Victor F. Lenzen

“ The 19th century witnessed the culmination of the ellipsoidal era of geodesy, but the rapid accumulation of data made possible a better approximation to the figure of the earth by the geoid. The geoid is defined as the average level of the sea, which is thought of as extended through the continents. The basis of geodetic calculations, however, is an ellipsoid of reference for which a gravity formula expresses the value of normal gravity at a point on the ellipsoid as a function of gravity at sea level at the equator, and of latitude. ”

“ 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. ”

“ 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 invariable pendulum with a single knife edge, however, is adequate for relative determinations. As we have seen, such invariable pendulums had been used by Bouguer and Kater, and after the experiences with the Repsold apparatus had been recommended again by Baeyer for relative determinations. But an invariable pendulum is subject to uncontrollable changes of length. Peirce proposed to detect such changes in an invariable pendulum in the field by combining the invariable and reversible principles. ”

“ For this purpose, one must reduce observed values of gravity to sea level by various reductions, such as free-air, Bouguer, isostatic reductions. If g0 is observed gravity reduced to sea level and γ is normal gravity obtained from the International Gravity Formula, then Δg = g0 - γ is the gravity anomaly. [112]
In 1849, Stokes derived a theorem whereby the distance N of the geoid from the ellipsoid of reference can be obtained from an integration of gravity anomalies over the surface of the earth. ”

“ Figure 16.—From a Design Left by Bessel, this portable apparatus was developed in 1862 by the firm of Repsold in Hamburg, whose founder had assisted Bessel in the construction of his pendulum apparatus of 1826. The pendulum is convertible, but differs from Kater’s in being geometrically symmetrical and, for this reason, Repsold’s is usually called “reversible.” Just to the right of the pendulum is a standard scale. To the left is a “vertical comparator” designed by Repsold to measure the distance between the knife edges of the pendulum. ”

“ 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 [Pg 310] 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) . ”

“ Early in the 19th century a systematic series of observations began to be conducted in order to determine the intensity of gravity at stations all over the world. Kater invariable pendulums, of which 13 examples have been mentioned in the literature, were used in surveys of gravity by Kater, Sabine, Goldingham, and other British pendulum swingers. ”

“ A new pattern of the reversible pendulum has been invented, having its surface as nearly as convenient in the form of an elongated ellipsoid. Three of these instruments have been constructed, two having a distance of one meter between the knife edges and the third a distance of one yard. It is proposed to swing one of the meter pendulums at a temperature near 32° F. at the same time that the yard is swung at 60° F., in order to determine anew the relation between the yard and the meter. [74]
The report for 1881-1882 mentions four of these Peirce pendulums. ”

“ 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 observing telescope, left, has a diaphragm with a vertical slit of such width that its view is just filled by the tail of the Kater pendulum when it is at rest. When the two pendulums are swinging, the white spot on the clock pendulum can be seen on each swing except that in which the two pendulums are in coincidence; thus, the coincidences are determined. (Portion of plate 5, Mémoires publiés par la Société française de Physique, vol. 4.) [Pg 317]
Figure 13.—This drawing accompanied John Goldingham’s report on the work done in India with Kater’s invariable pendulum. ”

“ Figure 17.—Repsold-Bessel reversible pendulum apparatus as made in 1875, and used in the gravity work of the U.S. Coast and Geodetic Survey. Continental geodesists continued to favor the general use of convertible pendulums and absolute determinations of gravity, while their English colleagues had turned to invariable pendulums and relative determinations, except for base stations. Perhaps the first important American contribution to gravity work was C. S. Peirce’s demonstration of the error inherent in the Repsold apparatus through flexure of the stand. ”

“ The relative intensity of gravity was determined by observations with the first Mendenhall apparatus at Washington, D.C., at stations on the Pacific Coast and in Alaska, and at the Stevens Institute, Hoboken, New Jersey, between March and October 1891.
Figure 26.—Vacuum receiver within which the Mendenhall pendulum is swung. The pressure is reduced to about 50 mm. to reduce the disturbing effect of air resistance. When the apparatus is sealed, the pendulum is lifted on the knife edge by the lever q and is started to swing by the lever r. The arc of swing is only about 1°. ”

“ Commandant Defforges, of France, also designed and used cylindrical reversible pendulums, but of different length so that the effect of flexure was eliminated in the reduction of observations. Maj. Robert von Sterneck, of Austria-Hungary, initiated a new era in gravity research by the invention of an apparatus with a short pendulum for relative determinations of gravity. ”

“ Picard proposed to the Academy the measurement of the meridian of Paris through all of France, and this project was supported by Colbert, who obtained the approval of the King. In 1684, Giovanni-Domenico Cassini and De la Hire commenced a trigonometrical measure of an arc south of Paris; subsequently, Jacques Cassini, the son [Pg 307] of Giovanni-Domenico, added the arc to the north of Paris. The project was completed in 1718. The length of a degree of arc south of Paris was found to be greater than the length north of Paris. ”

“ He became the first superintendent of the Prussian observatory established at Königsberg in 1810, and remained there during the remainder of his life. So important were his many contributions to precise measurement and calculation in astronomy that he is often considered the founder of the “modern” age in that science. This characteristic also shows in his venture into geodesy, 1826-1830, one product of which was the pendulum experiment reported in this article. ”

“ On April 1, 1866, the Central Bureau of Die Europäische Gradmessung was opened in Berlin under the presidency of Baeyer, and in 1868 there was founded at Berlin, also under his presidency, the Royal Prussian Geodetic Institute, which obtained regular budgetary status on January 1, 1870. A reversible pendulum for the Institute was ordered from A. Repsold and Sons, and it was delivered in the spring of 1869. ”

“ Dr. Schumann, in 1899, gave a mathematical theory of such determinations, [88] and in his paper cited the mathematical methods of Peirce and Cellérier for the theory of Faye’s proposal at Stuttgart in 1877 to swing two similar pendulums on the same support with equal amplitudes and in opposite phases. [Pg 337]
Figure 29.—The Gulf pendulum is about 10.7 inches long, and has a period of .89 second. It is made of fused quartz which is resistant to the influence of temperature change and to the earth’s magnetism. ”

“ The observations were made with the ordinary Stückrath apparatus, in which four Von Sterneck pendulums swung two by two in planes perpendicular to each other. This successful application of the method—which had been proposed by Faye and had been demonstrated theoretically to be sound by Peirce, who also published a design for its [Pg 338] application—was rapidly followed for pendulum apparatus for relative determinations by Potsdam, [91] Cambridge (England) , [92] Gulf Oil and Development Company, [93] and the Dominion Observatory at Ottawa. ”

“ This memoir and others by Cellérier and Plantamour confirming Peirce’s work were published as appendices to the proceedings of the conference. As appendices to Peirce’s contribution were published also two notes by Prof. von Oppolzer. At the second session on September 29, 1877, when Plantamour reported that the work of Hirsch and himself had confirmed experimentally the independent theoretical work of Cellérier and the theoretical and experimental work of Peirce on flexure, Peirce described his Hoboken experiments. ”

“ Within the box was an electromagnet whose coils were connected with a chronometer circuit and whose armature carried a long arm that moved two shutters, in both of which were horizontal slits of the same size. The shutters were behind the front face of the box, which also had a horizontal slit. A flash of light from an oil lamp or an electric spark was emitted from the box when the circuit was broken, but not when it was closed. ”

“ This pause was broken simultaneously in three different ways. Two pendulums of the Kater pattern were sent to India; two after Bessel’s design were set to work in Russia; and at Geneva, Plantamour’s zealous experiments with a pendulum of the same kind mark the commencement of an era of renewed activity on the European continent. [41]
With the statement that Kater invariable pendulums nos. ”