deg. ' " Ft.
Punnea 8 9 31.132 ..
Putchapolliam 10 59 42.276 1029174.9
Dodagunta 12 59 52.165 1756562.0
Namthabad 15 5 53.562 2518376.3
Daumergida 18 3 15.292 3591788.4
Takalkhera 21 5 51.532 4697329.5
Kalianpur 24 7 11.262 5794695.7
Kaliana 29 30 48.322 7755835.9
The data of the Russian arc (long. 26 deg. 40') taken from Struve's work are
as below:--
deg. ' " Ft.
Staro Nekrasovsk 45 20 2.94 ..
Vodu-Luy 47 1 24.98 616529.81
Suprunkovzy 48 45 3.04 1246762.17
Kremenets 50 5 49.95 1737551.48
Byelin 52 2 42.16 2448745.17
Nemesh 54 39 4.16 3400312.63
Jacobstadt 56 30 4.97 4076412.28
Dorpat 58 22 47.56 4762421.43
Hogland 60 5 9.84 5386135.39
Kilpi-maki 62 38 5.25 6317905.67
Tornea 65 49 44.57 7486789.97
Stuor-oivi 68 40 58.40 8530517.90
Fuglenaes 70 40 11.23 9257921.06
From the are measured in Cape Colony by Sir Thomas Maclear in long. 18 deg.
30', we have
deg. ' " Ft.
North End 29 44 17.66 ..
Heerenlogement Berg 31 58 9.11 811507.7
Royal Observatory 33 56 3.20 1526386.8
Zwart Kop 34 13 32.13 1632583.3
Cape Point 34 21 6.26 1678375.7
And, finally, for the Peruvian arc, in long. 281 deg. 0',
deg. ' " Ft.
Tarqui 3 4 32.068 ..
Cotchesqui 0 2 31.387 1131036.3
Having now stated the data of the problem, we may seek that oblate
ellipsoid (spheroid) which best represents the observations. Whatever
the real figure may be, it is certain that if we suppose it an ellipsoid
with three unequal axes, the arithmetical process will bring out an
ellipsoid, which will agree better with all the observed latitudes than
any spheroid would, therefore we do not _prove_ that it is an ellipsoid;
to prove this, arcs of longitude would be required. The result for the
spheroid may be expressed thus:--
a = 20926062 ft. = 6378206.4 metres.
b = 20855121 ft. = 6356583.8 metres.
b : a = 293.98 : 294.98.
As might be expected, the sum of the squares of the 40 latitude
corrections, viz. 153.99, is greater in this figure than in that of
three axes, where it amounts to 138.30. For this case, in the Indian arc
the largest corrections are at Dodagunta, + 3.87", and at Kalianpur, -
3.68". In the Russian arc the largest corrections are + 3.76", at
Tornea, and - 3.31", at Staro Nekrasovsk. Of the whole 40 corrections,
16 are under 1.0", 10 between 1.0" and 2.0", 10 between 2.0" and
3.0", and 4 over 3.0". The probable error of an observed latitude is +-
1.42"; for the spheroidal it would be very slightly larger. This
quantity may be taken therefore as approximately the probable amount of
local deflection.
If [rho] be the radius of curvature of the meridian in latitude [phi],
[rho]' that perpendicular to the meridian, D the length of a degree of
the meridian, D' the length of a degree of longitude, r the radius drawn
from the centre of the earth, V the angle of the vertical with the
radius-vector, then
Ft.
[rho] = 20890606.6 - 106411.5 cos 2[phi] + 225.8 cos 4[phi]
[rho]' = 20961607.3 - 35590.9 cos 2[phi] + 45.2 cos 4[phi]
D = 364609.87 - 1857.14 cos 2[phi] + 3.94 cos 4[phi]
D' = 365538.48 cos [phi] - 310.17 cos 3[phi] + 0.39 cos 5[phi]
Log r/a = 9.9992645 + .0007374 cos 2[phi] - .0000019 cos 4[phi]
V = 700.44" sin 2[phi] - 1.19" sin 4[phi].
A.R. Clarke has recalculated the elements of the ellipsoid of the earth;
his values, derived in 1880, in which he utilized the measurements of
parallel arcs in India, are particularly in practice. These values
are:--
a = 20926202 ft. = 6378249 metres,
b = 20854895 ft. = 6356515 metres,
b : a = 292.465 : 293.465.
The calculation of the elements of the ellipsoid of rotation from
measurements of the curvature of arcs in any given azimuth by means of
geographical longitudes, latitudes and azimuths is indicated in the
article GEODESY; reference may be made to _Principal Triangulation_,
Helmert's _Geodasie_, and the publications of the Kgl. Preuss. Geod.
Inst.:--_Lotabweichungen_ (1886), and _Die europ. Laengengradmessung in
52 deg. Br._ (1893). For the calculation of an ellipsoid with three
unequal axes see _Comparison of Standards_, preface; and for
non-elliptical meridians, _Principal Triangulation_, p. 733.
_Gravitation-Measurements._
According to Clairault's theorem (see above) the ellipticity e of the
mathematical surface of the earth is equal to the difference (5/2)m -ss,
where m is the ratio of the centrifugal force at the equator to gravity
at the equator, and ss is derived from the formula G = g(1 + ss
sin squared[phi]). Since the beginning of the 19th century many efforts have
been made to determine the constants of this formula, and numerous
expeditions undertaken to investigate the intensity of gravity in
different latitudes. If m be known, it is only necessary to determine ss
for the evaluation of e; consequently it is unnecessary to determine G
absolutely, for the relative values of G at two known latitudes suffice.
Such relative measurements are easier and more exact than absolute ones.
In some cases the ordinary thread pendulum, i.e. a spherical bob
suspended by a wire, has been employed; but more often a rigid metal
rod, bearing a weight and a knife-edge on which it may oscillate, has
been adopted. The main point is the constancy of the pendulum. From the
formula for the time of oscillation of the mathematically ideal
pendulum, t = 2 [pi] [root](l/G), l being the length, it follows that
for two points G1/G2 = t2 squared/t1 squared.
In 1808 J.B. Biot commenced his pendulum observations at several
stations in western Europe; and in 1817-1825 Captain Louis de Freycinet
and L.I. Duperrey prosecuted similar observations far into the southern
hemisphere. Captain Henry Kater confined himself to British stations
(1818-1819); Captain E. Sabine, from 1819 to 1829, observed similarly,
with Kater's pendulum, at seventeen stations ranging from the West
Indies to Greenland and Spitsbergen; and in 1824-1831, Captain Henry
Foster (who met his death by drowning in Central America) experimented
at sixteen stations; his observations were completed by Francis Baily in
London. Of other workers in this field mention may be made of F.B. Luetke
(1826-1829), a Russian rear-admiral, and Captains J.B. Basevi and W.T.
Heaviside, who observed during 1865 to 1873 at Kew and at 29 Indian
stations, particularly at More in the Himalayas at a height of 4696
metres. Of the earlier absolute determinations we may mention those of
Biot, Kater, and Bessel at Paris, London and Koenigsberg respectively.
The measurements were particularly difficult by reason of the length of
the pendulums employed, these generally being second-pendulums over 1
metre long. In about 1880, Colonel Robert von Sterneck of Austria
introduced the half-second pendulum, which permitted far quicker and
more accurate work. The use of these pendulums spread in all countries,
and the number of gravity stations consequently increased: in 1880 there
were about 120, in 1900 there were about 1600, of which the greater
number were in Europe. Sir E. Sabine[6] calculated the ellipticity to be
1/288.5, a value shown to be too high by Helmert, who in 1884, with the
aid of 120 stations, gave the value 1/299.26,[7] and in 1901, with about
1400 stations, derived the value 1/298.3.[8] The reason for the
excessive estimate of Sabine is that he did not take into account the
systematic difference between the values of G for continents and
islands; it was found that in consequence of the constitution of the
earth's crust (Pratt) G is greater on small islands of the ocean than
on continents by an amount which may approach to 0.3 cm. Moreover,
stations in the neighbourhood of coasts shelving to deep seas have a
surplus, but a little smaller. Consequently, Helmert conducted his
calculations of 1901 for continents and coasts separately, and obtained
G for the coasts 0.036 cm. greater than for the continents, while the
value of ss remained the same. The mean value, reduced to continents, is
G = 978.03(1 + 0.005302 sin squared[phi] - 0.000007 sin squared 2[phi])cm/sec squared.
The small term involving sin squared 2[phi] could not be calculated with
sufficient exactness from the observations, and is therefore taken from
the theoretical views of Sir G.H. Darwin and E. Wiechert. For the
constant g = 978.03 cm. another correction has been suggested (1906) by
the absolute determinations made by F. Kuehnen and Ph. Furtwaengler at
Potsdam.[9]
A report on the pendulum measurements of the 19th century has been
given by Helmert in the _Comptes rendus des seances de la 13^e
conference generale de l'Association Geod. Internationale a Paris_
(1900), ii. 139-385.
A difficulty presents itself in the case of the application of
measurements of gravity to the determination of the figure of the earth
by reason of the extrusion or standing out of the land-masses
(continents, &c.) above the sea-level. The potential of gravity has a
different mathematical expression outside the masses than inside. The
difficulty is removed by assuming (with Sir G.G. Stokes) the vertical
condensation of the masses on the sea-level, without its form being
considerably altered (scarcely 1 metre radially). Further, the value of
gravity (g) measured at the height H is corrected to sea-level by +
2gH/R, where R is the radius of the earth. Another correction, due to P.
Bouguer, is -(3/2)g[delta]H/[rho]R, where [delta] is the density of the
strata of height H, and [rho] the mean density of the earth. These two
corrections are represented in "Bouguer's Rule": g_H = g_s(1 - 2H/R +
3[delta]H/2[rho]R), where g_H is the gravity at height H, and g_s the
value at sea-level. This is supposed to take into account the attraction
of the elevated strata or plateau; but, from the analytical method, this
is not correct; it is also disadvantageous since, in general, the
land-masses are compensated subterraneously, by reason of the isostasis
of the earth's crust.
In 1849 Stokes showed that the normal elevations N of the geoid towards
the ellipsoid are calculable from the deviations [Delta]g of the
acceleration of gravity, i.e. the differences between the observed g and
the value calculated from the normal G formula. The method assumes that
gravity is measured on the earth's surface at a sufficient number of
points, and that it is conformably reduced. In order to secure the
convergence of the expansions in spherical harmonics, it is necessary to
assume all masses outside a surface parallel to the surface of the sea
at a depth of 21 km. (= R x ellipticity) to be condensed on this surface
(Helmert, _Geod._ ii. 172). In addition to the reduction with 2gH/R,
there still result small reductions with mountain chains and coasts, and
somewhat larger ones for islands. The sea-surface generally varies but
very little by this condensation. The elevation (N) of the geoid is then
equal to
_
/[pi]
N = R | FG^(-1) [Delta]g_[psi] d[psi],
_/
where [psi] is the spherical distance from the point N, and
[Delta]g_[psi] denotes the mean value of [Delta]g for all points in the
same distance [psi] around; F is a function of [psi], and has the
following values:--
+-------+-------+
| [Psi]=| F= |
+-------+-------+
| 0 deg. | 1 |
| 10 deg. | 1.22 |
| 20 deg. | 0.94 |
| 30 deg. | 0.47 |
| 40 deg. | -0.06 |
| 50 deg. | -0.54 |
| 60 deg. | -0.90 |
| 70 deg. | -1.08 |
| 80 deg. | -1.08 |
| 90 deg. | -0.91 |
| 100 deg. | -0.62 |
| 110 deg. | -0.27 |
| 120 deg. | +0.08 |
| 130 deg. | 0.36 |
| 140 deg. | 0.53 |
| 150 deg. | 0.56 |
| 160 deg. | 0.46 |
| 170 deg. | 0.26 |
| 180 deg. | 0 |
+-------+-------+
H. Poincare (_Bull. Astr._, 1901, p. 5) has exhibited N by means of
Lame's functions; in this case the condensation is effected on an
ellipsoidal surface, which approximates to the geoid. This condensation
is, in practice, the same as to the geoid itself.
If we imagine the outer land-masses to be condensed on the sea-level,
and the inner masses (which, together with the outer masses, causes the
deviation of the geoid from the ellipsoid) to be compensated in the
sea-level by a disturbing stratum (which, according to Gauss, is
possible), and if these masses of both kinds correspond at the point N
to a stratum of thickness D and density [delta], then, according to
Helmert (_Geod._ ii. 260) we have approximately
3 g /[delta]D \
[Delta]g = -- -- ( -------- - N ).
2 R \ [rho] /
Since N slowly varies empirically, it follows that in restricted regions
(of a few 100 km. in diameter) [Delta]g is a measure of the variation of
D. By applying the reduction of Bouguer to g, D is diminished by H and
only gives the thickness of the ideal disturbing mass which corresponds
to the perturbations due to subterranean masses. [Delta]g has positive
values on coasts, small islands, and high and medium mountain chains,
and occasionally in plains; while in valleys and at the foot of mountain
ranges it is negative (up to 0.2 cm.). We conclude from this that the
masses of smaller density existing under high mountain chains lie not
only vertically underneath but also spread out sideways.
_The European Arc of Parallel in 52 deg. Lat._
Many measurements of degrees of longitudes along central parallels in
Europe were projected and partly carried out as early as the first half
of the 19th century; these, however, only became of importance after the
introduction of the electric telegraph, through which calculations of
astronomical longitudes obtained a much higher degree of accuracy. Of
the greatest moment is the measurement near the parallel of 52 deg. lat.,
which extended from Valentia in Ireland to Orsk in the southern Ural
mountains over 69 deg. long, (about 6750 km.). F.G.W. Struve, who is to be
regarded as the father of the Russo-Scandinavian latitude-degree
measurements, was the originator of this investigation. Having made the
requisite arrangements with the governments in 1857, he transferred
them to his son Otto, who, in 1860, secured the co-operation of England.
A new connexion of England with the continent, via the English Channel,
was accomplished in the next two years; whereas the requisite
triangulations in Prussia and Russia extended over several decennaries.
The number of longitude stations originally arranged for was 15; and the
determinations of the differences in longitude were uniformly commenced
by the Russian observers E.I. von Forsch, J.I. Zylinski, B. Tiele and
others; Feaghmain (Valentia) being reserved for English observers. With
the concluding calculation of these operations, newer determinations of
differences of longitudes were also applicable, by which the number of
stations was brought up to 29. Since local deflections of the plumb-line
were suspected at Feaghmain, the most westerly station, the longitude
(with respect to Greenwich) of the trigonometrical station Killorglin at
the head of Dingle Bay was shortly afterwards determined.
The results (1891-1894) are given in volumes xlvii. and l. of the
memoirs (Zapiski) of the military topographical division of the
Russian general staff, volume li. contains a reconnexion of Orsk. The
observations made west of Warsaw are detailed in the _Die europ.
Laengengradmessung in 52 deg. Br._, i. and ii., 1893, 1896, published by
the Kgl. Preuss. Geod. Inst.
The following figures are quoted from Helmert's report "Die Groesse der
Erde" (_Sitzb. d. Berl. Akad. d. Wiss._, 1906, p. 535):--
_Easterly Deviation of the Astronomical Zenith_.
Name. Longitude.
deg. ' "
Feaghmain -10 21 -3.3
Killorglin - 9 47 +2.8
Haverfordwest - 4 58 +1.6
Greenwich 0 0 +1.5
Rosendael-Nieuport + 2 35 -1.7
Bonn + 7 6 -4.4
Goettingen + 9 57 -2.4
Brocken +10 37 +2.3
Leipzig +12 23 +2.7
Rauenberg-Berlin +13 23 +1.7
Grossenhain +13 33 -2.9
Schneekoppe +15 45 +0.1
Springberg +16 37 +0.8
Breslau-Rosenthal +17 2 +3.5
Trockenberg +18 53 -0.5
Schoensee +18 54 -2.9
Mirov +19 18 +2.2
Warsaw +21 2 +1.9
Grodno +23 50 -2.8
Bobruisk +29 14 +0.5
Orel +36 4 +4.4
Lipetsk +39 36 +0.2
Saratov +46 3 +6.4
Samara +50 5 -2.6
Orenburg +55 7 +1.7
Orsk +58 34 -8.0
These deviations of the plumb-line correspond to an ellipsoid having an
equatorial radius (a) of nearly 6,378,000 metres (prob. error +- 70
metres) and an ellipticity 1/299.15. The latter was taken for granted;
it is nearly equal to the result from the gravity-measurements; the
value for a then gives [Sigma][eta] squared a minimum (nearly). The
astronomical values of the geographical longitudes (with regard to
Greenwich) are assumed, according to the compensation of longitude
differences carried out by van de Sande Bakhuyzen (_Comp. rend, des
seances de la commission permanente de l'Association Geod.
Internationale a Geneve, 1893, annexe A.I._). Recent determinations
(Albrecht, _Astr. Nach._, 3993/4) have introduced only small alterations
in the deviations, a being slightly increased.
Of considerable importance in the investigation of the great arc was
the representation of the linear lengths found in different countries,
in terms of the same unit. The necessity for this had previously
occurred in the computation of the figure of the earth from
latitude-degree-measurements. A.R. Clarke instituted an extensive
series of comparisons at Southampton (see _Comparisons of Standards of
Length of England, France, Belgium, Prussia, Russia, India and
Australia, made at the Ordnance Survey Office, Southampton, 1866_, and
a paper in the _Philosophical Transactions_ for 1873, by Lieut.-Col.
A.R. Clarke, C.B., R.E., on the further comparisons of the standards
of Austria, Spain, the United States, Cape of Good Hope and Russia) and
found that 1 toise = 6.39453348 ft., 1 metre = 3.28086933 ft.
In 1875 a number of European states concluded the metre convention, and
in 1877 an international weights-and-measures bureau was established at
Breteuil. Until this time the metre was determined by the end-surfaces
of a platinum rod (_metre des archives_); subsequently, rods of
platinum-iridium, of cross-section H, were constructed, having engraved
lines at both ends of the bridge, which determine the distance of a
metre. There were thirty of the rods which gave as accurately as
possible the length of the metre; and these were distributed among the
different states (see WEIGHTS AND MEASURES). Careful comparisons with
several standard toises showed that the metre was not exactly equal to
443,296 lines of the toise, but, in round numbers, 1/75000 of the length
smaller. The metre according to the older relation is called the "legal
metre," according to the new relation the "international metre." The
values are (see _Europ. Laengengradmessung_, i. p. 230):--
Legal metre = 3.28086933 ft., International metre = 3.2808257 ft.
The values of a given above are in terms of the international metre; the
earlier ones in legal metres, while the gravity formulae are in
international metres.
_The International Geodetic Association (Internationale Erdmessung)._
On the proposition of the Prussian lieutenant-general, Johann Jacob
Baeyer, a conference of delegates of several European states met at
Berlin in 1862 to discuss the question of a "Central European
degree-measurement." The first general conference took place at Berlin
two years later; shortly afterwards other countries joined the movement,
which was then named "The European degree-measurement." From 1866 till
1886 Prussia had borne the expense incident to the central bureau at
Berlin; but when in 1886 the operations received further extension and
the title was altered to "The International Earth-measurement" or
"International Geodetic Association," the co-operating states made
financial contributions to this purpose. The central bureau is
affiliated with the Prussian Geodetic Institute, which, since 1892, has
been situated on the Telegraphenberg near Potsdam. After Baeyer's death
Prof. Friedrich Robert Helmert was appointed director. The funds are
devoted to the advancement of such scientific works as concern all
countries and deal with geodetic problems of a general or universal
nature. During the period 1897-1906 the following twenty-one countries
belonged to the association:--Austria, Belgium, Denmark, England,
France, Germany, Greece, Holland, Hungary, Italy, Japan, Mexico, Norway,
Portugal, Rumania, Russia, Servia, Spain, Sweden, Switzerland and the
United States of America. At the present time general conferences take
place every three years.[10]
Baeyer projected the investigation of the curvature of the meridians and
the parallels of the mathematical surface of the earth stretching from
Christiania to Palermo for 12 degrees of longitude; he sought to
co-ordinate and complete the network of triangles in the countries
through which these meridians passed, and to represent his results by a
common unit of length. This proposition has been carried out, and
extended over the greater part of Europe; as a matter of fact, the
network has, with trifling gaps, been carried over the whole of western
and central Europe, and, by some chains of triangles, over European
Russia. Through the co-operation of France, the network has been
extended into north Africa as far as the geographical latitude of 32 deg.;
in Greece a network, united with those of Italy and Bosnia, has been
carried out by the Austrian colonel, Heinrich Hartl; Servia has
projected similar triangulations; Rumania has begun to make the triangle
measurements, and three base lines have been measured by French
officers with Brunner's apparatus. At present, in Rumania, there is
being worked a connexion between the arc of parallel in lat. 47 deg./48 deg. in
Russia (stretching from Astrakan to Kishinev) with Austria-Hungary. In
the latter country and in south Bavaria the connecting triangles for
this parallel have been recently revised, as well as the French chain on
the Paris parallel, which has been connected with the German net by the
co-operation of German and French geodesists. This will give a long arc
of parallel, really projected in the first half of the 19th century. The
calculation of the Russian section gives, with an assumed ellipticity of
1/299.15, the value a = 6377350 metres; this is rather uncertain, since
the arc embraces only 19 deg. in longitude.
We may here recall that in France geodetic studies have recovered their
former expansion under the vigorous impulse of Colonel (afterwards
General) Francois Perrier. When occupied with the triangulation of
Algeria, Colonel Perrier had conceived the possibility of the geodetic
junction of Algeria to Spain, over the Mediterranean; therefore the
French meridian line, which was already connected with England, and was
thus produced to the 60th parallel, could further be linked to the
Spanish triangulation, cross thence into Algeria and extend to the
Sahara, so as to form an arc of about 30 deg. in length. But it then became
urgent to proceed to a new measurement of the French arc, between
Dunkirk and Perpignan. In 1869 Perrier was authorized to undertake that
revision. He devoted himself to that work till the end of his career,
closed by premature death in February 1888, at the very moment when the
_Depot de la guerre_ had just been transformed into the Geographical
Service of the Army, of which General F. Perrier was the first director.
His work was continued by his assistant, Colonel (afterwards General)
J.A.L. Bassot. The operations concerning the revision of the French arc
were completed only in 1896. Meanwhile the French geodesists had
accomplished the junction of Algeria to Spain, with the help of the
geodesists of the Madrid Institute under General Carlos Ibanez (1879),
and measured the meridian line between Algiers and El Aghuat (1881).
They have since been busy in prolonging the meridians of El Aghuat and
Biskra, so as to converge towards Wargla, through Ghardaia and Tuggurt.
The fundamental co-ordinates of the Pantheon have also been obtained
anew, by connecting the Pantheon and the Paris Observatory with the five
stations of Bry-sur-Marne, Morlu, Mont Valerien, Chatillon and
Montsouris, where the observations of latitude and azimuth have been
effected.[11]
According to the calculations made at the central bureau of the
international association on the great meridian arc extending from the
Shetland Islands, through Great Britain, France and Spain to El Aghuat in
Algeria, a = 6377935 metres, the ellipticity being assumed as 1/299.15.
The following table gives the difference: astronomical-geodetic latitude.
The net does not follow the meridian exactly, but deviates both to the
west and to the east; actually, the meridian of Greenwich is nearer the
mean than that of Paris (Helmert, _Groesse d. Erde_).
_West Europe-Africa Meridian-arc._[12]
Name. Latitude. A.-G.
deg. ' "
Saxavord 60 49.6 -4.0
Balta 60 45.0 -6.1
Ben Hutig 58 33.1 +0.3
Cowhythe 57 41.1 +7.3
Great Stirling 57 27.8 -2.3
Kellie Law 56 14.9 -3.7
Calton Hill 55 57.4 +3.5
Durham 54 46.1 -0.9
Burleigh Moor 54 34.3 +2.1
Clifton Beacon 53 27.5 +1.3
Arbury Hill 52 13.4 -3.0
Greenwich 51 28.6 -2.5
Nieuport 51 7.8 -0.4
Rosendael 51 2.7 -0.9
Lihons 49 49.9 +0.5
Pantheon 48 50.8 -0.0
Chevry 48 0.5 +2.2
Saligny le Vif 47 2.7 +3.0
Arpheuille 46 13.7 +6.3
Puy de Dome 45 46.5 +7.0
Rodez 44 21.4 +1.7
Carcassonne 43 13.3 +0.7
Rivesaltes 42 45.2 -0.7
Montolar 41 38.5 +3.6
Lerida 41 37.0 -0.2
Javalon 40 13.8 -0.2
Desierto 40 5.0 -4.5
Chinchilla 38 55.2 +2.2
Mola de Formentera 38 39.9 -1.2
Tetica 37 15.2 +3.5
Roldan 36 56.6 -6.0
Conjuros 36 44.4 -12.6
Mt. Sabiha 35 39.6 +6.5
Nemours 35 5.8 +7.4
Bouzareah 36 48.0 +2.9
Algiers (Voirol) 36 45.1 -9.1
Guelt es Stel 35 7.8 -1.0
El Aghuat 33 48.0 -2.8
[Illustration]
While the radius of curvature of this arc is obviously not uniform
(being, in the mean, about 600 metres greater in the northern than in
the southern part), the Russo-Scandinavian meridian arc (from 45 deg. to
70 deg.), on the other hand, is very uniformly curved, and gives, with an
ellipticity of 1/299.15, a = 6378455 metres; this arc gives the
plausible value 1/298.6 for the ellipticity. But in the case of this arc
the orographical circumstances are more favourable.
The west-European and the Russo-Scandinavian meridians indicate another
anomaly of the geoid. They were connected at the Central Bureau by means
of east-to-west triangle chains (principally by the arc of parallel
measurements in lat. 52 deg.); it was shown that, if one proceeds from the
west-European meridian arcs, the differences between the astronomical
and geodetic latitudes of the Russo-Scandinavian arc become some 4"
greater.[13]
The central European meridian, which passes through Germany and the
countries adjacent on the north and south, is under review at Potsdam
(see the publications of the Kgl. Preuss. Geod. Inst., _Lotabweichungen_,
Nos. 1-3). Particular notice must be made of the Vienna meridian, now
carried southwards to Malta. The Italian triangulation is now complete,
and has been joined with the neighbouring countries on the north, and
with Tunis on the south.
The United States Coast and Geodetic Survey has published an account of
the transcontinental triangulation and measurement of an arc of the
parallel of 39 deg., which extends from Cape May (New Jersey), on the
Atlantic coast, to Point Arena (California), on the Pacific coast, and
embraces 48 deg. 46' of longitude, with a linear development of about 4225
km. (2625 miles). The triangulation depends upon ten base-lines, with an
aggregate length of 86 km. the longest exceeding 17 km. in length, which
have been measured with the utmost care. In crossing the Rocky
Mountains, many of its sides exceed 100 miles in length, and there is
one side reaching to a length of 294 km., or 183 miles; the altitude of
many of the stations is also considerable, reaching to 4300 metres, or
14,108 ft., in the case of Pike's Peak, and to 14,421 ft. at Elbert
Peak, Colo. All geometrical conditions subsisting in the triangulation
are satisfied by adjustment, inclusive of the required accord of the
base-lines, so that the same length for any given line is found, no
matter from what line one may start.[14]
Over or near the arc were distributed 109 latitude stations, occupied
with zenith telescopes; 73 azimuth stations; and 29 telegraphically
determined longitudes. It has thus been possible to study in a very
complete manner the deviations of the vertical, which in the mountainous
regions sometimes amount to 25 seconds, and even to 29 seconds.
With the ellipticity 1/299.15, a = 6377897 +- 65 metres (prob. error); in
this calculation, however, some exceedingly perturbed stations are
excluded; for the employed stations the mean perturbation in longitude
is +- 4.9" (zenith-deflection east-to-west +- 3.8").
The computations relative to another arc, the "eastern oblique arc of
the United States," are also finished.[15] It extends from Calais
(Maine) in the north-east, to the Gulf of Mexico, and terminates at New
Orleans (Louisiana), in the south. Its length is 2612 km. (1623 miles),
the difference of latitude 15 deg. 1', and of longitude 22 deg. 47'. In the
main, the triangulation follows the Appalachian chain of mountains,
bifurcating once, so as to leave an oval space between the two branches.
It includes among its stations Mount Washington (1920 metres) and Mount
Mitchell (2038 metres). It depends upon six base-lines, and the
adjustment is effected in the same manner as for the arc of the
parallel. The astronomical data have been afforded by 71 latitude
stations, 17 longitude stations, and 56 azimuth stations, distributed
over the whole extent of the arc. The resulting dimensions of an
osculating spheroid were found to be
a = 6378157 metres +- 90 (prob. error),
e(ellipticity) = 1/304.5 +- 1.9 (prob. error).
With the ellipticity 1/399.15, a = 6378041 metres +- 80 (prob. er.).
During the years 1903-1906 the United States Coast and Geodetic Survey,
under the direction of O.H. Tittmann and the special management of John
F. Hayford, executed a calculation of the best ellipsoid of rotation for
the United States. There were 507 astronomical determinations employed,
all the stations being connected through the net-work of triangles. The
observed latitudes, longitude and azimuths were improved by the
attractions of the earth's crust on the hypothesis of isostasis for
three depths of the surface of 114, 121 and 162 km., where the isostasis
is complete. The land-masses, within the distance of 4126 km., were
taken into consideration. In the derivation of an ellipsoid of rotation,
the first case proved itself the most favourable, and there resulted:--
a = 6378283 metres +- 74 (prob. er.), ellipticity
= 1/297.8 +- 0.9 (prob. er.).
The most favourable value for the depth of the isostatic surface is
approximately 114 km.
The measurement of a great meridian arc, in long. 98 deg. W., has been
commenced; it has a range of latitude of 23 deg., and will extend over 50 deg.
when produced southwards and northwards by Mexico and Canada. It may
afterwards be connected with the arc of Quito. A new measurement of the
meridian arc of Quito was executed in the years 1901-1906 by the
_Service geographique_ of France under the direction of the Academie des
Sciences, the ground having been previously reconnoitred in 1899. The
new arc has an amplitude in latitude of 5 deg. 53' 33", and stretches from
Tulcan (lat. 0 deg. 48' 25") on the borders of Columbia and Ecuador, through
Columbia to Payta (lat. -5 deg. 5' 8") in Peru. The end-points, at which the
chain of triangles has a slight north-easterly trend, show a longitude
difference of 3 deg.. Of the 74 triangle points, 64 were latitude stations;
6 azimuths and 8 longitude-differences were measured, three base-lines
were laid down, and gravity was determined from six points, in order to
maintain indications over the general deformation of the geoid in that
region. Computations of the attraction of the mountains on the
plumb-line are also being considered. The work has been much delayed by
the hardships and difficulties encountered. It was conducted by
Lieut.-Colonel Robert Bourgeois, assisted by eleven officers and
twenty-four soldiers of the geodetic branch of the _Service
geographique_. Of these officers mention may be made of Commandant E.
Maurain, who retired in 1904 after suffering great hardships; Commandant
L. Massenet, who died in 1905; and Captains I. Lacombe, A. Lallemand,
and Lieut. Georges Perrier (son of General Perrier). It is conceivable
that the chain of triangles in longitude 98 deg. in North America may be
united with that of Ecuador and Peru: a continuous chain over the whole
of America is certainly but a question of time. During the years
1899-1902 the measurement of an arc of meridian was made in the extreme
north, in Spitzbergen, between the latitudes 76 deg. 38' and 80 deg. 50',
according to the project of P.G. Rosen. The southern part was determined
by the Russians--O. Baecklund, Captain D.D. Sergieffsky, F.N.
Tschernychev, A. Hansky and others--during 1899-1901, with the aid of 1
base-line, 15 trigonometrical, 11 latitude and 5 gravity stations. The
northern part, which has one side in common with the southern part, has
been determined by Swedes (Professors Rosen, father and son, E. Jaederin,
T. Rubin and others), who utilized 1 base-line, 9 azimuth measurements,
18 trigonometrical, 17 latitude and 5 gravity stations. The party worked
under excessive difficulties, which were accentuated by the arctic
climate. Consequently, in the first year, little headway was made.[16]
Sir David Gill, when director of the Royal Observatory, Cape Town,
instituted the magnificent project of working a latitude-degree
measurement along the meridian of 30 deg. long. This meridian passes through
Natal, the Transvaal, by Lake Tanganyika, and from thence to Cairo;
connexion with the Russo-Scandinavian meridian arc of the same longitude
should be made through Asia Minor, Turkey, Bulgaria and Rumania. With
the completion of this project a continuous arc of 105 deg. in latitude will
have been measured.[17]
Extensive triangle chains, suitable for latitude-degree measurements,
have also been effected in Japan and Australia.
Besides, the systematization of gravity measurements is of importance,
and for this purpose the association has instituted many reforms. It has
ensured that the relative measurements made at the stations in different
countries should be reduced conformably with the absolute determinations
made at Potsdam; the result was that, in 1906, the intensities of
gravitation at some 2000 stations had been co-ordinated. The intensity
of gravity on the sea has been determined by the comparison of
barometric and hypsometric observations (Mohn's method). The
association, at the proposal of Helmert, provided the necessary funds
for two expeditions:--English Channel--Rio de Janeiro, and the Red
Sea--Australia--San Francisco--Japan. Dr O. Hecker of the central bureau
was in charge; he successfully overcame the difficulties of the work,
and established the tenability of the isostatic hypothesis, which
necessitates that the intensity of gravity on the deep seas has, in
general, the same value as on the continents (without regard to the
proximity of coasts).[18]
As the result of the more recent determinations, the ellipticity,
compression or flattening of the ellipsoid of the earth may be assumed
to be very nearly 1/298.3; a value determined in 1901 by Helmert from
the measurements of gravity. The semi-major axis, a, of the meridian
ellipse may exceed 6,378,000 inter. metres by about 200 metres. The
central bureau have adopted, for practical reasons, the value 1/299.15,
after Bessel, for which tables exist; and also the value a =
6377397.155(1 + 0.0001).
The methods of theoretical astronomy also permit the evaluation of these
constants. The semi-axis a is calculable from the parallax of the moon
and the acceleration of gravity on the earth; but the results are
somewhat uncertain: the ellipticity deduced from lunar perturbations is
1/297.8 +- 2 (Helmert, _Geodaesie_, ii. pp. 460-473); William Harkness
(_The Solar Parallax and its related Constants_, 1891) from all possible
data derived the values: ellipticity = 1/300.2 +- 3, a = 6377972 +- 125
metres. Harkness also considered in this investigation the relation of
the ellipticity to precession and nutation; newer investigations of the
latter lead to the limiting values 1/296, 1/298 (Wiechert). It was
clearly noticed in this method of determination that the influence of
the assumption as to the density of the strata in the interior of the
earth was but very slight (Radau, _Bull. astr._ ii. (1885) 157). The
deviations of the geoid from the flattened ellipsoid of rotation with
regard to the heights (the directions of normals being nearly the same)
will scarcely exceed +- 100 metres (Helmert).[19]
The basis of the degree- and gravity-measurements is actually formed by
a stationary sea-surface, which is assumed to be level. However, by the
influence of winds and ocean currents the mean surface of the sea near
the coasts (which one assumes as the fundamental sea-surface) can
deviate somewhat from a level surface. According to the more recent
levelling it varies at the most by only some decimeters.[20]
It is well known that the masses of the earth are continually undergoing
small changes; the earth's crust and sea-surface reciprocally oscillate,
and the axis of rotation vibrates relatively to the body of the earth.
The investigation of these problems falls in the programme of the
Association. By continued observations of the water-level on sea-coasts,
results have already been obtained as to the relative motions of the
land and sea (cf. GEOLOGY); more exact levelling will, in the course of
time, provide observations on countries remote from the sea-coast. Since
1900 an international service has been organized between some
astronomical stations distributed over the north parallel of 39 deg. 8', at
which geographical latitudes are observed whenever possible. The
association contributes to all these stations, supporting four entirely:
two in America, one in Italy, and one in Japan; the others partially
(Tschardjui in Russia, and Cincinnati observatory). Some observatories,
especially Pulkowa, Leiden and Tokyo, take part voluntarily. Since 1906
another station for South America and one for Australia in latitude -31 deg.
55' have been added. According to the existing data, geographical
latitudes exhibit variations amounting to +-0.25", which, for the greater
part, proceed from a twelve- and a fourteen-month period.[21]
(A. R. C; F. R. H.)