Part 2

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, _Theorie 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

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 = [pi][sqrt](l/g), it follows that (T_{1})^{2}/(T_{2})^{2} = g_{2}/g_{1}. 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 T_{1} is the time of swing at a station where the intensity of gravity is g, and T_{2} is the time at a station where the intensity is g + [Delta]g, then [Delta]g/g = (T_{1})^{2}/(T_{2})^{2} - 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]

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.

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 Koenigsberg 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.

Bessel's experiments were significant in view of the care with which he determined the corrections. He corrected for the stiffness of the wire and for the lack of rigidity of connection between the bob and wire. The necessity for the latter correction had been pointed out by Laplace, who showed that through the circumstance that the pull of the wire is now on one side and now on the other side of the center of gravity, the bob acquires angular momentum about its center of gravity, which cannot be accounted for if the line of the wire, and therefore the force that it exerts, always passed through the center. In addition to a correction for buoyancy of the air considered by his predecessors, Bessel also took account of the inertia of the air set in motion by the pendulum.

The latter effect had been discovered by Du Buat in 1786,[23] but his work was unknown to Bessel. The length of the seconds pendulum at Koenigsberg, reduced to sea level, was found by Bessel to be 440.8179 lines. In 1835, Bessel determined the intensity of gravity at a site in Berlin where observations later were conducted in the Imperial Office of Weights and Measures by Charles S. Peirce of the U.S. Coast Survey.

Kater's Convertible and Invariable Pendulums

The systematic survey of the gravity field of the earth was given a great impetus by the contributions of Capt. Henry Kater, F.R.S. In 1817, he designed, constructed, and applied a convertible compound pendulum for the absolute determination of gravity at the house of Henry Browne, F.R.S., in Portland Place, London.[24] Kater's convertible pendulum (fig. 11) consisted of a brass rod to which were attached a flat circular bob of brass and two adjustable weights, the smaller of which was adjusted by a screw. The convertibility of the pendulum was constituted by the provision of two knife edges turned inwards on opposite sides of the center of gravity. The pendulum was swung on each knife edge, and the adjustable weights were moved until the times of swing were the same about each knife edge. When the times were judged to be the same, the distance between the knife edges was inferred to be the length of the equivalent simple pendulum, in accordance with Huygens' theorem on conjugate points of a compound pendulum. Kater determined the time of swing by the method of coincidences (fig. 12). He corrected for the buoyancy of the air. The final value of the length of the seconds pendulum at Browne's house in London, reduced to sea level, was determined to be 39.13929 inches.

The convertible compound pendulum had been conceived prior to its realization by Kater. In 1792, on the occasion of the proposal in Paris to establish the standard of length as the length of the seconds pendulum, Baron de Prony had proposed the employment of a compound pendulum with three axes of oscillation.[25] In 1800, he proposed the convertible compound pendulum with knife edges about which the pendulum could complete swings in equal times. De Prony's proposals were not accepted and his papers remained unpublished until 1889, at which time they were discovered by Defforges. The French decision was to experiment with the ball pendulum, and the determination of the length of the seconds pendulum was carried out by Borda and Cassini by methods previously described. Bohnenberger in his _Astronomie_ (1811),[26] made the proposal to employ a convertible pendulum for the absolute determination of gravity; thus, he has received credit for priority in publication. Capt. Kater independently conceived of the convertible pendulum and was the first to design, construct, and swing one.

After his observations with the convertible pendulum, Capt. Kater designed an invariable compound pendulum with a single knife edge but otherwise similar in external form to the convertible pendulum[27] (fig. 13). Thirteen of these Kater invariable pendulums have been reported as constructed and swung at stations throughout the world.[28] Kater himself swung an invariable pendulum at a station in London and at various other stations in the British Isles. Capt. Edward Sabine, between 1820 and 1825, made voyages and swung Kater invariable pendulums at stations from the West Indies to Greenland and Spitzbergen.[29] In 1820, Kater swung a Kater invariable pendulum at London and then sent it to Goldingham, who swung it in 1821 at Madras, India.[30] Also in 1820, Kater supplied an invariable pendulum to Hall, who swung it at London and then made observations near the equator and in the Southern Hemisphere, and at London again in 1823.[31] The same pendulum, after its knives were reground, was delivered to Adm. Luetke of Russia, who observed gravity with it on a trip around the world between 1826 and 1829.[32]

While the British were engaged in swinging the Kater invariable pendulums to determine relative values of the length of the seconds pendulum, or of gravity, the French also sent out expeditions. Capt. de Freycinet made initial observations at Paris with three invariable brass pendulums and one wooden one, and then carried out observations at Rio de Janeiro, Cape of Good Hope, Ile de France, Rawak (near New Guinea), Guam, Maui, and various other places.[33] A similar expedition was conducted in 1822-1825 by Captain Duperry.[34]