pendulum. Of a number of extraneous effects which tend to disturb the
accuracy of pendulum observations the most important is air resistance.
Experiments reported by the Greenwich (England) observatory in 1829 led
to the development of a vacuum chamber within which the pendulum was
swung.]
The unusual feature of these pendulums was in their symmetry of mass as
well as of form. They were made of bars, of iron in one case, and of
brass in the other, and each had two knife edges at opposite ends
equidistant from the center. Thus, although they resembled reversible
pendulums, their symmetry of mass prevented their use as such, and they
were rather equivalent to four separate invariable pendulums.[36]
Wilkes was taught the use of the pendulum by Baily, and conducted
experiments at Baily's house, where the latter had carried out the work
reported on in 1832. The subsequent experiments made on the U.S.
Exploring Expedition were under the charge of Wilkes, himself, who made
observations on 11 separate occasions, beginning with that in London
(1836) and followed by others in New York, Washington, D.C., Rio de
Janeiro, Sydney, Honolulu, "Pendulum Peak" (Mauna Loa), Mount Kanoha,
Nesqually (Oregon Territory), and, finally, two more times in
Washington, D.C. (1841 and 1845).
Wilkes' results were communicated to Baily, who appears to have found
the work defective because of insufficient attention to the maintenance
of temperature constancy and to certain alterations made to the
pendulums.[37] The results were also to have been included in the
publications of the Expedition, but were part of the unpublished 24th
volume. Fortunately they still exist, in what appears to be a printer's
proof.[38]
The Kater invariable pendulums were used to investigate the internal
constitution of the earth. Airy sought to determine the density of the
earth by observing the times of swing of pendulums at the top and bottom
of a mine. The first experiments were made in 1826 at the Dolcoath
copper mine in Cornwall, and failed when the pendulum fell to the
bottom. In 1854, the experiments were again undertaken in the Harton
coalpit, near Sunderland.[39] Gravity at the surface was greater than
below, because of the attraction of a shell equal to the depth of the
pit. From the density of the shell as determined from specimens of rock,
Airy found the density of the earth to be 6-1/2 times greater than that
of water. T. C. Mendenhall, in 1880, used a Kater convertible pendulum
in an invariable manner to compare values of gravity on Fujiyama and at
Tokyo, Japan.[40] He used a "simple" pendulum of the Borda type to
determine the absolute value of gravity at Tokyo. From the values of
gravity on the mountain and at Tokyo, and an estimate of the volume of
the mountain, he estimated the mean density of the earth as 5.77 times
greater than that of water.
In 1879, Maj. J. Herschel, R.E., stated:
The years from 1840 to 1865 are a complete blank, if we except
Airy's relative density experiments in 1854. This pause was
broken simultaneously in three different ways. Two pendulums of
the Kater pattern were sent to India; two after Bessel's design
were set to work in Russia; and at Geneva, Plantamour's zealous
experiments with a pendulum of the same kind mark the
commencement of an era of renewed activity on the European
continent.[41]
With the statement that Kater invariable pendulums nos. 4 and 6 (1821)
were used in India between 1865 and 1873, we now consider the other
events mentioned by Herschel.
[Illustration: Figure 15.--ONE OF FRANCIS BAILY'S PENDULUMS (62-1/2
inches long), shown on the left, is now in the possession of the Science
Museum, London, and, right, two views of a similar pendulum (37-5/8
inches long) made in the late 19th century by Edward Kuebel, Washington,
D.C., which is no. 316,876 in the collection of the U.S. National
Museum. Among a large number of pendulums tried by Baily in London
(1827-1840), was one which resembles the reversible pendulum
superficially, but which is actually an invariable pendulum having knife
edges at both ends. The purpose was apparently economy, since it is
equivalent to two separate invariable pendulums. This is the type of
pendulum used on the U.S. Exploring Expedition of 1838-1842. It is not
known what use was made of the Kuebel pendulum.]
Repsold-Bessel Reversible Pendulum
As we have noted, Bessel made determinations of gravity with a ball
("simple") pendulum in the period 1825-1827 and in 1835 at Koenigsberg
and Berlin, respectively. In the memoir on his observations at
Koenigsberg, he set forth the theory of the symmetrical compound pendulum
with interchangeable knife edges.[42] Bessel demonstrated theoretically
that if the pendulum were symmetrical with respect to its geometrical
center, if the times of swing about each axis were the same, the effects
of buoyancy and of air set in motion would be eliminated. Laplace had
already shown that the knife edge must be regarded as a cylinder and not
as a mere line of support. Bessel then showed that if the knife edges
were equal cylinders, their effects were eliminated by inverting the
pendulum; and if the knife edges were not equal cylinders, the
difference in their effects was canceled by interchanging the knives and
again determining the times of swing in the so-called erect and inverted
positions. Bessel further showed that it is unnecessary to make the
times of swing exactly equal for the two knife edges.
The simplified discussion for infinitely small oscillations in a vacuum
is as follows: If T_{1} and T_{2} are the times of swing about the knife
edges, and if h_{1} and h_{2} are distances of the knife edges from the
center of gravity, and if k is the radius of gyration about an axis
through the center of gravity, then from the equation of motion of a
rigid body oscillating about a fixed axis under gravity
(T_{1})^{2} = [pi]^{2}(k^{2} + (h_{1})^{2})/gh_{1},
(T_{2})^{2} = [pi]^{2}(k^{2} + (h_{2})^{2})/gh_{2}.
Then
(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.