(h_{1}(T_{1})^{2} - h_{2}(T_{2})^{2})/(h_{1} - h_{2})
= ([pi]^{2}/g)(h_{1} + h_{2})
= [tau]^{2}.
[tau] is then the time of swing of a simple pendulum of length h_{1} +
h_{2}. If the difference T_{1} - T_{2} is sufficiently small,
[tau] = (h_{1}T_{1} - h_{2}T_{2})/(h_{1} - h_{2}).
Prior to its publication by Bessel in 1828, the formula for the time of
swing of a simple pendulum of length h_{1} + h_{2} in terms of T_{1},
T_{2} had been given by C. F. Gauss in a letter to H. C. Schumacher
dated November 28, 1824.[43]
The symmetrical compound pendulum with interchangeable knives, for which
Bessel gave a posthumously published design and specifications,[44] has
been called a reversible pendulum; it may thereby be distinguished from
Kater's unsymmetrical convertible pendulum. In 1861, the Swiss Geodetic
Commission was formed, and in one of its first sessions in 1862 it was
decided to add determinations of gravity to the operations connected
with the measurement--at different points in Switzerland--of the arc of
the meridian traversing central Europe.[45] It was decided further to
employ a reversible pendulum of Bessel's design and to have it
constructed by the firm of A. Repsold and Sons, Hamburg. It was also
decided to make the first observations with the pendulum in Geneva;
accordingly, the Repsold-Bessel pendulum (fig. 16) was sent to Prof. E.
Plantamour, director of the observatory at Geneva, in the autumn of
1864.[46]
The Swiss reversible pendulum was about 560 mm. in length (distance
between the knife edges) and the time of swing was approximately 3/4
second. At the extremities of the stem of the pendulum were movable
cylindrical disks, one of which was solid and heavy, the other hollow
and light. It was intended by the mechanicians that equality of times of
oscillation about the knife edges would be achieved by adjusting the
position of a movable disk. The pendulum was hung by a knife edge on a
plate supported by a tripod and having an attachment from which a
measuring rod could be suspended so that the distance between the knife
edges could be measured by a comparator. Plantamour found it
impracticable to adjust a disk until the times of swing about each knife
edge were equal. His colleague, Charles Cellerier,[47] then showed that
if (T_{1} - T_{2})/T_{1} is sufficiently small so that one can neglect
its square, one can determine the length of the seconds pendulum from
the times of swing about the knife edges by a theory which uses the
distances of the center of gravity from the respective knife edges.
Thus, a role for the position of the center of gravity in the theory of
the reversible pendulum, which had been set forth earlier by Bessel, was
discovered independently by Cellerier for the Swiss observers of
pendulums.
In 1866, Plantamour published an extensive memoir "Experiences faites a
Geneve avec le pendule a reversion." Another memoir, published in 1872,
presented further results of determinations of gravity in Switzerland.
Plantamour was the first scientist in western Europe to use a
Repsold-Bessel reversible pendulum and to work out methods for its
employment.
The Russian Imperial Academy of Sciences acquired two Repsold-Bessel
pendulums, and observations with them were begun in 1864 by Prof.
Sawitsch, University of St. Petersburg, and others.[48] In 1869, the
Russian pendulums were loaned to the India Survey in order to enable
members of the Survey to supplement observations with the Kater
invariable pendulums nos. 4 and 6 (1821). During the transport of the
Russian apparatus to India, the knives became rusted and the apparatus
had to be reconditioned. Capt. Heaviside of the India Survey observed
with both pendulums at Kew Observatory, near London, in the spring of
1874, after which the Russian pendulums were sent to Pulkowa (Russia)
and were used for observations there and in the Caucasus.
The introduction of the Repsold-Bessel reversible pendulum for the
determination of gravity was accompanied by the creation of the first
international scientific association, one for geodesy. In 1861, Lt. Gen.
J. J. Baeyer, director of the Prussian Geodetic Survey, sent a
memorandum to the Prussian minister of war in which he proposed that the
independent geodetic surveys of the states of central Europe be
coordinated by the creation of an international organization.[49] In
1862, invitations were sent to the various German states and to other
states of central Europe. The first General Conference of the
association, initially called _Die Mittel-Europaeische Gradmessung_, also
_L'Association Geodesique Internationale_, was held from the 15th to
the 22d of October 1864 in Berlin.[50] The Conference decided upon
questions of organization: a general conference was to be held
ordinarily every three years; a permanent commission initially
consisting of seven members was to be the scientific organ of the
association and to meet annually; a central bureau was to be established
for the reception, publication, and distribution of reports from the
member states.
[Illustration: Figure 16.--FROM A DESIGN LEFT BY BESSEL, this portable
apparatus was developed in 1862 by the firm of Repsold in Hamburg, whose
founder had assisted Bessel in the construction of his pendulum
apparatus of 1826. The pendulum is convertible, but differs from Kater's
in being geometrically symmetrical and, for this reason, Repsold's is
usually called "reversible." Just to the right of the pendulum is a
standard scale. To the left is a "vertical comparator" designed by
Repsold to measure the distance between the knife edges of the pendulum.
To make this measurement, two micrometer microscopes which project
horizontally through the comparator are alternately focused on the knife
edges and on the standard scale.]
Under the topic "Astronomical Questions," the General Conference of 1864
resolved that there should be determinations of the intensity of gravity
at the greatest possible number of points of the geodetic network, and
recommended the reversible pendulum as the instrument of
observation.[51] At the second General Conference, in Berlin in 1867, on
the basis of favorable reports by Dr. Hirsch, director of the
observatory at Neuchatel, of Swiss practice with the Repsold-Bessel
reversible pendulum, this instrument was specifically recommended for
determinations of gravity.[52] The title of the association was changed
to _Die Europaeische Gradmessung_; in 1886, it became _Die Internationale
Erdmessung_, under which title it continued until World War I.
On April 1, 1866, the Central Bureau of _Die Europaeische Gradmessung_
was opened in Berlin under the presidency of Baeyer, and in 1868 there
was founded at Berlin, also under his presidency, the Royal Prussian
Geodetic Institute, which obtained regular budgetary status on January
1, 1870. A reversible pendulum for the Institute was ordered from A.
Repsold and Sons, and it was delivered in the spring of 1869. The
Prussian instrument was symmetrical geometrically, as specified by
Bessel, but different in form from the Swiss and Russian pendulums. The
distance between the knife edges was 1 meter, and the time of swing
approximately 1 second. The Prussian Repsold-Bessel pendulum was swung
at Leipzig and other stations in central Europe during the years
1869-1870 by Dr. Albrecht under the direction of Dr. Bruhns, director of
the observatory at Leipzig and chief of the astronomical section of the
Geodetic Institute. The results of these first observations appeared in
a publication of the Royal Prussian Geodetic Institute in 1871.[53]
Results of observations with the Russian Repsold-Bessel pendulums were
published by the Imperial Academy of Sciences. In 1872, Prof. Sawitsch
reported the work for western Europeans in "Les variations de la
pesanteur dans les provinces occidentales de l'Empire russe."[48] In
November 1873, the Austrian Geodetic Commission received a
Repsold-Bessel reversible pendulum and on September 24, 1874, Prof.
Theodor von Oppolzer reported on observations at Vienna and other
stations to the Fourth General Conference of _Die Europaeische
Gradmessung_ in Dresden.[54] At the fourth session of the Conference, on
September 28, 1874, a Special Commission, consisting of Baeyer, as
chairman, and Bruhns, Hirsch, von Oppolzer, Peters, and Albrecht, was
appointed to consider (under Topic 3 of the program): "Observations for
the determination of the intensity of gravity," the question, "Which
Pendulum-apparatuses are preferable for the determination of many
points?"
After the adoption of the Repsold-Bessel reversible pendulum for gravity
determinations in Europe, work in the field was begun by the U.S. Coast
Survey under the superintendency of Prof. Benjamin Peirce. There is
mention in reports of observations with pendulums prior to Peirce's
direction to his son Charles on November 30, 1872, "to take charge of
the Pendulum Experiments of the Coast Survey and to direct and inspect
all parties engaged in such experiments and as often as circumstances
will permit, to take the field with a party...."[55] Systematic and
important gravity work by the Survey was begun by Charles Sanders
Peirce. Upon receiving notice of his appointment, the latter promptly
ordered from the Repsolds a pendulum similar to the Prussian instrument.
Since the firm of mechanicians was engaged in making instruments for
observations of the transit of Venus in 1874, the pendulum for the
Coast Survey could not be constructed immediately. Meanwhile, during the
years 1873-1874, Charles Peirce conducted a party which made
observations of gravity in the Hoosac Tunnel near North Adams, and at
Northampton and Cambridge, Massachusetts. The pendulums used were
nonreversible, invariable pendulums with conical bobs. Among them was a
silver pendulum, but similar pendulums of brass were used also.[56]
[Illustration: Figure 17.--REPSOLD-BESSEL REVERSIBLE PENDULUM apparatus
as made in 1875, and used in the gravity work of the U.S. Coast and
Geodetic Survey. Continental geodesists continued to favor the general
use of convertible pendulums and absolute determinations of gravity,
while their English colleagues had turned to invariable pendulums and
relative determinations, except for base stations. Perhaps the first
important American contribution to gravity work was C. S. Peirce's
demonstration of the error inherent in the Repsold apparatus through
flexure of the stand.]
[Illustration: Figure 18.--CHARLES SANDERS PEIRCE (1839-1914), son of
Benjamin Peirce, Perkins Professor of Astronomy and Mathematics at
Harvard College. C. S. Peirce graduated from Harvard in 1859. From 1873
to 1891, as an assistant at the U.S. Coast and Geodetic Survey, he
accomplished the important gravimetric work described in this article.
Peirce was also interested in many other fields, but above all in the
logic, philosophy, and history of science, in which he wrote
extensively. His greatest fame is in philosophy, where he is regarded as
the founder of pragmatism.]
In 1874, Charles Peirce expressed the desire to be sent to Europe for at
least a year, beginning about March 1, 1875, "to learn the use of the
new convertible pendulum and to compare it with those of the European
measure of a Degree and the Swiss and to compare" his "invariable
pendulums in the manner which has been used by swinging them in London
and Paris."[57]
Charles S. Peirce, assistant, U.S. Coast Survey, sailed for Europe on
April 3, 1875, on his mission to obtain the Repsold-Bessel reversible
pendulum ordered for the Survey and to learn the methods of using it for
the determination of gravity. In England, he conferred with Maxwell,
Stokes, and Airy concerning the theory and practice of research with
pendulums. In May, he continued on to Hamburg and obtained delivery from
the Repsolds of the pendulum for the Coast Survey (fig. 17). Peirce then
went to Berlin and conferred with Gen. Baeyer, who expressed doubts of
the stability of the Repsold stand for the pendulum. Peirce next went to
Geneva, where, under arrangements with Prof. Plantamour, he swung the
newly acquired pendulum at the observatory.[58]
In view of Baeyer's expressed doubts of the rigidity of the Repsold
stand, Peirce performed experiments to measure the flexure of the stand
caused by the oscillations of the pendulum. His method was to set up a
micrometer in front of the pendulum stand and, with a microscope, to
measure the displacement caused by a weight passing over a pulley, the
friction of which had been determined. Peirce calculated the correction
to be applied to the length of the seconds pendulum--on account of the
swaying of the stand during the swings of the pendulum--to amount to
over 0.2 mm. Although Peirce's measurements of flexure in Geneva were
not as precise as his later measurements, he believed that failure to
correct for flexure of the stand in determinations previously made with
Repsold pendulums was responsible for appreciable errors in reported
values of the length of the seconds pendulum.
The Permanent Commission of _Die Europaeische Gradmessung_ met in Paris,
September 20-29, 1875. In conjunction with this meeting, there was held
on September 21 a meeting of the Special Commission on the Pendulum. The
basis of the discussion by the Special Commission was provided by
reports which had been submitted in response to a circular sent out by
the Central Bureau to the members on February 26, 1874.[59]
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.
[Illustration: Figure 19.--THREE PENDULUMS USED IN EARLY WORK at the
U.S. Coast and Geodetic Survey. Shown on the left is the Peirce
invariable; center, the Peirce reversible; and, right, the Repsold
reversible. Peirce designed the cylindrical pendulum in 1881-1882 to
study the effect of air resistance according to the theory of G. G.
Stokes on the motion of a pendulum in a viscous field. Three examples of
the Peirce pendulums are in the U.S. National Museum.]