[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.
EARTH, FIGURE OF THE. 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 deg. 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 500 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[1] 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 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 11th of June 1633 the sun's meridian altitude in London
as 62 deg. 1', and on the 6th of June 1635, his meridian altitude in York as
59 deg. 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 deg. 22' 55" 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 {1/12}th 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 deg. 12' 9",
obtained for the length of a degree 56,960 toises; while from the
southern portion, of which the amplitude was 6 deg. 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 deg. 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 deg. 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 [alpha] and [delta] 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 deg. 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 [alpha],
[delta], and [lambda] 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 10' or 15', 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 deg.
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 deg. 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 deg. 9', and, following generally the
methods of the English survey, he carried his triangulation as far north
as 20 deg. 30'. The work was continued by Sir George (then Captain) Everest,
who carried it to the latitude of 29 deg. 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.E., 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 descriptions 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., 1906, latitudes; vols. ix., 1883, x., 1887,
xv., 1893, longitudes; vol. xvii., 1901, 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 deg. 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 +- 0".53.
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'- g 5
----- + e = -- m,
g 2
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 [rho] the density,
dp = [rho](Xdx + Ydy + Zdz);
and for equilibrium the necessary conditions are, that [rho](Xdx + Ydy
+ Zdz) be a complete differential, and at the free surface Xdx + Ydy +
Zdz = 0. 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
dV dV dV
Xdx + Ydy + Zdz = --dx + --dy + --dz,
dx dy 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 = [rho]dU be a
complete differential; that is, [rho] 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 squared + y squared + z squared(1 + [epsilon] squared) = c squared(1 + [epsilon] squared); upon a particle P
of its mass at x, y, z has for components