Figure 6.—The result of early pendulum experiments was often expressed in terms of the length of a pendulum which would have a period of one second and was called “the seconds pendulum.” In 1792, J. C. Borda and J. D. Cassini determined the length of the seconds pendulum at Paris with this apparatus. The pendulum consists of a platinum ball about 1-1/2 inches in diameter, suspended by a fine iron wire. The length, about 12 feet, was such that its period would be nearly twice as long as that of the pendulum of the clock (A). The interval between coincidences was determined by observing, through the telescope at the left, the times when the two pendulums emerge together from behind the screen (M). The exact length of the pendulum was measured by a platinum scale (not shown) equipped with a vernier and an auxiliary copper scale for temperature correction.
When, at the end of the 18th century, the French revolutionary government established the metric system of weights and measures, the length of the seconds pendulum at Paris was considered, but not adopted, as the unit of length. (Plate 2, Mémoires publiés par la Société française de Physique, vol. 4.)
The practice with the “simple” pendulum on the part of Picard, Bouguer, La Condamine and others in France culminated in the work of Borda and Cassini in 1792 at the observatory in Paris[20] (fig. [6]). The experiments were undertaken to determine whether or not the length of the seconds pendulum should be adopted as the standard of length by the new government of France. The bob consisted of a platinum ball 16-1/6 Paris lines in diameter, and 9,911 grains (slightly more than 17 ounces) in weight. The bob was held to a brass cup covering about one-fifth of its surface by the interposition of a small quantity of grease. The cup with ball was hung by a fine iron wire about 12 Paris feet long. The upper end of the wire was attached to a cylinder which was part of a wedge-shaped knife edge, on the upper surface of which was a stem on which a small adjustable weight was held by a screw thread. The knife edge rested on a steel plate. The weight on the knife-edge apparatus was adjusted so that the apparatus would vibrate with the same period as the pendulum. Thus, the mass of the suspending apparatus could be neglected in the theory of motion of the pendulum about the knife edge.
Figure 7.—Results of experiments in the determination of the length of the seconds pendulum at Königsberg by a new method were reported by F. W. Bessel in 1826 and published in 1828. With this apparatus, he obtained two sets of data from the same pendulum, by using two different points of suspension. The pendulum was about 10 feet long. The distance between the two points of suspension (a and b) was 1 toise (about six feet). A micrometric balance (c) below the bob was used to determine the increase in length due to the weight of the bob. He projected the image of the clock pendulum (not shown) onto the gravity pendulum by means of a lens, thus placing the clock some distance away and eliminating the disturbing effect of its motion. (Portion of plate 6, Mémoires publiés par la Société française de Physique, vol. 4.)
In the earlier suspension from jaws there was uncertainty as to the point about which the pendulum oscillated. Borda and Cassini hung their pendulum in front of a seconds clock and determined the time of swing by the method of coincidences. The times on the clock were observed when the clock gained or lost one complete vibration (two swings) on the pendulum. Suppose that the wire pendulum makes n swings while the clock makes 2n + 2. If the clock beats seconds exactly, the time of one complete vibration is 2 seconds, and the time of swing of the wire pendulum is T = (2n + 2)/n = 2(1 + 1/n). An error in the time caused by uncertainty in determining the coincidence of clock and wire pendulum is reduced by employing a long interval of observation 2n. The whole apparatus was enclosed in a box, in order to exclude disturbances from currents of air. Corrections were made for buoyancy, for amplitude of swing and for variations in length of the wire with temperature. The final result was that the length of the seconds pendulum at the observatory in Paris was determined to be 440.5593 Paris lines, or 993.53 mm., reduced to sea level 993.85 mm. Some years later the methods of Borda were used by other French investigators, among whom was Biot who used the platinum ball of Borda suspended by a copper wire 60 cm. long.
Another historic “simple” pendulum was the one swung by Bessel (fig. [7]) for the determination of gravity at Königsberg 1825-1827.[21] The pendulum consisted of a ball of brass, copper, or ivory that was suspended by a fine wire, the upper end of which was wrapped and unwrapped on a horizontal cylinder as support. The pendulum was swung first from one point and then from another, exactly a “toise de Peru”[22] higher up, the bob being at the same level in each case (fig. [7]). Bessel found the period of vibration of the pendulum by the method of coincidences; and in order to avoid disturbances from the comparison clock, it was placed at some distance from the pendulum under observation.