The pendulum employed in observations of gravity prior to the 19th
century usually consisted of a small weight suspended by a filament
(figs. 4-6). The pioneer experimenters with "simple" pendulums changed
the length of the suspension until the pendulum beat seconds. Picard in
1669 determined the length of the seconds pendulum at Paris with a
"simple" pendulum which consisted of a copper ball an inch in diameter
suspended by a fiber of pite from jaws (pite was a preparation of the
leaf of a species of aloe and was not affected appreciably by moisture).
A celebrated set of experiments with a "simple" pendulum was conducted
by Bouguer[16] in 1737 in the Andes, as part of the expedition to
measure the Peruvian arc. The bob of the pendulum was a double
truncated cone, and the length was measured from the jaw suspension to
the center of oscillation of the thread and bob. Bouguer allowed for
change of length of his measuring rod with temperature and also for the
buoyancy of the air. He determined the time of swing by an elementary
form of the method of coincidences. The thread of the pendulum was swung
in front of a scale and Bouguer observed how long it took the pendulum
to lose a number of vibrations on the seconds clock. For this purpose,
he noted the time when the beat of the clock was heard and,
simultaneously, the thread moved past the center of the scale. A
historic aspect of Bouguer's method was that he employed an "invariable"
pendulum, that is, the length was maintained the same at the various
stations of observation, a procedure that has been described as having
been invented by Bouguer.
Since T = [pi][sqrt](l/g), it follows that (T_{1})^{2}/(T_{2})^{2} =
g_{2}/g_{1}. Thus, if the absolute value of gravity is known at one
station, the value at any other station can be determined from the ratio
of the squares of times of swing of an invariable pendulum at the two
stations. From the above equation, if T_{1} is the time of swing at a
station where the intensity of gravity is g, and T_{2} is the time at a
station where the intensity is g + [Delta]g, then [Delta]g/g =
(T_{1})^{2}/(T_{2})^{2} - 1.
Bouguer's investigations with his invariable pendulum yielded methods
for the determination of the internal structure of the earth. On the
Peruvian expedition, he determined the length of the seconds pendulum at
three stations, including one at Quito, at varying distances above sea
level. If values of gravity at stations of different elevation are to be
compared, they must be reduced to the same level, usually to sea level.
Since gravity decreases with height above sea level in accordance with
the law of gravitation, a free-air reduction must be applied to values
of gravity determined above the level of the sea. Bouguer originated the
additional reduction for the increase in gravity on a mountain or
plateau caused by the attraction of the matter in a plate. From the
relative values of gravity at elevated stations in Peru and at sea
level, Bouguer calculated that the mean density of the earth was 4.7
times greater than that of the _cordilleras_.[17] For greater accuracy
in the study of the internal structure of the earth, in the 19th century
the Bouguer plate reduction came to be supplemented by corrections for
irregularities of terrain and by different types of isostatic reduction.
La Condamine, who like Bouguer was a member of the Peruvian expedition,
conducted his own pendulum experiments (fig. 4). He experimented in 1735
at Santo Domingo en route to South America,[18] then at various stations
in South America, and again at Paris upon his return to France. His
pendulum consisted of a copper ball suspended by a thread of pite. For
experimentation the length initially was about 12 feet, and the time of
swing 2 seconds, but then the length was reduced to about 3 feet with
time of swing 1 second. Earlier, when it was believed that gravity was
constant over the earth, Picard and others had proposed that the length
of the seconds pendulum be chosen as the standard. La Condamine in 1747
revived the proposal in the form that the length of the seconds pendulum
at the equator be adopted as the standard of length. Subsequently, he
investigated the expansion of a toise of iron from the variation in the
period of his pendulum. In 1755, he observed the pendulum at Rome with
Boscovich. La Condamine's pendulum was used by other observers and
finally was lost at sea on an expedition around the world. The knowledge
of the pendulum acquired by the end of the 18th century was summarized
in 1785 in a memoir by Boscovich.[19]
[Illustration: Figure 5.--AN APPARATUS FOR THE PRACTICE MEASUREMENT of
the length of the pendulum devised on the basis of a series of
preliminary experiments by C. M. de la Condamine who, in the course of
the French geodetic expedition to Peru in 1735, devoted a 3-month
sojourn on the island of Santo Domingo to pendulum observations by
Mairan's Method. In this arrangement, shown here, a vertical rod of
ironwood is used both as the scale and as the support for the apparatus,
having at its top the brass pendulum support (A) and, below, a
horizontal mirror (O) which serves to align the apparatus vertically
through visual observation of the reflection of the pointer projecting
from A. The pendulum, about 37 inches long, consists of a thread of pite
(a humidity-resistant, natural fiber) and a copper ball of about 6
ounces. Its exact length is determined by adjusting the micrometer (S)
so that the ball nearly touches the mirror. It will be noted that the
clock pendulum would be obscured by the scale. La Condamine seems to
have determined the times of coincidence by visual observation of the
occasions on which "the pendulums swing parallel." (Portion of plate 1,
_Memoires publies par la Societe francaise de Physique_, vol. 4.)]
[Illustration: Figure 6.--THE RESULT of early pendulum experiments was
often expressed in terms of the length of a pendulum which would have a
period of one second and was called "the seconds pendulum." In 1792, J.
C. Borda and J. D. Cassini determined the length of the seconds pendulum
at Paris with this apparatus. The pendulum consists of a platinum ball
about 1-1/2 inches in diameter, suspended by a fine iron wire. The
length, about 12 feet, was such that its period would be nearly twice as
long as that of the pendulum of the clock (A). The interval between
coincidences was determined by observing, through the telescope at the
left, the times when the two pendulums emerge together from behind the
screen (M). The exact length of the pendulum was measured by a platinum
scale (not shown) equipped with a vernier and an auxiliary copper scale
for temperature correction.
When, at the end of the 18th century, the French revolutionary
government established the metric system of weights and measures, the
length of the seconds pendulum at Paris was considered, but not adopted,
as the unit of length. (Plate 2, _Memoires publies par la Societe
francaise de Physique_, vol. 4.)]
The practice with the "simple" pendulum on the part of Picard, Bouguer,
La Condamine and others in France culminated in the work of Borda and
Cassini in 1792 at the observatory in Paris[20] (fig. 6). The
experiments were undertaken to determine whether or not the length of
the seconds pendulum should be adopted as the standard of length by the
new government of France. The bob consisted of a platinum ball 16-1/6
Paris lines in diameter, and 9,911 grains (slightly more than 17 ounces)
in weight. The bob was held to a brass cup covering about one-fifth of
its surface by the interposition of a small quantity of grease. The cup
with ball was hung by a fine iron wire about 12 Paris feet long. The
upper end of the wire was attached to a cylinder which was part of a
wedge-shaped knife edge, on the upper surface of which was a stem on
which a small adjustable weight was held by a screw thread. The knife
edge rested on a steel plate. The weight on the knife-edge apparatus was
adjusted so that the apparatus would vibrate with the same period as the
pendulum. Thus, the mass of the suspending apparatus could be neglected
in the theory of motion of the pendulum about the knife edge.
[Illustration: Figure 7.--RESULTS OF EXPERIMENTS in the determination of
the length of the seconds pendulum at Koenigsberg by a new method were
reported by F. W. Bessel in 1826 and published in 1828. With this
apparatus, he obtained two sets of data from the same pendulum, by using
two different points of suspension. The pendulum was about 10 feet long.
The distance between the two points of suspension (_a_ and _b_) was 1
toise (about six feet). A micrometric balance (_c_) below the bob was
used to determine the increase in length due to the weight of the bob.
He projected the image of the clock pendulum (not shown) onto the
gravity pendulum by means of a lens, thus placing the clock some
distance away and eliminating the disturbing effect of its motion.
(Portion of plate 6, _Memoires publies par la Societe francaise de
Physique_, vol. 4.)]
In the earlier suspension from jaws there was uncertainty as to the
point about which the pendulum oscillated. Borda and Cassini hung their
pendulum in front of a seconds clock and determined the time of swing by
the method of coincidences. The times on the clock were observed when
the clock gained or lost one complete vibration (two swings) on the
pendulum. Suppose that the wire pendulum makes n swings while the clock
makes 2n + 2. If the clock beats seconds exactly, the time of one
complete vibration is 2 seconds, and the time of swing of the wire
pendulum is T = (2n + 2)/n = 2(1 + 1/n). An error in the time caused by
uncertainty in determining the coincidence of clock and wire pendulum is
reduced by employing a long interval of observation 2n. The whole
apparatus was enclosed in a box, in order to exclude disturbances from
currents of air. Corrections were made for buoyancy, for amplitude of
swing and for variations in length of the wire with temperature. The
final result was that the length of the seconds pendulum at the
observatory in Paris was determined to be 440.5593 Paris lines, or
993.53 mm., reduced to sea level 993.85 mm. Some years later the methods
of Borda were used by other French investigators, among whom was Biot
who used the platinum ball of Borda suspended by a copper wire 60 cm.
long.
Another historic "simple" pendulum was the one swung by Bessel (fig. 7)
for the determination of gravity at Koenigsberg 1825-1827.[21] The
pendulum consisted of a ball of brass, copper, or ivory that was
suspended by a fine wire, the upper end of which was wrapped and
unwrapped on a horizontal cylinder as support. The pendulum was swung
first from one point and then from another, exactly a "toise de
Peru"[22] higher up, the bob being at the same level in each case (fig.
7). Bessel found the period of vibration of the pendulum by the method
of coincidences; and in order to avoid disturbances from the comparison
clock, it was placed at some distance from the pendulum under
observation.
Bessel's experiments were significant in view of the care with which he
determined the corrections. He corrected for the stiffness of the wire
and for the lack of rigidity of connection between the bob and wire. The
necessity for the latter correction had been pointed out by Laplace, who
showed that through the circumstance that the pull of the wire is now on
one side and now on the other side of the center of gravity, the bob
acquires angular momentum about its center of gravity, which cannot be
accounted for if the line of the wire, and therefore the force that it
exerts, always passed through the center. In addition to a correction
for buoyancy of the air considered by his predecessors, Bessel also took
account of the inertia of the air set in motion by the pendulum.
[Illustration: Figure 8.--MODE OF SUSPENSION of Bessel's pendulum is
shown here. The iron wire is supported by the thumbscrew and clamp at
the left, but passes over a pin at the center, which is actually the
upper terminal of the pendulum. Bessel found this "cylinder of
unrolling" superior to the clamps and knife edges of earlier pendulums.
The counterweight at the right is part of a system for supporting the
scale in such a way that it is not elongated by its own weight.
With this apparatus, Bessel determined the ratio of the lengths of the
two pendulums and their times of vibration. From this the length of the
seconds pendulum was calculated. His method eliminated the need to take
into account such sources of inaccuracy as flexure of the pendulum wire
and imperfections in the shape of the bob. (Portion of plate 7,
_Memoires publies par la Societe francaise de Physique_, vol. 4.)]
[Illustration: Figure 9.--FRIEDRICH WILHELM BESSEL (1784-1846), German
mathematician and astronomer. He became the first superintendent of the
Prussian observatory established at Koenigsberg in 1810, and remained
there during the remainder of his life. So important were his many
contributions to precise measurement and calculation in astronomy that
he is often considered the founder of the "modern" age in that science.
This characteristic also shows in his venture into geodesy, 1826-1830,
one product of which was the pendulum experiment reported in this
article.]
The latter effect had been discovered by Du Buat in 1786,[23] but his
work was unknown to Bessel. The length of the seconds pendulum at
Koenigsberg, reduced to sea level, was found by Bessel to be 440.8179
lines. In 1835, Bessel determined the intensity of gravity at a site in
Berlin where observations later were conducted in the Imperial Office of
Weights and Measures by Charles S. Peirce of the U.S. Coast Survey.