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.]
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]
[Illustration: Figure 20.--SUPPORT FOR THE PEIRCE PENDULUM, 1889. Much
of the work of C. S. Peirce was concerned with the determination of the
error introduced into observations made with the portable apparatus by
the vibration of the stand with the pendulum. He showed that the popular
Bessel-Repsold apparatus was subject to such an error. His own pendulums
were swung from a simple but rugged wooden frame to which a hardened
steel bearing was fixed.]
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.
[Illustration: Figure 21.--REVERSIBLE PENDULUM APPARATUS of Defforges,
as constructed by Brunner, Paris, about 1887. The clock and telescope
used to observe coincidences are not shown. The telescope shown is part
of an interferometer used to measure flexure of the support. One mirror
of the interferometer is attached to the pendulum support; the other to
the separate masonry pillar at the left.]
[Illustration: Figure 22.--BECAUSE OF THE GREATER SIMPLICITY of its use,
the invariable pendulum superseded the convertible pendulum towards the
end of the 19th century, except at various national base stations (Kew,
Paris, Potsdam, Washington, D.C., etc.). Shown here are, right to left,
a pendulum of the type used by Peirce at the Hoosac Tunnel in 1873-74,
the Mendenhall 1/2-second pendulum of 1890, and the pendulum designed by
Peirce in 1881-1882.]
[Illustration: Figure 23.--THE OVERALL SIZE of portable pendulum
apparatus was greatly reduced with the introduction of this 1/2-second
apparatus in 1887, by the Austrian military officer, Robert von
Sterneck. Used with a vacuum chamber not shown here, the apparatus is
only about 2 feet high. Coincidences are observed by the reflection of a
periodic electric spark in two mirrors, one on the support and the other
on the pendulum itself.]
[Illustration: Figure 24.--THOMAS C. MENDENHALL (1841-1924). Although
largely self-educated, he became the first professor of physics and
mechanics at the Ohio Agricultural and Mechanical College (later Ohio
State University), and was subsequently connected with several other
universities. In 1878, while teaching at the Tokyo Imperial University
in Japan, he made gravity measurements between Tokyo and Fujiyama from
which he calculated the mean density of the earth. While superintendent
of the U.S. Coast and Geodetic Survey, 1889-94, he developed the
pendulum apparatus which bears his name.]
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.
[Illustration: Figure 25.--MENDENHALL'S 1/4-METER (1/2-SECOND)
APPARATUS. Shown on the left is the flash apparatus and, on the right,
the vacuum chamber within which the pendulum is swung. The flash
apparatus consists of a kerosene lantern and a telescope, mounted on a
box containing an electromagnetically operated shutter. The operation of
the shutter is controlled by a chronograph (not shown), so that it emits
a slit of light at regular intervals. The telescope is focused on two
mirrors within the apparatus, one fixed, the other attached to the top
of the pendulum. It is used to observe the reflection of the flashes
from these mirrors. When the two reflections are aligned, a
"coincidence" is marked on the chronograph tape. The second telescope
attached to the bottom of the vacuum chamber is for observing the
amplitude of the pendulum swing.]
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.
[Illustration: Figure 26.--VACUUM RECEIVER within which the Mendenhall
pendulum is swung. The pressure is reduced to about 50 mm. to reduce the
disturbing effect of air resistance. When the apparatus is sealed, the
pendulum is lifted on the knife edge by the lever _q_ and is started to
swing by the lever _r_. The arc of swing is only about 1 deg. The
stationary mirror is shown at _g_. The pendulum shown in outline in the
center, is only about 9.7 inches long.]
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.