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.
Bouguer’s investigations with his invariable pendulum yielded methods for the determination of the internal structure of the earth. On the Peruvian expedition, he determined the length of the seconds pendulum at three stations, including one at Quito, at varying distances above sea level. If values of gravity at stations of different elevation are to be compared, they must be reduced to the same level, usually to sea level. Since gravity decreases with height above sea level in accordance with the law of gravitation, a free-air reduction must be applied to values of gravity determined above the level of the sea. Bouguer originated the additional reduction for the increase in gravity on a mountain or plateau caused by the attraction of the matter in a plate. From the relative values of gravity at elevated stations in Peru and at sea level, Bouguer calculated that the mean density of the earth was 4.7 times greater than that of the cordilleras.[17] For greater accuracy in the study of the internal structure of the earth, in the 19th century the Bouguer plate reduction came to be supplemented by corrections for irregularities of terrain and by different types of isostatic reduction.
La Condamine, who like Bouguer was a member of the Peruvian expedition, conducted his own pendulum experiments (fig. [4]). He experimented in 1735 at Santo Domingo en route to South America,[18] then at various stations in South America, and again at Paris upon his return to France. His pendulum consisted of a copper ball suspended by a thread of pite. For experimentation the length initially was about 12 feet, and the time of swing 2 seconds, but then the length was reduced to about 3 feet with time of swing 1 second. Earlier, when it was believed that gravity was constant over the earth, Picard and others had proposed that the length of the seconds pendulum be chosen as the standard. La Condamine in 1747 revived the proposal in the form that the length of the seconds pendulum at the equator be adopted as the standard of length. Subsequently, he investigated the expansion of a toise of iron from the variation in the period of his pendulum. In 1755, he observed the pendulum at Rome with Boscovich. La Condamine’s pendulum was used by other observers and finally was lost at sea on an expedition around the world. The knowledge of the pendulum acquired by the end of the 18th century was summarized in 1785 in a memoir by Boscovich.[19]
Figure 5.—An apparatus for the practice measurement of the length of the pendulum devised on the basis of a series of preliminary experiments by C. M. de la Condamine who, in the course of the French geodetic expedition to Peru in 1735, devoted a 3-month sojourn on the island of Santo Domingo to pendulum observations by Mairan’s Method. In this arrangement, shown here, a vertical rod of ironwood is used both as the scale and as the support for the apparatus, having at its top the brass pendulum support (A) and, below, a horizontal mirror (O) which serves to align the apparatus vertically through visual observation of the reflection of the pointer projecting from A. The pendulum, about 37 inches long, consists of a thread of pite (a humidity-resistant, natural fiber) and a copper ball of about 6 ounces. Its exact length is determined by adjusting the micrometer (S) so that the ball nearly touches the mirror. It will be noted that the clock pendulum would be obscured by the scale. La Condamine seems to have determined the times of coincidence by visual observation of the occasions on which “the pendulums swing parallel.” (Portion of plate 1, Mémoires publiés par la Société française de Physique, vol. 4.)