Chapter II: Part 2

Gen. Baeyer stated that the distance of 1 meter between the knife edges of the Prussian Repsold-Bessel pendulum made it unwieldy and unsuited for transport. He declared that the instability of the stand also was a source of error. Accordingly, Gen. Baeyer expressed the opinion that absolute determinations of gravity should be made at a control station by a reversible pendulum hung on a permanent, and therefore stable stand, and he said that relative values of gravity with respect to the control station should be obtained in the field by means of a Bouguer invariable pendulum. Dr. Bruhns and Dr. Peters agreed with Gen. Baeyer; however, the Swiss investigators, Prof. Plantamour and Dr. Hirsch reported in defense of the reversible pendulum as a field instrument, as did Prof. von Oppolzer of Vienna. The circumstance that an invariable pendulum is subject to changes in length was offered as an argument in favor of the reversible pendulum as a field instrument.

Peirce was present during these discussions by the members of the Special Commission, and he reported that his experiments at Geneva demonstrated that the oscillations of the pendulum called forth a flexure of the support which hitherto had been neglected. The observers who used the Swiss and Austrian Repsold pendulums contended, in opposition to Peirce, that the Repsold stand was stable.

The outcome of these discussions was that the Special Commission reported to the Permanent Commission that the Repsold-Bessel reversible pendulum, except for some small changes, satisfied all requirements for the determination of gravity. The Special Commission proposed that the Repsold pendulums of the several states be swung at the Prussian Eichungsamt in Berlin where, as Peirce pointed out, Bessel had made his determination of the intensity of gravity with a ball pendulum in 1835. Peirce was encouraged to swing the Coast Survey reversible pendulum at the stations in France, England, and Germany where Borda and Cassini, Kater, and Bessel, respectively, had made historic determinations. The Permanent Commission, in whose sessions Peirce also participated, by resolutions adopted the report of the Special Commission on the Pendulum.[60]

During the months of January and February 1876, Peirce conducted observations in the Grande Salle du Meridien at the observatory in Paris where Borda, Biot, and Capt. Edward Sabine had swung pendulums early in the 19th century. He conducted observations in Berlin from April to June 1876 and, by experiment, determined the correction for flexure to be applied to the value of gravity previously obtained with the Prussian instrument. Subsequent observations were made at Kew. After his return to the United States on August 26, 1876, Peirce conducted experiments at the Stevens Institute in Hoboken, New Jersey, where he made careful measurements of the flexure of the stand by statical and dynamical methods. In Geneva, he had secured the construction of a vacuum chamber in which the pendulum could be swung on a support which he called the Geneva support. At the Stevens Institute, Peirce swung the Repsold-Bessel pendulum on the Geneva support and determined the effect of different pressures and temperatures on the period of oscillation of the pendulum. These experiments continued into 1878.[61]

Meanwhile, the Permanent Commission met October 5-10, 1876, in Brussels and continued the discussion of the pendulum.[62] Gen. Baeyer reported on Peirce's experiments in Berlin to determine the flexure of the stand. The difference of 0.18 mm. in the lengths of the seconds pendulum as determined by Bessel and as determined by the Repsold instrument agreed with Peirce's estimate of error caused by neglect of flexure of the Repsold stand. Dr. Hirsch, speaking for the Swiss survey, and Prof. von Oppolzer, speaking for the Austrian survey, contended, however, that their stands possessed sufficient stability and that the results found by Peirce applied only to the stands and bases investigated by him. The Permanent Commission proposed further study of the pendulum.

The Fifth General Conference of _Die Europaeische Gradmessung_ was held from September 27 to October 2, 1877, in Stuttgart.[63] Peirce had instructions from Supt. Patterson of the U.S. Coast Survey to attend this conference, and on arrival presented a letter of introduction from Patterson requesting that he, Peirce, be permitted to participate in the sessions. Upon invitation from Prof. Plantamour, as approved by Gen. Ibanez, president of the Permanent Commission, Peirce had sent on July 13, 1877, from New York, the manuscript of a memoir titled "De l'Influence de la flexibilite du trepied sur l'oscillation du pendule a reversion." This memoir and others by Cellerier 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 Cellerier and the theoretical and experimental work of Peirce on flexure, Peirce described his Hoboken experiments.

During the discussions at Stuttgart on the flexure of the Repsold stand, Herve Faye, president of the Bureau of Longitudes, Paris, suggested that the swaying of the stand during oscillations of the pendulum could be overcome by the suspension from one support of two similar pendulums which oscillated with equal amplitudes and in opposite phases. This proposal was criticized by Dr. Hirsch, who declared that exact observation of passages of a "double pendulum" would be difficult and that two pendulums swinging so close together would interfere with each other. The proposal of the double pendulum came up again at the meeting of the Permanent Commission at Geneva in 1879.[64] On February 17, 1879, Peirce had completed a paper "On a Method of Swinging Pendulums for the Determination of Gravity, Proposed by M. Faye." In this paper, Peirce presented the results of an analytical mechanical investigation of Faye's proposal. Peirce set up the differential equations, found the solutions, interpreted them physically, and arrived at the conclusion "that the suggestion of M. Faye ... is as sound as it is brilliant and offers some peculiar advantages over the existing method of swinging pendulums."

In a report to Supt. Patterson, dated July 1879, Peirce stated: "I think it is important before making a new pendulum apparatus to experiment with Faye's proposed method."[65] He wrote further: "The method proves to be perfectly sound in theory, and as it would greatly facilitate the work it is probably destined eventually to prevail. We must unfortunately leave to other surveys the merit of practically testing and introducing the new method, as our appropriations are insufficient for us to maintain the leading position in this matter, which we otherwise might take." Copies of the published version of Peirce's remarks were sent to Europe. At a meeting of the Academy of Sciences in Paris on September 1, 1879, Faye presented a report on Peirce's findings.[66] The Permanent Commission met September 16-20, 1879, in Geneva. At the third session on September 19, by action of Gen. Baeyer, copies of Peirce's paper on Faye's proposed method of swinging pendulums were distributed. Dr. Hirsch again commented adversely on the proposal, but moved that the question be investigated and reported on at the coming General Conference. The Permanent Commission accepted the proposal of Dr. Hirsch, and Prof. Plantamour was named to report on the matter at the General Conference. At Plantamour's request, Charles Cellerier was appointed to join him, since the problem essentially was a theoretical one.

The Sixth General Conference of _Die Europaeische Gradmessung_ met September 13-16, 1880, in Munich.[67] Topic III, part 7 of the program was entitled "On Determinations of Gravity through pendulum observations. Which construction of a pendulum apparatus corresponds completely to all requirements of science? Special report on the pendulum."

The conference received a memoir by Cellerier[68] on the theory of the double pendulum and a report by Plantamour and Cellerier.[69] Cellerier's mathematical analysis began with the equations of Peirce and used the latter's notation as far as possible. His general discussion included the results of Peirce, but he stated that the difficulties to be overcome did not justify the employment of the "double pendulum." He presented an alternative method of correcting for flexure based upon a theory by which the flexure caused by the oscillation of a given reversible pendulum could be determined from the behavior of an auxiliary pendulum of the same length but of different weight. This method of correcting for flexure was recommended to the General Conference by Plantamour and Cellerier in their joint report. At the fourth session of the conference on September 16, 1880, the problem of the pendulum was discussed and, in consequence, a commission consisting of Faye, Helmholtz, Plantamour (replaced in 1882 by Hirsch), and von Oppolzer was appointed to study apparatus suitable for relative determinations of gravity.

The Permanent Commission met September 11-15, 1882, at The Hague,[70] and at its last session appointed Prof. von Oppolzer to report to the Seventh General Conference on different forms of apparatus for the determination of gravity. The Seventh Conference met October 15-24, 1883, in Rome,[71] and, at its eighth session, on October 22, received a comprehensive, critical review from Prof. von Oppolzer entitled "Ueber die Bestimmung der Schwere mit Hilfe verschiedener Apparate."[72] Von Oppolzer especially expounded the advantages of the Bessel reversible pendulum, which compensated for air effects by symmetry of form if the times of swing for both positions were maintained between the same amplitudes, and compensated for irregular knife edges by making them interchangeable. Prof. von Oppolzer reviewed the problem of flexure of the Repsold stand and stated that a solution in the right direction was the proposal--made by Faye and theoretically pursued by Peirce--to swing two pendulums from the same stand with equal amplitudes and in opposite phases, but that the proposal was not practicable. He concluded that for absolute determinations of gravity, the Bessel reversible pendulum was highly appropriate if one swung two exemplars of different weight from the same stand for the elimination of flexure. Prof. von Oppolzer's important report recognized that absolute determinations were less accurate than relative ones, and should be conducted only at special places.

The discussions initiated by Peirce's demonstration of the flexure of the Repsold stand resulted, finally, in the abandonment of the plan to make absolute determinations of gravity at all stations with the reversible pendulum.

Peirce and Defforges Invariable, Reversible Pendulums

The Repsold-Bessel reversible pendulum was designed and initially used to make absolute determinations of gravity not only at initial stations such as Kew, the observatory in Paris, and the Smithsonian Institution in Washington, D.C., but also at stations in the field. 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. He explained his proposal to Faye in a letter dated July 23, 1880, and he presented it on September 16, 1880, at the fourth session of the sixth General Conference of _Die Europaeische Gradmessung_, in Munich.[73]

As recorded in the Proceedings of the Conference, Peirce wrote:

But I obviate it in making my pendulum both invariable and
reversible. Every alteration of the pendulum will be revealed
immediately by the change in the difference of the two periods
of oscillation in the two positions. Once discovered, it will be
taken account of by means of new measures of the distance
between the two supports.

Peirce added that it seemed to him that if the reversible pendulum perhaps is not the best instrument to determine absolute gravity, it is, on condition that it be truly invariable, the best to determine relative gravity. Peirce further stated that he would wish that the pendulum be formed of a tube of drawn brass with heavy plugs of brass equally drawn. The cylinder would be terminated by two hemispheres; the knives would be attached to tongues fixed near the ends of the cylinder.

During the years 1881 and 1882, four invariable, reversible pendulums were made after the design of Peirce at the office of the U.S. Coast and Geodetic Survey in Washington, D.C. The report of the superintendent for the year 1880-1881 states:

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 deg.
F. at the same time that the yard is swung at 60 deg. 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.

A description of the Peirce invariable, reversible pendulums was given by Assistant E. D. Preston in "Determinations of Gravity and the Magnetic Elements in Connection with the United States Scientific Expedition to the West Coast of Africa, 1889-90."[75] The invariable, reversible pendulum, Peirce no. 4, now preserved in the Smithsonian Institution's Museum of History and Technology (fig. 34), may be taken as typical of the meter pendulums: In the same memoir, Preston gives the diameter of the tube as 63.7 mm., thickness of tube 1.5 mm., weight 10.680 kilograms, and distance between the knives 1.000 meter.

The combination of invariability and reversibility in the Peirce pendulums was an innovation for relative determinations. Indeed, the combination was criticized by Maj. J. Herschel, R.E., of the Indian Survey, at a conference on gravity held in Washington in May 1882 on the occasion of his visit to the United States for the purpose of connecting English and American stations by relative determinations with three Kater invariable pendulums. These three pendulums have been designated as nos. 4, 6 (1821), and 11.[76]

Another novel characteristic of the Peirce pendulums was the mainly cylindrical form. Prof. George Gabriel Stokes, in a paper "On the Effect of the Internal Friction of Fluids on the Motion of Pendulums"[77] that was read to the Cambridge Philosophical Society on December 9, 1850, had solved the hydrodynamical equations to obtain the resistance to the motions of a sphere and a cylinder in a viscous fluid. Peirce had studied the effect of viscous resistance on the motion of his Repsold-Bessel pendulum, which was symmetrical in form but not cylindrical. The mainly cylindrical form of his pendulums (fig. 19) permitted Peirce to predict from Stokes' theory the effect of viscosity and to compare the results with experiment. His report of November 20, 1889, in which he presented the comparison of experimental results with the theory of Stokes, was not published.[78]

Peirce used his pendulums in 1883 to establish a station at the Smithsonian Institution that was to serve as the base station for the Coast and Geodetic Survey for some years. Pendulum Peirce no. 1 was swung at Washington in 1881 and was then taken by the party of Lieutenant Greely, U.S.A., on an expedition to Lady Franklin Bay where it was swung in 1882 at Fort Conger, Grinnell Land, Canada. Peirce nos. 2 and 3 were swung by Peirce in 1882 at Washington, D.C.; Hoboken, New Jersey; Montreal, Canada; and Albany, New York. Assistant Preston took Peirce no. 3 on a U.S. eclipse expedition to the Caroline Islands in 1883. Peirce in 1885 swung pendulums nos. 2 and 3 at Ann Arbor, Michigan; Madison, Wisconsin; and Ithaca, New York. Assistant Preston in 1887 swung Peirce nos. 3 and 4 at stations in the Hawaiian Islands, and in 1890 he swung Peirce nos. 3 and 4 at stations on the west coast of Africa.[79]

The new pattern of pendulum designed by Peirce was also adopted in France, after some years of experience with a Repsold-Bessel pendulum. Peirce in 1875 had swung his Repsold-Bessel pendulum at the observatory in Paris, where Borda and Cassini, and Biot, had made historic observations and where Sabine also had determined gravity by comparison with Kater's value at London. During the spring of 1880, Peirce made studies of the supports for the pendulums of these earlier determinations and calculated corrections to those results for hydrodynamic effects, viscosity, and flexure. On June 14, 1880, Peirce addressed the Academy of Sciences, Paris, on the value of gravity at Paris, and compared his results with the corrected results of Borda and Biot and with the transferred value of Kater.[80]

In the same year the French Geographic Service of the Army acquired a Repsold-Bessel reversible pendulum of the smaller type, and Defforges conducted experiments with it.[81] He introduced the method of measuring flexure from the movement of interference fringes during motion of the pendulum. He found an appreciable difference between dynamical and statical coefficients of flexure and concluded that the "correction formula of Peirce and Cellerier is suited perfectly to practice and represents exactly the variation of period caused by swaying of the support, on the condition that one uses the statical coefficient." Defforges developed a theory for the employment of two similar pendulums of the same weight, but of different length, and hung by the same knives. This theory eliminated the flexure of the support and the curvature of the knives from the reduction of observations.

Pendulums of 1-meter and of 1/2-meter distance between the knife edges were constructed from Defforges' design by Brunner Brothers in Paris (fig. 21). These Defforges pendulums were cylindrical in form with hemispherical ends like the Peirce pendulums, and were hung on knives that projected from the sides of the pendulum, as in some unfinished Gautier pendulums designed by Peirce in 1883 in Paris.

Von Sterneck and Mendenhall Pendulums

While scientists who had used the Repsold-Bessel pendulum apparatus discussed its defects and limitations for gravity surveys, Maj. Robert von Sterneck of Austria-Hungary began to develop an excellent apparatus for the rapid determination of relative values of gravity.[82] Maj. von Sterneck's apparatus contained a nonreversible pendulum 1/4-meter in length, and 1/2-second time of swing. The pendulum was hung by a single knife edge, which rested on a plate that was supported by a tripod. The pendulum was swung in a chamber from which air was exhausted and which could be maintained at any desired temperature. Times of swing were determined by the observation of coincidences of the pendulum with chronometer signals. In the final form a small mirror was attached to the knife edge perpendicular to the plane of vibration of the pendulum and a second fixed mirror was placed close to it so that the two mirrors were parallel when the pendulum was at rest. The chronometer signals worked a relay that gave a horizontal spark which was reflected into the telescope from the mirrors. When the pendulum was at rest, the image of the spark in both mirrors appeared on the horizontal cross wire in the telescope, and during oscillation of the pendulum the two images appeared in that position upon coincidence. In view of the reduced size of the pendulum, the chamber in which it was swung was readily portable, and with an improved method of observing coincidences, relative determinations of gravity could be made with rapidity and accuracy.

By 1887 Maj. von Sterneck had perfected his apparatus, and it was widely adopted in Europe for relative determinations of gravity. He used his apparatus in extensive gravity surveys and also applied it in the silver mines in Saxony and Bohemia, by the previously described methods of Airy, for investigations into the internal constitution of the earth.

On July 1, 1889, Thomas Corwin Mendenhall became superintendent of the U.S. Coast and Geodetic Survey. Earlier, he had been professor of physics at the University of Tokyo and had directed observations of pendulums for the determination of gravity on Fujiyama and at Tokyo. Supt. Mendenhall, with the cooperation of members of his staff in Washington, designed a new pendulum apparatus of the Von Sterneck type, and in October 1890 he ordered construction of the first model.[83]

Like the Von Sterneck apparatus, the Mendenhall pendulum apparatus employed a nonreversible, invariable pendulum 1/4-meter in length and of slightly more than 1/2-second in time of swing. Initially, the knife edge was placed in the head of the pendulum and hung on a fixed plane support, but after some experimentation Mendenhall attached the plane surface to the pendulum and hung it on a fixed knife edge. An apparatus was provided with a set of three pendulums, so that if discrepancies appeared in the results, the pendulum at fault could be detected. There was also a dummy pendulum which carried a thermometer. A pendulum was swung in a receiver in which the pressure and temperature of the air were controlled. The time of swing was measured by coincidences with the beat of a chronometer. The coincidences were determined by an optical method with the aid of a flash apparatus.

The flash apparatus was contained in a light metal box which supported an observing telescope and which was mounted on a stand. 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. When the circuit was broken a spring caused the arm to rise, and the shutters were actuated so that the three slits came into line and a flash of light was emitted. A small circular mirror was set in each side of the pendulum head, so that from either face of the pendulum the image of the illuminated slit could be reflected into the field of the observing telescope. A similar mirror was placed parallel to these two mirrors and rigidly attached to the support. The chronometer signals broke the circuit, causing the three slits momentarily to be in line, and when the images of the slit in the two mirrors coincided, a coincidence was observed. A coincidence occurred whenever the pendulum gained or lost one oscillation on the beat of the chronometer. 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.

Under Supt. Mendenhall's direction a smaller, 1/4-second, pendulum apparatus was also constructed and tested, but did not offer advantages over the 1/2-second apparatus, which therefore continued in use.

In accordance with Peirce's theory of the flexure of the stand under oscillations of the pendulum, determinations of the displacement of the receiver of the Mendenhall apparatus were part of a relative determination of gravity by members of the Coast and Geodetic Survey. Initially, a statical method was used, but during 1908-1909 members of the Survey adapted the Michelson interferometer for the determinations of flexure during oscillations from the shift of fringes.[84] The first Mendenhall pendulums were made of bronze, but about 1920 invar was chosen because of its small coefficient of expansion. About 1930, Lt. E. J. Brown of the Coast and Geodetic Survey made significant improvements in the Mendenhall apparatus, and the new form came to be known as the Brown Pendulum Apparatus.[85]

In 1907, the Michelson interferometer was adapted to this purpose by the U.S. Coast and Geodetic Survey. As shown here, the interferometer, resting on a wooden beam, is introduced into the path of a light beam reflected from a mirror on the vacuum chamber. Movement of that mirror causes a corresponding movement in the interference fringes in the interferometer, which can be measured.]

The original Von Sterneck apparatus and that of Mendenhall provided for the oscillation of one pendulum at a time. After the adoption of the Von Sterneck pendulum in Europe, there were developed stands on which two or four pendulums hung at the same time. This procedure provided a convenient way to observe more than one invariable pendulum at a station for the purpose of detecting changes in length. Prof. M. Haid of Karlsruhe in 1896 described a four-pendulum apparatus,[86] and Dr. Schumann of Potsdam subsequently described a two-pendulum apparatus.[87]

The multiple-pendulum apparatus then provided a method of determining the flexure of the stand from the action of one pendulum upon a second pendulum hung on the same stand. This method of determining the correction for flexure was a development from a "Wippverfahren" invented at the Geodetic Institute in Potsdam. A dynamometer was used to impart periodic impulses to the stand, and the effect was observed upon a pendulum initially at rest. Refinements of this method led to the development of a method used by Lorenzoni in 1885-1886 to determine the flexure of the stand by action of an auxiliary pendulum upon the principal pendulum. Dr. Schumann, in 1899, gave a mathematical theory of such determinations,[88] and in his paper cited the mathematical methods of Peirce and Cellerier 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.

In 1902, Dr. P. Furtwaengler[89] presented the mathematical theory of coupled pendulums in a paper in which he referred to Faye's proposal of 1877 and reported that the difficulties predicted upon its application had been found not to occur. Finally, during the gravity survey of Holland in the years 1913-1921, in view of instability of supports caused by the mobility of the soil, F. A. Vening Meinesz adopted Faye's proposed method of swinging two pendulums on the same support.[90] The observations were made with the ordinary Stueckrath 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 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.[94] Heiskanen and Vening Meinesz state:

The best way to eliminate the effect of flexure is to use two
synchronized pendulums of the same length swinging on the same
apparatus in the same plane and with the same amplitudes but in
opposite phases; it is clear then the flexure is zero.[95]

In view of the fact that the symmetrical reversible pendulum is named for Bessel, who created the theory and a design for its application by Repsold, it appears appropriate to call the method of eliminating flexure by swinging two pendulums on the same support the Faye-Peirce method. Its successful application was made possible by Maj. von Sterneck's invention of the short, 1/4-meter pendulum.

Absolute Value of Gravity at Potsdam

The development of the reversible pendulum in the 19th century culminated in the absolute determination of the intensity of gravity at Potsdam by Kuehnen and Furtwaengler of the Royal Prussian Geodetic Institute, which then became the world base for gravity surveys.[96]

We have previously seen that in 1869 the Geodetic Institute--founded by Lt. Gen. Baeyer--had acquired a Repsold-Bessel reversible pendulum which was swung by Dr. Albrecht under the direction of Dr. Bruhns. Dissatisfaction with this instrument was expressed by Baeyer in 1875 to Charles S. Peirce, who then, by experiment and mathematical analysis of the flexure of the stand under oscillations of the pendulum, determined that previously reported results with the Repsold apparatus required correction. Dr. F. R. Helmert, who in 1887 succeeded Baeyer as director of the Institute, secured construction of a building for the Institute in Potsdam, and under his direction the scientific study of the intensity of gravity was pursued with vigor. In 1894, it was discovered in Potsdam that a pendulum constructed of very flexible material yielded results which differed markedly from those obtained with pendulums of greater stiffness. Dr. Kuehnen of the Institute discovered that the departure from expectations was the result of the flexure of the pendulum staff itself during oscillations.[97]

Peirce, in 1883, had discovered that the recesses cut in his pendulums for the insertion of tongues that carried the knives had resulted in the flexure of the pendulum staff.[98] By experiment, he also found an even greater flexure for the Repsold pendulum. In order to eliminate this source of error, Peirce designed a pendulum with knives that extended from each side of the cylindrical staff, and he received authorization from the superintendent of the Coast and Geodetic Survey to arrange for the construction of such pendulums by Gautier in Paris. Peirce, who had made his plans in consultation with Gautier, was called home before the pendulums were completed, and these new instruments remained undelivered.

In a memoir titled "Effect of the flexure of a pendulum upon its period of oscillation,"[99] Peirce determined analytically the effect on the period of a pendulum with a single elastic connection between two rigid parts of the staff. Thus, Peirce discovered experimentally the flexure of the staff and derived for a simplified case the effect on the period. It is not known if he ever found the integrated effect of the continuum of elastic connections in the pendulum. Lorenzoni, in 1896, offered a solution to the problem, and Almansi, in 1899, gave an extended analysis. After the independent discovery of the problem at the Geodetic Institute, Dr. Helmert took up the problem and criticized the theories of Peirce and Lorenzoni. He then presented his own theory of flexure in a comprehensive memoir.[100] In view of the previous neglect of the flexure of the pendulum staff in the reduction of observations, Helmert directed that the Geodetic Institute make a new absolute determination of the intensity of gravity at Potsdam. For this purpose, Kuehnen and Furtwaengler used the following reversible pendulums which had been constructed by the firm of A. Repsold and Sons in Hamburg:

1. The seconds pendulum of the Geodetic Institute procured in
1869.

2. A seconds pendulum from the Astronomical Observatory, Padua.

3. A heavy, seconds pendulum from the Imperial and Royal
Military-Geographical Institute, Vienna.

4. A light, seconds pendulum from the Imperial and Royal
Military-Geographical Institute.

5. A 1/2-second, reversible pendulum of the Geodetic Institute
procured in 1892.

Work was begun in 1898, and in 1906 Kuehnen and Furtwaengler published their monumental memoir, "Bestimmung der Absoluten Groesze der Schwerkraft zu Potsdam mit Reversionspendeln."

The acceleration of gravity in the pendulum room of the Geodetic Institute was determined to be 981.274 +- 0.003 cm/sec^{2}. In view of the exceptionally careful and thorough determination at the Institute, Potsdam was accepted as the world base for the absolute value of the intensity of gravity. The absolute value of gravity at some other station on the Potsdam system was determined from the times of swing of an invariable pendulum at the station and at Potsdam by the relation (T_{1})^{2}/(T_{2})^{2} = g_{2}/g_{1}. Thus, in 1900, Assistant G. R. Putnam of the Coast and Geodetic Survey swung Mendenhall pendulums at the Washington base and at Potsdam, and by transfer from Potsdam determined the intensity of gravity at the Washington base to be 980.112 cm/sec^{2}.[101] In 1933, Lt. E. J. Brown made comparative measurements with improved apparatus and raised the value at the Washington base to 980.118 cm/sec^{2}.[102]

In view of discrepancies between the results of various relative determinations, the Coast and Geodetic Survey in 1928 requested the National Bureau of Standards to make an absolute determination for Washington. Heyl and Cook used reversible pendulums made of fused silica having a period of approximately 1 second. Their result, published in 1936, was interpreted to indicate that the value at Potsdam was too high by 20 parts in 1 million.[103] This estimate was lowered slightly by Sir Harold Jeffreys of Cambridge, England, who recomputed the results of Heyl and Cook by different methods.[104]

In 1939, J. S. Clark published the results of a determination of gravity with pendulums of a non-ferrous Y-alloy[105] at the National Physical Laboratory at Teddington, England, and, after recomputation of results by Jeffreys, the value was found to be 12.8 parts in 1 million less than the value obtained by transfer from Potsdam. Dr. Hugh L. Dryden of the National Bureau of Standards, and Dr. A. Berroth of the Geodetic Institute at Potsdam, have recomputed the Potsdam data by different methods of adjustment and concluded that the Potsdam value was too high by about 12 parts in a million.[106] Determination of gravity at Leningrad by Russian scientists likewise has indicated that the 1906 Potsdam value is too high. In the light of present information, it appears justifiable to reduce the Potsdam value of 981.274 by .013 cm/sec^{2} for purposes of comparison. If the Brown transfer from Potsdam in 1933 was taken as accurate, the value for the Washington base would be 980.105 cm/sec^{2}. In this connection, it is of interest to note that the value given by Charles S. Peirce for the comparable Smithsonian base in Washington, as determined by him from comparative methods in the 1880's and reported in the _Annual Report of the Superintendent of the Coast and Geodetic Survey for the year 1890-1891_, was 980.1017 cm/sec^{2}.[107] This value would appear to indicate that Peirce's pendulums, observations, and methods of reduction of data were not inferior to those of the scientists of the Royal Prussian Geodetic Institute at Potsdam.

Doubts concerning the accuracy of the Potsdam value of gravity have stimulated many new determinations of the intensity of gravity since the end of World War II. In a paper published in June 1957, A. H. Cook, Metrology Division, National Physical Laboratory, Teddington, England, stated:

At present about a dozen new absolute determinations are in
progress or are being planned. Heyl and Cook's reversible
pendulum apparatus is in use in Buenos Aires and further
reversible pendulum experiments have been made in the All Union
Scientific Research Institute of Metrology, Leningrad (V N I I M)
and are planned at Potsdam. A method using a very long pendulum
was tried out in Russia about 1910 and again more recently and
there are plans for similar work in Finland. The first
experiment with a freely falling body was that carried out by
Volet who photographed a graduated scale falling in an enclosure
at low air pressure. Similar experiments have been completed in
Leningrad and are in progress at the Physikalisch-Technische
Bundesanstalt (Brunswick) and at the National Research Council
(Ottawa), and analogous experiments are being prepared at the
National Physical Laboratory and at the National Bureau of
Standards. Finally, Professor Medi, Director of the Istituto
Nazionale di Geofisica (Rome), is attempting to measure the
focal length of the paraboloidal surface of a liquid in a
rotating dish.[108]

Application of Gravity Surveys

We have noted previously that in the ancient and early modern periods, the earth was presupposed to be spherical in form. Determination of the figure of the earth consisted in the measurement of the radius by the astronomical-geodetic method invented by Eratosthenes. Since the earth was assumed to be spherical, gravity was inferred to be constant over the surface of the earth. This conclusion appeared to be confirmed by the determination of the length of the seconds pendulum at various stations in Europe by Picard and others. The observations of Richer in South America, the theoretical discussions of Newton and Huygens, and the measurements of degrees of latitude in Peru and Sweden demonstrated that the earth is an oblate spheroid.

The theory of gravitation and the theory of central forces led to the result that the intensity of gravity is variable over the surface of the earth. Accordingly, determinations of the intensity of gravity became of value to the geodesist as a means of determining the figure of the earth. Newton, on the basis of the meager data available to him, calculated the ellipticity of the earth to be 1/230 (the ellipticity is defined by (a-b)/a, where a is the equatorial radius and b the polar radius). Observations of the intensity of gravity were made on the historic missions to Peru and Sweden. Bouguer and La Condamine found that at the equator at sea level the seconds pendulum was 1.26 Paris-lines shorter than at Paris. Maupertuis found that in northern Sweden a certain pendulum clock gained 59.1 seconds per day on its rate in Paris. Then Clairaut, from the assumption that the earth is a spheroid of equilibrium, derived a theorem from which the ellipticity of the earth can be derived from values of the intensity of gravity.

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. As has been noted previously, a Kater invariable pendulum was used by Adm. Luetke of Russia on a trip around the world. The French also sent out expeditions to determine values of gravity. After several decades of relative inactivity, Capts. Basevi and Heaviside of the Indian Survey carried out an important series of observations from 1865 to 1873 with Kater invariable pendulums and the Russian Repsold-Bessel pendulums. In 1881-1882 Maj. J. Herschel swung Kater invariable pendulums nos. 4, 6 (1821), and 11 at stations in England and then brought them to the United States in order to make observations which would connect American and English base stations.[109]

The extensive sets of observations of gravity provided the basis of calculations of the ellipticity of the earth. Col. A. R. Clarke in his _Geodesy_ (London, 1880) calculated the ellipticity from the results of gravity surveys to be 1/(292.2 +- 1.5). Of interest is the calculation by Charles S. Peirce, who used only determinations made with Kater invariable pendulums and corrected for elevation, atmospheric effect, and expansion of the pendulum through temperature.[110] He calculated the ellipticity of the earth to be 1/(291.5 +- 0.9).

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 general assembly of the International Union of Geodesy and Geophysics, which was founded after World War I to continue the work of _Die Internationale Erdmessung_, adopted in 1924 an international reference ellipsoid,[111] of which the ellipticity, or flattening, is Hayford's value 1/297. In 1930, the general assembly adopted a correlated International Gravity Formula of the form

[gamma] = [gamma]_{E}(1 + [beta]sin^{2} [phi] + [epsilon]sin^{2} 2[phi])

where [gamma] is normal gravity at latitude [phi], [gamma]_{E} is the value of gravity at sea level at the equator, [beta] is a parameter which is computed on the basis of Clairaut's theorem from the flattening value of the meridian, and [epsilon] is a constant which is derived theoretically. The plumb line is perpendicular to the geoid, and the components of angle between the perpendiculars to geoid and reference ellipsoid are deflections of the vertical. The geoid is above the ellipsoid of reference under mountains and it is below the ellipsoid on the oceans, where the geoid coincides with mean sea level. In physical geodesy, gravimetric data are used for the determination of the geoid and components of deflections of the vertical. For this purpose, one must reduce observed values of gravity to sea level by various reductions, such as free-air, Bouguer, isostatic reductions. If g_{0} is observed gravity reduced to sea level and [gamma] is normal gravity obtained from the International Gravity Formula, then

[Delta]g = g_{0} - [gamma]

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. Vening Meinesz further derived formulae for the calculation of components of the deflection of the vertical.

Geometrical geodesy, which was based on astronomical-geodetic methods, could give information only concerning the external form of the figure of the earth. The gravimetric methods of physical geodesy, in conjunction with methods such as those of seismology, enable scientists to test hypotheses concerning the internal structure of the earth. Heiskanen and Vening Meinesz summarize the present-day achievements of the gravimetric method of physical geodesy by stating[113] that it alone can give:

1. The flattening of the reference ellipsoid.

2. The undulations N of the geoid.

3. The components of the deflection of the vertical [xi] and
[eta] at any point, oceans and islands included.

4. The conversion of existing geodetic systems to the same world
geodetic system.

5. The reduction of triangulation base lines from the geoid to
the reference ellipsoid.

6. The correction of errors in triangulation in mountainous
regions due to the effect of the deflections of the vertical.

7. Geophysical applications of gravity measurements, e.g., the
isostatic study of the earth's interior and the exploration of
oil fields and ore deposits.

With astronomical observations or with existing triangulations, the gravimetric method can accomplish further results. Heiskanen and Vening Meinesz state:

It is the firm conviction of the authors that the gravimetric
method is by far the best of the existing methods for solving
the main problems of geodesy, i.e., to determine the shape of
the geoid on the continents as well as at sea and to convert the
existing geodetic systems to the world geodetic system. It can
also give invaluable help in the computation of the reference
ellipsoid.[114]

Summary

Since the creation of classical mechanics in the 17th century, the pendulum has been a basic instrument for the determination of the intensity of gravity, which is expressed as the acceleration of a freely falling body. Basis of theory is the simple pendulum, whose time of swing under gravity is proportional to the square root of the length divided by the acceleration due to gravity. Since the length of a simple pendulum divided by the square of its time of swing is equal to the length of a pendulum that beats seconds, the intensity of gravity also has been expressed in terms of the length of the seconds pendulum. The reversible compound pendulum has served for the absolute determination of gravity by means of a theory developed by Huygens. Invariable compound pendulums with single axes also have been used to determine relative values of gravity by comparative times of swing.

The history of gravity pendulums begins with the ball or "simple" pendulum of Galileo as an approximation to the ideal simple pendulum. Determinations of the length of the seconds pendulum by French scientists culminated in a historic determination at Paris by Borda and Cassini, from the corrected observations with a long ball pendulum. In the 19th century, Bessel found the length of the seconds pendulum at Koenigsberg and Berlin by observations with a ball pendulum and by original theoretical considerations. During the century, however, the compound pendulum came to be preferred for absolute and relative determinations.

Capt. Henry Kater, at London, constructed the first convertible compound for an absolute determination of gravity, and then he designed an invariable compound pendulum, examples of which were used for relative determinations at various stations in Europe and elsewhere. Bessel demonstrated theoretically the advantages of a reversible compound pendulum which is symmetrical in form and is hung by interchangeable knives. The firm of A. Repsold and Sons in Hamburg constructed pendulums from the specifications of Bessel for European gravity surveys.

Charles S. Peirce in 1875 received delivery in Hamburg of a Repsold-Bessel pendulum for the U.S. Coast Survey and observed with it in Geneva, Paris, Berlin, and London. Upon an initial stimulation from Baeyer, founder of _Die Europaeische Gradmessung_, Peirce demonstrated by experiment and theory that results previously obtained with the Repsold apparatus required correction, because of the flexure of the stand under oscillations of the pendulum. At the Stuttgart conference of the geodetic association in 1877, Herve Faye proposed to solve the problem of flexure by swinging two similar pendulums from the same support with equal amplitudes and in opposite phases. Peirce, in 1879, demonstrated theoretically the soundness of the method and presented a design for its application, but the "double pendulum" was rejected at that time. Peirce also designed and had constructed four examples of a new type of invariable, reversible pendulum of cylindrical form which made possible the experimental study of Stokes' theory of the resistance to motion of a pendulum in a viscous fluid. 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. Stands were then constructed in Europe on which two or four pendulums were hung at the same time. Finally, early in the present century, Vening Meinesz found that the Faye-Peirce method of swinging pendulums hung on a Stueckrath four-pendulum stand solved the problem of instability due to the mobility of the soil in Holland.

The 20th century has witnessed increasing activity in the determination of absolute and relative values of gravity. Gravimeters have been perfected and have been widely used for rapid relative determinations, but the compound pendulums remain as indispensable instruments. Mendenhall's replacement of knives by planes attached to nonreversible pendulums has been used also for reversible ones. The Geodetic Institute at Potsdam is presently applying the Faye-Peirce method to the reversible pendulum.[115] Pendulums have been constructed of new materials, such as invar, fused silica, and fused quartz. Minimum pendulums for precise relative determinations have been constructed and used. Reversible pendulums have been made with "I" cross sections for better stiffness. With all these modifications, however, the foundations of the present designs of compound pendulum apparatus were created in the 19th century.

FOOTNOTES:

[1] The basic historical documents have been collected, with a bibliography of works and memoirs published from 1629 to the end of 1885, in _Collection de memoires relatifs a la physique, publies par la Societe francaise de Physique_ [hereinafter referred to as _Collection de memoires_]: vol. 4, _Memoires sur le pendule, precedes d'une bibliographie_ (Paris: Gauthier-Villars, 1889); and vol. 5, _Memoires sur le pendule_, part 2 (Paris: Gauthier-Villars, 1891). Important secondary sources are: C. WOLF, "Introduction historique," pp. 1-42 in vol. 4, above; and GEORGE BIDDELL AIRY, "Figure of the Earth," pp. 165-240 in vol. 5 of _Encyclopaedia metropolitana_ (London, 1845).

[2] Galileo Galilei's principal statements concerning the pendulum occur in his _Discourses Concerning Two New Sciences_, transl. from Italian and Latin into English by Henry Crew and Alfonso de Salvio (Evanston: Northwestern University Press, 1939), pp. 95-97, 170-172.

[3] P. MARIN MERSENNE, _Cogitata physico-mathematica_ (Paris, 1644), p. 44.

[4] CHRISTIAAN HUYGENS, _Horologium oscillatorium, sive de motu pendulorum ad horologia adaptato demonstrationes geometricae_ (Paris, 1673), proposition 20.

[5] The historical events reported in the present section are from AIRY, "Figure of the Earth."

[6] ABBE JEAN PICARD, _La Mesure de la terre_ (Paris, 1671). JOHN W. OLMSTED, "The 'Application' of Telescopes to Astronomical Instruments, 1667-1669," _Isis_ (1949), vol. 40, p. 213.

[7] The toise as a unit of length was 6 Paris feet or about 1,949 millimeters.

[8] JEAN RICHER, _Observations astronomiques et physiques faites en l'isle de Caienne_ (Paris, 1679). JOHN W. OLMSTED, "The Expedition of Jean Richer to Cayenne 1672-1673," _Isis_ (1942), vol. 34, pp. 117-128.

[9] The Paris foot was 1.066 English feet, and there were 12 lines to the inch.

[10] CHRISTIAAN HUYGENS, "De la cause de la pesanteur," _Divers ouvrages de mathematiques[mathematiques] et de physique par MM. de l'Academie Royale[Royal] des Sciences_ (Paris, 1693), p. 305.

[11] ISAAC NEWTON, _Philosophiae naturalis principia mathematica_ (London, 1687), vol. 3, propositions 18-20.

[12] PIERRE BOUGUER, _La figure de la terre, determinee par les observations de Messieurs Bouguer et de La Condamine, envoyes par ordre du Roy au Perou, pour observer aux environs de l'equateur_ (Paris, 1749).

[13] P. L. MOREAU DE MAUPERTUIS, _La figure de la terre determinee par les observations de Messieurs de Maupertuis, Clairaut, Camus, Le Monnier, l'Abbe Outhier et Celsius, faites par ordre du Roy au cercle polaire_ (Paris, 1738).

[14] Paris, 1743.

[15] GEORGE GABRIEL STOKES, "On Attraction and on Clairaut's Theorem," _Cambridge and Dublin Mathematical Journal_ (1849), vol. 4, p. 194.

[16] See _Collection de memoires_, vol. 4, p. B-34, and J. H. POYNTING and SIR J. J. THOMSON, _Properties of Matter_ (London, 1927), p. 24.

[17] POYNTING and THOMSON, ibid., p. 22.

[18] CHARLES M. DE LA CONDAMINE, "De la mesure du pendule a Saint Domingue," _Collection de memoires_, vol. 4, pp. 3-16.

[19] PERE R. J. BOSCOVICH, _Opera pertinentia ad Opticam et Astronomiam_ (Bassani, 1785), vol. 5, no. 3.

[20] J. C. BORDA and J. D. CASSINI DE THURY, "Experiences pour connaitre la longueur du pendule qui bat les secondes a Paris," _Collection de memoires_, vol. 4, pp. 17-64.

[21] F. W. BESSEL, "Untersuchungen ueber die Laenge des einfachen Secundenpendels," _Abhandlungen der Koeniglichen Akademie der Wissenschaften zu Berlin, 1826_ (Berlin, 1828).

[22] Bessel used as a standard of length a toise which had been made by Fortin in Paris and had been compared with the original of the "toise de Peru" by Arago.

[23] L. G. DU BUAT, _Principes d'hydraulique_ (Paris, 1786). See excerpts in _Collection de memoires_, pp. B-64 to B-67.

[24] CAPT. HENRY KATER, "An Account of Experiments for Determining the Length of the Pendulum Vibrating Seconds in the Latitude of London," _Philosophical Transactions of the Royal Society of London_ (1818), vol. 108, p. 33. [Hereinafter abbreviated _Phil. Trans._]

[25] M. G. DE PRONY, "Methode pour determiner la longueur du pendule simple qui bat les secondes," _Collection de memoires_, vol. 4, pp. 65-76.

[26] _Collection de memoires_, vol. 4, p. B-74.

[27] _Phil. Trans._ (1819), vol. 109, p. 337.

[28] JOHN HERSCHEL, "Notes for a History of the Use of Invariable Pendulums," _The Great Trigonometrical Survey of India_ (Calcutta, 1879), vol. 5.

[29] CAPT. EDWARD SABINE, "An Account of Experiments to Determine the Figure of the Earth," _Phil. Trans._ (1828), vol. 118, p. 76.

[30] JOHN GOLDINGHAM, "Observations for Ascertaining the Length of the Pendulum at Madras in the East Indies," _Phil. Trans._ (1822), vol. 112, p. 127.

[31] BASIL HALL, "Letter to Captain Kater Communicating the Details of Experiments made by him and Mr. Henry Foster with an Invariable Pendulum," _Phil. Trans._ (1823), vol. 113, p. 211.

[32] See _Collection de memoires_, vol. 4, p. B-103.

[33] Ibid., p. B-88.

[34] Ibid., p. B-94.

[35] FRANCIS BAILY, "On the Correction of a Pendulum for the Reduction to a Vacuum, Together with Remarks on Some Anomalies Observed in Pendulum Experiments," _Phil. Trans._ (1832), vol. 122, pp. 399-492. See also _Collection de memoires_, vol. 4, pp. B-105, B-112, B-115, B-116, and B-117.

[36] One was of case brass and the other of rolled iron, 68 in. long, 2 in. wide, and 1/2 in. thick. Triangular knife edges 2 in. long were inserted through triangular apertures 19.7 in. from the center towards each end. These pendulums seem not to have survived. There is, however, in the collection of the U.S. National Museum, a similar brass pendulum, 37-5/8 in. long (fig. 15) stamped with the name of Edward Kuebel (1820-96), who maintained an instrument business in Washington, D.C., from about 1849. The history of this instrument is unknown.

[37] See Baily's remarks in the _Monthly Notices of the Royal Astronomical Society_ (1839), vol. 4, pp. 141-143. See also letters mentioned in footnote 38.

[38] This document, together with certain manuscript notes on the pendulum experiments and six letters between Wilkes and Baily, is in the U.S. National Archives, Navy Records Gp. 37. These were the source materials for the information presented here on the Expedition. We are indebted to Miss Doris Ann Esch and Mr. Joseph Rudmann of the staff of the U.S. National Museum for calling our attention to this early American pendulum work.

[39] G. B. AIRY, "Account of Experiments Undertaken in the Harton Colliery, for the Purpose of Determining the Mean Density of the Earth," _Phil. Trans._ (1856), vol. 146, p. 297.

[40] T. C. MENDENHALL, "Measurements of the Force of Gravity at Tokyo, and on the Summit of Fujiyama," _Memoirs of the Science Department, University of Tokyo_ (1881), no. 5.

[41] J. T. WALKER, _Account of Operations of The Great Trigonometrical Survey of India_ (Calcutta, 1879), vol. 5, app. no. 2.

[42] BESSEL, op. cit. (footnote 21), article 31.

[43] C. A. F. PETERS, _Briefwechsel zwischen C. F. Gauss und H. C. Schumacher_ (Altona, Germany, 1860), _Band_ 2, p. 3. The correction required if the times of swing are not exactly the same is said to have been given also by Bohnenberger.

[44] F. W. BESSEL, "Construction eines symmetrisch geformten Pendels mit reciproken Axen, von Bessel," _Astronomische Nachrichten_ (1849), vol. 30, p. 1.

[45] E. PLANTAMOUR, "Experiences faites a Geneve avec le pendule a reversion," _Memoires de la Societe de Physique et d'histoire naturelle de Geneve, 1865_ (Geneva, 1866), vol. 18, p. 309.

[46] Ibid., pp. 309-416.

[47] C. CELLERIER, "Note sur la Mesure de la Pesanteur par le Pendule," _Memoires de la Societe de Physique et d'histoire naturelle de Geneve, 1865_ (Geneva, 1866), vol. 18, pp. 197-218.