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I am trying to understand triangulation, especially I am focused on the triangulation Méchain and Delambre achieved in the end of the 18th century, in order to measure the distance between Dunkirk and Barcelona.

I've seen that only one distance has been measured in their whole work, it corresponds to the basis. It is also said that measurement errors have been tracked, by making sure the network of triangles matches.

If I understand this method to track errors correctly, they:

  • First drew $D_1$ (a first version of $D$) by calculating the length $CD$ (measuring $\hat{A}$, $\hat{D}$) and drawing the intersection between the line $(CM)$ and the circle of radius $CD$.

  • Then drew $D_2$ (a second version of $D$) by using a protractor at $A$ (they'd have previously measured $\angle DAM$) and stopping by at $D$ thanks to the length $AD$ (calculated using $\hat D$, $\angle DAM$ and $AM$).

  • Made sure $D_1$ and $D_2$ were very close each one another.

I think my three steps are ok, but I am self-doubting what's in each bullet, because bullet $1$ and bullet $2$ both use the measure of $\hat D$ (measured in the field).

PS:

  • $D_1$ is a first drawing of the point $D$, it is based on the position of $M$ (which is found based on the basis $AC$, $\alpha$ and $\hat{C}$, by intersecting the circles of radius $CM$ and $AM$) and by taking the intersection of the circle of radius $CD$ and the extension of $(CM)$.

  • $D_2$ is a second drawing of the point $D$, we here use the measure of the angle $\angle DAM$ (recall $D$ is the place where they decided to stand) and that of $\hat{D}$ and $AM$.

triangulation, network of triangles

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  • $\begingroup$ In your description it is not clear to me how they decided on D1, D2 ,.. $\endgroup$ Commented Jul 30 at 13:44
  • $\begingroup$ I have edited my answer (added a Post Scriptum and changed $D_1$ for $D$ and $D_2$ for $D$ in the construction of $D_1$ and $D_2$, since $D$ is the physical position where these people decided to set $D$, and it's different from the points drawn on their scheme). $\endgroup$ Commented Jul 30 at 14:11
  • $\begingroup$ This is about the specific error controlling scheme that they were using; you need to copy and translate the whole text on that topic so that people would not have to hunt down the text. $\endgroup$ Commented Jul 31 at 1:16

2 Answers 2

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This is NOT an answer but a long comment!

This post should be closed on this site on the grounds that it "Needs details or clarity", and/or "Needs more focus" and migrated to the History of Science and Mathematics SE where there is already one post with answers and links of relevance, Data that motivated early discussions about the mean and about error distributions.

As one might expect the main sources of interest about this topic are written in French and the diagram the OP has displayed is in this document The shape of the earth in which there is a brief description of triangulation based on the diagram.

"Principle of Triangulation Points A, C, D, E, and F can be targeted by an observation (such as lights placed at the top of high points at night). To determine AM (parallel to the meridian), we must know four hypotheses:

  • one side, the base (distance between points AC)
  • in triangle ACD, the angles ^A and ^C
  • the position of the base relative to the meridian (whose length we wish to determine) angle α By applying the sines relationship, we deduce AM By repeating the method, we determine MN, NR, RB."

The document The method of measurement by triangulation has some diagrams to illustrate where correction were made.

An excellent source of original documents is the Gallica – The BnF digital library with a search engine in English.

Here is an example written by Delambre, Base of the decimal metric system, or Measurement of the arc of the meridian between the parallels of Dunkirk.

The book The Measure of All Things: The Seven-Year Odyssey and Hidden Error That Transformed the World by Ken Alder is an interesting account in English.

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In a plane, if one knows one side of a triangle, and can take sightings to determine two angles of that triangle, then one knows (can calculate) all three sides. It's a little more complex on a sphere, but measurement of angles and one distance can turn a dozen visible-from-afar landmark points into a completely solved dozen of position coordinates.

If you know all the coordinates, you know all the distances.

The estimation of distance errors is... an exercise in algebra, starting with known measurement precision.

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