Part 6

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.

[48] A. SAWITSCH, "Les variations de la pesanteur dans les provinces occidentales de l'Empire russe," _Memoirs of the Royal Astronomical Society_ (1872), vol. 39, p. 19.

[49] J. J. BAEYER, _Ueber die Groesse und Figur der Erde_ (Berlin, 1861).

[50] _Comptes-rendus de la Conference Geodesique Internationale reunie a Berlin du 15-22 Octobre 1864_ (Neuchatel, 1865).

[51] Ibid., part III, subpart E.

[52] _Bericht ueber die Verhandlungen der vom 30 September bis 7 October 1867 zu Berlin abgehaltenen allgemeinen Conferenz der Europaeischen Gradmessung_ (Berlin, 1868). See report of fourth session, October 3, 1867.

[53] C. BRUHNS and ALBRECHT, "Bestimmung der Laenge des Secundenpendels in Bonn, Leiden und Mannheim," _Astronomisch-Geodaetische Arbeiten im Jahre 1870_ (Leipzig: Veroeffentlichungen des Koeniglichen Preussischen Geodaetischen Instituts, 1871).

[54] _Bericht ueber die Verhandlungen der vom 23 bis 28 September 1874 in Dresden abgehaltenen vierten allgemeinen Conferenz der Europaeischen Gradmessung_ (Berlin, 1875). See report of second session, September 24, 1874.

[55] CAROLYN EISELE, "Charles S. Peirce--Nineteenth-Century Man of Science," _Scripta Mathematica_ (1959), vol 24, p. 305. For the account of the work of Peirce, the authors are greatly indebted to this pioneer paper on Peirce's work on gravity. It is worth noting that the history of pendulum work in North America goes back to the celebrated Mason and Dixon, who made observations of "the going rate of a clock" at "the forks of the river Brandiwine in Pennsylvania," in 1766-67. These observations were published in _Phil. Trans._ (1768), vol. 58, pp. 329-335.

[56] The pendulums with conical bobs are described and illustrated in E. D. PRESTON, "Determinations of Gravity and the Magnetic Elements in Connection with the United States Scientific Expedition to the West Coast of Africa, 1889-90," _Report of the Superintendent of the Coast and Geodetic Survey for 1889-90_ (Washington, 1891), app. no. 12.

[57] EISELE, op. cit. (footnote 55), p. 311.

[58] The record of Peirce's observations in Europe during 1875-76 is given in C. S. PEIRCE, "Measurements of Gravity at Initial Stations in America and Europe," _Report of the Superintendent of the Coast Survey for 1875-76_ (Washington, 1879), pp. 202-337 and 410-416. Peirce's report is dated December 13, 1878, by which time the name of the Survey had been changed to U.S. Coast and Geodetic Survey.

[59] _Verhandlungen der vom 20 bis 29 September 1875 in Paris Vereinigten Permanenten Commission der Europaeischen Gradmessung_ (Berlin, 1876).

[60] Ibid. See report for fifth session, September 25, 1875.

[61] The experiments at the Stevens Institute, Hoboken, were reported by Peirce to the Permanent Commission which met in Hamburg, September 4-8, 1878, and his report was published in the general _Bericht_ for 1878 in the _Verhandlungen der vom 4 bis 8 September 1878 in Hamburg Vereinigten Permanenten Commission der Europaeischen Gradmessung_ (Berlin, 1879), pp. 116-120. Assistant J. E. Hilgard attended for the U.S. Coast and Geodetic Survey. The experiments are described in detail in C. S. PEIRCE, "On the Flexure of Pendulum Supports," _Report of the Superintendent of the U.S. Coast and Geodetic Survey for 1880-81_ (Washington, 1883), app. no. 14, pp. 359-441.

[62] _Verhandlungen der vom 5 bis 10 Oktober 1876 in Brussels Vereinigten Permanenten Commission der Europaeischen Gradmessung_ (Berlin, 1877). See report of third session, October 7, 1876.

[63] _Verhandlungen der vom 27 September bis 2 Oktober 1877 zu Stuttgart abgehaltenen fuenften allgemeinen Conferenz der Europaeischen Gradmessung_ (Berlin, 1878).

[64] _Verhandlung der vom 16 bis 20 September 1879 in Genf Vereinigten Permanenten Commission der Europaeischen Gradmessung_ (Berlin, 1880).

[65] _Assistants' Reports, U.S. Coast and Geodetic Survey, 1879-80._ Peirce's paper was published in the _American Journal of Science_ (1879), vol. 18, p. 112.

[66] _Comptes-rendus de l'Academie des Sciences_ (Paris, 1879), vol. 89, p. 462.

[67] _Verhandlungen der vom 13 bis 16 September 1880 zu Muenchen abgehaltenen sechsten allgemeinen Conferenz der Europaeischen Gradmessung_ (Berlin, 1881).

[68] Ibid., app. 2.

[69] Ibid., app. 2a.

[70] _Verhandlungen der vom 11 bis zum 15 September 1882 im Haag Vereinigten Permanenten Commission der Europaeischen Gradmessung_ (Berlin, 1883).

[71] _Verhandlungen der vom 15 bis 24 Oktober 1883 zu Rom abgehaltenen siebenten allgemeinen Conferenz der Europaeischen Gradmessung_ (Berlin, 1884). Gen. Cutts attended for the U.S. Coast and Geodetic Survey.

[72] Ibid., app. 6. See also, _Zeitschrift fuer Instrumentenkunde_ (1884), vol. 4, pp. 303 and 379.

[73] Op. cit. (footnote 67).

[74] _Report of the Superintendent of the U.S. Coast and Geodetic Survey for 1880-81_ (Washington, 1883), p. 26.

[75] _Report of the Superintendent of the U.S. Coast and Geodetic Survey for 1889-90_ (Washington, 1891), app. no. 12.

[76] _Report of the Superintendent of the U.S. Coast and Geodetic Survey for 1881-82_ (Washington, 1883).

[77] _Transactions of the Cambridge Philosophical Society_ (1856), vol. 9, part 2, p. 8. Also published in _Mathematical and Physical Papers_ (Cambridge, 1901), vol. 3, p. 1.

[78] Peirce's comparison of theory and experiment is discussed in a report on the Peirce memoir by WILLIAM FERREL, dated October 19, 1890, Martinsburg, West Virginia. _U.S. Coast and Geodetic Survey, Special Reports, 1887-1891_ (MS, National Archives, Washington).

[79] The stations at which observations were conducted with the Peirce pendulums are recorded in the reports of the Superintendent of the U.S. Coast and Geodetic Survey from 1881 to 1890.

[80] _Comptes-rendus de l'Academie des Sciences_ (Paris, 1880), vol. 90, p. 1401. HERVE FAYE's report, dated June 21, 1880, is in the same _Comptes-rendus_, p. 1463.

[81] COMMANDANT C. DEFFORGES, "Sur l'Intensite absolue de la pesanteur," _Journal de Physique_ (1888), vol. 17, pp. 239, 347, 455. See also, DEFFORGES, "Observations du pendule," _Memorial du Depot general de la Guerre_ (Paris, 1894), vol. 15. In the latter work, Defforges described a pendulum "reversible inversable," which he declared to be truly invariable and therefore appropriate for relative determinations. The knives remained fixed to the pendulums, and the effect of interchanging knives was obtained by interchanging weights within the pendulum tube.

[82] Papers by MAJ. VON STERNECK in _Mitteilungen des K. u. K. Militaer-geographischen Instituts, Wien_, 1882-87; see, in particular, vol. 7 (1887).

[83] T. C. MENDENHALL, "Determinations of Gravity with the New Half-Second Pendulum...," _Report of the Superintendent of the U.S. Coast and Geodetic Survey for 1890-91_ (Washington, 1892), part 2, pp. 503-564.

[84] W. H. BURGER, "The Measurement of the Flexure of Pendulum Supports with the Interferometer," _Report of the Superintendent of the U.S. Coast and Geodetic Survey for 1909-10_ (Washington, 1911), app. no. 6.

[85] E. J. BROWN, _A Determination of the Relative Values of Gravity at Potsdam and Washington_ (Special Publication No. 204, U.S. Coast and Geodetic Survey; Washington, 1936).

[86] M. HAID, "Neues Pendelstativ," _Zeitschrift fuer Instrumentenkunde_ (July 1896), vol. 16, p. 193.

[87] DR. R. SCHUMANN, "Ueber eine Methode, das Mitschwingen bei relativen Schweremessungen zu bestimmen," _Zeitschrift fuer Instrumentenkunde_ (January 1897), vol. 17, p. 7. The design for the stand is similar to that of Peirce's of 1879.

[88] DR. R. SCHUMANN, "Ueber die Verwendung zweier Pendel auf gemeinsamer Unterlage zur Bestimmung der Mitschwingung," _Zeitschrift fuer Mathematik und Physik_ (1899), vol. 44, p. 44.