The Toise-Metre Problem in the Struve Arc
by Vitali Kaptüg
Key words: standard, conversion, change, evidence, GPS.
The Toise-Metre Problem (TMP) in the Struve arc is similar to that
in other European triangulations based on various national pre-metric
length standards. It has occurred much later than the arc measurement
had been completed (1852) owing to the new International Metre (IM)
standard brought into service since 1889.
To use the former results the related standards were to go through
special investigations in terms of the IM. Particular attention had to
be given to the possibility of geometrical change during the elapsed
2. The Arc Length Standards
The fundamental standard of the Struve arc was his double-toise bar
of iron denoted as "N". Made in 1827 it was determined in
terms of lines of the Toise of Peru by W. Struve himself. It is
precisely the Toise of Peru which is the unit of the arc results
published by Struve. The transported standards used in the expeditions
were: N (up to 1846), its two copies P and R, and K. Tenner's standard
T. The "family tree" of the bar N included a large number of
interconnected "end" and "line" copies extensively
used in 1847-1928. Five members of the "family", N included,
have been compared with some European standards in 1847-1902. Some of
these links can be traced up to the IM standards. Besides, the bar N
was in 1893 directly certificated in terms of the IM at the BIPM.
3. The Problem
During 1894-1905 the toise-to-metre Conversion Coefficient (CC) for
the Struve Arc was determined within the range [1.949057,…1.949073]
by various means based on one or another evidence. Thus the TMP was
the problem of choice. The most reliable value was considered 1.949067
independently by Wassiliew 1905, Helmert 1906 and Zhongolovich 1956.
Common to them was an assumption of the invariability of the bar N
between 1827 and 1893. It is the matter of importance that every
circumstantial evidence gave lower values, within the left third of
the above range.
4. New Evidence
Two Swedish sources are studied (Lindhagen 1863, Jäderin 1915)
which have never been considered with respect to the TMP. They submit
reliable evidence that the 1893 BIPM certification of N is hardly
valid for the Struve times as the bar has lengthened by 24 microns (6
ppm) in between 1862 and 1899. Besides, Wassiliew 1907 has rejected
the temperature correction made in 1893 and advocated the longer (by 3
ppm) value of the metric length of the bar. All this new evidence give
reason to restrict the range of CC dispersion to its lower values
which would manifest significantly better accordance with the
mentioned indirect derivations.
The reduction of the traditional value of the CC would make us
change to some extent our evaluations of the accuracy of the two
Struve arc segments (Kaptüg et al 1996). In the northern segment the
discrepancy between the GPS and Struve results would approach the
3-sigma limit (20 ppm) whilst in the southern one it would not exceed
the 1-sigma level (4 ppm).
Department of Mathematical Geography
Russian Geographical Society
Fax: + 7 812 315 5043
The Toise-Metre Problem in the Struve Arc
Geodetic results of the 1816-1852 meridian arc measurements have been
presented in 1857 and 1860 by head of that famous international enterprise F. G.
W. Struve (Struve 1860). The quantities were expressed in old French units
"toises", as it is described in the table (Fig.1). The Toise-Metre
Problem (TMP) in the Struve arc was similar to that in other European
triangulations based on various national pre-metric or "old-metric"
length standards. It occurred much later than the arc measurement had been
completed owing to the new International Metre (IM) standard brought into
service since 1889. Standards supporting former results had to be subjected to
special certification procedures in terms of the IM. Particular attention had to
be given to the possibility of geometrical change during the elapsed time. But
there seems to have been no "problem" so far regarding.
PRACTICAL USE of the Struve arc. A large number of geometricians from
Bessel, 1841 to Zhongolovich, 1956 have made successful use of the results
presented by Struve. That was because the arc was not used as a single geodetic
or separately but always as a composition of its 12 segments and together with
similar arcs throughout the world. The most probable average values levelled
some UNCERTAINTY in converting toises into metres. Turning to those great
measurements today one can feel free of practical limitations. For us those arcs
are merely masterly works of technological art, pieces of our cultural heritage
(Fig.2). Both their merits and dismerits are just interesting, and the mentioned
uncertainty is too.
2. THE ARC LENGTH STANDARDS
The principal standard of the Struve arc was the double-toise bar of iron
denoted as "N". Being made in 1827 it was certificated in 1828 in
relation to the Toise of Peru (TP) by Struve himself. That was made indirectly,
through the mediation of an identical Fortin's copy of the TP. As Struve put it,
"The length N= 1728.01249 lines of the Toise of Peru has been invariably
taken as the starting figure through all of our computations of linear
quantities" ( Struve 1860, translation ). Divided by 864 the cited quantity
is equivalent to:
N= 2.0000145 TP. (1)  or 2.8 ppm
The intermediate probable errors estimated by Struve accumulate in this value
in a possible maximum error below 3 ppm. But was there any independent evidence
of the accuracy obtained by Struve in his certification ? Yes, there were five.
In 1865-1871 Clarke re-determined five different bars of those which had been
investigated by Struve in relation to his standard N. Three of those bars were
found almost of the same (within 1 ppm) length, the other two exceeded Struve's
values by only 2 ppm (in 40 years!).
So it should firmly be stated that it is the length of the TP which is the
unit of the Struve results. The bar N presented the standard measure embodying
the unit, just a PRECISE SCALED COPY of the TP.
The transported standards used in the total of 10 base measurements were: N (
1827, 1844, 1845 ), its two copies R ( 1848 ) and P ( 1850, 1851, 1852 ), and K.
Tenner's standard "sazhen" T ( Russian fathom ) used on three southern
base-lines ( 1820, 1827, 1838 ). The "family tree" of the bar N
included a number of interconnected "end" and "line" copies
extensively used in 1847-1928. It is of importance that the five
"senior" members of the "family", N and its four primary
copies have been compared with European standards during the period of 1847 -
1902. Some of those relations can be traced up to the IM standards. This
material presents the multitude of INDIRECT links between the Struve standard
and that of the IM. Besides, in 1893 the bar N was DIRECTLY certificated in
terms of the IM by Benoit at the B.I.P.M. with the result (Sokoloff 1894):
N= 3.897760 IM at 7.20 Celsius. (2)  or 2.6 ppm
A possible maximum error was estimated by 10 microns. The equation (2) still
required an adjustment to the legal temperature of every toise standard, i.e. to
16.25 Celsius. For the purpose the thermic coefficient n = ~ 11.394 ppm (
determined in 1852 by Struve, Lindhagen et al ) was used.
This lead to the final result of the metric certification of the Struve
N= 3.898162 IM at 16.25 Celsius in 1893. (3a)
3. THE PROBLEM AND APPROACHES
In the same year, just before the above certification Helmert (1893) had
justified the most probable metric valueof Struve's standard:
N= 3.8981525 IM at 16.25 Celsius. (3b)
The author was basing on the most reliable INDIRECT evidence originating from
the Struve, Bayer and Clarke comparisons involving Russian, German and British
standards. Helmert used his result (3b) for processing the Russian part of the
52th parallel arc measurements. The former author has also furnished a variety
of indirect results:
N from 3.898141 to 3.898165 IM. (3c)
He made no critique about the range and concluded with the following words:
"The 1893 B.I.P.M. result might probably exceed by a few microns the true
length of the standard but, should it be the case, this incorrectness is of no
practical importance... It is difficult to say how many obstacles would
withstand the wish to obtain a higher accuracy, taking into consideration that
the terminating surfaces of the bar bear evidence of rust..." (loose
Since 1894 the value (3a) has been regarded as the most reliable. The
corresponding value of the toise-to-metre Conversion Coefficient ( CC ) was
calculated as follows:
(3a) / (1) = 1.949067 IM/TP. (CCa)
Evidently, the tacit assumption of the INVARIABILITY of the Struve standard
in between 1828 and 1893 was adopted at this calculation. The coefficient ( CCa
) was made geodetic use of independently by Wassiliew (1905), Helmert (1906),
Zhongolovich (1956) et al. There were only small differences between the authors
in the higher decimals of that value.
A typical example of practical approach to the TMP is presented by the
mentioned Wassiliew's paper. In 1901-1902 he performed a special research which
argued the above mentioned traditional value of the thermic coefficient of the
bar N and found:
n~ = 11.609 ppm + 13 ppb, against 11.394 ppm
as previously. This lead Wassiliew to revise the 1893 result of the metric
certification of the bar N and derive a larger one (Wassiliew 1905):
N = 3.898174 IM for 1893. (3d)
However, he took into consideration the lower value (very close to 3b)
supported by a reliable German source and chose the medium value (3a). In his
next work (Wassiliew 1907) he changed his mind and maintained his own result
(3d). However, he seems to have not been followed in that choice by anyone else
since then. Strangely enough, no attention was paid to a similar incident with
the thermic coefficient of the famous Bessel's iron toise made in 1823 . The
preceding Bessel's value 11.40 ppm was in 50 years changed for 11.60 ppm ( the
same quantity for iron ) by Benoit at the B.I.P.M. Therefore Wassiliew's result
(3d) cannot be considered less probable than (3a) for it has its own reason.
Thus there is a rather large uncertainty of the required metric value of the
length of the bar N which may be supposed valid for the times of Struve:
N from 3.898141 IM to 3.898174 IM (3e)
It corresponds to the following (CC) range:
CC from 1.949057 to 1.949073 IM/TP (CCe)
which covers values supported by either direct or indirect evidence. One can
notice that INDIRECT evidence comes presumably to LOWER values, within the left
third of the latter range. It should not be forgotten that the more popular
value (CCa) was based on the assumption of INVARIABILITY of the bar N between
1828 and 1893.
In literature regarding the TMP in the Struve arc one can find three more
values staying outside the range (CCe). Some authors use the value 1.949081 IM/TP
which proceeds, to my mind, from mere confusion in definitions. Basing on the
fact that the bar N was a DOUBLE-toise copied from an identical Fortin's copy of
the TP those authors came to recognize the "toise" of Struve's results
as a "SINGLE toise" of the Struve standard N. This speculation leads
to the equation:
1 "single Struve toise" = ( 3a ) / 2 , or:
CC = 1.949081 IM/TP.
Of course, such an approach should be considered definitely wrong as the unit
"Struve toise" was identical to the unit presented by the Toise of
Peru as showed in paragraph 2.
The second outsider is the Toise of Peru itself, as far as it was in
1887-1891 when its metric certification was made. The result can be presented
CC = 1.949090 IM/TP.
Available sources give reasons to assume that the TP has probably suffered
lengthening after 1823. A reliable (not single) evidence of that comes from the
well-known copy of the TP made in 1823 for Bessel. In that year the copy was
certificated as nearly identical but in 1891 it was found by 29 microns (14 ppm)
shorter than its original. At the same time the length of Bessel's toise
remained the same if referred to its own copy "T9" (Helmert 1893).
The third well-known value:
1 toise = 1.9490363 (or 1.9490366) "metres"
has no relation to the IM, but refers to the so-called "legal" metre
defined after the 1799 French Royal statute.
The difference between the two "metres" ("unfortunate
duplication" after Bomford) has much been highlighted previously and
therefore it is omitted here. Thus traditional source material related to the
"Russian version" of the TMP leaves the 8 ppm uncertainty (CCe). It
makes it NOT INTERESTING to compare between the Struve results and possible GPS
re-measures of remaining arc fragments. It can be supposed, however, that lack
of information is responsible for the problem.
4. ANOTHER EVIDENCE.
Two Swedish sources ( Lindhagen 1863, Jäderin 1915 ) have been studied which
have never been considered with respect to the TMP. They evidence that the 1893
B.I.P.M. certificate of the bar N (2) is unlikely to be valid for the Struve
times. The basic Struve standard has probably LENGTHENED some time in between
1862 and 1899. This supposition originates from the material of comparisons
performed in those years between the bar N and its last primary copy made in
1861 for Swedish Academy of sciences. The detected change of the preceding
length difference between the bars was 22 - 26 microns. At the same time the
Swedish bar remained practically unchanged if referred to its own subsequent
This new evidence presents the most serious reason so far to question the
validity of previous practical solutions of the TMP. Neither the value (3a) nor
its corrected version (3d) can undoubtedly match the starting expression (1).
Too many suppositions are needed to support the use of those DIRECT results.
INDIRECT links of the Struve bar mentioned in paragraph 2 referred to the
three European standards: the British yard (Y), the toise of Bessel (TB, a copy
of the TP) and the klafter of Vienna (K). Each one has been certificated
afterwards in terms of the IM. The lacking metric value of the bar N can be
obtained from the most reliable final metric figures. Using the necessary source
quantities, respectively, from (Struve 1860, Clarke 1866 and Bomford 1862), (Struve
1860 and Helmert 1893), (Struve 1860 and Allmer 1990) one can derive:
N<Y> = 0.9144025 (1728.01249 / 405.34622) = 3.898146 IM,
N<TB> = 2 TB + 31 microns = 3.898153 IM, cf. (3b),
N<K> = 1.8965092 (1728.01249 / 840.70370) = 3.898153 IM.
Three completely different standards agree to 2 ppm in reproducing the
required metric "identity" of the Struve standard:
N from 3.898146 to 3.898153 IM. (3f)
It looks like the upper two thirds in the questioned range (3e) cover the
least supported values in question. The latter range corresponds to apparently
the most reasonable CC value:
CC from 1.949059 to 1.949062 IM/TP, (CCf)
where the uncertainty leaves nothing to be desired.
The possibility of reduction of the inherited dispersion of the
toise-to-metre Conversion Coefficient related to the Struve arc has serious
reasons. Adoption of the most reasonable value ( CCf ) would SHIFT to some
extent the evaluations of the actual accuracy of the two Struve arc segments
derived previously (Kaptüg et al 1996). In the northern part of the arc
triangulation (Hogland-Fuglenaes, see Fig.1) the discrepancy between the 1994
GPS results and those by Struve would approach 3-sigma level or roughly 20 ppm
per 1189 km. In the southern part (Hogland-Staro-Nekrassowka) the discrepancy
would not exceed one sigma or 4 ppm per 1641 km. Thus a significant DIFFERENCE
of the TWO PARTS of the Struve arc seems to be true.
It is interesting that due to inverse signs of those discrepancies, the
general Struve's result:
1447787 toises, see Fig.1, is in a VERY GOOD AGREEMENT (within few metres per
2822 km) with the observed GPS value on the WGS-84 ellipsoid.
I wish to express my sincere gratitude for sending me the source material not
available in St.-Petersburg to: Mr. James R. Smith, Honorary Secretary to the
FIG's IIHSM, Dr.Suzanne Debarbat of Paris Observatory and Prof.Franz Allmer,
Honorary Senator of the Technological University in Graz, Austria. Special
thanks are given to the staff of the library of Pulkovo astronomical observatory
and that of Russian Geographical society for the access to some scientific
rarities. I also thank Prof.Josef Weigel and the Organizing Committee of this
Historical Symposium for kindly cooperation and help in making this paper
Allmer F. (1990) : Simon Stampfer.- Mitteilungen der geodätischen Institute
der Technischen Universität Graz, 82, October, p.57.
Bomford G. (1962) : Geodesy. - 2nd edition, Oxford, p.43.
Clarke A. R. (1866) : Comparisons of the standards of length of England,
France, &c. London, 287 p.
Helmert F. R. (1893) : Die Europäische Längengradmessung in 52 Grad Breite,
I Heft.- Veröffentlichung des K. Preussischen Geodätischen Institutes und
Centralbureaus der Internationalen Erdmessung, Berlin, 4 Kapitel, S. 225-263.
Helmert F. R. (1906) : Die Grösse der Erde, 1te Mitteilung.-
Sitzungsberichte der K. Preussischen Akademie der Wissenschaften, XXVIII,
Jäderin Edv. (1915) : Mesure de la base a Hegla Hook. - Missions
scientifiques pour la mesure d'un arc de meridien au Spitzberg, mission suedoise,
I.II.A, Stockholm, 234 p.
Kaptüg V. B. et al. (1996) : Struve's arc of the meridian agrees with the
first GPS results. - Zeitschrift für Vermessungswesen, 12, p. 572 - 576.
Lindhagen D. G. (1863) : Komparationer mellan Struves dubbel toise och den för
svenska vetenskapsakademiens räkning förfärdigade kopian af densamma.- K.
Svenska Vetenskapsakademiens Handlingar, Ny följd, 4, 4, Stockholm, p.1-10.
Sokoloff A. (1894) : Comparaison de la double-toise N de l'Observatoire de
Poulkovo avec le metre international. - Izvestija Imperat. Akad. Nauk, V ser.,
I, 1, St.-Petersburg, p. 87 - 100.
Struve F.G.W. (1960): Arc du meridien de 25' 20'' entre le Danube et la Mer
Glaciale, I, 334 p., II, 485 p., St.-Petersbourg.
Wassiliew A.S. (1905): The Pulkovo Long baseline (in Russian).- Izvestija
Imp. Akad. nauk, V ser., XXIII, 3, St.-Petersburg, p. 173 - 194.
Wassiliew A.S. (1907): Mensuration de la base avec l'appareil de Struve.-
Missions scientifiques pour la mesure d'un arc de meridien au Spitzberg, mission
russe, I.III.A.a, St.-Petersbourg, 136 p.
Zhongolovich Iv.D. (1956) : On the determination of the size of the mean
terrestrial ellipsoid (in Russian). - Trudy Instituta teoreticheskoj astronomii,
VI, Moscow-Leningrad, p. 1 -66.
- Figure 1. Geodetic results of the 1816-1852 arc measurements presented by
- Figure 2. A measurement with the Struve base apparatus (by
permission of the Library of Pulkovo astronomical observatory).
Department of Mathematical Geography
Russian Geographical Society
19 March 2000