A principal contribution of the pendulum as a physical instrument has
been the determination of the figure of the earth.[5] That the earth
is spherical in form was accepted doctrine among the ancient Greeks.
Pythagoras is said to have been the first to describe the earth as a
sphere, and this view was adopted by Eudoxus and Aristotle.
The Alexandrian scientist Eratosthenes made the first estimate of the
diameter and circumference of a supposedly spherical earth by an
astronomical-geodetic method. He measured the angle between the
directions of the rays of the sun at Alexandria and Syene (Aswan),
Egypt, and estimated the distance between these places from the length
of time required by a caravan of camels to travel between them. From the
central angle corresponding to the arc on the surface, he calculated the
radius and hence the circumference of the earth. A second measurement
was undertaken by Posidonius, who measured the altitudes of stars at
Alexandria and Rhodes and estimated the distance between them from the
time required to sail from one place to the other.
With the decline of classical antiquity, the doctrine of the spherical
shape of the earth was lost, and only one investigation, that by the
Arabs under Calif Al-Mamun in A.D. 827, is recorded until the 16th
century. In 1525, the French mathematician Fernel measured the length of
a degree of latitude between Paris and Amiens by the revolutions of the
wheels of his carriage, the circumference of which he had determined. In
England, Norwood in 1635 measured the length of an arc between London
and York with a chain. An important forward step in geodesy was the
measurement of distance by triangulation, first by Tycho Brahe, in
Denmark, and later, in 1615, by Willebrord Snell, in Holland.
Of historic importance, was the use of telescopes in the triangulation
for the measurement of a degree of arc by the Abbe Jean Picard in
1669.[6] He had been commissioned by the newly established Academy of
Sciences to measure an arc corresponding to an angle of 1 deg., 22', 55"
of the meridian between Amiens and Malvoisine, near Paris. Picard proposed
to the Academy the measurement of the meridian of Paris through all of
France, and this project was supported by Colbert, who obtained the
approval of the King. In 1684, Giovanni-Domenico Cassini and De la Hire
commenced a trigonometrical measure of an arc south of Paris;
subsequently, Jacques Cassini, the son of Giovanni-Domenico, added the
arc to the north of Paris. The project was completed in 1718. The length
of a degree of arc south of Paris was found to be greater than the
length north of Paris. From the difference, 57,097 toises[7] minus
56,960 toises, it was concluded that the polar diameter of the earth is
larger than the equatorial diameter, i.e., that the earth is a prolate
spheroid (fig. 3).
[Illustration: Figure 3.--MEASUREMENTS OF THE LENGTH of a degree of
latitude which were completed in different parts of France in 1669 and
1718 gave differing results which suggested that the shape of the earth
is not a sphere but a prolate spheroid (1). But Richer's pendulum
observation of 1672, as explained by Huygens and Newton, indicated that
its shape is that of an oblate spheroid (2). The disagreement is
reflected in this drawing. In the 1730's it was resolved in favor of the
latter view by two French geodetic expeditions for the measurement of
degrees of latitude in the equatorial and polar regions (Ecuador--then
part of Peru--and Lapland).]
Meanwhile, Richer in 1672 had been sent to Cayenne, French Guiana, to
make astronomical observations and to measure the length of the seconds
pendulum.[8] He took with him a pendulum clock which had been adjusted
to keep accurate time in Paris. At Cayenne, however, Richer found that
the clock was retarded by 2 minutes and 28 seconds per day (fig. 1). He
also fitted up a "simple" pendulum to vibrate in seconds and measured
the length of this seconds pendulum several times every week for 10
months. Upon his return to Paris, he found that the length of the
"simple" pendulum which beat seconds at Cayenne was 1-1/4 Paris lines[9]
shorter than the length of the seconds pendulum at Paris. Huygens
explained the reduction in the length of the seconds pendulum--and,
therefore, the lesser intensity of gravity at the equator with respect
to the value at Paris--in terms of his theory of centripetal force as
applied to the rotation of the earth and pendulum.[10]
A more complete theory was given by Newton in the _Principia_.[11]
Newton showed that if the earth is assumed to be a homogeneous, mutually
gravitating fluid globe, its rotation will result in a bulging at the
equator. The earth will then have the form of an oblate spheroid, and
the intensity of gravity as a form of universal gravitation will vary
with position on the surface of the earth. Newton took into account
gravitational attraction and centrifugal action, and he calculated the
ratio of the axes of the spheroid to be 230:229. He calculated and
prepared a table of the lengths of a degree of latitude and of the
seconds pendulum for every 5 deg. of latitude from the equator to the
pole. A discrepancy between his predicted length of the seconds pendulum
at the equator and Richer's measured length was explained by Newton in
terms of the expansion of the scale with higher temperatures near the
equator.
Newton's theory that the earth is an oblate spheroid was confirmed by
the measurements of Richer, but was rejected by the Paris Academy of
Sciences, for it contradicted the results of the Cassinis, father and
son, whose measurements of arcs to the south and north of Paris had led
to the conclusion that the earth is a prolate spheroid. Thus, a
controversy arose between the English scientists and the Paris Academy.
The conflict was finally resolved by the results of expeditions sent by
the Academy to Peru and Sweden. The first expedition, under Bouguer, La
Condamine, and Godin in 1735, went to a region in Peru, and, with the
help of the Spaniard Ullo, measured a meridian arc of about 3 deg. 7'
near Quito, now in Ecuador.[12] The second expedition, with Maupertuis
and Clairaut in 1736, went to Lapland within the Arctic Circle and
measured an arc of about 1 deg. in length.[13] The northern arc of 1 deg.
was found to be longer than the Peruvian arc of 1 deg., and thus it was
confirmed that the earth is an oblate spheroid, that is, flattened at
the poles, as predicted by the theory of Newton.
[Illustration: Figure 4.--THE DIRECT USE OF A CLOCK to measure the force
of gravity was found to be limited in accuracy by the necessary
mechanical connection of the pendulum to the clock, and by the
unavoidable difference between the characteristics of a clock pendulum
and those of a theoretical (usually called "simple") pendulum, in which
the mass is concentrated in the bob, and the supporting rod is
weightless.
After 1735, the clock was used only to time the swing of a detached
pendulum, by the method of "coincidences." In this method, invented by
J. J. Mairan, the length of the detached pendulum is first accurately
measured, and the clock is corrected by astronomical observation. The
detached pendulum is then swung before the clock pendulum as shown here.
The two pendulums swing more or less out of phase, coming into
coincidence each time one has gained a vibration. By counting the number
of coincidences over several hours, the period of the detached pendulum
can be very accurately determined. The length and period of the detached
pendulum are the data required for the calculation of the force of
gravity.]
The period from Eratosthenes to Picard has been called the spherical era
of geodesy; the period from Picard to the end of the 19th century has
been called the ellipsoidal period. During the latter period the earth
was conceived to be an ellipsoid, and the determination of its
ellipticity, that is, the difference of equatorial radius and polar
radius divided by the equatorial radius, became an important geodetic
problem. A significant contribution to the solution of this problem was
made by determinations of gravity by the pendulum.
An epoch-making work during the ellipsoidal era of geodesy was
Clairaut's treatise, _Theorie de la figure de la terre_.[14] On the
hypothesis that the earth is a spheroid of equilibrium, that is, such
that a layer of water would spread all over it, and that the internal
density varies so that layers of equal density are coaxial spheroids,
Clairaut derived a historic theorem: If [gamma]_{E}, [gamma]_{P} are the
values of gravity at the equator and pole, respectively, and c the
centrifugal force at the equator divided by [gamma]_{E}, then the
ellipticity [alpha] = (5/2)c - ([gamma]_{P} - [gamma]_{E})/[gamma]_{E}.
Laplace showed that the surfaces of equal density might have any nearly
spherical form, and Stokes showed that it is unnecessary to assume any
law of density as long as the external surface is a spheroid of
equilibrium.[15] It follows from Clairaut's theorem that if the earth is
an oblate spheroid, its ellipticity can be determined from relative
values of gravity and the absolute value at the equator involved in c.
Observations with nonreversible, invariable compound pendulums have
contributed to the application of Clairaut's theorem in its original and
contemporary extended form for the determination of the figure and
gravity field of the earth.