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Smithsonian Institution
United States National Museum
Bulletin 240

[Illustration]

Smithsonian Press

Museum of History and Technology
Contributions from the Museum of History and Technology
  _Papers 34-44_
  _On Science and Technology_
Smithsonian Institution · Washington, D.C. 1966

       *       *       *       *       *

_Publications of the United States National Museum_


The scholarly and scientific publications of the United States National
Museum include two series, _Proceedings of the United States National
Museum_ and _United States National Museum Bulletin_.

In these series, the Museum publishes original articles and monographs
dealing with the collections and work of its constituent museums--The
Museum of Natural History and the Museum of History and
Technology--setting forth newly acquired facts in the fields of
anthropology, biology, history, geology, and technology. Copies of each
publication are distributed to libraries, to cultural and scientific
organizations, and to specialists and others interested in the different
subjects.

The _Proceedings_, begun in 1878, are intended for the publication, in
separate form, of shorter papers from the Museum of Natural History.
These are gathered in volumes, octavo in size, with the publication date
of each paper recorded in the table of contents of the volume.

In the _Bulletin_ series, the first of which was issued in 1875, appear
longer, separate publications consisting of monographs (occasionally in
several parts) and volumes in which are collected works on related
subjects. _Bulletins_ are either octavo or quarto in size, depending on
the needs of the presentation. Since 1902 papers relating to the
botanical collections of the Museum of Natural History have been
published in the _Bulletin_ series under the heading _Contributions from
the United States National Herbarium_, and since 1959, in _Bulletins_
titled "Contributions from the Museum of History and Technology," have
been gathered shorter papers relating to the collections and research of
that Museum.

The present collection of Contributions, Papers 34-44, comprises
Bulletin 240. Each of these papers has been previously published in
separate form. The year of publication is shown on the last page of each
paper.

FRANK A. TAYLOR _Director, United States National Museum_

       *       *       *       *       *

Contributions from the Museum of History and Technology:
Paper 44

DEVELOPMENT OF GRAVITY PENDULUMS IN THE 19TH CENTURY

by

Victor F. Lenzen and Robert P. Multhauf


    GALILEO, HUYGENS, AND NEWTON                               304

    FIGURE OF THE EARTH                                        306

    EARLY TYPES OF PENDULUMS                                   309

    KATER'S CONVERTIBLE AND INVARIABLE PENDULUMS               314

    REPSOLD-BESSEL REVERSIBLE PENDULUM                         320

    PEIRCE AND DEFFORGES INVARIABLE, REVERSIBLE PENDULUMS      327

    VON STERNECK AND MENDENHALL PENDULUMS                      331

    ABSOLUTE VALUE OF GRAVITY AT POTSDAM                       338

    APPLICATION OF GRAVITY SURVEYS                             342

    SUMMARY                                                    346




VICTOR F. LENZEN AND ROBERT P. MULTHAUF

DEVELOPMENT OF GRAVITY PENDULUMS IN THE 19th CENTURY


[Illustration: Figure 1.--A STUDY OF THE FIGURE OF THE EARTH WAS one of
the earliest projects of the French Academy of Sciences. In order to
test the effect of the earth's rotation on its gravitational force, the
Academy in 1672 sent Jean Richer to the equatorial island of Cayenne to
compare the rate of a clock which was known to have kept accurate time
in Paris. Richer found that the clock lost 2 minutes and 28 seconds at
Cayenne, indicating a substantial decrease in the force of gravity on
the pendulum. Subsequent pendulum experiments revealed that the period
of a pendulum varied not only with the latitude but also regionally,
under the influence of topographical features such as mountains. It
became clear that the measurement of gravity should be made a part of
the work of the geodetic surveyor.]


    _The history of gravity pendulums dates back to the time of
    Galileo. After the discovery of the variation of the force of
    gravity over the surface of the earth, gravity measurement
    became a major concern of physics and geodesy. This article
    traces the history of the development of instruments for this
    purpose._

    THE AUTHORS: _Victor F. Lenzen is Professor of Physics,
    Emeritus, at the University of California at Berkeley and Robert
    P. Multhauf is Chairman of the Department of Science and
    Technology in the Smithsonian Institution's Museum of History
    and Technology._


The intensity of gravity, or the acceleration of a freely falling body,
is an important physical quantity for the several physical sciences. The
intensity of gravity determines the weight of a standard pound or
kilogram as a standard or unit of force. In physical experiments, the
force on a body may be measured by determining the weight of a known
mass which serves to establish equilibrium against it. Thus, in the
absolute determination of the ampere with a current balance, the force
between two coils carrying current is balanced by the earth's
gravitational force upon a body of determinable mass. The intensity of
gravity enters into determinations of the size of the earth from the
angular velocity of the moon, its distance from the earth, and Newton's
inverse square law of gravitation and the laws of motion. Prediction
of the motion of an artificial satellite requires an accurate knowledge
of gravity for this astronomical problem.

The gravity field of the earth also provides data for a determination of
the figure of the earth, or geoid, but for this problem of geodesy
relative values of gravity are sufficient. If g is the intensity of
gravity at some reference station, and [Delta]g is the difference
between intensities at two stations, the values of gravity in geodetic
calculations enter as ratios ([Delta]g)/g over the surface of the earth.
Gravimetric investigations in conjunction with other forms of
geophysical investigation, such as seismology, furnish data to test
hypotheses concerning the internal structure of the earth.

Whether the intensity of gravity is sought in absolute or relative
measure, the most widely used instrument for its determination since the
creation of classical mechanics has been the pendulum. In recent
decades, there have been invented gravity meters based upon the
principle of the spring, and these instruments have made possible the
rapid determination of relative values of gravity to a high degree of
accuracy. The gravity meter, however, must be calibrated at stations
where the absolute value of gravity has been determined by other means
if absolute values are sought. For absolute determinations of gravity,
the pendulum historically has been the principal instrument employed.
Although alternative methods of determining absolute values of gravity
are now in use, the pendulum retains its value for absolute
determinations, and even retains it for relative determinations, as is
exemplified by the Cambridge Pendulum Apparatus and that of the Dominion
Observatory at Ottawa, Ontario.

The pendulums employed for absolute or relative determinations of
gravity have been of two basic types. The first form of pendulum used as
a physical instrument consisted of a weight suspended by a fiber, cord,
or fine wire, the upper end of which was attached to a fixed support.
Such a pendulum may be called a "simple" pendulum; the enclosure of the
word simple by quotation marks is to indicate that such a pendulum is an
approximation to a simple, or mathematical pendulum, a conceptual object
which consists of a mass-point suspended by a weightless inextensible
cord. If l is the length of the simple pendulum, the time of swing
(half-period in the sense of physics) for vibrations of infinitely small
amplitude, as derived from Newton's laws of motion and the hypothesis
that weight is proportional to mass, is T = [pi][sqrt](l/g).

The second form of pendulum is the compound, or physical, pendulum. It
consists of an extended solid body which vibrates about a fixed axis
under the action of the weight of the body. A compound pendulum may be
constituted to oscillate about one axis only, in which case it is
nonreversible and applicable only for relative measurements. Or a
compound pendulum may be constituted to oscillate about two axes, in
which case it is reversible (or "convertible") and may be used to
determine absolute values of gravity. Capt. Henry Kater, F.R.S., during
the years 1817-1818 was the first to design, construct, and use a
compound pendulum for the absolute determination of gravity. He
constructed a convertible pendulum with two knife edges and with it
determined the absolute value of gravity at the house of Henry Browne,
F.R.S., in Portland Place, London. He then constructed a similar
compound pendulum with only one knife edge, and swung it to determine
relative values of gravity at a number of stations in the British Isles.
The 19th century witnessed the development of the theory and practice of
observations with pendulums for the determination of absolute and
relative values of gravity.




Galileo, Huygens, and Newton


The pendulum has been both an objective and an instrument of physical
investigation since the foundations of classical mechanics were
fashioned in the 17th century.[1] It is tradition that the youthful
Galileo discovered that the period of oscillation of a pendulum is
constant by observations of the swings of the great lamp suspended from
the ceiling in the cathedral of Pisa.[2] The lamp was only a rough
approximation to a simple pendulum, but Galileo later performed more
accurate experiments with a "simple" pendulum which consisted of a heavy
ball suspended by a cord. In an experiment designed to confirm his laws
of falling bodies, Galileo lifted the ball to the level of a given
altitude and released it. The ball ascended to the same level on the
other side of the vertical equilibrium position and thereby confirmed a
prediction from the laws. Galileo also discovered that the period of
vibration of a "simple" pendulum varies as the square root of its
length, a result which is expressed by the formula for the time of
swing of the ideal simple pendulum. He also used a pendulum to measure
lapse of time, and he designed a pendulum clock. Galileo's experimental
results are important historically, but have required correction in the
light of subsequent measurements of greater precision.

Mersenne in 1644 made the first determination of the length of the
seconds pendulum,[3] that is, the length of a simple pendulum that beats
seconds (half-period in the sense of physics). Subsequently, he proposed
the problem to determine the length of the simple pendulum equivalent in
period to a given compound pendulum. This problem was solved by Huygens,
who in his famous work _Horologium oscillatorium_ ... (1673) set forth
the theory of the compound pendulum.[4]

Huygens derived a theorem which has provided the basis for the
employment of the reversible compound pendulum for the absolute
determination of the intensity of gravity. The theorem is that a given
compound pendulum possesses conjugate points on opposite sides of the
center of gravity; about these points, the periods of oscillation are
the same. For each of these points as center of suspension the other
point is the center of oscillation, and the distance between them is the
length of the equivalent simple pendulum. Earlier, in 1657, Huygens
independently had invented and patented the pendulum clock, which
rapidly came into use for the measurement of time. Huygens also created
the theory of centripetal force which made it possible to calculate the
effect of the rotation of the earth upon the observed value of gravity.

The theory of the gravity field of the earth was founded upon the laws
of motion and the law of gravitation by Isaac Newton in his famous
_Principia_ (1687). It follows from the Newtonian theory of gravitation
that the acceleration of gravity as determined on the surface of the
earth is the resultant of two factors: the principal factor is the
gravitational attraction of the earth upon bodies, and the subsidiary
factor is the effect of the rotation of the earth. A body at rest on the
surface of the earth requires some of the gravitational attraction for
the centripetal acceleration of the body as it is carried in a circle
with constant speed by the rotation of the earth about its axis. If the
rotating earth is used as a frame of reference, the effect of the
rotation is expressed as a centrifugal force which acts to diminish the
observed intensity of gravity.

       *       *       *       *       *

GLOSSARY OF GRAVITY TERMINOLOGY

ABSOLUTE GRAVITY: the value of the acceleration of gravity, also
expressed by the length of the seconds pendulum.

RELATIVE GRAVITY: the value of the acceleration of gravity relative to
the value at some standard point.

SIMPLE PENDULUM: see theoretical pendulum.

THEORETICAL PENDULUM: a heavy bob (point-mass) at the end of a
weightless rod.

SECONDS PENDULUM: a theoretical or simple pendulum of such length that
its time of swing (half-period) is one second. (This length is about one
meter.)

GRAVITY PENDULUM: a precisely made pendulum used for the measurement of
gravity.

COMPOUND PENDULUM: a pendulum in which the supporting rod is not
weightless; in other words, any actual pendulum.

CONVERTIBLE PENDULUM: a compound pendulum having knife edges at
different distances from the center of gravity. Huygens demonstrated
(1673) that if such a pendulum were to swing with equal periods from
either knife edge, the distance between those knife edges would be equal
to the length of a theoretical or simple pendulum of the same period.

REVERSIBLE PENDULUM: a convertible pendulum which is also symmetrical in
form.

INVARIABLE PENDULUM: a compound pendulum with only one knife edge, used
for relative measurement of gravity.

       *       *       *       *       *

From Newton's laws of motion and the hypothesis that weight is
proportional to mass, the formula for the half-period of a simple
pendulum is given by T = [pi][sqrt](l/g). If a simple pendulum beats
seconds, 1 = [pi][sqrt]([lambda]/g), where [lambda] is the length of the
seconds pendulum. From T = [pi][sqrt](l/g) and 1 = [pi][sqrt]([lambda]/g),
it follows that [lambda] = l/T^{2}. Then g = [pi]^{2}[lambda]. Thus, the
intensity of gravity can be expressed in terms of the length of the
seconds pendulum, as well as by the acceleration of a freely falling
body. During the 19th century, gravity usually was expressed in terms of
the length of the seconds pendulum, but present practice is to express
gravity in terms of g, for which the unit is the gal, or one centimeter
per second per second.

[Illustration: Figure 2.--THIS DRAWING, FROM RICHER'S _Observations
astronomiques et physiques faites en l'isle de Caïenne_ (Paris, 1679),
shows most of the astronomical instruments used by Richer, namely, one
of the two pendulum clocks made by Thuret, the 20-foot and the 5-foot
telescopes and the large quadrant. The figure may be intended as a
portrait of Richer. This drawing was done by Sebastian Le Clerc, a young
illustrator who made many illustrations of the early work of the Paris
Academy.]




Figure of the Earth


A principal contribution of the pendulum as a physical instrument has
been the determination of the figure of the earth.[5] That the earth
is spherical in form was accepted doctrine among the ancient Greeks.
Pythagoras is said to have been the first to describe the earth as a
sphere, and this view was adopted by Eudoxus and Aristotle.

The Alexandrian scientist Eratosthenes made the first estimate of the
diameter and circumference of a supposedly spherical earth by an
astronomical-geodetic method. He measured the angle between the
directions of the rays of the sun at Alexandria and Syene (Aswan),
Egypt, and estimated the distance between these places from the length
of time required by a caravan of camels to travel between them. From the
central angle corresponding to the arc on the surface, he calculated the
radius and hence the circumference of the earth. A second measurement
was undertaken by Posidonius, who measured the altitudes of stars at
Alexandria and Rhodes and estimated the distance between them from the
time required to sail from one place to the other.

With the decline of classical antiquity, the doctrine of the spherical
shape of the earth was lost, and only one investigation, that by the
Arabs under Calif Al-Mamun in A.D. 827, is recorded until the 16th
century. In 1525, the French mathematician Fernel measured the length of
a degree of latitude between Paris and Amiens by the revolutions of the
wheels of his carriage, the circumference of which he had determined. In
England, Norwood in 1635 measured the length of an arc between London
and York with a chain. An important forward step in geodesy was the
measurement of distance by triangulation, first by Tycho Brahe, in
Denmark, and later, in 1615, by Willebrord Snell, in Holland.

Of historic importance, was the use of telescopes in the triangulation
for the measurement of a degree of arc by the Abbé Jean Picard in
1669.[6] He had been commissioned by the newly established Academy of
Sciences to measure an arc corresponding to an angle of 1°, 22', 55" of
the meridian between Amiens and Malvoisine, near Paris. Picard proposed
to the Academy the measurement of the meridian of Paris through all of
France, and this project was supported by Colbert, who obtained the
approval of the King. In 1684, Giovanni-Domenico Cassini and De la Hire
commenced a trigonometrical measure of an arc south of Paris;
subsequently, Jacques Cassini, the son of Giovanni-Domenico, added the
arc to the north of Paris. The project was completed in 1718. The length
of a degree of arc south of Paris was found to be greater than the
length north of Paris. From the difference, 57,097 toises[7] minus
56,960 toises, it was concluded that the polar diameter of the earth is
larger than the equatorial diameter, i.e., that the earth is a prolate
spheroid (fig. 3).

[Illustration: Figure 3.--MEASUREMENTS OF THE LENGTH of a degree of
latitude which were completed in different parts of France in 1669 and
1718 gave differing results which suggested that the shape of the earth
is not a sphere but a prolate spheroid (1). But Richer's pendulum
observation of 1672, as explained by Huygens and Newton, indicated that
its shape is that of an oblate spheroid (2). The disagreement is
reflected in this drawing. In the 1730's it was resolved in favor of the
latter view by two French geodetic expeditions for the measurement of
degrees of latitude in the equatorial and polar regions (Ecuador--then
part of Peru--and Lapland).]

Meanwhile, Richer in 1672 had been sent to Cayenne, French Guiana, to
make astronomical observations and to measure the length of the seconds
pendulum.[8] He took with him a pendulum clock which had been adjusted
to keep accurate time in Paris. At Cayenne, however, Richer found that
the clock was retarded by 2 minutes and 28 seconds per day (fig. 1). He
also fitted up a "simple" pendulum to vibrate in seconds and measured
the length of this seconds pendulum several times every week for 10
months. Upon his return to Paris, he found that the length of the
"simple" pendulum which beat seconds at Cayenne was 1-1/4 Paris lines[9]
shorter than the length of the seconds pendulum at Paris. Huygens
explained the reduction in the length of the seconds pendulum--and,
therefore, the lesser intensity of gravity at the equator with respect
to the value at Paris--in terms of his theory of centripetal force as
applied to the rotation of the earth and pendulum.[10]

A more complete theory was given by Newton in the _Principia_.[11]
Newton showed that if the earth is assumed to be a homogeneous, mutually
gravitating fluid globe, its rotation will result in a bulging at the
equator. The earth will then have the form of an oblate spheroid, and
the intensity of gravity as a form of universal gravitation will vary
with position on the surface of the earth. Newton took into account
gravitational attraction and centrifugal action, and he calculated the
ratio of the axes of the spheroid to be 230:229. He calculated and
prepared a table of the lengths of a degree of latitude and of the
seconds pendulum for every 5° of latitude from the equator to the pole.
A discrepancy between his predicted length of the seconds pendulum at
the equator and Richer's measured length was explained by Newton in
terms of the expansion of the scale with higher temperatures near the
equator.

Newton's theory that the earth is an oblate spheroid was confirmed by
the measurements of Richer, but was rejected by the Paris Academy of
Sciences, for it contradicted the results of the Cassinis, father and
son, whose measurements of arcs to the south and north of Paris had led
to the conclusion that the earth is a prolate spheroid. Thus, a
controversy arose between the English scientists and the Paris Academy.
The conflict was finally resolved by the results of expeditions sent by
the Academy to Peru and Sweden. The first expedition, under Bouguer, La
Condamine, and Godin in 1735, went to a region in Peru, and, with the
help of the Spaniard Ullo, measured a meridian arc of about 3°7' near
Quito, now in Ecuador.[12] The second expedition, with Maupertuis and
Clairaut in 1736, went to Lapland within the Arctic Circle and measured
an arc of about 1° in length.[13] The northern arc of 1° was found to be
longer than the Peruvian arc of 1°, and thus it was confirmed that the
earth is an oblate spheroid, that is, flattened at the poles, as
predicted by the theory of Newton.

[Illustration: Figure 4.--THE DIRECT USE OF A CLOCK to measure the force
of gravity was found to be limited in accuracy by the necessary
mechanical connection of the pendulum to the clock, and by the
unavoidable difference between the characteristics of a clock pendulum
and those of a theoretical (usually called "simple") pendulum, in which
the mass is concentrated in the bob, and the supporting rod is
weightless.

After 1735, the clock was used only to time the swing of a detached
pendulum, by the method of "coincidences." In this method, invented by
J. J. Mairan, the length of the detached pendulum is first accurately
measured, and the clock is corrected by astronomical observation. The
detached pendulum is then swung before the clock pendulum as shown here.
The two pendulums swing more or less out of phase, coming into
coincidence each time one has gained a vibration. By counting the number
of coincidences over several hours, the period of the detached pendulum
can be very accurately determined. The length and period of the detached
pendulum are the data required for the calculation of the force of
gravity.]

The period from Eratosthenes to Picard has been called the spherical era
of geodesy; the period from Picard to the end of the 19th century has
been called the ellipsoidal period. During the latter period the earth
was conceived to be an ellipsoid, and the determination of its
ellipticity, that is, the difference of equatorial radius and polar
radius divided by the equatorial radius, became an important geodetic
problem. A significant contribution to the solution of this problem was
made by determinations of gravity by the pendulum.

An epoch-making work during the ellipsoidal era of geodesy was
Clairaut's treatise, _Théorie de la figure de la terre_.[14] On the
hypothesis that the earth is a spheroid of equilibrium, that is, such
that a layer of water would spread all over it, and that the internal
density varies so that layers of equal density are coaxial spheroids,
Clairaut derived a historic theorem: If [gamma]_{E}, [gamma]_{P} are the
values of gravity at the equator and pole, respectively, and c the
centrifugal force at the equator divided by [gamma]_{E}, then the
ellipticity [alpha] = (5/2)c - ([gamma]_{P} - [gamma]_{E})/[gamma]_{E}.

Laplace showed that the surfaces of equal density might have any nearly
spherical form, and Stokes showed that it is unnecessary to assume any
law of density as long as the external surface is a spheroid of
equilibrium.[15] It follows from Clairaut's theorem that if the earth is
an oblate spheroid, its ellipticity can be determined from relative
values of gravity and the absolute value at the equator involved in c.
Observations with nonreversible, invariable compound pendulums have
contributed to the application of Clairaut's theorem in its original and
contemporary extended form for the determination of the figure and
gravity field of the earth.




Early Types of Pendulums


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,
_Mémoires publiés par la Société française 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, _Mémoires publiés par la Société
française 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 Königsberg 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, _Mémoires publiés par la Société française 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 Königsberg 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,
_Mémoires publiés par la Société française 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 Königsberg 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
Königsberg, 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.




Kater's Convertible and Invariable Pendulums


The systematic survey of the gravity field of the earth was given a
great impetus by the contributions of Capt. Henry Kater, F.R.S. In 1817,
he designed, constructed, and applied a convertible compound pendulum
for the absolute determination of gravity at the house of Henry Browne,
F.R.S., in Portland Place, London.[24] Kater's convertible pendulum
(fig. 11) consisted of a brass rod to which were attached a flat
circular bob of brass and two adjustable weights, the smaller of which
was adjusted by a screw. The convertibility of the pendulum was
constituted by the provision of two knife edges turned inwards on
opposite sides of the center of gravity. The pendulum was swung on each
knife edge, and the adjustable weights were moved until the times of
swing were the same about each knife edge. When the times were judged to
be the same, the distance between the knife edges was inferred to be the
length of the equivalent simple pendulum, in accordance with Huygens'
theorem on conjugate points of a compound pendulum. Kater determined the
time of swing by the method of coincidences (fig. 12). He corrected for
the buoyancy of the air. The final value of the length of the seconds
pendulum at Browne's house in London, reduced to sea level, was
determined to be 39.13929 inches.

The convertible compound pendulum had been conceived prior to its
realization by Kater. In 1792, on the occasion of the proposal in Paris
to establish the standard of length as the length of the seconds
pendulum, Baron de Prony had proposed the employment of a compound
pendulum with three axes of oscillation.[25] In 1800, he proposed the
convertible compound pendulum with knife edges about which the pendulum
could complete swings in equal times. De Prony's proposals were not
accepted and his papers remained unpublished until 1889, at which time
they were discovered by Defforges. The French decision was to experiment
with the ball pendulum, and the determination of the length of the
seconds pendulum was carried out by Borda and Cassini by methods
previously described. Bohnenberger in his _Astronomie_ (1811),[26] made
the proposal to employ a convertible pendulum for the absolute
determination of gravity; thus, he has received credit for priority in
publication. Capt. Kater independently conceived of the convertible
pendulum and was the first to design, construct, and swing one.

[Illustration: Figure 10.--HENRY KATER (1777-1835), English army officer
and physicist. His scientific career began during his military service
in India, where he assisted in the "great trigonometrical survey."
Returned to England because of bad health, and retired in 1814, he
pioneered (1818) in the development of the convertible pendulum as an
alternative to the approximation of the "simple" pendulum for the
measurement of the "seconds pendulum." Kater's convertible pendulum and
the invariable pendulum introduced by him in 1819 were the basis of
English pendulum work. (_Photo courtesy National Portrait Gallery,
London._)]

After his observations with the convertible pendulum, Capt. Kater
designed an invariable compound pendulum with a single knife edge but
otherwise similar in external form to the convertible pendulum[27] (fig.
13). Thirteen of these Kater invariable pendulums have been reported as
constructed and swung at stations throughout the world.[28] Kater
himself swung an invariable pendulum at a station in London and at
various other stations in the British Isles. Capt. Edward Sabine,
between 1820 and 1825, made voyages and swung Kater invariable pendulums
at stations from the West Indies to Greenland and Spitzbergen.[29] In
1820, Kater swung a Kater invariable pendulum at London and then sent it
to Goldingham, who swung it in 1821 at Madras, India.[30] Also in 1820,
Kater supplied an invariable pendulum to Hall, who swung it at London
and then made observations near the equator and in the Southern
Hemisphere, and at London again in 1823.[31] The same pendulum, after
its knives were reground, was delivered to Adm. Lütke of Russia, who
observed gravity with it on a trip around the world between 1826 and
1829.[32]

[Illustration: Figure 11.--THE ATTEMPT TO APPROXIMATE the simple
(theoretical) pendulum in gravity experiments ended in 1817-18 when
Henry Kater invented the compound convertible pendulum, from which the
equivalent simple pendulum could be obtained according to the method of
Huygens (see text, p. 314). Developed in connection with a project to
fix the standard of English measure, Kater's pendulum was called
"compound" because it was a solid bar rather than the fine wire or
string with which earlier experimenters had tried to approximate a
"weightless" rod. It was called convertible because it is alternately
swung from the two knife edges (_a_ and _b_) at opposite ends. The
weights (_f_ and _g_) are adjusted so that the period of the pendulum is
the same from either knife edge. The distance between the two knife
edges is then equal to the length of the equivalent simple pendulum.]

[Illustration: Figure 12.--THE KATER CONVERTIBLE PENDULUM in use is
placed before a clock, whose pendulum bob is directly behind the
extended "tail" of the Kater pendulum. A white spot is painted on the
center of the bob of the clock pendulum. The observing telescope, left,
has a diaphragm with a vertical slit of such width that its view is just
filled by the tail of the Kater pendulum when it is at rest. When the
two pendulums are swinging, the white spot on the clock pendulum can be
seen on each swing except that in which the two pendulums are in
coincidence; thus, the coincidences are determined. (Portion of plate 5,
_Mémoires publiés par la Société française de Physique_, vol. 4.)]

[Illustration: Figure 13.--THIS DRAWING ACCOMPANIED John Goldingham's
report on the work done in India with Kater's invariable pendulum. The
value of gravity obtained, directly or indirectly, in terms of the
simple pendulum, is called "absolute." Once absolute values of gravity
were established at a number of stations, it became possible to use the
much simpler "relative" method for the measurement of gravity at new
stations. Because it has only one knife edge, and does not involve the
adjustments of the convertible pendulum, this one is called
"invariable." In use, it is first swung at a station where the absolute
value of gravity has been established, and this period is then compared
with its period at one or more new stations. Kater developed an
invariable pendulum in 1819, which was used in England and in Madras,
India, in 1821.]

While the British were engaged in swinging the Kater invariable
pendulums to determine relative values of the length of the seconds
pendulum, or of gravity, the French also sent out expeditions. Capt. de
Freycinet made initial observations at Paris with three invariable brass
pendulums and one wooden one, and then carried out observations at Rio
de Janeiro, Cape of Good Hope, Île de France, Rawak (near New Guinea),
Guam, Maui, and various other places.[33] A similar expedition was
conducted in 1822-1825 by Captain Duperry.[34]

During the years from 1827 to 1840, various types of pendulum were
constructed and swung by Francis Baily, a member of the Royal
Astronomical Society, who reported in 1832 on experiments in which no
less than 41 different pendulums were swung in vacuo, and their
characteristics determined.[35] In 1836, Baily undertook to advise the
American Lt. Charles Wilkes, who was to head the United States
Exploring Expedition of 1838-1842, on the procurement of pendulums for
this voyage. Wilkes ordered from the London instrument maker, Thomas
Jones, two unusual pendulums, which Wilkes described as "those
considered the best form by Mr. Baily for traveling pendulums," and
which Baily, himself, described as "precisely the same as the two
invariable pendulums belonging to this [Royal Astronomical] Society,"
except for the location of the knife edges.

[Illustration: Figure 14.--VACUUM CHAMBER FOR USE with the Kater
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 Kübel, 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 Kübel 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 Königsberg
and Berlin, respectively. In the memoir on his observations at
Königsberg, 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 Cellérier,[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 Cellérier for the Swiss observers of
pendulums.

In 1866, Plantamour published an extensive memoir "Expériences faites à
Genève avec le pendule à réversion." 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-Europäische Gradmessung_, also
_L'Association Géodésique 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 Neuchâtel, 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 Europäische 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 Europäische 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 Europäische
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 Europäische 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 Europäische 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.
Ibañez, 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 flexibilité du trépied sur l'oscillation du pendule à
réversion." This memoir and others by Cellérier 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
Cellérier 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,
Hervé 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
Cellérier was appointed to join him, since the problem essentially was a
theoretical one.

The Sixth General Conference of _Die Europäische 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 Cellérier[68] on the theory of the
double pendulum and a report by Plantamour and Cellérier.[69]
Cellérier'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 Cellérier 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 "Über
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 Europäische
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° F.
    at the same time that the yard is swung at 60° 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 Cellérier 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°. 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.

In accordance with Peirce's theory of the flexure of the stand under
oscillations of the pendulum, determinations of the displacement of the
receiver of the Mendenhall apparatus were part of a relative
determination of gravity by members of the Coast and Geodetic Survey.
Initially, a statical method was used, but during 1908-1909 members of
the Survey adapted the Michelson interferometer for the determinations
of flexure during oscillations from the shift of fringes.[84] The first
Mendenhall pendulums were made of bronze, but about 1920 invar was
chosen because of its small coefficient of expansion. About 1930, Lt. E.
J. Brown of the Coast and Geodetic Survey made significant improvements
in the Mendenhall apparatus, and the new form came to be known as the
Brown Pendulum Apparatus.[85]

[Illustration: Figure 27.--THE MICHELSON INTERFEROMETER. The horizontal
component of the force acting on the knife edge through the swinging
pendulum causes the support to move in unison with the pendulum, and
thereby affects the period of the oscillation. This movement is the
so-called flexure of the pendulum support, and must be taken into
account in the most accurate observations.

In 1907, the Michelson interferometer was adapted to this purpose by the
U.S. Coast and Geodetic Survey. As shown here, the interferometer,
resting on a wooden beam, is introduced into the path of a light beam
reflected from a mirror on the vacuum chamber. Movement of that mirror
causes a corresponding movement in the interference fringes in the
interferometer, which can be measured.]

The original Von Sterneck apparatus and that of Mendenhall provided for
the oscillation of one pendulum at a time. After the adoption of the Von
Sterneck pendulum in Europe, there were developed stands on which two or
four pendulums hung at the same time. This procedure provided a
convenient way to observe more than one invariable pendulum at a station
for the purpose of detecting changes in length. Prof. M. Haid of
Karlsruhe in 1896 described a four-pendulum apparatus,[86] and Dr.
Schumann of Potsdam subsequently described a two-pendulum
apparatus.[87]

[Illustration: Figure 28.--APPARATUS WHICH WAS DEVELOPED IN 1929 by the
Gulf Research and Development Company, Harmarville, Pennsylvania. It was
designed to achieve an accuracy within one ten-millionth of the true
value of gravity, and represents the extreme development of pendulum
apparatus for relative gravity measurement. The pendulum was designed so
that the period would be a minimum. The case (the top is missing in this
photograph) is dehumidified and its temperature and electrostatic
condition are controlled. Specially designed pendulum-lifting and
-starting mechanisms are used. The problem of flexure of the case is
overcome by the Faye-Peirce method (see text) in which two dynamically
matched pendulums are swung simultaneously, 180° apart in phase.]

The multiple-pendulum apparatus then provided a method of determining
the flexure of the stand from the action of one pendulum upon a second
pendulum hung on the same stand. This method of determining the
correction for flexure was a development from a "Wippverfahren" invented
at the Geodetic Institute in Potsdam. A dynamometer was used to impart
periodic impulses to the stand, and the effect was observed upon a
pendulum initially at rest. Refinements of this method led to the
development of a method used by Lorenzoni in 1885-1886 to determine the
flexure of the stand by action of an auxiliary pendulum upon the
principal pendulum. Dr. Schumann, in 1899, gave a mathematical theory of
such determinations,[88] and in his paper cited the mathematical methods
of Peirce and Cellérier for the theory of Faye's proposal at Stuttgart
in 1877 to swing two similar pendulums on the same support with equal
amplitudes and in opposite phases.

[Illustration: Figure 29.--THE GULF PENDULUM is about 10.7 inches long,
and has a period of .89 second. It is made of fused quartz which is
resistant to the influence of temperature change and to the earth's
magnetism. Quartz pendulums are subject to the influence of
electrostatic charge, and provision is made to counteract this through
the presence of a radium salt in the case. The bearings are made of
Pyrex glass.]

In 1902, Dr. P. Furtwängler[89] presented the mathematical theory of
coupled pendulums in a paper in which he referred to Faye's proposal of
1877 and reported that the difficulties predicted upon its application
had been found not to occur. Finally, during the gravity survey of
Holland in the years 1913-1921, in view of instability of supports
caused by the mobility of the soil, F. A. Vening Meinesz adopted Faye's
proposed method of swinging two pendulums on the same support.[90] The
observations were made with the ordinary Stückrath apparatus, in which
four Von Sterneck pendulums swung two by two in planes perpendicular to
each other. This successful application of the method--which had been
proposed by Faye and had been demonstrated theoretically to be sound by
Peirce, who also published a design for its application--was rapidly
followed for pendulum apparatus for relative determinations by
Potsdam,[91] Cambridge (England),[92] Gulf Oil and Development
Company,[93] and the Dominion Observatory at Ottawa.[94] Heiskanen and
Vening Meinesz state:

    The best way to eliminate the effect of flexure is to use two
    synchronized pendulums of the same length swinging on the same
    apparatus in the same plane and with the same amplitudes but in
    opposite phases; it is clear then the flexure is zero.[95]

In view of the fact that the symmetrical reversible pendulum is named
for Bessel, who created the theory and a design for its application by
Repsold, it appears appropriate to call the method of eliminating
flexure by swinging two pendulums on the same support the Faye-Peirce
method. Its successful application was made possible by Maj. von
Sterneck's invention of the short, 1/4-meter pendulum.

[Illustration: Figure 30.--THE ACCUMULATED DATA OF GRAVITY observations
over the earth's surface have indicated that irregularities such as
mountains do not have the effect which would be expected in modifying
gravity, but are somehow compensated for. The most satisfactory solution
to this still unanswered question has been the theory of isostasy,
according to which variations in the density of the material in the
earth's crust produce a kind of hydrostatic equilibrium between its
higher and lower parts, as they "float" on the earth's fluid core. The
metals of different density floating in mercury in this diagram
illustrate isostasy according to the theory of Pratt and Hayford.]




Absolute Value of Gravity at Potsdam


The development of the reversible pendulum in the 19th century
culminated in the absolute determination of the intensity of gravity at
Potsdam by Kühnen and Furtwängler of the Royal Prussian Geodetic
Institute, which then became the world base for gravity surveys.[96]

We have previously seen that in 1869 the Geodetic Institute--founded by
Lt. Gen. Baeyer--had acquired a Repsold-Bessel reversible pendulum which
was swung by Dr. Albrecht under the direction of Dr. Bruhns.
Dissatisfaction with this instrument was expressed by Baeyer in 1875 to
Charles S. Peirce, who then, by experiment and mathematical analysis of
the flexure of the stand under oscillations of the pendulum, determined
that previously reported results with the Repsold apparatus required
correction. Dr. F. R. Helmert, who in 1887 succeeded Baeyer as director
of the Institute, secured construction of a building for the Institute
in Potsdam, and under his direction the scientific study of the
intensity of gravity was pursued with vigor. In 1894, it was discovered
in Potsdam that a pendulum constructed of very flexible material yielded
results which differed markedly from those obtained with pendulums of
greater stiffness. Dr. Kühnen of the Institute discovered that the
departure from expectations was the result of the flexure of the
pendulum staff itself during oscillations.[97]

Peirce, in 1883, had discovered that the recesses cut in his pendulums
for the insertion of tongues that carried the knives had resulted in the
flexure of the pendulum staff.[98] By experiment, he also found an even
greater flexure for the Repsold pendulum. In order to eliminate this
source of error, Peirce designed a pendulum with knives that extended
from each side of the cylindrical staff, and he received authorization
from the superintendent of the Coast and Geodetic Survey to arrange for
the construction of such pendulums by Gautier in Paris. Peirce, who had
made his plans in consultation with Gautier, was called home before the
pendulums were completed, and these new instruments remained
undelivered.

In a memoir titled "Effect of the flexure of a pendulum upon its period
of oscillation,"[99] Peirce determined analytically the effect on the
period of a pendulum with a single elastic connection between two rigid
parts of the staff. Thus, Peirce discovered experimentally the flexure
of the staff and derived for a simplified case the effect on the period.
It is not known if he ever found the integrated effect of the continuum
of elastic connections in the pendulum. Lorenzoni, in 1896, offered a
solution to the problem, and Almansi, in 1899, gave an extended
analysis. After the independent discovery of the problem at the Geodetic
Institute, Dr. Helmert took up the problem and criticized the theories
of Peirce and Lorenzoni. He then presented his own theory of flexure in
a comprehensive memoir.[100] In view of the previous neglect of the
flexure of the pendulum staff in the reduction of observations, Helmert
directed that the Geodetic Institute make a new absolute determination
of the intensity of gravity at Potsdam. For this purpose, Kühnen and
Furtwängler used the following reversible pendulums which had been
constructed by the firm of A. Repsold and Sons in Hamburg:

    1. The seconds pendulum of the Geodetic Institute procured in
    1869.

    2. A seconds pendulum from the Astronomical Observatory, Padua.

    3. A heavy, seconds pendulum from the Imperial and Royal
    Military-Geographical Institute, Vienna.

    4. A light, seconds pendulum from the Imperial and Royal
    Military-Geographical Institute.

    5. A 1/2-second, reversible pendulum of the Geodetic Institute
    procured in 1892.

Work was begun in 1898, and in 1906 Kühnen and Furtwängler published
their monumental memoir, "Bestimmung der Absoluten Grösze der
Schwerkraft zu Potsdam mit Reversionspendeln."

The acceleration of gravity in the pendulum room of the Geodetic
Institute was determined to be 981.274 ± 0.003 cm/sec^{2}. In view of the
exceptionally careful and thorough determination at the Institute,
Potsdam was accepted as the world base for the absolute value of the
intensity of gravity. The absolute value of gravity at some other
station on the Potsdam system was determined from the times of swing of
an invariable pendulum at the station and at Potsdam by the relation
(T_{1})^{2}/(T_{2})^{2} = g_{2}/g_{1}. Thus, in 1900, Assistant G. R.
Putnam of the Coast and Geodetic Survey swung Mendenhall pendulums at
the Washington base and at Potsdam, and by transfer from Potsdam
determined the intensity of gravity at the Washington base to be 980.112
cm/sec^{2}.[101] In 1933, Lt. E. J. Brown made comparative measurements
with improved apparatus and raised the value at the Washington base to
980.118 cm/sec^{2}.[102]

In view of discrepancies between the results of various relative
determinations, the Coast and Geodetic Survey in 1928 requested the
National Bureau of Standards to make an absolute determination for
Washington. Heyl and Cook used reversible pendulums made of fused silica
having a period of approximately 1 second. Their result, published in
1936, was interpreted to indicate that the value at Potsdam was too high
by 20 parts in 1 million.[103] This estimate was lowered slightly by Sir
Harold Jeffreys of Cambridge, England, who recomputed the results of
Heyl and Cook by different methods.[104]

[Illustration: Figure 31.--MAP SHOWING THE DISTRIBUTION of gravity
stations throughout the United States as of December 1908.]

[Illustration: Figure 32.--MAP SHOWING THE DISTRIBUTION of gravity
stations throughout the United States in 1923.]

In 1939, J. S. Clark published the results of a determination of gravity
with pendulums of a non-ferrous Y-alloy[105] at the National Physical
Laboratory at Teddington, England, and, after recomputation of results
by Jeffreys, the value was found to be 12.8 parts in 1 million less than
the value obtained by transfer from Potsdam. Dr. Hugh L. Dryden of the
National Bureau of Standards, and Dr. A. Berroth of the Geodetic
Institute at Potsdam, have recomputed the Potsdam data by different
methods of adjustment and concluded that the Potsdam value was too high
by about 12 parts in a million.[106] Determination of gravity at
Leningrad by Russian scientists likewise has indicated that the 1906
Potsdam value is too high. In the light of present information, it
appears justifiable to reduce the Potsdam value of 981.274 by .013
cm/sec^{2} for purposes of comparison. If the Brown transfer from
Potsdam in 1933 was taken as accurate, the value for the Washington base
would be 980.105 cm/sec^{2}. In this connection, it is of interest to
note that the value given by Charles S. Peirce for the comparable
Smithsonian base in Washington, as determined by him from comparative
methods in the 1880's and reported in the _Annual Report of the
Superintendent of the Coast and Geodetic Survey for the year 1890-1891_,
was 980.1017 cm/sec^{2}.[107] This value would appear to indicate that
Peirce's pendulums, observations, and methods of reduction of data were
not inferior to those of the scientists of the Royal Prussian Geodetic
Institute at Potsdam.

Doubts concerning the accuracy of the Potsdam value of gravity have
stimulated many new determinations of the intensity of gravity since the
end of World War II. In a paper published in June 1957, A. H. Cook,
Metrology Division, National Physical Laboratory, Teddington, England,
stated:

    At present about a dozen new absolute determinations are in
    progress or are being planned. Heyl and Cook's reversible
    pendulum apparatus is in use in Buenos Aires and further
    reversible pendulum experiments have been made in the All Union
    Scientific Research Institute of Metrology, Leningrad (V N I I M)
    and are planned at Potsdam. A method using a very long pendulum
    was tried out in Russia about 1910 and again more recently and
    there are plans for similar work in Finland. The first
    experiment with a freely falling body was that carried out by
    Volet who photographed a graduated scale falling in an enclosure
    at low air pressure. Similar experiments have been completed in
    Leningrad and are in progress at the Physikalisch-Technische
    Bundesanstalt (Brunswick) and at the National Research Council
    (Ottawa), and analogous experiments are being prepared at the
    National Physical Laboratory and at the National Bureau of
    Standards. Finally, Professor Medi, Director of the Istituto
    Nazionale di Geofisica (Rome), is attempting to measure the
    focal length of the paraboloidal surface of a liquid in a
    rotating dish.[108]




Application of Gravity Surveys


We have noted previously that in the ancient and early modern periods,
the earth was presupposed to be spherical in form. Determination of the
figure of the earth consisted in the measurement of the radius by the
astronomical-geodetic method invented by Eratosthenes. Since the earth
was assumed to be spherical, gravity was inferred to be constant over
the surface of the earth. This conclusion appeared to be confirmed by
the determination of the length of the seconds pendulum at various
stations in Europe by Picard and others. The observations of Richer in
South America, the theoretical discussions of Newton and Huygens, and
the measurements of degrees of latitude in Peru and Sweden demonstrated
that the earth is an oblate spheroid.

[Illustration: Figure 33.--GRAVITY CHARACTERISTICS OF THE GLOBE.
Deductions as to the distribution of matter in the earth can be made
from gravity measurements. This globe shows worldwide variations in
gravity as they now appear from observations at sea (in submarines) as
well as on land. It is based on data from the Institute of Geodesy at
Ohio State University.]

The theory of gravitation and the theory of central forces led to the
result that the intensity of gravity is variable over the surface of the
earth. Accordingly, determinations of the intensity of gravity became
of value to the geodesist as a means of determining the figure of the
earth. Newton, on the basis of the meager data available to him,
calculated the ellipticity of the earth to be 1/230 (the ellipticity is
defined by (a-b)/a, where a is the equatorial radius and b the polar
radius). Observations of the intensity of gravity were made on the
historic missions to Peru and Sweden. Bouguer and La Condamine found
that at the equator at sea level the seconds pendulum was 1.26
Paris-lines shorter than at Paris. Maupertuis found that in northern
Sweden a certain pendulum clock gained 59.1 seconds per day on its rate
in Paris. Then Clairaut, from the assumption that the earth is a
spheroid of equilibrium, derived a theorem from which the ellipticity of
the earth can be derived from values of the intensity of gravity.

[Illustration: Figure 34.--AN EXHIBIT OF GRAVITY APPARATUS at the
Smithsonian Institution. Suspended on the wall, from left to right, are
the invariable pendulums of Mendenhall (1/2-second), Peirce (1873-1874),
and Peirce (1881-1882); the double pendulum of Edward Kübel (see fig.
15, p. 319), and the reversible pendulum of Peirce. On the display
counter, from left to right, are the vacuum chamber, telescope and flash
apparatus for the Mendenhall 1/4-second apparatus. Shown below these are
the four pendulums used with the Mendenhall apparatus, the one on the
right having a thermometer attached. At bottom, right, is the Gulf
apparatus (cover removed) mentioned in the text, shown with one quartz
pendulum.]

Early in the 19th century a systematic series of observations began to
be conducted in order to determine the intensity of gravity at stations
all over the world. Kater invariable pendulums, of which 13 examples
have been mentioned in the literature, were used in surveys of gravity
by Kater, Sabine, Goldingham, and other British pendulum swingers. As
has been noted previously, a Kater invariable pendulum was used by Adm.
Lütke of Russia on a trip around the world. The French also sent out
expeditions to determine values of gravity. After several decades of
relative inactivity, Capts. Basevi and Heaviside of the Indian Survey
carried out an important series of observations from 1865 to 1873 with
Kater invariable pendulums and the Russian Repsold-Bessel pendulums. In
1881-1882 Maj. J. Herschel swung Kater invariable pendulums nos. 4, 6
(1821), and 11 at stations in England and then brought them to the
United States in order to make observations which would connect American
and English base stations.[109]

The extensive sets of observations of gravity provided the basis of
calculations of the ellipticity of the earth. Col. A. R. Clarke in his
_Geodesy_ (London, 1880) calculated the ellipticity from the results of
gravity surveys to be 1/(292.2 ± 1.5). Of interest is the calculation by
Charles S. Peirce, who used only determinations made with Kater
invariable pendulums and corrected for elevation, atmospheric effect,
and expansion of the pendulum through temperature.[110] He calculated
the ellipticity of the earth to be 1/(291.5 ± 0.9).

The 19th century witnessed the culmination of the ellipsoidal era of
geodesy, but the rapid accumulation of data made possible a better
approximation to the figure of the earth by the geoid. The geoid is
defined as the average level of the sea, which is thought of as extended
through the continents. The basis of geodetic calculations, however, is
an ellipsoid of reference for which a gravity formula expresses the
value of normal gravity at a point on the ellipsoid as a function of
gravity at sea level at the equator, and of latitude. The general
assembly of the International Union of Geodesy and Geophysics, which was
founded after World War I to continue the work of _Die Internationale
Erdmessung_, adopted in 1924 an international reference ellipsoid,[111]
of which the ellipticity, or flattening, is Hayford's value 1/297. In
1930, the general assembly adopted a correlated International Gravity
Formula of the form

[gamma] = [gamma]_{E}(1 + [beta]sin^{2} [phi] + [epsilon]sin^{2} 2[phi])

where [gamma] is normal gravity at latitude [phi], [gamma]_{E} is the
value of gravity at sea level at the equator, [beta] is a parameter
which is computed on the basis of Clairaut's theorem from the flattening
value of the meridian, and [epsilon] is a constant which is derived
theoretically. The plumb line is perpendicular to the geoid, and the
components of angle between the perpendiculars to geoid and reference
ellipsoid are deflections of the vertical. The geoid is above the
ellipsoid of reference under mountains and it is below the ellipsoid on
the oceans, where the geoid coincides with mean sea level. In physical
geodesy, gravimetric data are used for the determination of the geoid
and components of deflections of the vertical. For this purpose, one
must reduce observed values of gravity to sea level by various
reductions, such as free-air, Bouguer, isostatic reductions. If g_{0} is
observed gravity reduced to sea level and [gamma] is normal gravity
obtained from the International Gravity Formula, then

  [Delta]g = g_{0} - [gamma]

is the gravity anomaly.[112]

In 1849, Stokes derived a theorem whereby the distance N of the geoid
from the ellipsoid of reference can be obtained from an integration of
gravity anomalies over the surface of the earth. Vening Meinesz further
derived formulae for the calculation of components of the deflection of
the vertical.

Geometrical geodesy, which was based on astronomical-geodetic methods,
could give information only concerning the external form of the figure
of the earth. The gravimetric methods of physical geodesy, in
conjunction with methods such as those of seismology, enable scientists
to test hypotheses concerning the internal structure of the earth.
Heiskanen and Vening Meinesz summarize the present-day achievements of
the gravimetric method of physical geodesy by stating[113] that it
alone can give:

    1. The flattening of the reference ellipsoid.

    2. The undulations N of the geoid.

    3. The components of the deflection of the vertical [xi] and
    [eta] at any point, oceans and islands included.

    4. The conversion of existing geodetic systems to the same world
    geodetic system.

    5. The reduction of triangulation base lines from the geoid to
    the reference ellipsoid.

    6. The correction of errors in triangulation in mountainous
    regions due to the effect of the deflections of the vertical.

    7. Geophysical applications of gravity measurements, e.g., the
    isostatic study of the earth's interior and the exploration of
    oil fields and ore deposits.

With astronomical observations or with existing triangulations, the
gravimetric method can accomplish further results. Heiskanen and Vening
Meinesz state:

    It is the firm conviction of the authors that the gravimetric
    method is by far the best of the existing methods for solving
    the main problems of geodesy, i.e., to determine the shape of
    the geoid on the continents as well as at sea and to convert the
    existing geodetic systems to the world geodetic system. It can
    also give invaluable help in the computation of the reference
    ellipsoid.[114]




Summary


Since the creation of classical mechanics in the 17th century, the
pendulum has been a basic instrument for the determination of the
intensity of gravity, which is expressed as the acceleration of a freely
falling body. Basis of theory is the simple pendulum, whose time of
swing under gravity is proportional to the square root of the length
divided by the acceleration due to gravity. Since the length of a simple
pendulum divided by the square of its time of swing is equal to the
length of a pendulum that beats seconds, the intensity of gravity also
has been expressed in terms of the length of the seconds pendulum. The
reversible compound pendulum has served for the absolute determination
of gravity by means of a theory developed by Huygens. Invariable
compound pendulums with single axes also have been used to determine
relative values of gravity by comparative times of swing.

The history of gravity pendulums begins with the ball or "simple"
pendulum of Galileo as an approximation to the ideal simple pendulum.
Determinations of the length of the seconds pendulum by French
scientists culminated in a historic determination at Paris by Borda and
Cassini, from the corrected observations with a long ball pendulum. In
the 19th century, Bessel found the length of the seconds pendulum at
Königsberg and Berlin by observations with a ball pendulum and by
original theoretical considerations. During the century, however, the
compound pendulum came to be preferred for absolute and relative
determinations.

Capt. Henry Kater, at London, constructed the first convertible compound
for an absolute determination of gravity, and then he designed an
invariable compound pendulum, examples of which were used for relative
determinations at various stations in Europe and elsewhere. Bessel
demonstrated theoretically the advantages of a reversible compound
pendulum which is symmetrical in form and is hung by interchangeable
knives. The firm of A. Repsold and Sons in Hamburg constructed pendulums
from the specifications of Bessel for European gravity surveys.

Charles S. Peirce in 1875 received delivery in Hamburg of a
Repsold-Bessel pendulum for the U.S. Coast Survey and observed with it
in Geneva, Paris, Berlin, and London. Upon an initial stimulation from
Baeyer, founder of _Die Europäische Gradmessung_, Peirce demonstrated by
experiment and theory that results previously obtained with the Repsold
apparatus required correction, because of the flexure of the stand under
oscillations of the pendulum. At the Stuttgart conference of the
geodetic association in 1877, Hervé Faye proposed to solve the problem
of flexure by swinging two similar pendulums from the same support with
equal amplitudes and in opposite phases. Peirce, in 1879, demonstrated
theoretically the soundness of the method and presented a design for its
application, but the "double pendulum" was rejected at that time. Peirce
also designed and had constructed four examples of a new type of
invariable, reversible pendulum of cylindrical form which made possible
the experimental study of Stokes' theory of the resistance to motion of
a pendulum in a viscous fluid. Commandant Defforges, of France, also
designed and used cylindrical reversible pendulums, but of different
length so that the effect of flexure was eliminated in the reduction of
observations. Maj. Robert von Sterneck, of Austria-Hungary, initiated a
new era in gravity research by the invention of an apparatus with a
short pendulum for relative determinations of gravity. Stands were then
constructed in Europe on which two or four pendulums were hung at the
same time. Finally, early in the present century, Vening Meinesz found
that the Faye-Peirce method of swinging pendulums hung on a Stückrath
four-pendulum stand solved the problem of instability due to the
mobility of the soil in Holland.

The 20th century has witnessed increasing activity in the determination
of absolute and relative values of gravity. Gravimeters have been
perfected and have been widely used for rapid relative determinations,
but the compound pendulums remain as indispensable instruments.
Mendenhall's replacement of knives by planes attached to nonreversible
pendulums has been used also for reversible ones. The Geodetic Institute
at Potsdam is presently applying the Faye-Peirce method to the
reversible pendulum.[115] Pendulums have been constructed of new
materials, such as invar, fused silica, and fused quartz. Minimum
pendulums for precise relative determinations have been constructed and
used. Reversible pendulums have been made with "I" cross sections for
better stiffness. With all these modifications, however, the foundations
of the present designs of compound pendulum apparatus were created in
the 19th century.




FOOTNOTES:

[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 mémoires relatifs a la physique, publiés par la
Société française de Physique_ [hereinafter referred to as _Collection
de mémoires_]: vol. 4, _Mémoires sur le pendule, précédés d'une
bibliographie_ (Paris: Gauthier-Villars, 1889); and vol. 5, _Mémoires
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] ABBÉ 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 Caïenne_ (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 mathématiques[mathematiques] et de physique par MM. de l'Académie
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, déterminée par les
observations de Messieurs Bouguer et de La Condamine, envoyés par ordre
du Roy au Pérou, pour observer aux environs de l'equateur_ (Paris,
1749).

[13] P. L. MOREAU DE MAUPERTUIS, _La figure de la terre déterminée par
les observations de Messieurs de Maupertuis, Clairaut, Camus, Le
Monnier, l'Abbé 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 mémoires_, 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 à Saint
Domingue," _Collection de mémoires_, vol. 4, pp. 3-16.

[19] PÈRE 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, "Expériences pour connaître
la longueur du pendule qui bat les secondes à Paris," _Collection de
mémoires_, vol. 4, pp. 17-64.

[21] F. W. BESSEL, "Untersuchungen über die Länge des einfachen
Secundenpendels," _Abhandlungen der Königlichen 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 mémoires_, 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, "Méthode pour déterminer la longueur du pendule
simple qui bat les secondes," _Collection de mémoires_, vol. 4, pp.
65-76.

[26] _Collection de mémoires_, 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 mémoires_, 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 mémoires_, 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 Kübel
(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, "Expériences faites à Genève avec le pendule à
réversion," _Mémoires de la Société de Physique et d'histoire naturelle
de Genève, 1865_ (Geneva, 1866), vol. 18, p. 309.

[46] Ibid., pp. 309-416.

[47] C. CELLÉRIER, "Note sur la Mesure de la Pesanteur par le Pendule,"
_Mémoires de la Société de Physique et d'histoire naturelle de Genève,
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, _Über die Grösse und Figur der Erde_ (Berlin, 1861).

[50] _Comptes-rendus de la Conférence Géodésique Internationale réunie à
Berlin du 15-22 Octobre 1864_ (Neuchâtel, 1865).

[51] Ibid., part III, subpart E.

[52] _Bericht über die Verhandlungen der vom 30 September bis 7 October
1867 zu Berlin abgehaltenen allgemeinen Conferenz der Europäischen
Gradmessung_ (Berlin, 1868). See report of fourth session, October 3,
1867.

[53] C. BRUHNS and ALBRECHT, "Bestimmung der Länge des Secundenpendels
in Bonn, Leiden und Mannheim," _Astronomisch-Geodätische Arbeiten im
Jahre 1870_ (Leipzig: Veröffentlichungen des Königlichen Preussischen
Geodätischen Instituts, 1871).

[54] _Bericht über die Verhandlungen der vom 23 bis 28 September 1874 in
Dresden abgehaltenen vierten allgemeinen Conferenz der Europäischen
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 Europäischen 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 Europäischen 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 Europäischen Gradmessung_
(Berlin, 1877). See report of third session, October 7, 1876.

[63] _Verhandlungen der vom 27 September bis 2 Oktober 1877 zu Stuttgart
abgehaltenen fünften allgemeinen Conferenz der Europäischen Gradmessung_
(Berlin, 1878).

[64] _Verhandlung der vom 16 bis 20 September 1879 in Genf Vereinigten
Permanenten Commission der Europäischen 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'Académie des Sciences_ (Paris, 1879), vol. 89,
p. 462.

[67] _Verhandlungen der vom 13 bis 16 September 1880 zu München
abgehaltenen sechsten allgemeinen Conferenz der Europäischen
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 Europäischen Gradmessung_
(Berlin, 1883).

[71] _Verhandlungen der vom 15 bis 24 Oktober 1883 zu Rom abgehaltenen
siebenten allgemeinen Conferenz der Europäischen Gradmessung_ (Berlin,
1884). Gen. Cutts attended for the U.S. Coast and Geodetic Survey.

[72] Ibid., app. 6. See also, _Zeitschrift für 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'Académie des Sciences_ (Paris, 1880), vol. 90,
p. 1401. HERVÉ FAYE's report, dated June 21, 1880, is in the same
_Comptes-rendus_, p. 1463.

[81] COMMANDANT C. DEFFORGES, "Sur l'Intensité absolue de la pesanteur,"
_Journal de Physique_ (1888), vol. 17, pp. 239, 347, 455. See also,
DEFFORGES, "Observations du pendule," _Mémorial du Dépôt général 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.
Militär-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 für Instrumentenkunde_
(July 1896), vol. 16, p. 193.

[87] DR. R. SCHUMANN, "Über eine Methode, das Mitschwingen bei relativen
Schweremessungen zu bestimmen," _Zeitschrift für 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, "Über die Verwendung zweier Pendel auf gemeinsamer
Unterlage zur Bestimmung der Mitschwingung," _Zeitschrift für Mathematik
und Physik_ (1899), vol. 44, p. 44.

[89] P. FURTWÄNGLER, "Über die Schwingungen zweier Pendel mit annähernd
gleicher Schwingungsdauer auf gemeinsamer Unterlage," _Sitzungsberichte
der Königlicher 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 für 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. KÜHNEN and P. FURTWÄNGLER, _Bestimmung der Absoluten
Grösze der Schwerkraft zu Potsdam mit Reversionspendeln_ (Berlin:
Veröffentlichungen des Königlichen Preussischen Geodätischen Instituts,
1906), new ser., no. 27.

[97] Reported by Dr. F. Kühnen 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, _Beiträge zur Theorie des Reversionspendels_
(Potsdam: Veröffentlichungen des Königlichen Preussischen Geodätischen
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 Géodésique_ (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 Géodésique_ (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 Géodésique_ (March 1, 1959),
no. 51, p.72.


    U.S. GOVERNMENT PRINTING OFFICE: 1965

    For sale by the Superintendent of Documents, U.S. Government Printing
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INDEX


    Airy, G. B., 319, 324, 332

    Albrecht, Karl Theodore, 322, 338

    Al-Mamun, seventh calif of Bagdad, 306

    Almansi, Emilio, 339

    Aristotle, 306


    Baeyer, J. J., 321, 322, 324, 338, 346

    Baily, Francis, 317

    Basevi, James Palladio, 345

    Berroth, A., 342

    Bessel, Friedrich Wilhelm, 313, 314, 319, 320, 324, 325, 338, 346

    Biot, Jean Baptiste, 325, 329

    Bohnenberger, Johann Gottlieb Friedrich, 315

    Borda, J. C., 311, 312, 315, 325, 329, 346

    Boscovitch, Père R. J., 310, 311

    Bouguer, Pierre, 307, 309, 327, 343, 345

    Brahe, Tycho, 306

    Brown, E. J., 334, 339

    Browne, Henry, 304, 314

    Bruhns, C., 322, 324, 338

    Brunner Brothers (Paris), 329


    Cassini, Giovanni-Domenico, 306, 307

    Cassini, Jacques, 306

    Cassini de Thury, J. D., 311, 312, 315, 325, 329, 346

    Cellérier, Charles, 320, 321, 325, 326, 329, 336

    Clairaut, Alexis Claude, 308, 309, 343, 345

    Clark, J. S., 342

    Clarke, A. R., 345

    Colbert, Jean Baptiste, 306

    Cook, A. H., 342

    Cook, Guy S., 339, 342


    Defforges, C., 314, 329, 346

    De Freycinet, Louis Claude de Saulses, 317

    De la Hire, Gabriel Philippe, 306

    De Prony, M. G., 314

    Dryden, Hugh L., 342

    Du Buat, L. G., 314

    Duperry, Capt. Louis Isidore, 317


    Eratosthenes, 306, 308, 342

    Eudoxus of Cnidus, 306


    Faye, Hervé, 325, 336, 346, 347

    Fernel, Jean, 306

    Furtwängler, P., 337


    Galilei, Galileo, 304, 305, 346

    Gauss, C. F., 320

    Gautier, P., 339

    Godin, Louis, 307

    Goldingham, John, 316, 345

    Greely, A. W., 329

    Gulf Oil and Development Company, 338


    Haid, M., 335

    Hall, Basil, 316

    Heaviside, W. J., 321, 345

    Heiskanen, W. A., 338, 345, 346

    Helmert, F. R., 338, 339

    Helmholtz, Hermann von, 326

    Herschel, John, 319, 328, 345

    Heyl, Paul R., 339, 342

    Hirsch, Adolph, 322, 324

    Huygens, Christiaan, 304, 305, 307, 314, 342, 346


    Ibañez, Carlos, 325


    Jeffreys, Sir Harold, 342

    Jones, Thomas, 318


    Kater, Henry, 304, 314, 325, 327, 329, 345, 346

    Kühnen, F., 338, 339


    La Condamine, Charles Marie de, 307, 310, 311, 343

    Laplace, Marquis Pierre Simon de, 309, 313, 320

    Lorenzoni, Giuseppe, 336, 339

    Lütke, Count Feodor Petrovich, 316, 345


    Maupertius, P. L. Moreau de, 308, 343

    Maxwell, James Clerk, 324

    Medi, Enrico, 342

    Mendenhall, Thomas Corwin, 319, 331, 332, 334, 347

    Mersenne, P. Marin, 305


    Newton, Sir Isaac, 303, 307, 308, 342, 343

    Norwood, Richard, 306


    Oppolzer, Theodor von, 322, 324


    Patterson, Carlile Pollock, 325, 326

    Peirce, Charles Sanders, 314, 322, 332, 336, 342, 345

    Peters, C. A. F., 322, 324

    Picard, Abbé Jean, 306, 308, 342

    Plantamour, E., 319, 324

    Posidonius, 306

    Preston, E. D., 328, 329

    Putnam, G. R., 339

    Pythagoras, 306


    Repsold, A., and Sons (Hamburg), 320, 322, 338, 339, 346

    Richer, Jean, 307, 342


    Sabine, Capt. Edward, 315, 325, 329, 345

    Sawitsch, A., 321, 322

    Schumacher, H. C., 320

    Schumann, R., 335, 336

    Snell, Willebrord, 306

    Sterneck, Robert von, 331, 332, 335, 338, 346

    Stokes, George Gabriel, 324, 328, 329, 345, 346


    Ulloa, Antonio de, 308


    Vening Meinesz, F. A., 337, 338, 345

    Volet, Charles, 342


    Wilkes, Charles, 317, 318




       *       *       *       *       *




Transcriber's note:

Footnotes have been moved to the end of the paper. Illustrations and the
GLOSSARY OF GRAVITY TERMINOLOGY section have been moved to avoid breaks
in paragraphs. Minor punctuation errors have been corrected without
note. Typographical errors and inconsistencies have been corrected as
follows:

  P. 320 'difference T_{1} - T_{2} is sufficiently'--had 'sufficlently.'
  P. 321 'faites à Genève avec le pendule à réversion'--had 'reversion.'
  P. 326 'Schwere mit Hilfe verschiedener Apparate'--had 'verschiedene.'
  P. 328 'between the yard and the meter.'--closing quote mark deleted.
  P. 334 'Mendenhall apparatus were part of'--'was' changed to 'were.'
  P. 342 'of the Geodetic Institute at Potsdam'--had 'Postdam.'
  P. 345 'The gravimetric methods of physical'--had 'mtehods.'
  Footnote 1 'Société française de Physique'--had 'Française.'
  Footnote 3 'Cogitata physico-mathematica'--had 'physica.'
  Footnote 10 'mathématiques et de physique par MM. de l'Académie
      Royale'--had 'mathematiques,' 'Royal.'
  Footnote 12 'par ordre du Roy au Pérou, pour observer'--had 'Perou,
      pour observir.'
  Footnote 19 'Opticam et Astronomiam'--had 'Astronomian.'
  Footnote 20 'connaître la longueur du pendule qui'--had 'connaitre la
      longuer.'
  Footnote 21 'Abhandlungen der Königlichen Akademie'--had 'Königliche.'
  Footnote 25 'pour déterminer la longueur du pendule'--had 'longeur.'
  Footnote 41 'Survey of India (Calcutta, 1879)'-- had 'Surey.'
  Footnotes 45 and 47 'Société de Physique et d'histoire'--had
      'd'historire.'
  Footnote 49 'Über die Grösse und Figur der Erde'--had 'Grosse.'
  Footnote 53 'Bestimmung der Länge'--had 'Lange';
      'Astronomisch-Geodätische Arbeiten'--had 'Astronomische';
      'Veröffentlichungen des Königlichen'--had 'Königliche.'
  Footnote 55 '(1768), vol. 58, pp. 329-335.'--had '329-235.'
  Footnote 66 'Comptes-rendus de l'Académie'--had 'L'Académie.'
  Footnote 81 'Sur l'Intensité absolue'--had 'l'Intensite.'
  Footnote 89 'Sitzungsberichte der Königlicher'--had 'Königliche.'
  Footnote 100 'Veröffentlichungen des Königlichen' had
      'Veröffentlichungen Königliche.'

Capitalisation of 'Von'/'von' has been regulaized to 'von' for all
personal names, except at the beginning of a sentence, and when
referring to the Von Sterneck pendulum.