What is the likely error of the triangulation of Gauss' survey?

Question

I have been trying to figure out the famous Gauss' big triangle measurement, but not being particularly well-versed in geodesy (I come at this from a physics perspective), I have been having some troubles with decyphering some of it.

The famous historical survey associated to the field of differential geometry is the Gauss land survey for the Kingdom of Hanover, between 1821 and 1825. Legend has it that he used this survey to check the accuracy of the triangle postulate, although this is likely a myth.

However, the survey can still be technically used for that purpose, and Gauss does indeed use it as an example of the triangle postulate.

However, one thing that I have not been able to find is, is the Gauss triangulation indeed summing up to 180° up to experimental error?

The "big" triangulation that people usually refer to is the largest triangle of the survey, formed by :

• Hoher Hagen (a hill near Göttingen)
• The Brocken lodge on top of the Brocken mountain
• Großer Inselsberg (another mountain).

Looking at google map, this is about the following triangle :

The published details of these measurements can be found in Gauss' Werke compilation, p. 297. The details are a little hard to make out :

As far as I can tell, considering that it should add up to (also hard to make out) 180°0'44.476'', the three values of those angles are probably 86°13'38.366'' for the angle situated at Hoher Hagen, 53°6'45.642'' for the angle situated at Brocken, and 40°39'44.473'' for the angle situated at Großer Inselsberg (I still can't get them all to add up to Gauss' result but that's about what my eyes can make out).

Gauss also adds a value of 0.680(?)'' to the total for not terribly clear reasons, at least for my poor German, and compute a total excess angle of 14.853''.

Is this kind of excess angle within the experimental range of such a triangulation? I could not find too many details on the process itself. According to this contemporary article, Gauss did this measurement using the famed heliotrope he constructed for this survey [p. 115] (apparently this one).

According to this book, Gauss used a "twelve-inch repeating theodolite". It also mentions quite a lot about Gauss' process regarding random error, but the values seem to be fairly small compared to the 14'' : always in the range of 0'' to 4'', which does seem to fit some of the smaller triangles up in the north of Hanover but not so much the big one, or some of the slightly smaller ones such as Falkenberg-Deister-Lichtenberg (~8''), making me think this may be more systematic error due to the greater distance.

Does this seem perhaps like a reasonable size of error for some effect like refraction?

Answer 1

This is a very good question for GIS.SE and one which many geodesists or surveyors should be able to answer. It does, however, appear to be based on a false premise, so I'll deal with that first.

Spherical Excess

In surveys covering small enough areas -- most "every day" land surveys, in fact -- it is perfectly OK to use a "flat Earth" model. That is, it is reasonable to assume that every vertical direction is perfectly normal to a perfectly flat reference surface. And that every theodolite, when correctly set up, is making measurements of horizontal angles about those vertical axes. In such cases, the rules of planar Euclidean geometry apply. For example, the sum of the three angles of any triangle on a plane equals a straight angle (ie, exactly 180°).

Now, when the areas or distances of surveys start to get large, and the accuracy requirements of the surveys remain stringent, the flat Earth model is inadequate, and must be replaced by a spherical Earth model. I'm going to ignore the further refinement of using a spheroidal (or ellipsoidal) model because it makes things unnecessarily complex for this discussion. In the spherical Earth model we assume that every vertical direction is perfectly normal to a perfectly spherical reference surface, a globe. Each vertical direction now radiates away from the centre of the globe. And every theodolite, when correctly set up, is still making measurements of horizontal angles about those vertical axes, but each has its own local "horizontal plane", tangent to the globe. In such cases, the rules of spherical Euclidean geometry apply.

A triangle on a sphere is known as a spherical triangle, and the sum of the three angles of any triangle on a sphere is always greater than a straight angle (ie, >180°). The extra amount is know as spherical excess, and the larger the triangle, the greater the excess. So, the sum of the angles of your example large triangle are expected to be in excess of 180°.

More on Spherical Triangles

Here's a spherical triangle (from John D Cook)

The vertices are A, B, and C.

The angles at each vertex are α, β, and γ, respectively. (For some reason, γ at C is not labeled.)

The arcs opposite each vertex are a, b, and c, respectively. Note, the arcs are arc angles, not arc lengths, subtended at the centre of the sphere.

There are a number of laws of spherical triangles, one is the law of cosines:

cos a = cos b cos c + sin b sin c cos α

which you can rearrange to get any vertex angle (eg, α) from the arc angles (a, b, and c).

You should be able to look up the arc lengths of your example by using the same mapping tool used for your diagram. I don't think you need exact values for those distances if all you need is a good approximation for the sum of α, β, and γ and hence the spherical excess.

You can convert the distances to arc angles via

arc angle = arc dist / R, where R is the sphere's (ie, Earth's) radius.

Survey Error

The expected random error of up to 4" for the theodolite you describe seems reasonable to me, for measuring horizontal angles. Presumably, multiple observations at each survey station would have been made in order to get more reliable values, possibly better even than 4".

Finally, you speculate on two other things.

Refraction is generally a problem in survey angular measuremnt, but probably not in this case. Layers of air of differing temperatures do cause significant problems when measuring vertical angles, but they have nothing to do with your example. When horizontal lines of sight happen to pass close to buildings or other solid objects that have temperatures different to that of the ambient air, then lateral refraction can be problematic. However, triangulation surveys, from hilltop to hilltop, did not generally have this problem.

You may know that systematic error is not necessarily a measurement error, but an error in the underlying mathematical model. What you perceive to be an error, in the range of 15", is probably the error from using a planar trigonometry model of the survey, rather than a spherical trigonometry model.