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Transcriber's note:
This is Paper 44 from the _Smithsonian Institution United
States National Museum Bulletin 240_, comprising Papers 34-44,
which will also be available as a complete e-book.
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from the _Bulletin_ are included in each single-paper e-book.
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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
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The _Proceedings_, begun in 1878, are intended for the publication, in
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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
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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 Caienne_ (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 Abbe 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 deg., 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 deg. 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 deg. 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 deg. in length.[13] The northern arc of 1 deg.
was found to be longer than the Peruvian arc of 1 deg., 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, _Theorie 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,
_Memoires publies par la Societe francaise de Physique_, vol. 4.)]
[Illustration: Figure 6.--THE RESULT of early pendulum experiments was
often expressed in terms of the length of a pendulum which would have a
period of one second and was called "the seconds pendulum." In 1792, J.
C. Borda and J. D. Cassini determined the length of the seconds pendulum
at Paris with this apparatus. The pendulum consists of a platinum ball
about 1-1/2 inches in diameter, suspended by a fine iron wire. The
length, about 12 feet, was such that its period would be nearly twice as
long as that of the pendulum of the clock (A). The interval between
coincidences was determined by observing, through the telescope at the
left, the times when the two pendulums emerge together from behind the
screen (M). The exact length of the pendulum was measured by a platinum
scale (not shown) equipped with a vernier and an auxiliary copper scale
for temperature correction.
When, at the end of the 18th century, the French revolutionary
government established the metric system of weights and measures, the
length of the seconds pendulum at Paris was considered, but not adopted,
as the unit of length. (Plate 2, _Memoires publies par la Societe
francaise de Physique_, vol. 4.)]
The practice with the "simple" pendulum on the part of Picard, Bouguer,
La Condamine and others in France culminated in the work of Borda and
Cassini in 1792 at the observatory in Paris[20] (fig. 6). The
experiments were undertaken to determine whether or not the length of
the seconds pendulum should be adopted as the standard of length by the
new government of France. The bob consisted of a platinum ball 16-1/6
Paris lines in diameter, and 9,911 grains (slightly more than 17 ounces)
in weight. The bob was held to a brass cup covering about one-fifth of
its surface by the interposition of a small quantity of grease. The cup
with ball was hung by a fine iron wire about 12 Paris feet long. The
upper end of the wire was attached to a cylinder which was part of a
wedge-shaped knife edge, on the upper surface of which was a stem on
which a small adjustable weight was held by a screw thread. The knife
edge rested on a steel plate. The weight on the knife-edge apparatus was
adjusted so that the apparatus would vibrate with the same period as the
pendulum. Thus, the mass of the suspending apparatus could be neglected
in the theory of motion of the pendulum about the knife edge.
[Illustration: Figure 7.--RESULTS OF EXPERIMENTS in the determination of
the length of the seconds pendulum at Koenigsberg by a new method were
reported by F. W. Bessel in 1826 and published in 1828. With this
apparatus, he obtained two sets of data from the same pendulum, by using
two different points of suspension. The pendulum was about 10 feet long.
The distance between the two points of suspension (_a_ and _b_) was 1
toise (about six feet). A micrometric balance (_c_) below the bob was
used to determine the increase in length due to the weight of the bob.
He projected the image of the clock pendulum (not shown) onto the
gravity pendulum by means of a lens, thus placing the clock some
distance away and eliminating the disturbing effect of its motion.
(Portion of plate 6, _Memoires publies par la Societe francaise de
Physique_, vol. 4.)]
In the earlier suspension from jaws there was uncertainty as to the
point about which the pendulum oscillated. Borda and Cassini hung their
pendulum in front of a seconds clock and determined the time of swing by
the method of coincidences. The times on the clock were observed when
the clock gained or lost one complete vibration (two swings) on the
pendulum. Suppose that the wire pendulum makes n swings while the clock
makes 2n + 2. If the clock beats seconds exactly, the time of one
complete vibration is 2 seconds, and the time of swing of the wire
pendulum is T = (2n + 2)/n = 2(1 + 1/n). An error in the time caused by
uncertainty in determining the coincidence of clock and wire pendulum is
reduced by employing a long interval of observation 2n. The whole
apparatus was enclosed in a box, in order to exclude disturbances from
currents of air. Corrections were made for buoyancy, for amplitude of
swing and for variations in length of the wire with temperature. The
final result was that the length of the seconds pendulum at the
observatory in Paris was determined to be 440.5593 Paris lines, or
993.53 mm., reduced to sea level 993.85 mm. Some years later the methods
of Borda were used by other French investigators, among whom was Biot
who used the platinum ball of Borda suspended by a copper wire 60 cm.
long.
Another historic "simple" pendulum was the one swung by Bessel (fig. 7)
for the determination of gravity at Koenigsberg 1825-1827.[21] The
pendulum consisted of a ball of brass, copper, or ivory that was
suspended by a fine wire, the upper end of which was wrapped and
unwrapped on a horizontal cylinder as support. The pendulum was swung
first from one point and then from another, exactly a "toise de
Peru"[22] higher up, the bob being at the same level in each case (fig.
7). Bessel found the period of vibration of the pendulum by the method
of coincidences; and in order to avoid disturbances from the comparison
clock, it was placed at some distance from the pendulum under
observation.
Bessel's experiments were significant in view of the care with which he
determined the corrections. He corrected for the stiffness of the wire
and for the lack of rigidity of connection between the bob and wire. The
necessity for the latter correction had been pointed out by Laplace, who
showed that through the circumstance that the pull of the wire is now on
one side and now on the other side of the center of gravity, the bob
acquires angular momentum about its center of gravity, which cannot be
accounted for if the line of the wire, and therefore the force that it
exerts, always passed through the center. In addition to a correction
for buoyancy of the air considered by his predecessors, Bessel also took
account of the inertia of the air set in motion by the pendulum.
[Illustration: Figure 8.--MODE OF SUSPENSION of Bessel's pendulum is
shown here. The iron wire is supported by the thumbscrew and clamp at
the left, but passes over a pin at the center, which is actually the
upper terminal of the pendulum. Bessel found this "cylinder of
unrolling" superior to the clamps and knife edges of earlier pendulums.
The counterweight at the right is part of a system for supporting the
scale in such a way that it is not elongated by its own weight.
With this apparatus, Bessel determined the ratio of the lengths of the
two pendulums and their times of vibration. From this the length of the
seconds pendulum was calculated. His method eliminated the need to take
into account such sources of inaccuracy as flexure of the pendulum wire
and imperfections in the shape of the bob. (Portion of plate 7,
_Memoires publies par la Societe francaise de Physique_, vol. 4.)]
[Illustration: Figure 9.--FRIEDRICH WILHELM BESSEL (1784-1846), German
mathematician and astronomer. He became the first superintendent of the
Prussian observatory established at Koenigsberg in 1810, and remained
there during the remainder of his life. So important were his many
contributions to precise measurement and calculation in astronomy that
he is often considered the founder of the "modern" age in that science.
This characteristic also shows in his venture into geodesy, 1826-1830,
one product of which was the pendulum experiment reported in this
article.]
The latter effect had been discovered by Du Buat in 1786,[23] but his
work was unknown to Bessel. The length of the seconds pendulum at
Koenigsberg, reduced to sea level, was found by Bessel to be 440.8179
lines. In 1835, Bessel determined the intensity of gravity at a site in
Berlin where observations later were conducted in the Imperial Office of
Weights and Measures by Charles S. Peirce of the U.S. Coast Survey.
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. Luetke 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,
_Memoires publies par la Societe francaise 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, Ile 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.