca. 2930/Sqr(6370),
which comes out to 15 arcseconds. My understanding is that Gauss'
concern was how to split the excess between the three individual angles,
considering that the curvature of the earth changes with latitude.
On the other hand, the sum of the three angles in the PLANE of the
three mountain tops was 180 degrees. Thus, curvature of space was
either non existent or below measurability on terrestrial triangles.
Gauss must have left both possibilities open. In a letter to Taurinus
from 1824, he said "If the non-Euclidean geometry should be true, if
the constant [of curvature] is not altogether too large with respect
to those quantities that can be obtained by our measurements of the
earth OR THE HEAVENS [my emphasis], then this constant could be
determined a posteriori". (Quoted after Tord, "C. F. Gauss", MIT Press
1970). But to my knowledge, Gauss never gave any indication as to how
this constant could be obtained from measurements in the heavens.
Hence my interest in Lobachevski's proposal. It turned out he was
right in the long run. Curvature of space has been measured in the
heavens first (by Eddington). I don't think it has been measured on
the earth yet.
Best regards
Herbert Prinz
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