[1] The basic historical documents have been collected, with a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.
[89] P. FURTWAENGLER, "Ueber die Schwingungen zweier Pendel mit
annaehernd gleicher Schwingungsdauer auf gemeinsamer Unterlage,"
_Sitzungsberichte der Koeniglicher Preussischen Akademie der
Wissenschaften zu Berlin_ (Berlin, 1902) pp. 245-253. Peirce
investigated the plan of swinging two pendulums on the same
stand (_Report of the Superintendent of the U.S. Coast and
Geodetic Survey for 1880-81_, Washington, 1883, p. 26; also in
CHARLES SANDERS PEIRCE, _Collected Papers_, 6.273). At a
conference on gravity held in Washington during May 1882, Peirce
again advanced the method of eliminating flexure by hanging two
pendulums on one support and oscillating them in antiphase
("Report of a conference on gravity determinations held in
Washington, D.C., in May, 1882," _Report of the Superintendent
of the U.S. Coast and Geodetic Survey for 1881-82_, Washington,
1883, app. no. 22, pp. 503-516).
[90] F. A. VENING MEINESZ, _Observations de pendule dans les Pays-Bas_
(Delft, 1923).
[91] A. BERROTH, "Schweremessungen mit zwei und vier gleichzeitig auf
demselben Stativ schwingenden Pendeln," _Zeitschrift fuer
Geophysik_, vol. 1 (1924-25), no. 3, p. 93.
[92] "Pendulum Apparatus for Gravity Determinations," _Engineering_
(1926), vol. 122, pp. 271-272.
[93] MALCOLM W. GAY, "Relative Gravity Measurements Using Precision
Pendulum Equipment," _Geophysics_ (1940), vol. 5, pp. 176-191.
[94] L. G. D. THOMPSON, "An Improved Bronze Pendulum Apparatus for
Relative Gravity Determinations," [published by] _Dominion
Observatory_ (Ottawa, 1959), vol. 21, no. 3, pp. 145-176.
[95] W. A. HEISKANEN and F. A. VENING MEINESZ, _The Earth and its
Gravity Field_ (McGraw: New York, 1958).
[96] F. KUEHNEN and P. FURTWAENGLER, _Bestimmung der Absoluten Groesze
der Schwerkraft zu Potsdam mit Reversionspendeln_ (Berlin:
Veroeffentlichungen des Koeniglichen Preussischen Geodaetischen
Instituts, 1906), new ser., no. 27.
[97] Reported by Dr. F. Kuehnen to the fifth session, October 9, 1895,
of the Eleventh General Conference, _Die Internationale
Erdmessung_, held in Berlin from September 25 to October 12,
1895. A footnote states that Assistant O. H. Tittmann, who
represented the United States, subsequently reported Peirce's
prior discovery of the influence of the flexure of the pendulum
itself upon the period (_Report of the Superintendent of the
U.S. Coast and Geodetic Survey for 1883-84_, Washington, 1885,
app. 16, pp. 483-485).
[98] _Assistants' Reports, U.S. Coast and Geodetic Survey, 1883-84_
(MS, National Archives, Washington).
[99] C. S. PEIRCE, "Effect of the Flexure of a Pendulum Upon its
Period of Oscillation," _Report of the Superintendent of the
U.S. Coast and Geodetic Survey for 1883-84_ (Washington, 1885),
app. no. 16.
[100] F. R. HELMERT, _Beitraege zur Theorie des Reversionspendels_
(Potsdam: Veroeffentlichungen des Koeniglichen Preussischen
Geodaetischen Instituts, 1898).
[101] J. A. DUERKSEN, _Pendulum Gravity Data in the United States_
(Special Publication No. 244, U.S. Coast and Geodetic Survey;
Washington, 1949).
[102] Ibid., p. 2. See also, E. J. BROWN, loc. cit. (footnote 85).
[103] PAUL R. HEYL and GUY S. COOK, "The Value of Gravity at
Washington," _Journal of Research, National Bureau of Standards_
(1936), vol. 17, p. 805.
[104] SIR HAROLD JEFFREYS, "The Absolute Value of Gravity," _Monthly
Notices of the Royal Astronomical Society, Geophysical
Supplement_ (London, 1949), vol. 5, p. 398.
[105] J. S. CLARK, "The Acceleration Due to Gravity," _Phil. Trans._
(1939), vol. 238, p. 65.
[106] HUGH L. DRYDEN, "A Reexamination of the Potsdam Absolute
Determination of Gravity," _Journal of Research, National Bureau
of Standards_ (1942), vol. 29, p. 303; and A. BERROTH, "Das
Fundamentalsystem der Schwere im Lichte neuer
Reversionspendelmessungen," _Bulletin Geodesique_ (1949), no.
12, pp. 183-204.
[107] T. C. MENDENHALL, op. cit. (footnote 83), p. 522.
[108] A. H. COOK, "Recent Developments in the Absolute Measurement of
Gravity," _Bulletin Geodesique_ (June 1, 1957), no. 44, pp.
34-59.
[109] See footnote 89.
[110] C. S. PEIRCE, "On the Deduction of the Ellipticity of the Earth,
from Pendulum Experiments," _Report of the Superintendent of the
U.S. Coast and Geodetic Survey for 1880-81_ (Washington, 1883),
app. no. 15, pp. 442-456.
[111] HEISKANEN and VENING MEINESZ, op. cit. (footnote 95), p. 74.
[112] Ibid., p. 76.
[113] Ibid., p. 309.
[114] Ibid., p. 310.
[115] K. REICHENEDER, "Method of the New Measurements at Potsdam by
Means of the Reversible Pendulum," _Bulletin Geodesique_ (March
1, 1959), no. 51, p.72.
* * * * *