[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, _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 [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. Early Types of Pendulums