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, Théorie 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 γE, γP are the values of gravity at the equator and pole, respectively, and c the centrifugal force at the equator divided by γE, then the ellipticity α = (5/2)c - (γP - γE)/γE.
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.
Early Types of Pendulums
The pendulum employed in observations of gravity prior to the 19th century usually consisted of a small weight suspended by a filament (figs. [4]-[6]). The pioneer experimenters with “simple” pendulums changed the length of the suspension until the pendulum beat seconds. Picard in 1669 determined the length of the seconds pendulum at Paris with a “simple” pendulum which consisted of a copper ball an inch in diameter suspended by a fiber of pite from jaws (pite was a preparation of the leaf of a species of aloe and was not affected appreciably by moisture).
A celebrated set of experiments with a “simple” pendulum was conducted by Bouguer [16] in 1737 in the Andes, as part of the expedition to measure the Peruvian arc. The bob of the pendulum was a double truncated cone, and the length was measured from the jaw suspension to the center of oscillation of the thread and bob. Bouguer allowed for change of length of his measuring rod with temperature and also for the buoyancy of the air. He determined the time of swing by an elementary form of the method of coincidences. The thread of the pendulum was swung in front of a scale and Bouguer observed how long it took the pendulum to lose a number of vibrations on the seconds clock. For this purpose, he noted the time when the beat of the clock was heard and, simultaneously, the thread moved past the center of the scale. A historic aspect of Bouguer’s method was that he employed an “invariable” pendulum, that is, the length was maintained the same at the various stations of observation, a procedure that has been described as having been invented by Bouguer.
Since T = π√(l/g), it follows that T12/T22 = g2/g1. Thus, if the absolute value of gravity is known at one station, the value at any other station can be determined from the ratio of the squares of times of swing of an invariable pendulum at the two stations. From the above equation, if T1 is the time of swing at a station where the intensity of gravity is g, and T2 is the time at a station where the intensity is g + Δg, then (Δg)/g = (T12/T22) - 1.