We have noted previously that in the ancient and early modern periods,
the earth was presupposed to be spherical in form. Determination of the
figure of the earth consisted in the measurement of the radius by the
astronomical-geodetic method invented by Eratosthenes. Since the earth
was assumed to be spherical, gravity was inferred to be constant over
the surface of the earth. This conclusion appeared to be confirmed by
the determination of the length of the seconds pendulum at various
stations in Europe by Picard and others. The observations of Richer in
South America, the theoretical discussions of Newton and Huygens, and
the measurements of degrees of latitude in Peru and Sweden demonstrated
that the earth is an oblate spheroid.
[Illustration: Figure 33.--GRAVITY CHARACTERISTICS OF THE GLOBE.
Deductions as to the distribution of matter in the earth can be made
from gravity measurements. This globe shows worldwide variations in
gravity as they now appear from observations at sea (in submarines) as
well as on land. It is based on data from the Institute of Geodesy at
Ohio State University.]
The theory of gravitation and the theory of central forces led to the
result that the intensity of gravity is variable over the surface of the
earth. Accordingly, determinations of the intensity of gravity became
of value to the geodesist as a means of determining the figure of the
earth. Newton, on the basis of the meager data available to him,
calculated the ellipticity of the earth to be 1/230 (the ellipticity is
defined by (a-b)/a, where a is the equatorial radius and b the polar
radius). Observations of the intensity of gravity were made on the
historic missions to Peru and Sweden. Bouguer and La Condamine found
that at the equator at sea level the seconds pendulum was 1.26
Paris-lines shorter than at Paris. Maupertuis found that in northern
Sweden a certain pendulum clock gained 59.1 seconds per day on its rate
in Paris. Then Clairaut, from the assumption that the earth is a
spheroid of equilibrium, derived a theorem from which the ellipticity of
the earth can be derived from values of the intensity of gravity.
[Illustration: Figure 34.--AN EXHIBIT OF GRAVITY APPARATUS at the
Smithsonian Institution. Suspended on the wall, from left to right, are
the invariable pendulums of Mendenhall (1/2-second), Peirce (1873-1874),
and Peirce (1881-1882); the double pendulum of Edward Kuebel (see fig.
15, p. 319), and the reversible pendulum of Peirce. On the display
counter, from left to right, are the vacuum chamber, telescope and flash
apparatus for the Mendenhall 1/4-second apparatus. Shown below these are
the four pendulums used with the Mendenhall apparatus, the one on the
right having a thermometer attached. At bottom, right, is the Gulf
apparatus (cover removed) mentioned in the text, shown with one quartz
pendulum.]
Early in the 19th century a systematic series of observations began to
be conducted in order to determine the intensity of gravity at stations
all over the world. Kater invariable pendulums, of which 13 examples
have been mentioned in the literature, were used in surveys of gravity
by Kater, Sabine, Goldingham, and other British pendulum swingers. As
has been noted previously, a Kater invariable pendulum was used by Adm.
Luetke of Russia on a trip around the world. The French also sent out
expeditions to determine values of gravity. After several decades of
relative inactivity, Capts. Basevi and Heaviside of the Indian Survey
carried out an important series of observations from 1865 to 1873 with
Kater invariable pendulums and the Russian Repsold-Bessel pendulums. In
1881-1882 Maj. J. Herschel swung Kater invariable pendulums nos. 4, 6
(1821), and 11 at stations in England and then brought them to the
United States in order to make observations which would connect American
and English base stations.[109]
The extensive sets of observations of gravity provided the basis of
calculations of the ellipticity of the earth. Col. A. R. Clarke in his
_Geodesy_ (London, 1880) calculated the ellipticity from the results of
gravity surveys to be 1/(292.2 +- 1.5). Of interest is the calculation by
Charles S. Peirce, who used only determinations made with Kater
invariable pendulums and corrected for elevation, atmospheric effect,
and expansion of the pendulum through temperature.[110] He calculated
the ellipticity of the earth to be 1/(291.5 +- 0.9).
The 19th century witnessed the culmination of the ellipsoidal era of
geodesy, but the rapid accumulation of data made possible a better
approximation to the figure of the earth by the geoid. The geoid is
defined as the average level of the sea, which is thought of as extended
through the continents. The basis of geodetic calculations, however, is
an ellipsoid of reference for which a gravity formula expresses the
value of normal gravity at a point on the ellipsoid as a function of
gravity at sea level at the equator, and of latitude. The general
assembly of the International Union of Geodesy and Geophysics, which was
founded after World War I to continue the work of _Die Internationale
Erdmessung_, adopted in 1924 an international reference ellipsoid,[111]
of which the ellipticity, or flattening, is Hayford's value 1/297. In
1930, the general assembly adopted a correlated International Gravity
Formula of the form
[gamma] = [gamma]_{E}(1 + [beta]sin^{2} [phi] + [epsilon]sin^{2} 2[phi])
where [gamma] is normal gravity at latitude [phi], [gamma]_{E} is the
value of gravity at sea level at the equator, [beta] is a parameter
which is computed on the basis of Clairaut's theorem from the flattening
value of the meridian, and [epsilon] is a constant which is derived
theoretically. The plumb line is perpendicular to the geoid, and the
components of angle between the perpendiculars to geoid and reference
ellipsoid are deflections of the vertical. The geoid is above the
ellipsoid of reference under mountains and it is below the ellipsoid on
the oceans, where the geoid coincides with mean sea level. In physical
geodesy, gravimetric data are used for the determination of the geoid
and components of deflections of the vertical. For this purpose, one
must reduce observed values of gravity to sea level by various
reductions, such as free-air, Bouguer, isostatic reductions. If g_{0} is
observed gravity reduced to sea level and [gamma] is normal gravity
obtained from the International Gravity Formula, then
[Delta]g = g_{0} - [gamma]
is the gravity anomaly.[112]
In 1849, Stokes derived a theorem whereby the distance N of the geoid
from the ellipsoid of reference can be obtained from an integration of
gravity anomalies over the surface of the earth. Vening Meinesz further
derived formulae for the calculation of components of the deflection of
the vertical.
Geometrical geodesy, which was based on astronomical-geodetic methods,
could give information only concerning the external form of the figure
of the earth. The gravimetric methods of physical geodesy, in
conjunction with methods such as those of seismology, enable scientists
to test hypotheses concerning the internal structure of the earth.
Heiskanen and Vening Meinesz summarize the present-day achievements of
the gravimetric method of physical geodesy by stating[113] that it
alone can give: