EARTH (a word common to
Teutonic languages, cf.
Ger.
Erde, Dutch
aarde, Swed. and
Dan.
jord; outside Teutonic it appears
only in the Gr. g paq"€, on the ground; it has been connected by
some etymologists with the
Aryan
root
ar-, to
plough, which is seen in the Lat.
arare, obsolete Eng. "
ear," and Gr. apouv, but this is now considered
very doubtful; see G. Curtius,
Greek Etymology, Eng. trans., i. 426; Max
Milner,
Lectures, 8th ed. i. 294). From early times the
word " earth " has been used in several connexions - from that of
soil or ground to that of the
planet which we inhabit, but it is difficult to
trace the exact historic sequence of the diverse usages. In the
cosmogony of the
Pythagoreans, Platonists and other philosophers, the term or its
equivalent denoted an element or fundamental quality which
conferred upon matter the character of earthiness; and in the
subsequent development of theories as to the ultimate composition
of matter by the alchemists, iatrochemists, and early
phlogistonists an element of the same name was retained (see
Element). In modern
chemistry, the common term " earth " is
applied to certain oxides: - the
alkaline earths " are the oxides of
calcium (lime),
barium. (baryta) and
strontium (strontia); the "
rare earths " are
the oxides of a certain class of rare metals.
THE Earth The terrestrial globe is a member of the
Solar system, the
third in distance from the Sun, and the largest within the
orbit of
Jupiter. In the wider sense it may be regarded
as composed of a gaseous
atmosphere (see
Meteorology), which encircles the crust or
lithosphere (see
Geography), and surface
waters or
hydrosphere (see
Ocean
And Oceanography). The description of the surface features is a
branch of
Geography, and
the discussions as to their origin and permanence belongs to
Physiography (in the narrower sense), physiographical
geology, or physical geography.
The investigation of the crust belongs to geology and of rocks in
particular to
petrology.
In the present article we shall treat the subject matter of the
Earth as a planet under the following headings: (1) Figure and
Size, (2) Mass and
Density,
(3) Astronomical Relations, (4)
Evolution and Age. These subjects will be
treated summarily,. readers being referred to the article
Astronomy and to the
cross-references for details.
1.
Figure and Size. - To primitive man the Earth was a
flat disk with its surface diversified by mountains, rivers and
seas. In many cosmogonies this disk was encircled by waters,
unmeasurable by man and extending to a junction with the
sky; and the disk stood as an island
rising up through the waters from the floor of the universe, or was
borne as an immovable ship on the surface. Of such a nature was the
cosmogony of the Babylonians and Hebrews;
Homer states the same idea, naming the encircling
waters 'f21cEavos; and
Hesiod
regarded it as a disk midway between the sky and the infernal
regions. The theory that the Earth extended downwards to the limit
of the universe was subjected to modification when it was seen that
the same sun and stars reappeared in the east after their setting
in the west. But man slowly realized that the earth was isolated in
space, floating freely as a
balloon, and much
speculation was associated about that which
supported the Earth. Tunnels. in the foundations to permit the
passage of the sun and stars were suggested; the Greeks considered
twelve columns to support the heavens, and in their
mythology the god
Atlas appears condemned to support
the columns; while the Egyptians had the Earth supported by four
elephants, which themselves stood on a
tortoise swimming on a sea. Earthquakes were. regarded
as due to a movement of these foundations; in
Japan this was considered to be due to the motion
of a great spider, an animal subsequently replaced by a
cat-fish; in
Mongolia it is a hog; in
India, a
mole; in some parts of
South America, a
whale; and among some of the
North
American Indians,. a
giant
tortoise.
The doctrine of the spherical form has been erroneously assigned
to
Thales; but he accepted the Semitic
conception of the disk, and regarded the production of springs
after earthquakes as due to the inrushing of the waters under the
Earth into fissures in the surface. His pupil,
Anaximander (610-547),
according to
Diogenes
Laertius, believed it to be spherical (see
The Observatory, 1894,
P. 208); and
Anaximenes probably held a similar view. The
spherical form is undoubtedly a discovery of
Pythagoras, and was taught by the
Pythagoreans and by the Eleatic Parmenides. The expositor of
greatest moment was
Aristotle; his arguments are those which we
employ to-day:- the ship gradually disappearing from
hull to
mast as it recedes from the harbour to the
horizon; the circular
shadow cast by the Earth on the
Moon during an
eclipse, and the alteration in the appearance
of the heavens as one passes from point to point on the Earth's
surface.' He records attempts made to determine the circumference;
but the first scientific investigation in this 1 Aristotle regarded
the Earth as having an upper inhabited half and a lower uninhabited
one, and the
air on the lower half
as tending to flow upwards through the Earth. The obstruction of
this passage brought about an
accumulation of air within the Earth, and
the increased pressure may occasion oscillations of the surface,
which may be so intense as to cause earthquakes.
direction was made 150 years later by Eratosthenes. The
spherical form, however, only became generally accepted after the
Earth's circumnavigation (see
Geography).
The historical development of the methods for determining the
figure of
the Earth (by which we mean a theoretical surface in part
indicated by the ocean at rest, and in other parts by the level to
which water freely communicating with the oceans by canals
traversing the land masses would rise) and the mathematical
investigation of this problem are treated in the articles
Figure of
the Earth, and
Geodesy;
here the results are summarized.
Sir Isaac Newton deduced from the
mechanical consideration of the figure of
equilibrium of a mass of rotating fluid,
the form of an oblate
spheroid, the
ellipticity of a
meridian section being 1/231, and the axes in
the ratio 230:231. Geodetic measurements by the Cassinis and other
French astronomers pointed to a prolate form, but the Newtonian
figure was proved to be correct by the measurement of meridional
arcs in
Peru and
Lapland by the expeditions
organized by the French Academy of Sciences. More recent work
points to an elliptical equatorial section, thus making the earth
pear-shaped. The position of the
longer axis is somewhat uncertain; it is certainly in
Africa, Clarke placing it in
longitude 8° 15' W., and
Schubert in longitude 41° 4' E.; W. J. Sollas, arguing from
terrestrial symmetry, has chosen the position lat. 6° N., long. 28°
E.,
i.e. between Clarke's and Schubert's positions. For
the lengths of the axes and the ellipticity of the Earth, see
Figure of
the Earth.
2. Mass and Density
The earliest scientific investigation on the density and mass of
the Earth (the problem is really single if the volume of the Earth
be known) was made by
Newton,
who, mainly from astronomical considerations, suggested the
limiting densities 5 and 6; it is remarkable that this prophetic
guess should be realized, the mean value from subsequent researches
being about 5.2, which gives for the mass the value 6 X io 2 '
tons. The
density
of the Earth has been determined by several experimenters
within recent years by methods described in the article
Gravitation; the most
probable value is there stated to be 5.527.
3. Astronomical Relations
The grandest achievements of astronomical science are
undoubtedly to be associated with the elucidation of the complex
motion of our planet. The notion that the Earth was fixed and
immovable at the centre of an immeasurable universe long possessed
the minds of men; and we find the illustrious
Ptolemy accepting this view in the 2nd century
A.D.. and rejecting the notion of a rotating Earth - a theory which
had been proposed as early as the 5th century B.C. by
Philolaus on philosophical
grounds, and in the 3rd century B.C. by the astronomer
Aristarchus of
Samos. He argued that if the Earth
rotated then points at the
equator had the enormous velocity of about 1000
m. per hour, and as a consequence there should be terrific gales
from the east; the fact that there were no such gales invalidated,
in his opinion, the theory. The Ptolemaic theory was unchallenged
until 1543, in which year the
De Revolutionibus orbium
Celestium of
Copernicus was published. In this work it
was shown that the common astronomical phenomena could be more
simply explained by regarding the Earth as annually revolving about
a fixed Sun, and daily rotating about itself. A clean sweep was
made of the
geocentric
epicyclic motions of the planets which Ptolemy's theory demanded,
and in place there was substituted a
procession of planets about the Sun at
different distances. The development of the Copernican theory - the
corner-stone of modern
astronomy - by
Johann Kepler and Sir Isaac Newton is
treated in the article
Astronomy:
History; here we shall
summarily discuss the motions of our planet and its relation to the
solar system.
The Earth has two principal motions - revolution about the Sun,
rotation about its axis; there are in addition a number of secular
motions.
Revolution
The Earth revolves about the Sun in an elliptical orbit having
the Sun at one
focus. The plane
of the orbit is termed the
ecliptic; it is inclined to the Earth's
equator at an angle termed the obliquity, and the points of
intersection of the equator and ecliptic are termed the equinoctial
points. The major axis of the
ellipse is the line of apsides; when the Earth
is nearest the Sun it is said to be in
perihelion, when farthest it is in
aphelion. The mean distance
of the Earth from the Sun is a most important astronomical
constant, since it is the unit of linear measurement; its value is
about 93,000,000 m., and the difference between the perihelion and
aphelion distances is about 3,000,000 m. The eccentricity of the
orbit is o 016751. A tabular comparison of the orbital constants of
the Earth and the other planets is given in the article
Planet. The period of revolution
with regard to the Sun, or, in other words, the time taken by the
Sun apparently to pass from one
equinox to the same equinox, is the tropical or
equinoctial year; its length is 365 d. 5 hrs. 48 m. 46 secs. It is
about 20 minutes shorter than the true or sidereal year, which is
the time taken for the Sun apparently to travel from one
star to it again. The difference in
these two years is due to the secular variation termed precession
(see below). A third year is named the
anomalistic year,
which is the time occupied in the passage from perihelion to
perihelion; it is a little longer than the sidereal.
Rotation
The Earth rotates about an axis terminating at the north and
south geographical poles, and perpendicular to the equator; the
period of rotation is termed the
day, of which several kinds are distinguished
according to the body or point of reference. The rotation is
performed from west to east; this daily rotation occasions the
diurnal
motion of the celestial sphere, the rising of the Sun and stars
in the east and their setting in the west, and also the phenomena
of day and night. The inclination of the axis to the ecliptic
brings about the presentation of places in different latitudes to
the more direct rays of the sun; this is revealed in the variation
in the length of daylight with the time of the year, and the
phenomena of seasons.Although the rotation of the Earth was an accepted fact soon
after its suggestion by Copernicus, an experimental proof was
wanting until 1851, when Foucault performed his celebrated pendulum
experiment at the
Pantheon,
Paris. A pendulum about 200 ft. long, composed of
a flexible wire carrying a heavy
iron bob, was suspended so as to be free to
oscillate in any direction. The bob was provided with a
style which passed over a table
strewn with fine
sand, so that the
style traced the direction in which the bob was swinging. It was
found that the oscillating pendulum never retraced its path, but at
each swing it was apparently deviated to the right, and moreover
the deviations in equal times were themselves equal. This means
that the floor of the Pantheon was moving, and therefore the Earth
was rotating. If the pendulum were swung in the southern
hemisphere, the deviation would be to the left; if at the equator
it would not deviate, while at the poles the plane of oscillation
would
traverse a complete
circle in 24 hours.
The rotation of the Earth appears to be perfectly uniform,
comparisons of the times of transits, eclipses, &c., point to a
variation of less than yhth of a second since the time of Ptolemy.
Theoretical investigations on the phenomena of tidal
friction point, however, to a
retardation, which may to some extent be diminished by the
accelerations occasioned by the shrinkage of the globe, and some
other factors difficult to evaluate (see
Tide).
We now proceed to the secular variations.
Precession
The axis of the earth does not preserve an invariable direction
in space, but in a certain time it describes a
cone, in much the same manner as the axis of a top
spinning out of the
vertical. The equator, which preserves approximately the same
inclination to the ecliptic (there is a slight variation in the
obliquity which we shall mention later), must move so that its
intersections with the ecliptic, or equinoctial points, pass in a
retrograde direction,
i.e. opposite to that of the Earth. This motion is termed
the
precession of the
equinoxes, and was observed by
Hipparchus in the 2nd century B.C.; Ptolemy
corrected the catalogue of Hipparchus for precession by adding 2°
40' to the longitudes, the latitudes being unaltered by this
motion, which at the present time is 50.26" annually, the complete
circuit being made in about
26,000 years. Owing to precession the signs of the
zodiac are traversing paths
through the constellations, or, in other words, the constellations
are continually shifting with regard to the equinoctial points; at
one time the vernal equinox
Aries was in the constellations of that name; it
is now in
Pisces, and will
then pass into
Aquarius.
The
pole star,
i.e. the
star towards which the Earth's axis points, is also shifting owing
to precession; in about 2700 B.C. the Chinese observed a Draconis
as the pole star (at present a Ursae minoris occupies this position
and will do so until 3500); in 13600 Vega (a Lyrae) the brightest
star in
the Northern
hemisphere, will be nearest.
Precession is the result of the Sun and the Moon's attraction on
the Earth not being a single force through its centre of gravity.
If the Earth were a homogeneous sphere the attractions would act
through the centre, and such forces would have no effect upon the
rotation about the centre of gravity, but the Earth being
spheroidal the equatorial band which stands up as it were beyond
the surface of a sphere is more strongly attracted, with the result
that the axis undergoes a tilting. The precession due to the Sun is
termed the
solar precession and that due to the Moon the
lunar precession; the joint effect (two-thirds of which is
due to the Moon) is the
luni-solar precession. Solar
precession is greatest at the solstices and
zero at the equinoxes; the part of luni-solar
precession due to the Moon varies with the position of the Moon in
its orbit. The obliquity is unchanged by precession (see
Precession Of The
Equinoxes).
In treating precession we have stated that the axis of the Earth
traces a cone, and it follows that the pole describes a circle
(approximately) on the celestial sphere, about the pole of the
ecliptic. This is not quite true. Irregularities in the attracting
forces which occasion precession also cause a slight oscillation
backwards and forwards over the mean precessional path of the pole,
the pole tracing a wavy line or nodding. Both the Sun and Moon
contribute to this effect. Solar nutation depends upon the position
of the Sun on the ecliptic; its period is therefore I year, and in
extent it is only I. 2"; lunar nutation depends upon the position
of the Moon's nodes; its period is therefore about 18.6 years, the
time of revolution of the nodes, and its extent is 9.
2". There is also given to the obliquity a small
oscillation to and fro. Nutation is one of the great discoveries of
James Bradley
(1747).
Planetary Precession
So far we have regarded the ecliptic as absolutely fixed, and
treated precession as a real motion of the equator. The
ecliptic, however, is itself
subject to a motion, due to the attractions of the planets on the
Earth. This effect also displaces the equinoctial points. Its
annual value is 0.13". The term General Precession in longitude is
given to the displacement of the intersection of the equator with
the apparent ecliptic on the latter. The standard value is
50.2453", which prevailed in 1850, and the value at 1850+t,
i.e. the constant of precession, is 5 0.2 453" +
0.0002225"
t. This value is also liable to a very small
change. The nutation of the obliquity at time 1850 + t is given by
the formula 23° 27' 32. o" -0.47" I. Complete expressions for these
functions are given in Newcomb's
Spherical Astronomy
(1908), and in the
Nautical Almanac. The variation of the
line of
apsides is the name given to the motion of the major axis of
the Earth's orbit along the ecliptic. It is due to the general
influence of the planets, and the revolution is effected in 21,000
years.
The variation of the eccentricity denotes an oscillation of the
form of the Earth's orbit between a circle and ellipse. This
followed the mathematical researches of Lagrange and Leverrier. It
was suggested by
Sir John
Herschel in 1830 that this variation might occasion great
climatic changes, and
James Croll developed the theory as
affording a solution of the glacial periods in
geology.
Another secular motion of the Earth is due to the fact that the
axis of rotation is not rigidly fixed within it, but its polar
extremities wander in a circle of about 50 ft. diameter. This
oscillation brings about a variability in terrestrial latitudes,
hence the name.
Euler showed mathematically that such an
oscillation existed, and, making certain assumptions as to the
rigidity of the Earth, deduced that its period was 305 days; S. C.
Chandler, from 1890 onwards, deduced from observations of the stars
a period of 428 days;. and
Simon Newcomb explained the deviation of
these periods by pointing out that Euler's assumption of a
perfectly rigid Earth is not in accordance with fact. For details
of this intricate subject see the articles
Latitude and
Figure of the Earth.
4.
Evolution and Age. - In its earliest history the
mass now consolidated as the Earth and Moon was part of a vast
nebulous aggregate, which in the course of time formed a central
nucleus - our Sun - which
shed its outer layers in such a
manner as to form the solar system (see
Nebular Theory). The moon may have been
formed from the Earth in a similar manner, but the theory of tidal
friction suggests the
elongation of the Earth along an equatorial
axis to form a pear-shaped figure, and that in the course of time
the protuberance shot off to form the Moon (see
Tide). The age of the Earth has been investigated
from several directions, as have also associated questions related
to climatic changes, internal temperature,
orientation of the land and water
(permanence of oceans and continents), &c. These problems are
treated in the articles
Geology and
Geography.
Figure of the Earth. The
determination of the figure of the earth is a problem of the
highest importance in astronomy, inasmuch as the diameter of the
earth is the unit to which all celestial distances must be
referred.
Historical. Reasoning from the uniform level appearance
of the horizon, the variations in
altitude of the circumpolar stars as one
travels towards the north or south, the disappearance of a ship
standing out to sea, and perhaps other phenomena, the earliest
astronomers regarded the earth as a sphere, and they endeavoured to
ascertain its dimensions. Aristotle relates that the mathematicians
had found the circumference to be 400,000 stadia (about 46,000
miles). But Eratosthenes (
c. 250 B.C.) appears to have
been the first who entertained an accurate idea of the principles
on which the determination of the figure of the earth really
depends, and attempted to reduce them to practice. His results were
very inaccurate, but his method is the same as that which is
followed at the present day - depending, in fact,on the comparison
of a line measured on the earth's surface with the corresponding
arc of the heavens. He observed that at Syene in Upper
Egypt, on the day of the summer
solstice, the sun was exactly
vertical, whilst at
Alexandria at the same season of the
year its
zenith distance was
7° 12', or one-fiftieth of the circumference of a circle. He
assumed that these places were on the same meridian; and, reckoning
their distance apart as
5000 stadia, he inferred that the
circumference of the earth was 250,000 stadia (about 29,000 miles).
A similar attempt was made by
Posidonius, who adopted a method which
differed from that of Eratosthenes only in using a star instead of
the sun. He obtained 240,000 stadia (about 27,600 miles) for the
circumference. Ptolemy in his
Geography assigns the length
of the degree as Soo stadia.
The
Arabs also investigated
the question of the earth's magnitude. The
caliph Abdallah al
Mamun (A.D. 814), having fixed on a spot in the
plains of
Mesopotamia, despatched one company of
astronomers northwards and another southwards, measuring the
journey by rods, until each found the altitude of the pole to have
changed one degree. But the result of this measurement does not
appear to have been very satisfactory. From this time the subject
seems to have attracted no attention until about 1500, when
Jean
Fernel (1497-1558), a Frenchman, measured a distance in the
direction of the meridian near Paris by counting the number of
revolutions of the
wheel of a
carriage. His astronomical
observations were made with a
triangle used as a quadrant, and his resulting
length of a degree was very near the truth.
Willebrord
Snell substituted a chain of triangles for actual linear
measurement. He measured his base line on the frozen surface of the
meadows near
Leiden, and
measured the angles of his triangles, which lay between
Alkmaar and
Bergen-op-Zoom,
with a quadrant and semicircles. He took the precaution of
Eratosthenes Batavus, seu de terrae ambitus ver y quantitate
suscitatus, a Willebrordo Snellio, Lugduni-Batavorum X1617).
VIII. 26 comparing his standard with that of the French, so that
his result was expressed in toises (the length of the toise is
about 6.39 English ft.). The work was recomputed and reobserved by
P. von Musschenbroek in 1729. In 1637 an Englishman, Richard
Norwood, published a
determination of the figure of the earth in a volume entitled
The Seaman's Practice, contayning a Fundamentall Probleme in
Navigation experimentally verified, namely, touching the Compasse
of the Earth and Sea and the quantity of a Degree in our English
Measures. He observed on the I rth of June 1633 the sun's
meridian altitude in
London as
62° 1', and on the 6th of June 1635, his meridian altitude in
York as 59° 33'. He measured the
distance between these places partly with a chain and partly by
pacing. By this means, through
compensation of errors, he arrived at
367,176 ft. for the degree - a very fair result.
The application of the
telescope to angular instruments was the next
important step. Jean Picard was the first who in 1669, with the
telescope, using such precautions as the nature of the operation
requires, measured an arc of meridian. He measured with wooden rods
a base line of 5663 toises, and a second or base of verification of
3902 toises; his triangulation extended from Malvoisine, near
Paris, to Sourdon, near
Amiens. The angles of the triangles were
measured with a quadrant furnished with a telescope having
cross-wires. The difference of latitude of the terminal stations
was determined by observations made with a sector on a star in
Cassiopeia, giving 1° 22'
S5" for the
amplitude.
The terrestrial measurement gave 78,850 toises, whence he inferred
for the length of the degree 57,060 toises.
Hitherto geodetic observations had been confined to the
determination of the magnitude of the earth considered as a sphere,
but a discovery made by Jean Richer (d. 1696) turned the attention
of mathematicians to its deviation from a spherical form. This
astronomer, having been sent by the Academy of Sciences of Paris to
the island of
Cayenne, in
South
America, for the
purpose of investigating the amount of astronomical
refraction and other
astronomical objects, observed that his
clock, which had been regulated at Paris to
beat seconds, lost about two minutes
and a half daily at Cayenne, and that in order to bring
it to measure mean solar time it was necessary to
shorten the pendulum by more than a line (about 2th of an in.).
This fact, which was scarcely credited till it had been confirmed
by the subsequent observations of Varin and
Deshayes on the coasts of Africa and America,
was first explained in the
third book of Newton's
Principia, who showed that it could only be referred to a
diminution of gravity arising either from a protuberance of the
equatorial parts of the earth and consequent increase of the
distance from the centre, or from the counteracting effect of the
centrifugal force. About the same time (1673) appeared Christian
Huygens'
De Horologio Oscillatorio, in which for the first
time were found correct notions on the subject of centrifugal
force. It does not, however, appear that they were applied to the
theoretical investigation of the figure of the earth before the
publication of Newton's
Principia. In 1690 Huygens
published his
De Causa Gravitatis, which contains an
investigation of the figure of the earth on the supposition that
the attraction of every particle is towards the centre.
Between 1684 and 1718 J. and D.
Cassini, starting from Picard's base, carried a
triangulation northwards from Paris to
Dunkirk and southwards from Paris to Collioure.
They measured a base of 7246 toises near
Perpignan, and a somewhat shorter base near
Dunkirk; and from the northern portion of the arc, which had an
amplitude of 2° 12' 9", obtained for the length of a degree 56,960
toises; while from the southern portion, of which the amplitude was
6° 18' 57", they obtained 57,097 toises. The immediate inference
from this was that, the degree diminishing with increasing
latitude, the earth must be a prolate spheroid. This conclusion was
totally opposed to the theoretical investigations of Newton and
Huygens, and accordingly the Academy of Sciences of Paris
determined to apply a decisive test by the measurement of arcs at a
great distance from each other - one in the neighbourhood of the
equator, the other in a high latitude. Thus arose the celebrated
expeditions of the French academicians. In May 1735 Louis Godin,
Pierre Bouguer
and
Charles Marie de la
Condamine, under the auspices of
Louis XV., proceeded to Peru, where, assisted
by two Spanish officers, after ten years of laborious exertion,
they measured an arc of 3° 7', the northern end near the equator.
The second party consisted of
Pierre Louis Moreau de
Maupertuis, Alexis Claude
Clairault,
Charles Etienne Louis
Camus,
Pierre Charles Lemonnier, and
Reginaud Outhier, who reached the
Gulf of Bothnia in July 1736; they were
in some respects more fortunate than the first party, inasmuch as
they completed the measurement of an arc near the polar circle of
57' amplitude and returned within sixteen months from the date of
their departure.
The measurement of Bouguer and De la Condamine was executed with
great care, and on account of the locality, as well as the manner
in which all the details were conducted, it has always been
regarded as a most valuable determination. The southern limit was
at Tarqui, the northern at Cotchesqui. A base of 6272 toises was
measured in the vicinity of
Quito, near the northern extremity of the arc,
and a second base of 5260 toises near the southern extremity. The
mountainous nature of the country made the work very laborious, in
some cases the difference of heights of two neighbouring stations
exceeding 1 mile; and they had much trouble with their instruments,
those with which they were to determine the latitudes proving
untrustworthy. But they succeeded by simultaneous observations of
the same star at the two extremities of the arc in obtaining very
fair results. The whole length of the arc amounted to 176,945
toises, while the difference of latitudes was 3° 7' 3". In
consequence of a misunderstanding that arose between De la
Condamine and Bouguer, their operations were conducted separately,
and each wrote a full account of the expedition. Bouguer's book was
published in 1749; that of De la Condamine in 1751. The toise used
in this measure was afterwards regarded as the standard toise, and
is always referred to as the
Toise of Peru. The party of
Maupertuis, though their work was quickly despatched, had also to
contend with great difficulties. Not being able to make use of the
small islands in the Gulf of Bothnia for the trigonometrical
stations, they were forced to penetrate into the forests of
Lapland, commencing operations at Tornea, a city situated on the
mainland near the extremity of the gulf. From this, the southern
extremity of their arc, they carried a chain of triangles northward
to
the mountain
Kittis, which they selected as the northern
terminus. The latitudes were determined by
observations with a sector (made by George Graham) of the zenith
distance of a and
b Draconis. The base line was measured
on the frozen surface of the river Tornea about the middle of the
arc; two parties measured it separately, and they differed by about
4 in. The result of the whole was that the difference of latitudes
of the terminal stations was
57' 29". 6, and the length of
the arc 55,023 toises. In this expedition, as well as in that to
Peru, observations were made with a pendulum to determine the force
of gravity; and these observations coincided with the geodetic
results in proving that the earth was an oblate and not prolate
spheroid.
In 1740 was published in the Paris
Memoires an account,
by Cassini de Thury, of a remeasurement by himself and
Nicolas Louis de Lacaille of
the meridian of Paris. With a view to determine more accurately the
variation of the degree along the meridian, they divided the
distance from Dunkirk to Collioure into four partial arcs of about
two degrees each, by observing the latitude at five stations. The
results previously obtained by J. and D. Cassini were not
confirmed, but, on the contrary, the length of the degree derived
from these partial arcs showed on the whole an increase with an
increasing latitude. Cassini and Lacaille also measured an arc of
parallel across the mouth of the
Rhone. The difference of time of the extremities
was determined by the observers at either end noting the instant of
a
signal given by flashing
gunpowder at a point near
the middle of the arc.
While at the Cape of Good Hope in 1752, engaged in various
astronomical observations, Lacaille measured an arc of meridian of
1° 13' 17", which gave him for the length of the degree 57,037
toises - an unexpected result, which has led to the remeasurement
of the arc by Sir Thomas Maclear (see
Geodesy).
Passing over the measurements made between
Rome and
Rimini and on the plains of
Piedmont by the
Jesuits Ruggiero Giuseppe Boscovich and
Giovanni Battista Beccaria,
and also the arc measured with deal rods in
North America by
Charles Mason and
Jeremiah
Dixon, we come to the
commencement of the English triangulation. In 1783, in consequence
of a representation from Cassini de Thury on the advantages that
would accrue from the geodetic connexion of Paris and
Greenwich, General
William Roy was, with
the king's approval, appointed by
the Royal Society to conduct the
operations on the part of
England, Count Cassini, Mechain and Delambre
being appointed on the French side. A precision previously unknown
was attained by the use of Ramsden's
theodolite, which was the first to make the
spherical excess of triangles measurable. The wooden rods with
which the first base was measured were replaced by
glass rods, which were afterwards rejected for
the
steel
chain of Ramsden. (For further details see
Account of the
Trigonometrical Survey of England and Wales.) Shortly after
this,
the National Convention of
France, having agreed to remodel
their system of
weights and measures, chose for
their unit of length the ten-millionth part of the meridian
quadrant. In order to obtain this length precisely, the
remeasurement of the French meridian was resolved on, and deputed
to J. B. J. Delambre and Pierre Francois Andre Mechain. The details
of this operation will be found in the
Base du systeme metrique
decimale. The arc was subsequently extended by
Jean Baptiste
Biot and
Dominique Francois Jean
Arago to the island of Iviza. Operations for the connexion of
England with the continent of
Europe were resumed in 1821 to 1823 by
Henry Kater and
Thomas Frederick Colby on the
English side, and F. J. D. Arago and Claude
Louis Mathieu on the
French.
The publication in 1838 of Friedrich Wilhelm Bessel's
Gradmessung in Ostpreussen marks an era in the science of
geodesy. Here we find the
method of least squares applied to the calculation of a network of
triangles and the reduction of the observations generally. The
systematic manner in which all the observations were taken with the
view of securing final results of extreme accuracy is admirable.
The triangulation, which was a small one, extended about a degree
and a half along the shores of the Baltic in a N.N.E. direction.
The angles were observed with theodolites of 12 and 15 in.
diameter, and the latitudes determined by means of the transit
instrument in the
prime vertical - a method much used in
Germany. (The base apparatus
is described in the article
Geodesy.) The principal triangulation
of Great
Britain and Ireland, which was commenced in 1783 under General
Roy, for the more immediate purpose of connecting the observatories
of Greenwich and Paris, had been gradually extended, under the
successive direction of Colonel E. Williams, General W. Mudge,
General T. F. Colby, Colonel L. A. Hall, and Colonel Sir
Henry James; it was
finished in 1851. The number of stations is about 250. At 32 of
these the latitudes were determined with Ramsden's and Airy's
zenith sectors. The theodolites used for this work were, in
addition to the two great theodolites of Ramsden which were used by
General Roy and Captain Kater, a smaller theodolite of 18 in.
diameter by the same mechanician, and another of 24 in. diameter by
Messrs Troughton and Simms. Observations for determination of
absolute
azimuth were made
with those instruments at a large number of stations; the stars a,
b, and X Ursae Minoris and 51 Cephei being those observed
always at the greatest azimuths. At six of these stations the
probable error of the result is under 0.4", at twelve under 0.5",
at thirty-four under 0.7": so that the absolute azimuth of the
whole network is determined with extreme accuracy. Of the seven
base lines which have been measured, five were by means of steel
chains and two with Colby's compensation bars (see
Geodesy). The triangulation was
computed by least squares. The total number of equations of
condition for the triangulation is 920; if therefore the whole had
been reduced in one mass, as it should have been, the solution of
an
equation of 920 unknown
quantities would have occurred as a part of the work. To avoid this
an approximation was resorted to; the triangulation was divided
into twenty-one parts or figures; four of these, not adjacent, were
first adjusted by the method explained, and the corrections thus
determined in these figures carried into the equations of condition
of the adjacent figures. The average number of equations in a
figure is 44; the largest equation is one of 77 unknown quantities.
The vertical
limb of Airy's zenith
sector is read by four microscopes, and in the complete observation
of a star there are 10
micrometer readings and 12 level readings.
The instrument is portable; and a complete determination of
latitude, affected with the mean of the
declination errors of two stars, is
effected by two micrometer readings and four level readings. The
observation consists in measuring with the telescope micrometer the
difference of zenith distances of two stars which cross the
meridian, one to the north and the other to the south of the
observer at zenith distances which differ by not much more than io'
or is', the interval of the times of transit being not less than
one nor more than twenty minutes. The advantages are that, with
simplicity in the construction of the instrument and facility in
the manipulation, refraction is eliminated (or nearly so, as the
stars are generally selected within 25° of the zenith), and there
is no large divided circle. The telescope, which is counterpoised
on one side of the vertical axis, has a small circle for finding,
and there is also a small horizontal circle. This instrument is
universally used in American geodesy.
The principal work containing the methods and results of these
operations was published in 1858 with the title
Ordnance Trigonometrical
Survey of Great Britain and Ireland. Account of the observations and
calculations of the principal triangulation and of the figure,
dimensions and mean specific gravity of the earth as derived
therefrom. Drawn up by Captain Alexander Ross Clarke, R.E., F.R.A.S.,
under the direction of Lieut.-Colonel H. James, R.E., F.R.S., M.R.I.A., &c. A
supplement appeared in 1862:
Extension of the Triangulation of
the Ordnance Survey into France and Belgium, with the measurement of an arc of
parallel in 52° N. from Valentia in Ireland to Mount Kemmel in
Belgium. Published by. Col. Sir Henry James. Extensive
operations for
surveying
India and determining the figure of the earth were commenced in
1800. Colonel W. Lambton started the great meridian arc at Punnae
in latitude 8° 9', and, following generally the methods of the
English survey, he carried his triangulation as far north as 20°
30'. The work was continued by Sir George (then Captain) Everest,
who carried it to the latitude of 29° 30'. Two admirable volumes by
Sir George
Everest, published in 1830 and in 1847, give the details of
this undertaking. The survey was afterwards prosecuted by Colonel
T. T. Walker, R.F., who made valuable contributions to geodesy. The
working out of the Indian chains of triangle by the method of least
squares presents peculiar difficulties, but, enormous in extent as
the work was, it has been thoroughly carried out. The ten base
lines on which the survey depends were measured with Colby's
compensation bars.
The survey is detailed in eighteen volumes, published at
Dehra Dun, and entitled
Account of the Operations of the Great Trigonometrical Survey
of India. Of these the first nine were published under the
direction of Colonel Walker; and the remainder by Colonels Strahan
and St G. C.
Gore, Major S. G.
Burrard and others. Vol. i., 1870, treats of the base lines; vol.
ii., 1879, history and general de3criptions of the principal
triangulation and of its reduction; vol. v., 1879, pendulum
operations (Captains T. P. Basevi and W. T. Heaviside); vols. xi.,
1890, and xviii., 5906, latitudes; vols. ix., 1883, x., 1887, xv.,
1893, longitudes; vol. xvii., 5905, the
Indo-European longitude-arcs
from
Karachi to Greenwich.
The other volumes contain the triangulations.
In 1860
Friedrich Georg Wilhelm
Struve published his
Arc du meridien de 25° 20' entre le Danube et la Mer Glaciale mesure
depuis 1816 jusqu'en 1855. The latitudes of the thirteen
astronomical stations of this arc were determined partly with
vertical circles and partly by means of the transit instrument in
the prime vertical. The triangulation, a great part of which,
however, is a simple chain of triangles, is reduced by the method
of least squares, and the probable errors of the resulting
distances of
parallels
is given; the probable error of the whole arc in length is 6.2
toises. Ten base lines were measured. The sum of the lengths of the
ten measured bases is 29,863 toises, so that the average length of
a base line is 19,100 ft. The azimuths were observed at fourteen
stations. In high latitudes the determination of the meridian is a
matter of great difficulty; nevertheless the azimuths at all the
northern stations were successfully determined, - the probable
error of the result at Fuglenaes being o".
Before proceeding with the modern developments of geodetic
measurements and their application to the figure of the earth, we
must discuss the " mechanical theory," which is indispensable for a
full understanding of the subject.
Mechanical Theory. Newton, by applying his theory of
gravitation, combined
with the so-called centrifugal force, to the earth, and assuming
that an oblate
ellipsoid
of rotation is a form of equilibrium for a homogeneous fluid
rotating with uniform angular velocity, obtained the ratio of the
axes 229:230, and the law of variation of gravity on the surface. A
few years later Huygens published an investigation of the figure of
the earth, supposing the attraction of every particle to be towards
the centre of the earth, obtaining as a result that the proportion
of the axes should be 578:579. In 1740
Colin Maclaurin, in his
De causa
physica fluxus et refluxus maris, demonstrated that the oblate
ellipsoid of revolution is a figure which satisfies the conditions
of equilibrium in the case of a revolving homogeneous fluid mass,
whose particles attract one another according to the law of the
inverse square of the distance; he gave the equation connecting the
ellipticity with the proportion of the centrifugal force at the
equator to gravity, and determined the attraction on a particle
situated anywhere on the surface of such a body. In 1743 Clairault
published his
Theorie de la figure de la terre, which
contains a remarkable theorem ("Clairault's Theorem"), establishing
a relation between the ellipticity of the earth and the variation
of gravity from the equator to the poles. Assuming that the earth
is composed of concentric ellipsoidal strata having a common axis
of rotation, each stratum homogeneous in itself, but the
ellipticities and densities of the successive strata varying
according to any law, and that the superficial stratum has the same
form as if it were fluid, he proved that _g
+e= 2
m, g where
g, g' are the amounts of gravity at
the equator and at the pole respectively,
e the
ellipticity of the meridian (or " flattening "), and m the ratio of
the centrifugal force at the equator to
g. He also proved
that the increase of gravity in proceeding from the equator to the
poles is as the square of the sine of the latitude. This, taken
with the former theorem, gives the means of determining the earth's
ellipticity from observation of the relative force of gravity at
any two places. P. S. Laplace, who devoted much attention to the
subject, remarks on Clairault's work that " the importance of all
his results and the elegance with which they are presented place
this work amongst the most beautiful of mathematical productions "
(Isaac Todhunter's
History of the Mathematical Theories of
Attraction and the Figure of the Earth, vol. i. p. 229).
The problem of the figure of the earth treated as a question of
mechanics or
hydrostatics is one
of great difficulty, and it would be quite impracticable but for
the circumstance that the surface differs but little from a sphere.
In order to express the forces at any point of the body arising
from the attraction of its particles, the form of the surface is
required, but this form is the very one which it is the object of
the investigation to discover; hence the complexity of the subject,
and even with all the present resources of mathematicians only a
partial and imperfect solution can be obtained.
We may here briefly indicate the line of reasoning by which some
of the most important results may be obtained. If X, Y, Z be the
components parallel to three rectangular axes of the forces acting
on a particle of a fluid mass at the point
x, y, z, then,
p being the pressure there, and
p the density,
dp = p(Xdx+Ydy+Zdz) and for equilibrium the necessary
conditions are, that
p(Xdx+ Ydy+Zdz) be a complete
differential, and at the free surface
Xdx+ Ydy+Zdz=o. This
equation implies that the resultant of the forces is normal to the
surface at every point, and in a homogeneous fluid it is obviously
the
differential equation of all
surfaces of equal pressure. If the fluid be heterogeneous then it
is to be remarked that for forces of attraction according to the
ordinary law of gravitation, if X, Y, Z be the components of the
attraction of a mass whose potential is V, then Xdx+Ydy+Zdz = dV dV
dx dx
d-Ty -
dy + dz dz, which is a
complete differential. And in the case of a fluid rotating with
uniform velocity, in which the so-called centrifugal force enters
as a force acting on each particle proportional to its distance
from the axis of rotation, the corresponding part of
Xdx+Ydy+Zdz is obviously a complete differential.
Therefore for the forces with which we are now concerned
Xdx+Ydy+Zdz=dU, where U is some function of x,
y,
z, and it is necessary for equilibrium that
dp = pdU be a
complete differential; that is,
p must be a function of U
or a function of
p, and so also
p a function of
U. So that dU=0 is the differential equation of surfaces of equal
pressure and density.
We may now show that a homogeneous fluid mass in the form of an
oblate ellipsoid of revolution having a uniform velocity of
rotation can be in equilibrium. It may be proved that the
attraction of the ellipsoid x 2 +y 2 +z 2 (1+E 2) =c 2 (I+E 2) upon
a particle P of its mass at x, y, z has for components X =
- Ax, Y = - Ay, Z = - Cz, where 1 +E2 I) A p / tan-le, 2 /1+€2 1 1
C =4?rk p 0 tan and k 2 the constant of attraction. Besides the
attraction of the mass of the ellipsoid, the centrifugal force at P
has for components d-xw 2, +yw 2 , o; then the condition
of fluid equilibrium is (A - w 2) xdx+ (A - w
2) ydy+Czdz =0, which by integration gives (A - w 2) (x 2 +y
2) +Cz 2 = constant.
This is the equation of an ellipsoid of rotation, and therefore
the equilibrium is possible. The equation coincides with that of
the surface of the fluid mass if we make A - w2= C /(1 +€2), which
gives k2p '3' E 2 In the case of the earth, which
is nearly spherical, we obtain by expanding the expression for 3) 2
in powers of E 2, rejecting the higher powers, and remarking that
the ellipticity e= w 2 /27rk 2 p =4E 2 /15 = 8e/15.
Now if m be the ratio of the centrifugal force to the
intensity of gravity at the equator, and a = c (1 +e),
then m =aw 2 /31rk 2 pa, . . 0/27rk 2 p = m.
In the case of the earth it is a matter of observation that m
=1/289, hence the ellipticity e=5m/4=1/231, so that the ratio of
the axes on the supposition of a homogeneous fluid earth is
230:231, as stated by Newton.
Now, to come to the case of a heterogeneous fluid, we shall
assume that its surfaces of equal density are spheroids, concentric
and having a common axis of rotation, and that the ellipticity of
these surfaces varies from the centre to the outer surface, the
density also varying. In other words, the body is composed of
homogeneous spheroidal shells of variable density and ellipticity.
On this supposition we shall express the attraction of the mass
upon a particle in its interior, and then, taking into account the
centrifugal force, form the equation expressing the condition of
fluid equilibrium. The attraction of the homogeneous spheroid
x 2+y2+z2 (1
+2e) = c 2 (1
+2e),
where
e is the ellipticity (of which the square is
neglected), on an internal particle, whose co-ordinates are
x =
f, y =o, z =
h, has for its x and z components X'=
pf(1 - fe), Z'=
- 431rk2ph(1+5e), the Y component
being of course zero. Hence we infer that the attraction of a
shell whose inner surface has an
ellipticity
e, and its outer surface an ellipticity
e+de, the density being
p, is expressed by dX' =
i rk 2 pf
de, dZ' = - s - irk 2 ph de. To apply this to
our heterogeneous spheroid; if we put
c 1 for the semiaxis
of that surface of equal density on which is situated the attracted
point P, and co for the semiaxis of the outer surface, the
attraction of that portion of the body which is exterior to P,
namely, of all the shells which enclose P, has for components X o =
i i rk2 f j °l op
e d c, Zo = - iIirk 2 h j Cl
o
p-dc, both
e and
p being functions of
c. Again the attraction of a homogeneous spheroid of
density
p on an
external point
f, h has
the components X" = - 3,rk 2 p fr 3 {c 3 (I
+2e) - Xec 5
l, Z" = - 3,rk 2 phr 3 {c 3 (I
+2e) - A'ec 5 }, where
A= t (4 h2 - f 2)
/r 4 , v= 5(Z h 2 -
3f 2)/ r4,
and r2=f
+h2. Now
e being considered a function
of
c, we can at once express the attraction of a shell
(density
p) contained between the surface defined by
c+dc, e+de and that defined by
c, e upon an
external point; the differentials with respect to
c, viz.
dX" dZ", must then be integrated with
p under the integral
sign as being a function of
c. The integration will extend
from
c = o to
c = c l . Thus the components of
the attraction of the heterogeneous spheroid upon a particle within
its mass, whose co-ordinates are
f, o,
h,
are ? fc X=-3,rk2f y3Jo
lp d{c
3 (I+2e)} -
y10l
pd(eG5) s cl o
pde Z=-3,rk'h
d{c3(I+2e))-3pd(eG5) +f
pde]. We take into
account the rotation of the earth by adding the centrifugal force
fw 2 = F to X. Now, the surface of constant density upon which the
point
f, o,
h is situated gives (I - 2e)
fdf+hdh= o; and the condition of equilibrium is that (X
+F)df +Zdh = o. Therefore, (X+F)h=Zf (I - 2e), which,
neglecting small quantities of the order e 2 and putting
w't
2 = 4,r2k2, gives 2 e c1 3 6
Cc' 5 6% 3,r
y3
o pd{c (I-}-2e)) - 5r5Jo
pd(ec5) SJci pde= 2 Here we must
now put c for c1, c for r; and I +2e under the first integral sign
may be replaced by unity, since small quantities of the second
order are neglected. Two differentiations
lead us to the following very important
differential equation (Clairault):
d 2 e 2pc 2 de (2 pc 6)
dc 2+ fpc 2 dc dc + f pc' dc c2 'e
=o.When p is expressed in terms of c, this
equation can be integrated. We infer then that a rotating spheroid
of very small ellipticity, composed of fluid homogeneous strata
such as we have specified, will be in equilibrium; and when the law
of the density is expressed, the law of the corresponding
ellipticities will follow.
If we put M for the mass of the spheroid, then M;andm= e 3
t 472 and putting
c=co in the equation expressing
the condition of equilibrium, we find M(2e
- m) =3,r.5 c 2
o
pd(ec5). Making these substitutions in the expressions
for the forces at the surface, and putting
r/c = I +e -
e(h/c)2, we get G
sin = ak2
I (52-m-2e)
h2 S h. Here G is gravity in
the latitude
4,, and a the
radius of the equator. Since G= Mksec
4, =
(c /f) {' + e + (
eh '/ c2)), t) 13 -m+ jm
- e)
sin e 4, an expression which contains the theorems we have
referred to as discovered by Clairault.
The theory of the figure of the earth as a rotating ellipsoid
has been especially investigated by Laplace in his
Mecanique
celeste. The principal English works are: -
Sir
George Airy,
Mathematical Tracts, a lucid treatment
without the use of Laplace's coefficients;
Archdeacon Pratt's
Attractions and
Figure of the Earth; and O'Brien's
Mathematical
Tracts; in the last two Laplace's coefficients are used.
In 1845 Sir G. G. Stokes (
Camb. Trans. viii.; see also
Camb. Dub. Math. 'Journ., ' 1849, iv.) proved that if the
external form of the sea - imagined to percolate the land by canals
- be a spheroid with small ellipticity, then the law of gravity is
that which we have shown above; his proof required no assumption as
to the ellipticity of the internal strata, or as to the past or
present fluidity of the earth. This investigation admits of being
regarded conversely, viz. as determining the elliptical form of the
earth from measurements of gravity; if G, the observed value of
gravity in latitude 4), be expressed in the form G = g(sine 0),
where g is the value at the equator and a coefficient. In this
investigation, the square and higher powers of the ellipticity are
neglected; the solution was completed by F. R. Helmert with regard
to the square of the ellipticity, who showed that a term with sin 2
24) appeared (see Helmert,
Geodasie, ii. 83). For the
coefficient of this term, the gravity measurements give a small but
not sufficiently certain value; we therefore assume a value which
agrees best with the
hypothesis of the fluid state of the entire
earth; this assumption is well supported, since even at a depth of
only 50 km. the pressure of the superincumbent crust is so great
that rocks become plastic, and behave approximately as fluids, and
consequently the crust of the earth floats, to some extent, on the
interior (even though this may not be fluid in the usual sense of
the word). This is the geological theory of " Isostasis " (cf.
Geology); it agrees with the
results of measurements of gravity (
vide infra), and was
brought forward in the middle of the 19th century by J. H. Pratt,
who deduced it from observations made in India.
The sin 2 24) term in the expression for G, and the
corresponding deviation of the meridian from an ellipse, have been
analytically established by Sir G. H. Darwin and E. Wiechert;
earlier and less complete investigations were made by Sir G. B.
Airy and O. Callandreau. In consequence of the sin 2 24) term, two
parameters of the level surfaces in the interior of the earth are
to be determined; for this purpose, Darwin develops two
differential equations in the place of the one by Clairault. By
assuming Roche's law for the variation of the density in the
interior of the Earth, viz.
p = p i - k (c/c,) 2, k being
a coefficient, it is shown that in latitude 45°, the meridian is
depressed about 34 metres from the ellipse, and the coefficient of
the term sin'4)
cos' 0(=1 sin224))
is - (3 0000295. According to Wiechert the earth is composed of a
kernel and a shell, the kernel
being composed of material, chiefly metallic iron, of density near
8.2, and the shell, about 900 miles thick, of silicates, &c.,
of density about 3.2. On this assumption the depression in latitude
45° is 24 metres, and the coefficient of sin e 4) cos 2 4) is, in
round numbers, - o 0000280.1 To this additional term in the formula
for G, there corresponds an extension of Clairault's formula for
the calculation of the flattening from (3 with terms of the higher
orders; this was first accomplished by Helmert.
For a long time the assumption of an ellipsoid with three
unequal axes has been held possible for the figure of the earth, in
consequence of an important theorem due to K. G. Jacobi, who proved
that for a homogeneous fluid in rotation a spheroid is not the only
form of equilibrium; an ellipsoid rotating round its least axis may
with certain proportions of the axes and a certain time of
revolution be a form of equilibrium. 2 It has been objected to the
figure of three unequal axes that it does not satisfy, in the
proportions of the axes, the conditions brought out in Jacobi's
theorem (
c: a<11 1 2). Admitting this, it has to be
noted, on the other hand, that Jacobi's theorem contemplates a
homogeneous fluid, and this is certainly far from the actual
condition of our globe; indeed the irregular distribution of
continents and oceans suggests the possibility of a sensible
divergence from a perfect surface of revolution. We may, however,
assume the ellipsoid with three unequal axes to be an
interpolation form.
More plausible forms are little adapted for computation.'
Consequently we now generally take the ellipsoid of rotation as a
basis, especially so because measurements of gravity have shown
that the deviation from it is but trifling.
Local Attraction. In speaking of the figure of the
earth, we mean the surface of the sea imagined to percolate the
continents by canals. That
1 O. Callendreau, "Memoire sur la theorie de la figure des
planetes,"
Ann. obs. de Paris (1889); G. H. Darwin, "The
Theory of the Figure of the Earth carried to the Second Order of
Small Quantities,"
Mon. Not. R.A.S., 1899; E. Wiechert,
"Über die Massenverteilung im Innern der Erde,"
Nach. d. hon.
G. d. W. zu
Gött., 1897. 2 See I. Todhunter,
Proc. Roy. Soc., 1870. 3 J. H. Jeans, "On the Vibrations
and Stability of a Gravitating Planet,"
Proc. Roy. Soc.
vol. 71; G. H. Darwin, "On the Figure and Stability of a liquid
Satellite,"
Phil.
Trans. 206, p. 161; A. E. H. Love, "The Gravitational
Stability of the Earth,"
Phil. Trans. 207, p. 237;
Proc. Roy. Soc. vol. 80.
G cos?
= 2 '-' -
e- 2m+ (
2m
-2e) c c this surface should turn out, after precise
measurements, to be exactly an ellipsoid of revolution is
a priori improbable.
Although it may be highly probable that originally the earth was a
fluid mass, yet in the cooling whereby the present crust has
resulted, the actual solid surface has been left most irregular in
form. It is clear that these irregularities of the visible surface
must be accompanied by irregularities in the mathematical figure of
the earth, and when we consider the general surface of our globe,
its irregular distribution of mountain masses, continents, with
oceans and islands, we are prepared to admit that the earth may not
be precisely any surface of revolution. Nevertheless, there must
exist some spheroid which agrees very closely with the mathematical
figure of the earth, and has the same axis of rotation. We must
conceive this figure as exhibiting slight departures from the
spheroid, the two surfaces cutting one another in various lines;
thus a point of the surface is defined by its latitude, longitude,
and its height above the " spheroid of reference." Calling this
height N, then of the actual magnitude of this quantity we can
generally have no information, it only obtrudes itself on our
notice by its variations. In the vicinity of mountains it may
change sign in the space of a few miles; N being regarded as a
function of the latitude and longitude, if its differential
coefficient with respect to the former be zero at a certain point,
the normals to the two surfaces then will lie in the prime
vertical; if the differential coefficient of N with respect to the
longitude be zero, the two normals will lie in the meridian; if
both coefficients are zero, the normals will coincide. The
comparisons of terrestrial measurements with the corresponding
astronomical observations have always been accompanied with
discrepancies. Suppose A and B to be two trigonometrical stations,
and that at A there is a disturbing force drawing the vertical
through an angle
S, then it is evident that the apparent
zenith of A will be really that of some other place A', whose
distance from A is
rS, when
r is the earth's
radius; and similarly if there be a disturbance at B of the amount
S', the apparent zenith of B will be really that of some
other place B', whose distance from B is
re'. Hence we
have the discrepancy that, while the geodetic measurements deal
with the points A and B, the astronomical observations belong to
the points A', B'. Should
S, S' be equal and parallel, the
displacements
AA', BB' will be equal
and parallel, and no discrepancy will appear. The non-recognition
of this circumstance often led to much perplexity in the early
history of geodesy. Suppose that, through the unknown variations of
N, the probable error of an observed latitude (that is, the angle
between the normal to the mathematical surface of the earth at the
given point and that of the corresponding point on the spheroid of
reference) be
e, then if we compare two arcs of a degree
each in mean latitudes, and near each other, say about five degrees
of latitude apart, the probable error of the resulting value of the
ellipticity will be approximately o 5 o oe, e being expressed in
seconds, so that if
e be so great as 2" the probable error
of the resulting ellipticity will be greater than the ellipticity
itself.
It is necessary at times to calculate the attraction of a
mountain, and the consequent disturbance of the astronomical
zenith, at any point within its influence. The deflection of the
plumb-line, caused by a local attraction whose amount is k2A3, is
measured by the ratio of k 2 A3 to the force of gravity at the
station. Expressed in seconds, the deflection A is A=
12 " 447AS/p, where
p is the mean density
of the earth,
S that of the attracting mass, and A = fs3
xdv, in which
dv is a volume element of the attracting
mass within the distance
s from the point of deflection,
and
x the
projection of
s on the horizontal
plane through this point, the linear unit in expressing A being a
mile. Suppose, for instance, a table-land whose form is a rectangle
of 12 miles by 8 miles, having a height of soo ft. and density half
that of the earth; let the observer be 2 miles distant from the
middle point of the longer side. The deflection then is I" 472; but
at z mile it increases to 2" 20.
At sixteen astronomical stations in the English survey the
disturbance of latitude due to the form of the ground has been
computed, and the following will give an idea of the results. At
six stations the deflection is under 2", at six others it is
between 2" and 4", and at four stations it exceeds 4". There is one
very exceptional station on the north coast of
Banffshire, near the
village of Portsoy, at which the deflection amounts to 10", so that
if that village were placed on a
map
in a position to correspond with its astronomical latitude, it
would be 1000 ft. out of position! There is the sea to the north
and an undulating country to the south, which, however, to a
spectator at the station does not suggest any great disturbance of
gravity. A somewhat rough estimate of the local attraction from
external causes gives a maximum limit of 5", therefore we have 5"
which must arise from unequal density in the underlying strata in
the surrounding country. In order to throw light on this remarkable
phenomenon, the
latitudes of a number of stations between
Nairn on the west,
Fraserburgh on the east, and
the Grampians on
the south, were observed, and the local deflections determined. It
is somewhat singular that the deflections diminish in all
directions, not
very regularly certainly, and most slowly
in a southwest direction, finally disappearing, and leaving the
maximum at the original station at Portsoy.
The method employed by Dr C. Hutton for computing the attraction
of masses of ground is so simple and effectual that it can hardly
be improved on. Let a horizontal plane pass through the given
station; let
r, 0 be the polar co-ordinates of any point
in this plane, and
r, 0, z, the co-ordinates of a particle
of the attracting mass; and let it be required to find the
attraction of a portion of the mass contained between the
horizontal planes z= o, z =
h, the cylindrical surfaces
r = r, r = r2, and the vertical planes
0=0 1 ,0=0 2
. The component of the attraction at the station or origin
along the line is e h r 2 cos0 k2 (82 f 2 of o (r2
+ z2)3
dr
dO dz J J =k 2 Sh (sin 02 - sin 0 1)
log {r2+(rz
+h 2)
1/2 /r i + (r 1
2 +h 2) By taking r 2 - r 1, sufficiently small, and supposing la
also small compared with r 1 +r 2 (as it usually is), the
attraction is
k 2 3(r 2 - r i) (sin 02
- sin 0,)
hir, where
r=1 (r i d-r 2). This form suggests
the following procedure. Draw on the contoured map a series of
equidistant circles, concentric with the station, intersected by
radial lines so disposed that the sines of their azimuths are in
arithmetical progression. Then, having estimated from the map the
mean heights of the various compartments, the calculation is
obvious.
In mountainous countries, as near the
Alps and in the
Caucasus, deflections have been observed to
the amount of as much as 30", while in the Himalayas deflections
amounting to 60" were observed. On the other hand, deflections have
been observed in flat countries, such as that noted by Professor K.
G. Schweizer, who has shown that, at certain stations in the
vicinity of
Moscow, within a
distance of 16 miles the plumb-line varies 16" in such a manner as
to indicate a vast deficiency of matter in the underlying strata;
deflections of 20" were observed in the level regions of north
Germany.
Since the attraction of a mountain mass is expressed as a
numerical multiple of
S :p the ratio of the density
of the mountain to that of the earth, if we have any independent
means of ascertaining the amount of the deflection, we have at once
the ratio
p :6, and thus we obtain the mean density
of the earth, as, for instance, at Schiehallion, and afterwards at
Arthur's Seat. Experiments of this kind for determining the mean
density of the earth have been made in greater numbers; but they
are not free from objection (see
Gravitation).
Let us now consider the perturbation attending a spherical
subterranean mass. A compact mass of great density at a small
distance under the surface of the earth will produce an elevation
of the mathematical surface which is expressed by the formula
y=aµ{(i - 2u cos 0-Fu2)4 - i}, where a is the radius of the
(spherical) earth, a(i - u) the distance of the disturbing
mass below the surface, 11 the ratio of the disturbing mass to the
mass of the earth, and a9 the distance of any point on the
surface from that point, say Q, which is vertically over the
disturbing mass. The maximum value of y is at Q, where it
is y=aµu(i - u). The deflection at the distance
aB is A= 11 u sin B(I - zu cos 9 + u 2) -1, or
since 0 is small, putting h+u= 1, we have A=µ
0(h 2 +0 2) 2 . The maximum deflection takes place at a
point whose distance from Q is to the depth of the mass as I: J 2,
and its amount is 211/3 h2.
If, for instance, the disturbing mass were a sphere a mile in
diameter, the excess of its density above that of the surrounding
country being equal to half the density of the earth, and the depth
of its centre half a mile, the greatest deflection would be 5", and
the greatest value of y only two inches. Thus a large
disturbance of gravity may arise from an irregularity in the
mathematical surface whose actual magnitude, as regards height at
least, is extremely small.
The effect of the disturbing mass 11 on the vibrations of a
pendulum would be a maximum at Q; if v be the number of
seconds of time gained per diem by the pendulum at Q, and a the
number of seconds of angle in the maximum deflection, then it may
be shown that v/a=7rd/3/Io.
The great Indian survey, and the attendant measurements of the
degree of latitude, gave occasion to elaborate investigations of
the deflection of the plumb-line in the neighbourhood of the high
plateaus and mountain chains of
Central Asia. Archdeacon Pratt
(
Phil. Trans.,18J5 and 1857), in instituting these
investigations, took into consideration the influence of the
apparent diminution of the mass of the earth's crust occasioned by
the neighbouring ocean-basins; he concluded that the accumulated
masses of mountain chains, &c., corresponded to subterranean
mass diminutions, so that over any level surface in a fixed depth
(perhaps loo miles or more) the masses of prisms of equal section
are equal. This is supported by the gravity measurements at More in
the Himalayas at a height of 4696 metres, which showed no
deflection due to the mountain chain (
Phil. Trans., 1871);
more recently, H. A. Faye (
Compt. rend., 1880) arrived at
the same conclusion for the entire continent.
This compensation, however, must only be regarded as a general
principle; in certain cases, the compensating masses show marked
horizontal displacements. Further investigations, especially of
gravity measurements, will undoubtedly establish other important
facts. Colonel S. G. Burrard has recently recalculated, with the
aid of more exact data, certain Indian deviations of the
plumb-line, and has established that in the region south of the
Himalayas (lat. 24°) there is a subterranean perturbing mass. The
extent of the compensation of the high mountain chains is difficult
to recognize from the latitude observations, since the same effect
may result from different causes; on the other hand, observations
of geographical longitude have established a strong compensation.'
Meridian Arcs. The astronomical stations for the
measurement of the degree of latitude will generally lie not
exactly on the same meridian; and it is therefore necessary to
calculate the arcs of meridian M which lie between the latitude of
neighbouring stations. If S be the geodetic line calculated from
the triangulation with the astronomically determined azimuths al
and a2, then M = S cos Zia I + i vd 2 s i n 2a. .. } in which 2a
=al+a2 - 180°, Oa=a2 - a1-180°.
The length of the arc of meridian between the latitudes 01 and
02 is M= (I sin where a 2 e 2 2 - 2; instead of using the
eccentricity e, put the ratio of the axes b: a= i
- n: 1+n, then
1
Survey of India, The Attraction of the
Himalaya Mountains upon the
Plumb Line in India" (1901), p. 98.
__ ? b (I+n)(I - n2)d?
M t (I +2ncos20
+n2) This, after integration, gives M/b= (I+fl+flh
+fl3) ao
- (3n+3n h +n h)
ai+ (24n3) where
ao=ti - al sin (02-01) ('+431) a 2 = sin 2(0 2 -4 j) cos 2
(02+(I)i) a 3 = sin 3(02-4))) cos 3(4)2+4,1).
The part of M which depends on n 3 is very small; in fact, if we
calculate it for one of the longest arcs measured, the
Russian arc, it amounts to only
an
inch and a half, therefore we
omit this term, and put for M/b the value I +n+4 7 / 2) am - (3n+3n
2) al+ (In) a2.
Now, if we suppose the observed latitudes to be affected with
errors, and that the true latitudes are 01+ xi, 02+x2; and if
further we suppose that ni+dn is the true value of a -
b: a+b, and that n, itself is merely a very
approximate numerical value, we get, on making these substitutions
and neglecting the influence of the corrections x on the
position of the arc in latitude, i.e. on 4)1+ 02,
M/ b= (i+ni +n) ao - (3 n i+3 n) ai+ (nl) a2 +
(I+Zn) ao - (3+6n) a.+ (fli) a 2 do -1-n i - 3ndao } dao
here dao = x 2 - x i; and as b is only known
approximately, put b=b l (I +u); then we get, after
dividing through by the co efficient of dao, which is = 11-n i -3n
1 cos (02-01) cos (02+4 m), an equation of the form x 2 =x l
+h+fu+gv, where for con venience we put v for dn. Now
in every measured arc there are not only the extreme stations
determined in latitude, but also a number of intermediate stations
so that if there be i+1 stations there will be i
equations x2 =xl+ f iu +gly+hl X3 = x l
-F-f2u+g2v+h2 x; =xl +fiu+gtiv +h, In combining a number of
different arcs of meridian, with the view of determining the figure
of the earth, each arc will supply a number of equations in
u and v and the corrections to its observed
latitudes. Then, according to the method of least squares, those
values of u and v are the most probable which
render the sum of the squares of all the errors x
a minimum. The corrections x which are here applied arise not from
errors of observation only. The mere uncertainty of a latitude, as
determined with modern instruments, does not exceed a very small
fraction of a second as far as errors of observation go, but no
accuracy in observing will remove the error that may arise from
local attraction. This, as we have seen, may amount to some
seconds, so that the corrections x to the observed
latitudes are attributable to local attraction. Archdeacon Pratt
objected to this mode of applying least squares first used by
Bessel; but Bessel was right, and the objection is groundless.
Bessel found, in 1841, from ten meridian arcs with a total
amplitude of 50° 6: a= 3272077 toises = 6 377397
metres.
e (ellipticity) = (a - b)/a= 1 /299.15 (prob.
error =3.2).
The probable error in the length of the earth's quadrant is = 33
6 m.
We now give a series of some meridian-arcs measurements, which
were utilized in 1866 by A. R. Clarke in the Comparisons of the
Standards of Length, pp. 280-287; details of the calculations
are given by the same author in his Geodesy (1880), pp.
311 et seq. The data of the French arc from Formentera to
Dunkirk are (L a2 a3r Stations. Astronomical Distance of
Latitudes. Parallels.
° Ft.
|
Formentera .
|
3 8
|
39
|
53.17
|
|
|
Mountjouy .
|
41
|
21
|
44.96
|
982671.04
|
|
Barcelona .
|
41
|
22
|
47.90
|
988701.92
|
|
Carcassonne .
|
43
|
12
|
54'3 0
|
1657287'93
|
|
Pantheon .
|
4 8
|
5 0
|
47'9 8
|
3710827.13
|
|
Dunkirk
|
51
|
2
|
8'41
|
4509790.84
|
The distance of the parallels of Dunkirk and Greenwich, deduced
from the extension of the triangulation of England into France, in
1862, is 161407.3 ft., which is 3.9 ft. greater than that obtained
from Captain Kater's triangulation, and 3.2 ft. less than the
distance calculated by Delambre from General Roy's triangulation.
The following table shows the data of the English arc with the
distances in standard feet from Formentera.
|
|
|
|
Ft.
|
|
Formentera
|
|
|
|
|
|
Greenwich
|
51
|
28
|
38'30
|
1198'
467'198.3
|
|
Arbury
|
52
|
13
|
26.59
|
4943837.6
|
|
Clifton
|
53
|
2 7
|
2 9'5 0
|
5394063.4
|
|
Kellie Law
|
56
|
1 4
|
53' 60
|
6413221.7
3221.
|
|
Stirling
|
|
2 7
|
49.12
|
6857323.3
|
|
Saxavord
|
60
|
49
|
37' 21
|
8086820.7
|
The latitude assigned in this table to Saxavord is not the
directly observed latitude, which is 60° 49' 38.58", for there are
here a cluster of three points, whose latitudes are astronomically
determined; and if we transfer, by means of the geodesic connexion,
the latitude of Gerth of Scaw to Saxavord, we get 60° 49' 36.59";
and if we similarly transfer the latitude of
Balta, we get 60° 49' 36.46". The mean of these
three is that entered in the above table.
For the Indian arc in long. 77° 40' we have the following data:
And, finally, for the Peruvian arc, in long. 281° o', Punnea.
Putchapolliam Dodagunta Namthabad. Daumergida. Takalkhera.
Kalianpur .
Kaliana The data of the Russian arc (long. 26° 40') taken from
Struve's work are as below Ft.
616529.81 1246762-17 1737551.48 2448745.17 3400312.63 4076412.28
4762421'43 5386135'39 631 79 05.67 7486789'97 8530517.90 9257921.06
Thomas Maclear North End. Heerenlogement
Berg Royal Observatory .
Zwart Kop.. Cape Point.. .
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 = 2092-6062 ft. = 6378206.4
metres.
b =2085-5121 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 ISO", 10 between 1.0" and 2.O",
10 between 2.0" and 3.O ", and 4 over 3.O". 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 p be the radius of curvature of the meridian in
latitude cb, p' 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.
P = 2089-0606.6 - 106411.5 cos 20 + 225.8 cos 40
2096-1607.3 - 3559 0.9 cos 20 + 4 5.2 cos 40 D = 364609.87 -
1857.14 cos 2 4) + 3'94 cos 40 D'= 3 6 553 8 '4 8 cos - 310.17 cos
30-f0.39 COS 50 Log rla= 9.9992645 -?- .0007374 cos 20 -
0000019 cos 44> V = 700.44" sin 20.- 1 19" sin 40.
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 = 2092-6202 ft. = 6378249 metres.
b = 2085-4895 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.
Langengradmessung in 52° 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 2m-13, where
m is the ratio of the centrifugal force at the equator to
gravity at the equator, and s is derived from the formula
G= g(1 +0 sin 2 0). 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 /3 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 knifeedge 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= 27r1/ 11G,
1 being the
length, it follows that for two points GI/G2=4/ti.
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 Ft.
|
29 44 31 58
33 56
34 13 34 21
|
17.66 9 11 3.20
32'13 6.26
|
Ft.
811507 7 1526386.8 1632583.3
1678375'7
|
1029174'9 17 5 6562.0 2518 37 6.3 3591788'4 4697329'5 5794695'7
7755835'9 ° 8 Io 12 151821 24 29 9 59 59535 7 30 31.132 42'276
52'165 53 562 15.292 51.532 11.262 48 322 Staro Nekrasovsk
Vodu-Luy. Suprunkovzy
Kremenets. Byelin. Nemesh. Jacobstadt Dorpat.
Hogland Kilpi-maki Tornea Stuor-oivi Fuglenaes From the arc
measured in
Cape
Colony by Sir in long. 18° 30', we have 45 47 48 50 52 54 56 58
60 62 65 68 70 20 455239 30 22 5 38 49 40 40 2.94 24.98 3.04 49.95
42.16 4.16 4.97 47.56 9' 84 5.25 44'57 58.40 11.23 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. Lake (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
Konigsberg respectively. The measurements
were particularly difficult by reason of the length of the
pendulums employed, these generally being second-pendulums over
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'
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, 2 and in 1901, with about 1400 stations,
derived the value 1/298.3. 3 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 I H, and
g, 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.
|
4, =
|
0°
|
Io°
|
20°
|
30°
|
40°
|
50°
|
60°
|
70°
|
80°
|
90°
|
100°
|
11O°
|
120°
|
130°
|
140°
|
150°
|
160°
|
170°
|
180°
|
|
F =
|
I
|
1.22
|
0.94
|
0.47
|
- 0.06
|
- 0.5 4
|
- 0.90
|
-1.08
|
-1.08
|
-0.91
|
-o 62
|
-0.27
|
+0.08
|
0.36
|
0.5 3
|
o 56
|
0.46
|
o 26
|
o
|
In 1849 Stokes showed that the normal elevations N of the
geoid towards the ellipsoid are
calculable from the deviations
Ag 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 N = R ' FG -l og
dI, where is the spherical distance from the point
N, and
Ag,, denotes the mean value of
Ag for all
points in the same distance (' around; F is a function of tp, and
has the following values: - 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 o 036 cm. greater than for the
continents, while the value of (
3 remained the same. The
mean value, reduced to continents, is G=978.03(1+0.005302 sin 2 4)
-0.000007 sin e 2q5)cm/sec2.
The small term involving sin 2 2c/) 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. Kiihnen and Ph.
Furtwangler at
Potsdam.4 A
report on the pendulum measurements of the 19th century has been
given by Helmert in the
Comptes rendus des seances de la 13 8
conference generale de l'Association Geod. Internationale d
Paris 0900), 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 landmasses
(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 i 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 -
2g8H/pR, where 8 is the density of the strata of height H,
and p the m..-,an density of the earth. These two
corrections are represented in " Bouguer's Rule ": g H = g s (I -
2H/R+38H/2pR), where gx is the gravity at height 1 Account of
Experiments to Determine the Figure of the Earth by means of a
Pendulum vibrating Seconds in Different Latitudes (1825).
2 Helmert,
Theorien d. hoheren Geod. ii.,
Leipzig, 1884.
Helmert,
Sitzber. d. kgl. preuss. Ak. d. Wiss. zu
Berlin (1901), P. 33
6 4 " Bestimmung der absoluten Grosse der Schwerkraft zu Potsdam
mit Reversionspendeln " (Veroffentlichung
des kgl. preuss.
Geod. Inst., N.F., No. 27).
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
8, then, according to Helmert (Geod. ii. 260) we
have approximately 3 gOD_N}.
g 2 R p J Since N slowly varies empirically, it follows that in
restricted regions (of a few roo km. in diameter) Ag 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. Ag 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°
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°
lat., which extended from Valentia in Ireland to Orsk in the
southern
Ural
mountains over 69° 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
VIII. 26 a 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 1. 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. Langengradmessung in 52° Br., i. and ii., 1893,
1896, published by the Kgl. Preuss. Geod. Inst.
|
Name.
|
|
Longitude.
o
|
|
|
Feaghmain .
|
|
- 10
|
21
|
- 3.3
|
|
Killorglin. .
|
|
- 9
|
47
|
+2.8
|
|
Haverfordwest .
|
|
- 4
|
58
|
+1.6
|
|
Greenwich. .
|
|
0
|
0
|
+ I.5
|
|
Rosendael-Nieuport
|
|
+ 2
|
35
|
- 1.7
|
|
Bonn. .
|
|
7
|
6
|
- 4.4
|
|
Göttingen. .
|
|
+9
|
57
|
- 2.4
|
|
Brocken .
|
|
+10
|
37
|
+2.3
|
|
Leipzig
|
|
+12
|
23
|
+2' 7
|
|
Rauenberg-Berlin
|
|
+13
|
23
|
+1.7
|
|
Grossenhain .
|
|
+ 1 3
|
33
|
- 2.9
|
|
Schneekoppe .
|
|
+15
|
45
|
+0.1
|
|
Springberg
|
|
+16
|
37
|
+0.8
|
|
Breslau-Rosenthal
|
|
+ 1 7
|
2
|
+3'5
|
|
Trockenberg
|
|
+ 18
|
53
|
- 0.5
|
|
Schonsee .
|
|
+ 18
|
54
|
- 2.9
|
|
Mirov
|
|
+19
|
18
|
+2.2
|
|
Warsaw
|
|
+21
|
2
|
+1 9
|
|
Grodno
|
|
+ 2 3
|
50
|
- 2.8
|
|
Bobruisk
|
|
+29
|
14
|
+0.5
|
|
Orel
|
|
+3 6
|
4
|
+4.4
|
|
Lipetsk
|
|
+39
|
3 6
|
+0.2
|
|
Saratov
|
|
+46
|
3
|
+6.4
|
|
Samara
|
|
+50
|
5
|
- 2.6
|
|
Orenburg .
|
|
+55
|
7
|
+1'7
|
|
Orsk
|
|
+5 8
|
34
|
- 8 o
|
The following figures are quoted from Helmert's report " Die
Grosse der Erde " (Sitzb. d. Berl. Akad. d. Wiss., 1906,
P. 535):- Easterly Deviation of the Astronomical Zenith.
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 /77 2 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 6-3945-3348 ft., 1 metre
=3.2808-6933 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.
Ldngengradmessung, i. p. 230): Legal metre =3.2808-6933 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.' 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 0; 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
1 Die KOnigl.
Observatorien fiir Astrophysik, Meteorologie and Geodasie bei
Potsdam (Berlin, 1890);
Verhandlungen der I. Allgemeinen
Conferenz der Bevollmachtigten zur mitteleurop. Gradmessung,
October, 1864, in Berlin (Berlin, 1865); A. Hirsch,
Verhandlungen der VIII. Allg. Conf. der Internationalen
Erdmessung, October, 1886, in Berlin (Berlin, 1887); and
Verhandlungen der XI. Allg. Conf. d. I. E., October, 1895,
in Berlin (1896).
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°/48° in Russia
(stretching from Astrakan to Kishinev) with
Austria-Hungary. In the latter country
and in southBavaria 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= 637735 0 metres; this is rather uncertain, since
the arc embraces only 19° 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° 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.' 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,
Grosse d. Erde). West Europe-Africa Meridian-arc.' Name.
Latitude. A.-G. o Saxavord 60 49.6 Balta. 60 45.0 Ben Hutig. 58
33'1 Cowhythe. 57 41.1 Great
Stirling 57 27.8 Kellie Law. 56 14'9 Calton
Hill. 55 57'4
Durham 54 46.1
Burleigh
Moor 54 34.3
Clifton Beacon 53 27'5 1 Ibanez and Perrier,
Jonction geod. et astr. de l'Algerie avec l'Espagne
(Paris, 1886);
Memorial du depot general de la guerre, t.
xii.:
Nouvelle meridienne de France (Paris, 1885, 1902,
2904);
Comptes rendus des seances de la 12 e -19 e conference
generale de l'Assoc. Geod. Internat., 1898 at
Stuttgart, 1900 at Paris,
1903 at
Copenhagen,
1906 at
Budapest (Berlin,
1899, 1901, 2904, 1908); A. Ferrero,
Rapport sur les
triangulations, pres. a la Il e conf. gen. 1898. 2 R.
Schumann,
C. r. de Budapest, p. 244.
|
Name.
Arbury Hill
Greenwich
Nieuport
Rosendael
Lihons .
Pantheon
|
.
.
|
Latitude.
0
52 1 3.4
51 28.6
51 7.8
51 2.7
49 49'9
48 50.8
|
A.-G.
-3.o
-2.5
-0.4
-0.9
+0'5
-0.o
|
|
Chevry
|
.
|
48
|
0 5
|
+2.2
|
|
Saligny le Vif
|
.
|
47
|
2'7
|
+3.0
|
|
Arpheuille
|
|
46
|
1 3 7
|
+6.3
|
|
Puy de Dome
|
|
45
|
4 6 '5
|
+7'o
|
|
Roden .
|
|
44
|
21 ' 4
|
+ 1 '7
|
|
Carcassonne
|
|
43
|
1 3'3
|
+0'7
|
|
Rivesalte
|
|
|
|
-0-7
|
|
Montolar
|
|
41
|
38'5
|
+3.6
|
|
Lerida .
|
|
41
|
37.0
|
-0 2
|
|
Javalon .
|
|
40
|
13 8
|
-0.2
|
|
Desierto
|
|
40
|
5' 0
513;2
|
-4.5
|
|
Chinchilla. .
|
|
38
|
5'2
|
+2'2
|
|
Mola de Formentera
|
|
38
|
39'9
|
- 1.2
|
|
Tetica. .
|
|
37
|
15 2
|
+3'5
|
|
Roldan. .
|
|
36
|
566
|
-60
|
|
Conjuros .
|
|
3 6
|
44'4
|
-12.6
|
|
Mt. Sabiha .
|
|
35
|
39' 6
|
+6'5
|
|
Nemours .
|
|
35
|
5' 8
|
+7.4
|
|
Bouzareah
|
|
3 6
|
4 8 ' 0
|
+2 '9
|
|
Algiers (Voirol)
|
|
3 6
|
45.1
|
-9.1
|
|
Guelt es Stel.
|
|
35
|
7.8
|
- 1.0
|
|
El Aghuat .
|
|
33
|
48'o
|
-2.8
|
West Europe-Africa Meridian-arc (contd.). 'sasavord' to
Hutig ?
Stirling ellie
Law C to ill ss Clif
t
on on urleigh Moor Arbury Hill reenwich
Nieuport Rosendai:l Lihons Pantheon Chevry
Saligny le
Vif Arpheuille
Puy de Dime ' ' Rodes
Carcassonne
Rivesaltes Montolar Javalon
Chinchilla > Desierto
Lerida
0 ' 'v Mole de Formentera TetfcaConjuros
Roldan
Algiers Wok Bouzareah Guelt es Stet
ElAghuat Mt. Sabiha
Nemours
4-0 -6.1 +0.3 +7'3
-2.3 -3'7 +3'5 - 0.9
+2' 1
+1.3 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° to 70°), on the other hand, is very
uniformly curved, and gives, with an ellipticity of 1/299.15,
a = 6 37 8 455 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°); 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.' 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°, which extends from
Cape May (
New Jersey), on the
Atlantic coast, to Point
Arena (California), on the Pacific coast, and
embraces 48° 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 ioo 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.2 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
t 3.8").
The computations relative to another arc, the " eastern oblique
arc of the United States," are also finished. 3 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° 1',
and of longitude 22° 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 1 O. and A.
Borsch, " Verbindung d. russ.-skandina y. mit der franz.-engl.
Breitengradmessung " (
Verhandlungen der 9. Allgem. Conf. d. I.
E. in Paris, 1889, Ann. xi.).
U.S. Coast and Geodetic Survey; H. S. Pritchett,
superintendent.
The Transcontinental Triangulation and the American Arc of the
Parallel, by C. A. Schott (Washington, 1900).
U.S. Coast and Geodetic Survey; O. H. Tittmann, superintendent.
The Eastern Oblique Arc of the United States, by C. A.
Schott (1902).
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 /3 0
4.5 =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° W., has
been commenced; it has a range of latitude of 23°, and will extend
over 50° 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° 53' 33",
and stretches from Tulcan (lat. o° 48' 25") on the borders of
Columbia and
Ecuador, through Columbia to
Payta (lat. - 5° 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°. 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° 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° 38'
and 80° 50', according to the project of P. G. Rosen. The southern
part was determined by the Russians - O. Backlund, Captain D. D.
Sergieffsky, F. N. Tschernychev, A. Hansky and others - during
1899-1901, with the aid of i base-line, 15 trigonometrical, II
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. Jaderin, T. Rubin and
others), who utilized i 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.4 4
Missions scientifiques pour
la mesure d'un arc de me'ridien au Spitzberg entreprises en
1899-1902 sous les auspices des gouvernements russe et suedois.
Mission russe (St Petersbourg, 1904);
Mission
suedoise (Stockholm, 1904).
Sir David Gill, when director of the Royal Observatory,
Cape Town, instituted the
magnificent project of working a latitudedegree measurement along
the meridian of 30° long. This meridian passes through
Natal, the
Transvaal, by Lake
Tanganyika, and from thence to
Cairo; connexion with the
RussoScandinavian 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° in latitude will have been
measured.' 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).2 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
semimajor 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=6
377397.1 55(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,
Geodeisie, 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 +loo metres (Helmert).3
The basis of the degreeand 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.4 ' Sir David Gill,
Report on the Geodetic Survey of
South Africa,
1833-1892 (Cape Town, 1896), vol. ii. 1901, vol. iii.
2905.
z O. Hecker, Bestimmung der Schwerkraft a. d.
Atlantischen Ozean (Veroffentl. d. Kgl. Preuss. Geod.
Inst. No. II), Berlin, 1903.
' F. R. Helmert, " Neuere Fortschritte in der Erkenntnis der
math. Erdgestalt " (Verhandl. des VII. Internationalen
GeographenKongresses, Berlin, 1899), London, 1901.
C. Lallemand, " Rapport sur les travaux du service du
nivellement general de la France, de 1900 a 1903 " (Comp. rend.
de la 14' .conf. gen. de l'Assoc. Geod. Intern., 1903, p.
178).
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° 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 -
3 1 ° 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 twelveand a
fourteenmonth period. 5 (A. R. C.; F. R. H.
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