CONVERTIBLE PENDULUM: a compound pendulum having knife edges at different distances from the center of gravity. Huygens demonstrated (1673) that if such a pendulum were to swing with equal periods from either knife edge, the distance between those knife edges would be equal to the length of a theoretical or simple pendulum of the same period. REVERSIBLE PENDULUM: a convertible pendulum which is also symmetrical in form. INVARIABLE PENDULUM: a compound pendulum with only one knife edge, used for relative measurement of gravity. * * * * * From Newton's laws of motion and the hypothesis that weight is proportional to mass, the formula for the half-period of a simple pendulum is given by T = [pi][sqrt](l/g). If a simple pendulum beats seconds, 1 = [pi][sqrt]([lambda]/g), where [lambda] is the length of the seconds pendulum. From T = [pi][sqrt](l/g) and 1 = [pi][sqrt]([lambda]/g), it follows that [lambda] = l/T^{2}. Then g = [pi]^{2}[lambda]. Thus, the intensity of gravity can be expressed in terms of the length of the seconds pendulum, as well as by the acceleration of a freely falling body. During the 19th century, gravity usually was expressed in terms of the length of the seconds pendulum, but present practice is to express gravity in terms of g, for which the unit is the gal, or one centimeter per second per second. [Illustration: Figure 2.--THIS DRAWING, FROM RICHER'S _Observations astronomiques et physiques faites en l'isle de Caïenne_ (Paris, 1679), shows most of the astronomical instruments used by Richer, namely, one of the two pendulum clocks made by Thuret, the 20-foot and the 5-foot telescopes and the large quadrant. The figure may be intended as a portrait of Richer. This drawing was done by Sebastian Le Clerc, a young illustrator who made many illustrations of the early work of the Paris Academy.] Figure of the Earth 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.