Investigation of the accuracy of Krayenhoff's
triangulation (1802-1811) in Belgium, The Netherlands and a part of North
Western Germany
N.D. Haasbroek
Nederlandse Commissie voor Geodesie 16,
Delft, 1972. 222 pages.
ISBN-13: 978 90 6132 023 4. ISBN-10: 90 6132 023 2. € 22.50
Introduction
Krayenhoff's triangulation in a part of Belgium, The Netherlands (
with the exception of the province of Limburg ), and a part of
northwestern Germany, carried out between 1802 and 1811 and published in
his Precis Historique was praised to the skies shortly after its
completion also because of the appreciative but rash judgments of
Delambre and Van Swinden.
The first who, in 1824-1825, in his letters to Schumacher, Bessel, and
Olbers criticized Krayenhoff's work was the great German mathematician
C.F. Gauss. To Bessel e.g. he writes: "Krayenhoff hat aus vielen
Winkelreihen immer nur diejenigen beibehalten die am besten zu passen
schienen, ohne anzugeben wieviel die anderen abweichen".
In the same strain he writes to his pupil and friend Schumacher: "Entweder
muss also Herr Krayenhoff seine Ausgleichungen nicht gehörig gemacht
haben oder seine Winkelmessungen involvieren versteckter Weise viel
grössere Fehler als man nach der Prüfung durch die Dreiecke und die
Gyruswinkel erwarten sollte und im letzten Fall ist man berechtigt zu
glauben dass die angegebenen Beobachtungswinkel wenigstens parteiisch
gewählt sind um diese Schliessung der einzelnen Dreiecke und Tours
d'horizon zu erzwingen".
In The Netherlands, Gauss' adverse criticism was borrowed by Verdam; on
the pages 206-214 of his "Methode der kleinste quadraten" (Method of the
least squares, Groningen, 1850) he reproduces in detail what Gauss had
said on the accuracy of the northeastern part of the triangulation
network ( the surroundings of Drachten, Leeuwarden, and Dokkum ). In
Jordan's "Handbuch der Vermessungskunde" this same part of the network
is discussed.
In 1864 appears, also in The Netherlands, the criticism of Kaiser and
Cohen Stuart: "De eischen der medewerking aan de ontworpen graadmeting
in Midden Europa voor het Koningrijk der Nederlanden". As the title: "Requirements
for the cooperation of the Kingdom of The Netherlands in the designed
Middle European Triangulation "already suggests, the motive for this
criticism was a request of the Prussian general Baeyer whether
Krayenhoff's observations could be used for such a triangulation.
Baeyer claimed that, if so, the measurements should be recomputed and
suggested that, like in other countries, some army officers should be
charged with this work under the supervision of Kaiser. The measurements
should be completed with new astronomical measurements. The inaccuracy
in length between two far distant points in the adjusted network should
not exceed the factor 1 to 20,000. For, such was the reasoning in those
days, if the inaccuracy of latitude determination is estimated at about
1/3" ( about 10 metres ), and the distance between two astronomical
stations at about 200 km, then the error in length on account of the
astronomical determination is about 1 to 20,000.
Apart from his introduction on the pages 4-16 of the booklet, Kaiser has
not collaborated with the investigation laid down in the latter part. As
the wellknown and elder astronomer he only gave his name to the
contents. All the work - and it was a thorough investigation indeed -
was done by Cohen Stuart. He concludes that Krayenhofffs measurements
should be rejected. His judgment agrees with Gauss' opinion: the far too
small closing errors (standard deviation m1) in the angles around a
central point, the also very small closing errors in the sum of the
angles of the triangles (standard deviation m2) and the often
considerable closing errors (standard deviation m3) in the sine
equations demonstrate that Krayenhoff made his observations look better
than they really are. In reality, according to Cohen Stuart, the
standard deviations m1, m2, and m3 should be alike when the observations
are independent of each other. By this judgment the sentence on
Krayenhofffs triangulation was passed.
After a new but unsuccessful attempt for a triangulation by Stamkart in
the years 1865-1881 it has been replaced in The Netherlands by the
network of the Rijksdriehoeksmeting. The first order measurements for
the network were carried out between 1885 and 1905 by the Rijkscommissie
voor Graadmeting en Waterpassing (government commission for
Triangulation and Levelling ). They
have a high accuracy as may be found in the publication of the first
order triangulation in "Triangulation des Pays Bas". Thanks to the
precautions during the measurements and the system of measuring the
angles on a station in all combinations, m1, m2, and m3 are about alike
as Cohen Stuart made it already his ideal. These results could be
attained by much self-control which made the art of measuring a waiting
for the most favourable circumstances. At Finsterwolde, e.g., one of the
first order points, the two engineers charged with the measurements,
remained for six weeks with the result that not even one angle could be
measured. In 1888 at only three stations the measurements could be
finished.
Cohen Stuart should have known, however, that even in his time not one
triangulation satisfied these conditions and that also his demands do not hold
for the very large English triangulation described in "Account of the Principal
Triangulation of the Ordnance Survey of Great Brittain and Ireland (London
1858)" .
Twenty five years later - in 1889 - Van der Plaats returned to the subject of
Krayenhoff's triangulation in an excellent paper in the Dutch professional
journal "Tijdschrift voor Kadaster en Landmeetkunde" [ 20 ] . Not, as Gauss and
Cohen Stuart did, to condemn the triangulation but to take it under protection
because the judgment of its opponents "is partial and based upon wrong
principles of justice and wrong considerations". In an often emotional manner
and certainly not free from a theatrical effect he reacts on Cohen Stuart's
judgment "that the measurements are far too inaccurate to be used for the new
Middle European Triangulation. It is even not possible, neither to judge
Krayenhoff's measurements, nor to recompute them as even his registers are not
the unchanged results of mutual independent observations" with the words:
"Let us suppose that (what would not have been impossible) this judgment would
have been given in 1818. With fervent indignation the then sixty years old
general would have answered to the waylayer of his honour: Judge, yes condemn my
geodetic work; an honest judgment is welcome to me and I will answer, give
information, and correct my work as much as I can. But don't attack my
personality. Are you a stranger in the national history of the past twenty years
that you think me capable of such a thing. I have concealed nothing in my
documents, nothing added to or withheld from the results of the observations. Go
and investigate the publications of others; test my work by theirs". I shall
have the opportunity to quote Van der Plaats' work several times. Here follows
already such a quotation in which he remarks that "the only effectual means to
judge a triangulation is to compare it with a later one with an uncontested
higher accuracy".
Up till now this was never done. This study will be an attempt. It can give an
answer to the question whether, according to the requirements of 1864, (a
relative length error of 1 to 20,000 between two far distant points in the
recomputed network) Krayenhoff's measurements could be used or had to be
rejected. Van der Plaats was convinced that they could be used, notwithstanding
the imperfection of the triangulation which he admits. For Krayenhoff's
rehabilitation as geodesist according to Van der Plaats the following lines of
poetry might then be used as an introduction to an eventual new (third) edition
of his Precis Historique. They are borrowed from Racine's tragedy Brittannicus (second
act, third scene) and they run:
"J'ose dire pourtant que je n'ai mérité
Ni cet excès d'honneur, ni cette indignité".
Contents
1. Introduction 9
2. Krayenhoff's biography 12
3. The motive for the triangulation 14
I. Geodetic part of the triangulation
4. General survey of the triangulation 16
5. Description of the instruments used 20
6. Execution of the angular measurements 24
7. Chronological order in which the stations were visited and survey of the
number of angles and series measured there 31
8. Accuracy of the angular measurement 37
9. Influence of the eccentricity of the lower telescope on the result of the
angular measurements 40
10. Reduction of the measured space angles to the horizon 42
11. Reduction of the measured angles to centre 43
12. Reduction of the spherical angles, reduced to horizon and centre, to angles
between the chords on the sphere 51
13. Conditions the angles of the triangulation network have to comply with
56
14. Analysis of the closing errors in the angles around the central points
84
15. Analysis of the closing errors in the triangles 87
16. Analysis of the closing errors in the side equations 110
17. Consideration on the rejection of series measured in the triangulation
112
18. Krayenhoff's computation of his triangulation network and his efforts to
make it a closing mathematical figure 118
19. Adjustment of the spherical angles of the triangulation network according to
the method of the least squares 129
20. Provisional adaptation of the adjusted network to the points Rhenen and
Gorinchem of the R.D.–triangulation 131
21. Final adaptation of Krayenhofft s adjusted triangulation to 65 identical
points of the R.D.–network 135
22. Comparison between the side lengths in tableau I11 of the Precis Historique
and those found from the adjustment according to the method of the least squares
140
23. Comparison of the angles and sides (chords) of the adjusted network with the
results of the R.D. 143
24. Final consideration on the geodetic part of the triangulation 154
II. Astronomical part of the triangulation
25. Introduction 156
26. Determination of the latitude in Amsterdam (station No. 40) 157
27. Determination of the latitude at Jever (station No. 102) 163
28. Determination of astronomical azimuths. General considerations and results
of the measurement of the azimuth Amsterdam-Utrecht 167
29. Measurement and computation of the azimuth Jever-Varel 178
30. Instrumental errors, affecting the accuracy of the determination of azimuths
186
31. Determination of standard deviations in azimuths 188
32. Survey of the geographical coordinates φ and λ of all the points of the
triangulation network and the azimuths of all the sides and, for the common
points and sides, a comparison with the R.D.–results 192
33. Conclusions 214
References 219