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+The Project Gutenberg eBook, Development of Gravity Pendulums in the 19th
+Century, by Victor Fritz Lenzen and Robert P. Multhauf
+
+
+This eBook is for the use of anyone anywhere at no cost and with
+almost no restrictions whatsoever.  You may copy it, give it away or
+re-use it under the terms of the Project Gutenberg License included
+with this eBook or online at www.gutenberg.org
+
+
+
+
+
+Title: Development of Gravity Pendulums in the 19th Century
+       Contributions from the Museum of History and Technology, Papers 34-44 On Science and Technology, Smithsonian Institution, 1966
+
+
+Author: Victor Fritz Lenzen and Robert P. Multhauf
+
+
+
+Release Date: January 21, 2011  [eBook #35024]
+
+Language: English
+
+Character set encoding: ISO-8859-1
+
+
+***START OF THE PROJECT GUTENBERG EBOOK DEVELOPMENT OF GRAVITY PENDULUMS
+IN THE 19TH CENTURY***
+
+
+E-text prepared by Chris Curnow, Joseph Cooper, Louise Pattison, and the
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+
+
+
+Note: Project Gutenberg also has an HTML version of this
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+      See 35024-h.htm or 35024-h.zip:
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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.
+
+      The front material, introduction and relevant index entries
+      from the _Bulletin_ are included in each single-paper e-book.
+
+      Mathematical notation used in this e-text:
+
+        1. Greek letters are represented by the name of the letter
+           in square brackets; _e.g._, [pi].
+
+        2. Subscripts are denoted by underscore followed by the
+           subscript in curly braces; _e.g._, T_{n}. Superscripts
+           are denoted by a caret followed by the superscript in
+           curly braces; _e.g._, T^{n}. To avoid possible confusion,
+           subscripted variables which are raised to a power are
+           enclosed in brackets, thus (T_{1})^{2} represents
+           'T one squared'.
+
+        3. Square root is denoted by [sqrt].
+
+        4. To improve readability, italic markup (underscores
+           enclosing text) has been omitted from letters used in
+           mathematical formulae, and some equations have been set
+           'out of line'.
+
+      Please see the end of the book for a list of corrections.
+
+
+
+
+
+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
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+
+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
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+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
+    Office Washington, D.C., 20402--Price 70 cents.
+
+
+
+
+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.
+
+
+
+***END OF THE PROJECT GUTENBERG EBOOK DEVELOPMENT OF GRAVITY PENDULUMS IN
+THE 19TH CENTURY***
+
+
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+<h1 class="pg">The Project Gutenberg eBook, Development of Gravity Pendulums in the 19th
+Century, by Victor Fritz Lenzen and Robert P. Multhauf</h1>
+<pre>
+This eBook is for the use of anyone anywhere at no cost and with
+almost no restrictions whatsoever.  You may copy it, give it away or
+re-use it under the terms of the Project Gutenberg License included
+with this eBook or online at <a href = "http://www.gutenberg.org">www.gutenberg.org</a></pre>
+<p>Title: Development of Gravity Pendulums in the 19th Century</p>
+<p>       Contributions from the Museum of History and Technology, Papers 34-44 On Science and Technology, Smithsonian Institution, 1966</p>
+<p>Author: Victor Fritz Lenzen and Robert P. Multhauf</p>
+<p>Release Date: January 21, 2011  [eBook #35024]</p>
+<p>Language: English</p>
+<p>Character set encoding: ISO-8859-1</p>
+<p>***START OF THE PROJECT GUTENBERG EBOOK DEVELOPMENT OF GRAVITY PENDULUMS IN THE 19TH CENTURY***</p>
+<p>&nbsp;</p>
+<h3>E-text prepared by Chris Curnow, Joseph Cooper, Louise Pattison,<br />
+    and the Online Distributed Proofreading Team<br />
+    (http://www.pgdp.net)</h3>
+<p>&nbsp;</p>
+
+<div class="tnote">
+<h3>Transcriber&rsquo;s Note:</h3>
+
+<p>This is Paper 44 from the <i>Smithsonian Institution United States
+National Museum Bulletin 240</i>, comprising Papers 34-44, which will
+also be available as a complete e-book.</p>
+
+<p>The front material, introduction and relevant index entries from
+the <i>Bulletin</i> are included in each single-paper e-book.</p>
+
+<p><a href="#corrections_44">Corrections</a> to typographical errors are underlined
+<ins class="mycorr" title="Original: like thsi">like this</ins>.
+Hover the cursor over the marked text to view the original text.</p>
+</div>
+<p>&nbsp;</p>
+<hr class="full" />
+<p>&nbsp;</p>
+<p>&nbsp;</p>
+<p>&nbsp;</p>
+
+<h1>SMITHSONIAN INSTITUTION<br />
+UNITED STATES NATIONAL MUSEUM<br />
+BULLETIN 240</h1>
+
+<div class="figright">
+  <img src="images/cover.png" alt="Smithsonian Press Logo" title="" />
+</div>
+
+<p class="right" style="clear:both;">SMITHSONIAN PRESS<br /></p>
+
+<p>MUSEUM OF HISTORY AND TECHNOLOGY</p>
+
+<p style="font-size: 2em; font-weight: bold;" class="smcap">Contributions<br />
+From the<br />
+Museum<br />
+of History and<br />
+Technology</p>
+
+<p style="font-size: 1.25em;"><em>Papers 34-44<br />
+On Science and Technology</em></p>
+
+<p>SMITHSONIAN INSTITUTION &middot; WASHINGTON, D.C. 1966</p>
+
+<hr style="width: 65%;" />
+
+<p class="center" style="font-size: 1.25em;"><em>Publications of the United States National Museum</em></p>
+
+<p>The scholarly and scientific publications of the United States National Museum
+include two series, <cite>Proceedings of the United States National Museum</cite> and <cite>United States
+National Museum Bulletin</cite>.</p>
+
+<p>In these series, the Museum publishes original articles and monographs dealing
+with the collections and work of its constituent museums&mdash;The Museum of Natural
+History and the Museum of History and Technology&mdash;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.</p>
+
+<p>The <cite>Proceedings</cite>, 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.</p>
+
+<p>In the <cite>Bulletin</cite> 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. <cite>Bulletins</cite> 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 <cite>Bulletin</cite> series under the heading <cite>Contributions from the United States
+National Herbarium</cite>, and since 1959, in <cite>Bulletins</cite> titled &ldquo;Contributions from the Museum
+of History and Technology,&rdquo; have been gathered shorter papers relating to the collections
+and research of that Museum.</p>
+
+<p>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.</p>
+
+<p class="right"><span class="smcap">Frank A. Taylor</span><br />
+<em>Director, United States National Museum</em></p>
+
+<hr style="width: 65%;" />
+<p><span class="pagenum"><a name="Page_301" id="Page_301">[Pg 301]</a></span></p>
+
+<h1><a name="Paper_44" id="Paper_44"></a><span class="smcap">Contributions from<br />
+The Museum of History and Technology</span>:<br />
+<span class="smcap">Paper</span> 44<br />
+<br /><br />
+<span class="smcap">Development of Gravity Pendulums in the 19th Century<br />
+<br /></span>
+</h1>
+<p><span class="rnum" style="font-size: larger;"><i>Victor F. Lenzen</i> and <i>Robert P. Multhauf</i></span><br /><br/>
+</p>
+<p><span class="rnum">GALILEO, HUYGENS, AND NEWTON &nbsp;<a href="#Page_304">304</a></span><br />
+</p>
+<p><span class="rnum">FIGURE OF THE EARTH &nbsp;<a href="#Page_306">306</a></span><br />
+</p>
+<p><span class="rnum">EARLY TYPES OF PENDULUMS &nbsp;<a href="#Page_309">309</a></span><br />
+</p>
+<p><span class="rnum">KATER&rsquo;S CONVERTIBLE AND INVARIABLE PENDULUMS &nbsp;<a href="#Page_314">314</a></span><br />
+</p>
+<p><span class="rnum">REPSOLD-BESSEL REVERSIBLE PENDULUM &nbsp;<a href="#Page_320">320</a></span><br />
+</p>
+<p><span class="rnum">PEIRCE AND DEFFORGES INVARIABLE, REVERSIBLE PENDULUMS &nbsp;<a href="#Page_327">327</a></span><br />
+</p>
+<p><span class="rnum">VON STERNECK AND MENDENHALL PENDULUMS &nbsp;<a href="#Page_331">331</a></span><br />
+</p>
+<p><span class="rnum">ABSOLUTE VALUE OF GRAVITY AT POTSDAM &nbsp;<a href="#Page_338">338</a></span><br />
+</p>
+<p><span class="rnum">APPLICATION OF GRAVITY SURVEYS &nbsp;<a href="#Page_342">342</a></span><br />
+</p>
+<p><span class="rnum">SUMMARY &nbsp;<a href="#Page_346">346</a></span><br />
+</p>
+<p><span class="pagenum"><a name="Page_302" id="Page_302">[Pg 302]</a></span></p>
+
+
+
+
+<p style="clear:both;"><br /><br />
+<i>Victor F. Lenzen</i> and<br />
+<i>Robert P. Multhauf</i><br />
+</p>
+
+<h2>
+DEVELOPMENT OF GRAVITY PENDULUMS<br />
+IN THE 19th CENTURY</h2>
+
+<div class="figcenter" style="width: 600px;"><a name="fig_1" id="fig_1"></a>
+<img src="images/i002.png" width="600" height="456" alt="Figure 1." title="Figure 1." />
+
+<p class="caption2">Figure 1.&mdash;<span class="smcap">A study of the figure of the earth was</span> one of the earliest projects of the
+French Academy of Sciences. In order to test the effect of the earth&rsquo;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.</p>
+</div>
+
+<p><span class="pagenum"><a name="Page_303" id="Page_303">[Pg 303]</a></span></p>
+
+<div class="blockquotn"><p><i>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.</i></p>
+
+<p><span class="smcap">THE AUTHORS</span>: <i>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&rsquo;s Museum of History and Technology.</i></p></div>
+
+<p>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&rsquo;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&rsquo;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.</p>
+
+<p>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 <i>g</i> is the intensity of gravity
+at some reference station, and &#916;<i>g</i> is the difference
+between intensities at two stations, the values of
+gravity in geodetic calculations enter as ratios (&#916;<i>g</i>)/<i>g</i>
+over the surface of the earth. Gravimetric investigations
+in conjunction with other forms of geophysical
+investigation, such as seismology, furnish data to test<span class="pagenum"><a name="Page_304" id="Page_304">[Pg 304]</a></span>
+hypotheses concerning the internal structure of the
+earth.</p>
+
+<p>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.</p>
+
+<p>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 &ldquo;simple&rdquo; 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 <i>l</i> 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&rsquo;s laws of motion and the
+hypothesis that weight is proportional to mass, is
+<i>T</i> = <span style="white-space: nowrap;">&#960;&#8730;(<i>l</i>/<i>g</i>).</span></p>
+
+<p>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
+&ldquo;convertible&rdquo;) 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.</p>
+
+
+
+<hr style="width: 65%;" />
+<h3>Galileo, Huygens, and Newton</h3>
+
+
+<p>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.<a name="FNanchor_1_1" id="FNanchor_1_1"></a><a href="#Footnote_1_1" class="fnanchor">[1]</a> 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.<a name="FNanchor_2_2" id="FNanchor_2_2"></a><a href="#Footnote_2_2" class="fnanchor">[2]</a> The lamp was only a rough
+approximation to a simple pendulum, but Galileo
+later performed more accurate experiments with a
+&ldquo;simple&rdquo; 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 &ldquo;simple&rdquo;
+pendulum varies as the square root of its length, a<span class="pagenum"><a name="Page_305" id="Page_305">[Pg 305]</a></span>
+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&rsquo;s experimental
+results are important historically, but have required
+correction in the light of subsequent measurements
+of greater precision.</p>
+
+<p>Mersenne in 1644 made the first determination of
+the length of the seconds pendulum,<a name="FNanchor_3_3" id="FNanchor_3_3"></a><a href="#Footnote_3_3" class="fnanchor">[3]</a> 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 <i>Horologium
+oscillatorium</i> ... (1673) set forth the theory of the
+compound pendulum.<a name="FNanchor_4_4" id="FNanchor_4_4"></a><a href="#Footnote_4_4" class="fnanchor">[4]</a></p>
+
+<p>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.</p>
+
+<p>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 <i>Principia</i>
+(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.</p>
+
+<div class="sidebar">
+<p class="center smcap">Glossary of Gravity Terminology</p>
+
+<p class="hang">ABSOLUTE GRAVITY: the value of the acceleration of gravity,
+also expressed by the length of the seconds pendulum.</p>
+
+<p class="hang">RELATIVE GRAVITY: the value of the acceleration of gravity
+relative to the value at some standard point.</p>
+
+<p class="hang">SIMPLE PENDULUM: see theoretical pendulum.</p>
+
+<p class="hang">THEORETICAL PENDULUM: a heavy bob (point-mass) at the
+end of a weightless rod.</p>
+
+<p class="hang">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.)</p>
+
+<p class="hang">GRAVITY PENDULUM: a precisely made pendulum used for
+the measurement of gravity.</p>
+
+<p class="hang">COMPOUND PENDULUM: a pendulum in which the supporting
+rod is not weightless; in other words, any actual pendulum.</p>
+
+<p class="hang">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.</p>
+
+<p class="hang">REVERSIBLE PENDULUM: a convertible pendulum which is
+also symmetrical in form.</p>
+
+<p class="hang">INVARIABLE PENDULUM: a compound pendulum with only
+one knife edge, used for relative measurement of gravity.</p>
+
+</div>
+
+<p>From Newton&rsquo;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
+<i>T</i> = <span class="nowrap">&#960;&#8730;(<i>l</i>/<i>g</i>)</span>. If a simple pendulum beats seconds,
+1 = <span class="nowrap">&#960;&#8730;(&#955;/<i>g</i>)</span>, where &#955; is the length of the seconds
+pendulum. From <i>T</i> = <span class="nowrap">&#960;&#8730;(<i>l</i>/<i>g</i>)</span> and 1 = <span class="nowrap">&#960;&#8730;(&#955;/<i>g</i>)</span>, it follows
+that &#955; = <span class="nowrap"><i>l</i>/<i>T</i><sup>2</sup></span>. Then <i>g</i> = <span class="nowrap">&#960;<sup>2</sup>&#955;</span>. 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 <i>g</i>, for which the unit is
+the gal, or one centimeter per second per second.<span class="pagenum"><a name="Page_306" id="Page_306">[Pg 306]</a></span></p>
+
+<div class="figcenter" style="width: 800px;"><a name="fig_2" id="fig_2"></a>
+<img src="images/i010.png" width="800" height="275" alt="Figure 2." title="Figure 2." />
+
+<p class="caption2">Figure 2.&mdash;<span class="smcap">This drawing, from Richer&rsquo;s</span> <i>Observations astronomiques et physiques faites en l&rsquo;isle de
+Caïenne</i> (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.</p>
+</div>
+
+
+
+<hr style="width: 65%;" />
+<h3>Figure of the Earth</h3>
+
+
+<p>A principal contribution of the pendulum as a
+physical instrument has been the determination of the
+figure of the earth.<a name="FNanchor_5_5" id="FNanchor_5_5"></a><a href="#Footnote_5_5" class="fnanchor">[5]</a> 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.</p>
+
+<p>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.</p>
+
+<p>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.</p>
+
+<p>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.<a name="FNanchor_6_6" id="FNanchor_6_6"></a><a href="#Footnote_6_6" class="fnanchor">[6]</a> He had been
+commissioned by the newly established Academy of
+Sciences to measure an arc corresponding to an angle
+of 1°, 22&#8242;, 55&#8243; 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<span class="pagenum"><a name="Page_307" id="Page_307">[Pg 307]</a></span>
+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<a name="FNanchor_7_7" id="FNanchor_7_7"></a><a href="#Footnote_7_7" class="fnanchor">[7]</a> 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. <a href="#fig_3">3</a>).</p>
+
+<div class="figright" style="width: 300px;"><a name="fig_3" id="fig_3"></a>
+<img src="images/i013.png" width="300" height="341" alt="Figure 3." title="Figure 3." />
+
+<p class="caption2">Figure 3.&mdash;<span class="smcap">Measurements of the length</span> 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&rsquo;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&rsquo;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&mdash;then part of Peru&mdash;and Lapland).</p>
+</div>
+<p>Meanwhile, Richer in 1672 had been sent to
+Cayenne, French Guiana, to make astronomical
+observations and to measure the length of the seconds
+pendulum.<a name="FNanchor_8_8" id="FNanchor_8_8"></a><a href="#Footnote_8_8" class="fnanchor">[8]</a> 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. <a href="#fig_1">1</a>). He also fitted up a &ldquo;simple&rdquo; 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 &ldquo;simple&rdquo; pendulum which beat
+seconds at Cayenne was 1-1/4 Paris lines<a name="FNanchor_9_9" id="FNanchor_9_9"></a><a href="#Footnote_9_9" class="fnanchor">[9]</a> shorter than
+the length of the seconds pendulum at Paris. Huygens
+explained the reduction in the length of the seconds
+pendulum&mdash;and, therefore, the lesser intensity of
+gravity at the equator with respect to the value at
+Paris&mdash;in terms of his theory of centripetal force as
+applied to the rotation of the earth and pendulum.<a name="FNanchor_10_10" id="FNanchor_10_10"></a><a href="#Footnote_10_10" class="fnanchor">[10]</a></p>
+
+<p>A more complete theory was given by Newton in
+the <i>Principia</i>.<a name="FNanchor_11_11" id="FNanchor_11_11"></a><a href="#Footnote_11_11" class="fnanchor">[11]</a> 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&rsquo;s measured length was explained
+by Newton in terms of the expansion of the scale
+with higher temperatures near the equator.</p>
+
+<p>Newton&rsquo;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<span class="pagenum"><a name="Page_308" id="Page_308">[Pg 308]</a></span>
+in Peru, and, with the help of the Spaniard Ullo,
+measured a meridian arc of about 3°7&#8242; near Quito,
+now in Ecuador.<a name="FNanchor_12_12" id="FNanchor_12_12"></a><a href="#Footnote_12_12" class="fnanchor">[12]</a> 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.<a name="FNanchor_13_13" id="FNanchor_13_13"></a><a href="#Footnote_13_13" class="fnanchor">[13]</a> 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.</p>
+
+<div class="figcenter" style="width: 600px;"><a name="fig_4" id="fig_4"></a>
+<img src="images/i015.png" width="600" height="390" alt="Figure 4." title="Figure 4." />
+
+<p class="caption2">Figure 4.&mdash;<span class="smcap">The direct use of a clock</span> 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 &ldquo;simple&rdquo;) pendulum,
+in which the mass is concentrated in the bob, and the supporting rod is weightless.</p>
+
+<p class="caption2">After 1735, the clock was used only to time the swing of a detached pendulum, by the method of &ldquo;coincidences.&rdquo;
+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.</p>
+</div>
+
+<p>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.</p>
+
+<p>An epoch-making work during the ellipsoidal era<span class="pagenum"><a name="Page_309" id="Page_309">[Pg 309]</a></span>
+of geodesy was Clairaut&rsquo;s treatise, <i>Théorie de la figure
+de la terre</i>.<a name="FNanchor_14_14" id="FNanchor_14_14"></a><a href="#Footnote_14_14" class="fnanchor">[14]</a> 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 &#947;<sub>E</sub>, &#947;<sub>P</sub> are the values of gravity at the
+equator and pole, respectively, and <i>c</i> the centrifugal
+force at the equator divided by &#947;<sub>E</sub>, then the ellipticity
+&#945; = <span class="nowrap">(5/2)<i>c</i> - (&#947;<sub>P</sub> - &#947;<sub>E</sub>)/&#947;<sub>E</sub>.</span></p>
+
+<p>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.<a name="FNanchor_15_15" id="FNanchor_15_15"></a><a href="#Footnote_15_15" class="fnanchor">[15]</a> It follows from Clairaut&rsquo;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 <i>c</i>.
+Observations with nonreversible, invariable compound
+pendulums have contributed to the application
+of Clairaut&rsquo;s theorem in its original and contemporary
+extended form for the determination of the figure and
+gravity field of the earth.</p>
+
+
+
+<hr style="width: 65%;" />
+<h3>Early Types of Pendulums</h3>
+
+
+<p>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 &ldquo;simple&rdquo; 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
+&ldquo;simple&rdquo; 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).</p>
+
+<p>A celebrated set of experiments with a &ldquo;simple&rdquo;
+pendulum was conducted by Bouguer<a name="FNanchor_16_16" id="FNanchor_16_16"></a><a href="#Footnote_16_16" class="fnanchor">[16]</a> 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&rsquo;s
+method was that he employed an &ldquo;invariable&rdquo;
+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.</p>
+
+<p>Since <i>T</i> = <span class="nowrap">&#960;&#8730;(<i>l</i>/<i>g</i>)</span>, it follows that <span class="nowrap"><i>T</i><sub>1</sub><sup>2</sup>/<i>T</i><sub>2</sub><sup>2</sup></span> = <span class="nowrap"><i>g</i><sub>2</sub>/<i>g</i><sub>1</sub></span>.
+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 <i>T</i><sub>1</sub> is the time of swing at a
+station where the intensity of gravity is <i>g</i>, and <i>T</i><sub>2</sub>
+is the time at a station where the intensity is <i>g</i> + &#916;<i>g</i>,
+then <span class="nowrap">(&#916;<i>g</i>)/<i>g</i></span> = <span class="nowrap">(<i>T</i><sub>1</sub><sup>2</sup>/<i>T</i><sub>2</sub><sup>2</sup>) - 1.</span></p>
+
+<p>Bouguer&rsquo;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 <i>cordilleras</i>.<a name="FNanchor_17_17" id="FNanchor_17_17"></a><a href="#Footnote_17_17" class="fnanchor">[17]</a> For greater
+accuracy in the study of the internal structure of the
+earth, in the 19th century the Bouguer plate reduction<span class="pagenum"><a name="Page_310" id="Page_310">[Pg 310]</a></span>
+came to be supplemented by corrections for irregularities
+of terrain and by different types of isostatic
+reduction.</p>
+
+<p>La Condamine, who like Bouguer was a member of
+the Peruvian expedition, conducted his own pendulum
+experiments (fig. <a href="#fig_4">4</a>). He experimented in 1735 at
+Santo Domingo en route to South America,<a name="FNanchor_18_18" id="FNanchor_18_18"></a><a href="#Footnote_18_18" class="fnanchor">[18]</a> 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&rsquo;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.<a name="FNanchor_19_19" id="FNanchor_19_19"></a><a href="#Footnote_19_19" class="fnanchor">[19]</a></p>
+
+<div class="figcenter" style="width: 740px;">
+<div class="figleft" style="width: 300px;"><a name="fig_5" id="fig_5"></a>
+<img src="images/i021.png" width="300" height="531" alt="Figure 5." title="Figure 5." />
+</div>
+<div class="figright" style="width: 400px;">
+<p class="caption2">Figure 5.&mdash;<span class="smcap">An apparatus for the practice measurement</span>
+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&rsquo;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 &ldquo;the pendulums swing parallel.&rdquo; (Portion of plate
+1, <i>Mémoires publiés par la Société française de Physique</i>, vol. 4.)</p>
+</div>
+</div>
+<p><span class="pagenum"><a name="Page_311" id="Page_311">[Pg 311]</a></span></p>
+
+<div class="figcenter" style="width: 600px;"><a name="fig_6" id="fig_6"></a>
+<img src="images/i023.png" width="600" height="675" alt="Figure 6." title="Figure 6." />
+
+<p class="caption2">Figure 6.&mdash;<span class="smcap">The result</span> 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 &ldquo;the seconds
+pendulum.&rdquo; 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.</p>
+
+<p class="caption2">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, <i>Mémoires publiés par la Société
+française de Physique</i>, vol. 4.)</p>
+</div>
+
+<p>The practice with the &ldquo;simple&rdquo; 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<a name="FNanchor_20_20" id="FNanchor_20_20"></a><a href="#Footnote_20_20" class="fnanchor">[20]</a> (fig. <a href="#fig_6">6</a>). The<span class="pagenum"><a name="Page_312" id="Page_312">[Pg 312]</a></span>
+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.</p>
+
+<div class="figright" style="width: 400px;"><a name="fig_7" id="fig_7"></a>
+<img src="images/i024.png" width="300" height="751" alt="Figure 7." title="Figure 7." />
+
+<p class="caption2">Figure 7.&mdash;<span class="smcap">Results of experiments</span> 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
+(<i>a</i> and <i>b</i>) was 1 toise (about six feet). A micrometric
+balance (<i>c</i>) 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, <i>Mémoires publiés
+par la Société française de Physique</i>, vol. 4.)</p>
+</div>
+
+<p>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 <i>n</i>
+swings while the clock makes <span class="nowrap">2<i>n</i> + 2</span>. 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 <i>T</i> = <span class="nowrap">(2<i>n</i> + 2)/<i>n</i></span> = <span class="nowrap">2(1 + 1/<i>n</i>)</span>. An error
+in the time caused by uncertainty in determining the
+coincidence of clock and wire pendulum is reduced<span class="pagenum"><a name="Page_313" id="Page_313">[Pg 313]</a></span>
+by employing a long interval of observation 2<i>n</i>. 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.</p>
+
+<p>Another historic &ldquo;simple&rdquo; pendulum was the one
+swung by Bessel (fig. <a href="#fig_7">7</a>) for the determination of
+gravity at Königsberg 1825-1827.<a name="FNanchor_21_21" id="FNanchor_21_21"></a><a href="#Footnote_21_21" class="fnanchor">[21]</a> 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 &ldquo;toise de Peru&rdquo;<a name="FNanchor_22_22" id="FNanchor_22_22"></a><a href="#Footnote_22_22" class="fnanchor">[22]</a> higher up, the bob being at
+the same level in each case (fig. <a href="#fig_7">7</a>). 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.</p>
+
+<p>Bessel&rsquo;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.</p>
+
+<div class="figcenter" style="width: 600px;"><a name="fig_8" id="fig_8"></a>
+<img src="images/i025.png" width="600" height="597" alt="Figure 8." title="Figure 8." />
+
+<p class="caption2">Figure 8.&mdash;<span class="smcap">Mode of suspension</span> of Bessel&rsquo;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 &ldquo;cylinder of unrolling&rdquo; 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.</p>
+
+<p class="caption2">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, <i>Mémoires publiés par la Société française
+de Physique</i>, vol. 4.)</p>
+</div>
+
+<p><span class="pagenum"><a name="Page_314" id="Page_314">[Pg 314]</a></span></p>
+
+<div class="figcenter" style="width: 600px;"><a name="fig_9" id="fig_9"></a>
+<img src="images/i027.png" width="300" height="379" alt="Figure 9." title="Figure 9." />
+
+<p class="caption2">Figure 9.&mdash;<span class="smcap">Friedrich Wilhelm Bessel</span> (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 &ldquo;modern&rdquo; 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.</p>
+</div>
+
+<p>The latter effect had been discovered by Du Buat in
+1786,<a name="FNanchor_23_23" id="FNanchor_23_23"></a><a href="#Footnote_23_23" class="fnanchor">[23]</a> 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.</p>
+
+
+
+<hr style="width: 65%;" />
+<h3>Kater&rsquo;s Convertible and Invariable Pendulums</h3>
+
+
+<div class="figright" style="width: 360px;"><a name="fig_10" id="fig_10"></a>
+<img src="images/i029.png" width="300" height="372" alt="Figure 10." title="Figure 10." />
+
+<p class="caption2">Figure 10.&mdash;<span class="smcap">Henry Kater</span> (1777-1835),
+English army officer and
+physicist. His scientific career began
+during his military service in
+India, where he assisted in the
+&ldquo;great trigonometrical survey.&rdquo;
+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 &ldquo;simple&rdquo; pendulum for
+the measurement of the &ldquo;seconds
+pendulum.&rdquo; Kater&rsquo;s convertible
+pendulum and the invariable pendulum
+introduced by him in 1819
+were the basis of English pendulum
+work. (<i>Photo courtesy National Portrait
+Gallery, London.</i>)</p>
+</div>
+
+<div class="figright" style="width: 360px;">
+<div class="figleft" style="width: 75px;"><a name="fig_11" id="fig_11"></a>
+<img src="images/i030.png" width="75" height="710" alt="Figure 11." title="Figure 11." />
+</div>
+
+<div class="figright" style="width: 250px;">
+<p class="caption2">Figure 11.&mdash;<span class="smcap">The attempt to approximate</span>
+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. <a href="#Page_314">314</a>). Developed in
+connection with a project to fix the
+standard of English measure, Kater&rsquo;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 (<i>a</i> and <i>b</i>) at opposite ends. The
+weights (<i>f</i> and <i>g</i>) 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.</p>
+</div>
+</div>
+
+<p>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.<a name="FNanchor_24_24" id="FNanchor_24_24"></a><a href="#Footnote_24_24" class="fnanchor">[24]</a> Kater&rsquo;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&rsquo; theorem on conjugate points of a compound
+pendulum. Kater determined the time of swing by
+the method of coincidences (fig. <a href="#fig_12">12</a>). He corrected
+for the buoyancy of the air. The final value of the
+length of the seconds pendulum at Browne&rsquo;s house in
+London, reduced to sea level, was determined to be
+39.13929 inches.</p>
+
+<p>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.<a name="FNanchor_25_25" id="FNanchor_25_25"></a><a href="#Footnote_25_25" class="fnanchor">[25]</a> In 1800, he proposed the convertible
+compound pendulum with knife edges about which
+the pendulum could complete swings in equal times.
+De Prony&rsquo;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<span class="pagenum"><a name="Page_315" id="Page_315">[Pg 315]</a></span>
+pendulum was carried out by Borda and Cassini by
+methods previously described. Bohnenberger in his
+<i>Astronomie</i> (1811),<a name="FNanchor_26_26" id="FNanchor_26_26"></a><a href="#Footnote_26_26" class="fnanchor">[26]</a> 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.</p>
+
+<p>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<a name="FNanchor_27_27" id="FNanchor_27_27"></a><a href="#Footnote_27_27" class="fnanchor">[27]</a>
+(fig. <a href="#fig_13">13</a>). Thirteen of these Kater invariable pendulums
+have been reported as constructed and swung
+at stations throughout the world.<a name="FNanchor_28_28" id="FNanchor_28_28"></a><a href="#Footnote_28_28" class="fnanchor">[28]</a> 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.<a name="FNanchor_29_29" id="FNanchor_29_29"></a><a href="#Footnote_29_29" class="fnanchor">[29]</a> In 1820, Kater swung a Kater invariable
+pendulum at London and then sent it to Goldingham,
+who swung it in 1821 at Madras, India.<a name="FNanchor_30_30" id="FNanchor_30_30"></a><a href="#Footnote_30_30" class="fnanchor">[30]</a> 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.<a name="FNanchor_31_31" id="FNanchor_31_31"></a><a href="#Footnote_31_31" class="fnanchor">[31]</a> 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.<a name="FNanchor_32_32" id="FNanchor_32_32"></a><a href="#Footnote_32_32" class="fnanchor">[32]</a></p>
+
+<p><span class="pagenum"><a name="Page_316" id="Page_316">[Pg 316]</a></span></p>
+
+<div class="figcenter" style="width: 600px;"><a name="fig_12" id="fig_12"></a>
+<img src="images/i031.png" width="600" height="562" alt="Figure 12." title="Figure 12." />
+
+<p class="caption2">Figure 12.&mdash;<span class="smcap">The Kater convertible pendulum</span> in use is placed before a clock, whose
+pendulum bob is directly behind the extended &ldquo;tail&rdquo; 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, <i>Mémoires publiés par la Société française de Physique</i>, vol. 4.)<span class="pagenum"><a name="Page_317" id="Page_317">[Pg 317]</a></span></p>
+</div>
+
+<div class="figcenter" style="width: 600px;"><a name="fig_13" id="fig_13"></a>
+<img src="images/i034.png" width="600" height="420" alt="Figure 13." title="Figure 13." />
+
+<p class="caption2">Figure 13.&mdash;<span class="smcap">This drawing accompanied</span> John Goldingham&rsquo;s report on the work done in India with Kater&rsquo;s
+invariable pendulum. The value of gravity obtained, directly or indirectly, in terms of the simple pendulum,
+is called &ldquo;absolute.&rdquo; Once absolute values of gravity were established at a number of stations, it became
+possible to use the much simpler &ldquo;relative&rdquo; 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 &ldquo;invariable.&rdquo; 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.</p>
+</div>
+
+<div class="figright" style="width: 350px;"><a name="fig_14" id="fig_14"></a>
+<img src="images/i037.png" width="300" height="400" alt="Figure 14." title="Figure 14." />
+
+<p class="caption2">Figure 14.&mdash;<span class="smcap">Vacuum chamber for use</span> 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.</p>
+</div>
+
+<p>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.<a name="FNanchor_33_33" id="FNanchor_33_33"></a><a href="#Footnote_33_33" class="fnanchor">[33]</a> A similar expedition was conducted
+in 1822-1825 by Captain Duperry.<a name="FNanchor_34_34" id="FNanchor_34_34"></a><a href="#Footnote_34_34" class="fnanchor">[34]</a></p>
+
+<p>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.<a name="FNanchor_35_35" id="FNanchor_35_35"></a><a href="#Footnote_35_35" class="fnanchor">[35]</a> In 1836,
+Baily undertook to advise the American Lt. Charles
+Wilkes, who was to head the United States Exploring<span class="pagenum"><a name="Page_318" id="Page_318">[Pg 318]</a></span>
+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 &ldquo;those
+considered the best form by Mr. Baily for traveling
+pendulums,&rdquo; and which Baily, himself, described as
+&ldquo;precisely the same as the two invariable pendulums
+belonging to this [Royal Astronomical] Society,&rdquo;
+except for the location of the knife edges.</p>
+
+<div class="figright" style="width: 350px;"><a name="fig_15" id="fig_15"></a>
+<img src="images/i039.png" width="200" height="680" alt="Figure 15." title="Figure 15." />
+
+<p class="caption2">Figure 15.&mdash;<span class="smcap">One of Francis Baily&rsquo;s pendulums</span>
+(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.</p>
+</div>
+
+<p>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.<a name="FNanchor_36_36" id="FNanchor_36_36"></a><a href="#Footnote_36_36" class="fnanchor">[36]</a></p>
+
+<p>Wilkes was taught the use of the pendulum by
+Baily, and conducted experiments at Baily&rsquo;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, &ldquo;Pendulum Peak&rdquo;
+(Mauna Loa), Mount Kanoha, Nesqually (Oregon
+Territory), and, finally, two more times in Washington,
+D.C. (1841 and 1845).</p>
+
+<p>Wilkes&rsquo; 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.<a name="FNanchor_37_37" id="FNanchor_37_37"></a><a href="#Footnote_37_37" class="fnanchor">[37]</a> 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&rsquo;s proof.<a name="FNanchor_38_38" id="FNanchor_38_38"></a><a href="#Footnote_38_38" class="fnanchor">[38]</a></p>
+
+<p>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.<span class="pagenum"><a name="Page_319" id="Page_319">[Pg 319]</a></span>
+In 1854, the experiments were again undertaken in the
+Harton coalpit, near Sunderland.<a name="FNanchor_39_39" id="FNanchor_39_39"></a><a href="#Footnote_39_39" class="fnanchor">[39]</a> 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.<a name="FNanchor_40_40" id="FNanchor_40_40"></a><a href="#Footnote_40_40" class="fnanchor">[40]</a> He used a &ldquo;simple&rdquo;
+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.</p>
+
+<p>In 1879, Maj. J. Herschel, R.E., stated:</p>
+
+<blockquote><p>The years from 1840 to 1865 are a complete blank, if
+we except Airy&rsquo;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&rsquo;s design were set to work in
+Russia; and at Geneva, Plantamour&rsquo;s zealous experiments
+with a pendulum of the same kind mark the
+commencement of an era of renewed activity on the
+European continent.<a name="FNanchor_41_41" id="FNanchor_41_41"></a><a href="#Footnote_41_41" class="fnanchor">[41]</a></p></blockquote>
+
+<p>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.</p>
+
+
+<hr style="width: 65%;" /><p><span class="pagenum"><a name="Page_320" id="Page_320">[Pg 320]</a></span></p>
+<h3>Repsold-Bessel Reversible Pendulum</h3>
+
+
+<p>As we have noted, Bessel made determinations of
+gravity with a ball (&ldquo;simple&rdquo;) 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.<a name="FNanchor_42_42" id="FNanchor_42_42"></a><a href="#Footnote_42_42" class="fnanchor">[42]</a> 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.</p>
+
+<p>The simplified discussion for infinitely small oscillations
+in a vacuum is as follows: If <i>T</i><sub>1</sub> and <i>T</i><sub>2</sub>
+are the times of swing about the knife edges, and if
+<i>h</i><sub>1</sub> and <i>h</i><sub>2</sub> are distances of the knife edges from the
+center of gravity, and if <i>k</i> 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 <span class="nowrap"><i>T</i><sub>1</sub><sup>2</sup></span> = <span class="nowrap">&#960;<sup>2</sup>(<i>k</i><sup>2</sup> + <i>h</i><sub>1</sub><sup>2</sup>)/<i>g</i><i>h</i><sub>1</sub></span>,
+<span class="nowrap"><i>T</i><sub>2</sub><sup>2</sup></span> = <span class="nowrap">&#960;<sup>2</sup>(<i>k</i><sup>2</sup> + <i>h</i><sub>2</sub><sup>2</sup>)/<i>g</i><i>h</i><sub>2</sub></span>. Then <span class="nowrap">(<i>h</i><sub>1</sub><i>T</i><sub>1</sub><sup>2</sup>
+- <i>h</i><sub>2</sub><i>T</i><sub>2</sub><sup>2</sup>)/(<i>h</i><sub>1</sub> - <i>h</i><sub>2</sub>)</span> = <span class="nowrap"><span class="nowrap">(&#960;<sup>2</sup>/<i>g</i>)(<i>h</i><sub>1</sub> + <i>h</i><sub>2</sub>) = &#964;<sup>2</sup>.</span></span></p>
+
+<p>&#964; is then the time of swing of a simple pendulum
+of length <span class="nowrap"><i>h</i><sub>1</sub> + <i>h</i><sub>2</sub></span>. If the difference <span class="nowrap"><i>T</i><sub>1</sub> - <i>T</i><sub>2</sub></span> is <a name="corr_44_01" id="corr_44_01"></a><ins class="mycorr" title="Original: sufficlently">sufficiently</ins>
+small, &#964; = <span class="nowrap">(<i>h</i><sub>1</sub><i>T</i><sub>1</sub> - <i>h</i><sub>2</sub><i>T</i><sub>2</sub>)/(<i>h</i><sub>1</sub> - <i>h</i><sub>2</sub>)</span>. Prior to its publication
+by Bessel in 1828, the formula for the time of
+swing of a simple pendulum of length <span class="nowrap"><i>h</i><sub>1</sub> + <i>h</i><sub>2</sub></span> in terms
+of <i>T</i><sub>1</sub>, <i>T</i><sub>2</sub> had been given by C. F. Gauss in a letter
+to H. C. Schumacher dated November 28, 1824.<a name="FNanchor_43_43" id="FNanchor_43_43"></a><a href="#Footnote_43_43" class="fnanchor">[43]</a></p>
+
+<p>The symmetrical compound pendulum with interchangeable
+knives, for which Bessel gave a posthumously
+published design and specifications,<a name="FNanchor_44_44" id="FNanchor_44_44"></a><a href="#Footnote_44_44" class="fnanchor">[44]</a> has
+been called a reversible pendulum; it may thereby
+be distinguished from Kater&rsquo;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&mdash;at
+different points in Switzerland&mdash;of the arc
+of the meridian traversing central Europe.<a name="FNanchor_45_45" id="FNanchor_45_45"></a><a href="#Footnote_45_45" class="fnanchor">[45]</a> It was
+decided further to employ a reversible pendulum of
+Bessel&rsquo;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. <a href="#fig_16">16</a>) was sent to Prof. E. Plantamour,
+director of the observatory at Geneva, in the autumn
+of 1864.<a name="FNanchor_46_46" id="FNanchor_46_46"></a><a href="#Footnote_46_46" class="fnanchor">[46]</a></p>
+
+<p>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,<a name="FNanchor_47_47" id="FNanchor_47_47"></a><a href="#Footnote_47_47" class="fnanchor">[47]</a>
+<span class="pagenum"><a name="Page_321" id="Page_321">[Pg 321]</a></span>then showed that if <span class="nowrap">(<i>T</i><sub>1</sub> - <i>T</i><sub>2</sub>)/<i>T</i><sub>1</sub></span> 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.</p>
+
+<p>In 1866, Plantamour published an extensive memoir
+&ldquo;Expériences faites à Genève avec le pendule à <a name="corr_44_02" id="corr_44_02"></a><ins class="mycorr" title="Original: reversion">réversion</ins>.&rdquo;
+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.</p>
+
+<p>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.<a href="#Footnote_48_48" class="fnanchor">[48]</a> 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.</p>
+
+<p>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.<a name="FNanchor_49_49" id="FNanchor_49_49"></a><a href="#Footnote_49_49" class="fnanchor">[49]</a> 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 <i>Die Mittel-Europäische Gradmessung</i>,
+also <i>L&rsquo;Association Géodésique Internationale</i>, was<span class="pagenum"><a name="Page_322" id="Page_322">[Pg 322]</a></span>
+held from the 15th to the 22d of October 1864 in
+Berlin.<a name="FNanchor_50_50" id="FNanchor_50_50"></a><a href="#Footnote_50_50" class="fnanchor">[50]</a> 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.</p>
+
+<div class="figright" style="width: 350px;"><a name="fig_16" id="fig_16"></a>
+<img src="images/i043.png" width="300" height="415" alt="Figure 16." title="Figure 16." />
+
+<p class="caption2">Figure 16.&mdash;<span class="smcap">From a Design Left by Bessel</span>,
+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&rsquo;s in being geometrically
+symmetrical and, for this
+reason, Repsold&rsquo;s is usually called &ldquo;reversible.&rdquo;
+Just to the right of the pendulum
+is a standard scale. To the left is a
+&ldquo;vertical comparator&rdquo; 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.</p>
+</div>
+
+<p>Under the topic &ldquo;Astronomical Questions,&rdquo; 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.<a name="FNanchor_51_51" id="FNanchor_51_51"></a><a href="#Footnote_51_51" class="fnanchor">[51]</a> 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.<a name="FNanchor_52_52" id="FNanchor_52_52"></a><a href="#Footnote_52_52" class="fnanchor">[52]</a> The title of the association was
+changed to <i>Die Europäische Gradmessung</i>; in 1886, it
+became <i>Die Internationale Erdmessung</i>, under which
+title it continued until World War I.</p>
+
+<p>On April 1, 1866, the Central Bureau of <i>Die Europäische
+Gradmessung</i> 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.<a name="FNanchor_53_53" id="FNanchor_53_53"></a><a href="#Footnote_53_53" class="fnanchor">[53]</a></p>
+
+<p>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 &ldquo;Les
+variations de la pesanteur dans les provinces occidentales
+de l&rsquo;Empire russe.&rdquo;<a name="FNanchor_48_48" id="FNanchor_48_48"></a><a href="#Footnote_48_48" class="fnanchor">[48]</a> 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 <i>Die Europäische Gradmessung</i> in
+Dresden.<a name="FNanchor_54_54" id="FNanchor_54_54"></a><a href="#Footnote_54_54" class="fnanchor">[54]</a> 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): &ldquo;Observations
+for the determination of the intensity of
+gravity,&rdquo; the question, &ldquo;Which Pendulum-apparatuses
+are preferable for the determination of many points?&rdquo;</p>
+
+<p>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&rsquo;s direction to his son Charles on November
+30, 1872, &ldquo;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....&rdquo;<a name="FNanchor_55_55" id="FNanchor_55_55"></a><a href="#Footnote_55_55" class="fnanchor">[55]</a> Systematic and important gravity
+work by the Survey was begun by Charles Sanders<span class="pagenum"><a name="Page_323" id="Page_323">[Pg 323]</a></span>
+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.<a name="FNanchor_56_56" id="FNanchor_56_56"></a><a href="#Footnote_56_56" class="fnanchor">[56]</a></p>
+
+<div class="figcenter" style="width: 640px;">
+<div class="figleft" style="width: 300px;"><a name="fig_17" id="fig_17"></a>
+<img src="images/i046.png" width="300" height="524" alt="Figure 17." title="Figure 17." />
+
+<p class="caption2">Figure 17.&mdash;<span class="smcap">Repsold-Bessel reversible pendulum</span> 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&rsquo;s demonstration of the
+error inherent in the Repsold apparatus through flexure
+of the stand.</p>
+</div>
+
+<div class="figright" style="width: 300px;"><a name="fig_18" id="fig_18"></a>
+<img src="images/i047.png" width="300" height="390" alt="Figure 18." title="Figure 18." />
+
+<p class="caption2">Figure 18.&mdash;<span class="smcap">Charles Sanders Peirce</span> (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.</p>
+</div>
+</div>
+
+<p style="clear:both;"><span class="pagenum"><a name="Page_324" id="Page_324">[Pg 324]</a></span>In 1874, Charles Peirce expressed the desire to be
+sent to Europe for at least a year, beginning about
+March 1, 1875, &ldquo;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&rdquo;
+his &ldquo;invariable pendulums in the manner which
+has been used by swinging them in London and
+Paris.&rdquo;<a name="FNanchor_57_57" id="FNanchor_57_57"></a><a href="#Footnote_57_57" class="fnanchor">[57]</a></p>
+
+<p>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. <a href="#fig_17">17</a>). 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.<a name="FNanchor_58_58" id="FNanchor_58_58"></a><a href="#Footnote_58_58" class="fnanchor">[58]</a></p>
+
+<p>In view of Baeyer&rsquo;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&mdash;on account of the swaying of the stand
+during the swings of the pendulum&mdash;to amount to
+over 0.2 mm. Although Peirce&rsquo;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.</p>
+
+<p>The Permanent Commission of <i>Die Europäische
+Gradmessung</i> 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.<a name="FNanchor_59_59" id="FNanchor_59_59"></a><a href="#Footnote_59_59" class="fnanchor">[59]</a></p>
+
+<p>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.</p>
+
+<p>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.</p>
+
+<p>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<span class="pagenum"><a name="Page_325" id="Page_325">[Pg 325]</a></span>
+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.<a name="FNanchor_60_60" id="FNanchor_60_60"></a><a href="#Footnote_60_60" class="fnanchor">[60]</a></p>
+
+<p>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.<a name="FNanchor_61_61" id="FNanchor_61_61"></a><a href="#Footnote_61_61" class="fnanchor">[61]</a></p>
+
+<p>Meanwhile, the Permanent Commission met October
+5-10, 1876, in Brussels and continued the discussion
+of the pendulum.<a name="FNanchor_62_62" id="FNanchor_62_62"></a><a href="#Footnote_62_62" class="fnanchor">[62]</a> Gen. Baeyer reported on
+Peirce&rsquo;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&rsquo;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.</p>
+
+<p>The Fifth General Conference of <i>Die Europäische
+Gradmessung</i> was held from September 27 to October 2,
+1877, in Stuttgart.<a name="FNanchor_63_63" id="FNanchor_63_63"></a><a href="#Footnote_63_63" class="fnanchor">[63]</a> 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 &ldquo;De
+l&rsquo;Influence de la flexibilité du trépied sur l&rsquo;oscillation
+du pendule à réversion.&rdquo; This memoir and others
+by Cellérier and Plantamour confirming Peirce&rsquo;s
+work were published as appendices to the proceedings
+of the conference. As appendices to Peirce&rsquo;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.</p>
+
+<p>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 &ldquo;double pendulum&rdquo;
+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.<a name="FNanchor_64_64" id="FNanchor_64_64"></a><a href="#Footnote_64_64" class="fnanchor">[64]</a> On February 17, 1879, Peirce had completed
+a paper &ldquo;On a Method of Swinging Pendulums
+for the Determination of Gravity, Proposed by M.<span class="pagenum"><a name="Page_326" id="Page_326">[Pg 326]</a></span>
+Faye.&rdquo; In this paper, Peirce presented the results
+of an analytical mechanical investigation of Faye&rsquo;s
+proposal. Peirce set up the differential equations,
+found the solutions, interpreted them physically,
+and arrived at the conclusion &ldquo;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.&rdquo;</p>
+
+<p>In a report to Supt. Patterson, dated July 1879,
+Peirce stated: &ldquo;I think it is important before making
+a new pendulum apparatus to experiment with Faye&rsquo;s
+proposed method.&rdquo;<a name="FNanchor_65_65" id="FNanchor_65_65"></a><a href="#Footnote_65_65" class="fnanchor">[65]</a> He wrote further: &ldquo;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.&rdquo; Copies of the published version of Peirce&rsquo;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&rsquo;s findings.<a name="FNanchor_66_66" id="FNanchor_66_66"></a><a href="#Footnote_66_66" class="fnanchor">[66]</a> 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&rsquo;s paper on
+Faye&rsquo;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&rsquo;s request, Charles Cellérier
+was appointed to join him, since the problem
+essentially was a theoretical one.</p>
+
+<p>The Sixth General Conference of <i>Die Europäische
+Gradmessung</i> met September 13-16, 1880, in Munich.<a name="FNanchor_67_67" id="FNanchor_67_67"></a><a href="#Footnote_67_67" class="fnanchor">[67]</a>
+Topic III, part 7 of the program was entitled &ldquo;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.&rdquo;</p>
+
+<div class="figright" style="width: 350px;"><a name="fig_19" id="fig_19"></a>
+<img src="images/i054.png" width="200" height="524" alt="Figure 19." title="Figure 19." />
+
+<p class="caption2">Figure 19.&mdash;<span class="smcap">Three pendulums used in early work</span>
+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.</p>
+</div>
+
+<p>The conference received a memoir by Cellérier<a name="FNanchor_68_68" id="FNanchor_68_68"></a><a href="#Footnote_68_68" class="fnanchor">[68]</a> on
+the theory of the double pendulum and a report by
+Plantamour and Cellérier.<a name="FNanchor_69_69" id="FNanchor_69_69"></a><a href="#Footnote_69_69" class="fnanchor">[69]</a> Cellérier&rsquo;s mathematical
+analysis began with the equations of Peirce and used
+the latter&rsquo;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 &ldquo;double pendulum.&rdquo;
+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.</p>
+
+<p>The Permanent Commission met September 11-15,
+1882, at The Hague,<a name="FNanchor_70_70" id="FNanchor_70_70"></a><a href="#Footnote_70_70" class="fnanchor">[70]</a> 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,<a name="FNanchor_71_71" id="FNanchor_71_71"></a><a href="#Footnote_71_71" class="fnanchor">[71]</a> and, at its
+eighth session, on October 22, received a comprehensive,
+critical review from Prof. von Oppolzer entitled
+&ldquo;Über die Bestimmung der Schwere mit Hilfe <a name="corr_44_16" id="corr_44_16"></a><ins class="mycorr" title="Original: verschiedene">verschiedener</ins>
+Apparate.&rdquo;<a name="FNanchor_72_72" id="FNanchor_72_72"></a><a href="#Footnote_72_72" class="fnanchor">[72]</a> 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<span class="pagenum"><a name="Page_327" id="Page_327">[Pg 327]</a></span>
+the proposal&mdash;made by Faye and theoretically
+pursued by Peirce&mdash;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&rsquo;s important report recognized
+that absolute determinations were less accurate
+than relative ones, and should be conducted only at
+special places.</p>
+
+<p>The discussions initiated by Peirce&rsquo;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.</p>
+
+
+
+
+<hr style="width: 65%;" />
+<h3>Peirce and Defforges Invariable, Reversible Pendulums</h3>
+
+<p>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 <i>Die Europäische Gradmessung</i>,
+in Munich.<a name="FNanchor_73_73" id="FNanchor_73_73"></a><a href="#Footnote_73_73" class="fnanchor">[73]</a></p>
+
+<p>As recorded in the Proceedings of the Conference,
+Peirce wrote:</p>
+
+<blockquote><p>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.</p></blockquote><p><span class="pagenum"><a name="Page_328" id="Page_328">[Pg 328]</a></span></p>
+
+<p>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.</p>
+
+<p>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:</p>
+
+<blockquote><p>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
+<a name="corr_44_03" id="corr_44_03"></a><ins class="mycorr" title="Original: meter.&rdquo;">meter.</ins><a name="FNanchor_74_74" id="FNanchor_74_74"></a><a href="#Footnote_74_74" class="fnanchor">[74]</a></p></blockquote>
+
+<p>The report for 1881-1882 mentions four of these
+Peirce pendulums.</p>
+
+<p>A description of the Peirce invariable, reversible
+pendulums was given by Assistant E. D. Preston in
+&ldquo;Determinations of Gravity and the Magnetic Elements
+in Connection with the United States Scientific
+Expedition to the West Coast of Africa, 1889-90.&rdquo;<a name="FNanchor_75_75" id="FNanchor_75_75"></a><a href="#Footnote_75_75" class="fnanchor">[75]</a>
+The invariable, reversible pendulum, Peirce
+no. 4, now preserved in the Smithsonian Institution&rsquo;s
+Museum of History and Technology (fig. <a href="#fig_34">34</a>), 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.</p>
+
+<p>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.<a name="FNanchor_76_76" id="FNanchor_76_76"></a><a href="#Footnote_76_76" class="fnanchor">[76]</a></p>
+
+<div class="figright" style="width: 350px;"><a name="fig_20" id="fig_20"></a>
+<img src="images/i057.png" width="300" height="442" alt="Figure 20." title="Figure 20." />
+
+<p class="caption2">Figure 20.&mdash;<span class="smcap">Support for the Peirce pendulum</span>,
+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.</p>
+</div>
+
+<p>Another novel characteristic of the Peirce pendulums
+was the mainly cylindrical form. Prof. George
+Gabriel Stokes, in a paper &ldquo;On the Effect of the<span class="pagenum"><a name="Page_329" id="Page_329">[Pg 329]</a></span>
+Internal Friction of Fluids on the Motion of Pendulums&rdquo;<a name="FNanchor_77_77" id="FNanchor_77_77"></a><a href="#Footnote_77_77" class="fnanchor">[77]</a>
+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. <a href="#fig_19">19</a>) permitted Peirce to predict
+from Stokes&rsquo; 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.<a name="FNanchor_78_78" id="FNanchor_78_78"></a><a href="#Footnote_78_78" class="fnanchor">[78]</a></p>
+
+<p>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.<a name="FNanchor_79_79" id="FNanchor_79_79"></a><a href="#Footnote_79_79" class="fnanchor">[79]</a></p>
+
+<p>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&rsquo;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.<a name="FNanchor_80_80" id="FNanchor_80_80"></a><a href="#Footnote_80_80" class="fnanchor">[80]</a></p>
+
+<p>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.<a name="FNanchor_81_81" id="FNanchor_81_81"></a><a href="#Footnote_81_81" class="fnanchor">[81]</a> 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 &ldquo;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.&rdquo; 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.</p>
+
+<p>Pendulums of 1-meter and of 1/2-meter distance
+between the knife edges were constructed from
+Defforges&rsquo; design by Brunner Brothers in Paris
+(fig. <a href="#fig_21">21</a>). 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.<span class="pagenum"><a name="Page_330" id="Page_330">[Pg 330]</a></span></p>
+
+<div class="figcenter" style="width: 600px;"><a name="fig_21" id="fig_21"></a>
+<img src="images/i060.png" width="600" height="688" alt="Figure 21." title="Figure 21." />
+
+<p class="caption2">Figure 21.&mdash;<span class="smcap">Reversible pendulum apparatus</span> 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.</p>
+</div>
+<p><span class="pagenum"><a name="Page_331" id="Page_331">[Pg 331]</a></span></p>
+
+<div class="figcenter" style="width: 615px;">
+<div class="figleft" style="width: 250px;"><a name="fig_22" id="fig_22"></a>
+<img src="images/i061-a.png" width="250" height="579" alt="Figure 22." title="Figure 22." />
+
+<p class="caption2">Figure 22.&mdash;<span class="smcap">Because of the greater simplicity</span> 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.</p>
+</div>
+
+<div class="figright" style="width: 300px;"><a name="fig_23" id="fig_23"></a>
+<img src="images/i061-b.png" width="300" height="395" alt="Figure 23." title="Figure 23." />
+
+<p class="caption2">Figure 23.&mdash;<span class="smcap">The overall size</span> 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.</p>
+</div>
+</div>
+
+<div class="figcenter" style="width: 615px;"><a name="fig_24" id="fig_24"></a>
+<img style="margin-left:45px;" src="images/i063.png" width="300" height="316" alt="Figure 24." title="Figure 24." />
+
+<p class="caption2">Figure 24.&mdash;<span class="smcap">Thomas C. Mendenhall</span> (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.</p>
+
+</div>
+
+
+<hr style="width: 65%;" />
+<h3>Von Sterneck and Mendenhall Pendulums</h3>
+
+
+<p>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.<a name="FNanchor_82_82" id="FNanchor_82_82"></a><a href="#Footnote_82_82" class="fnanchor">[82]</a> Maj. von Sterneck&rsquo;s apparatus
+contained a nonreversible pendulum 1/4-meter in<span class="pagenum"><a name="Page_332" id="Page_332">[Pg 332]</a></span>
+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.</p>
+
+<p>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.</p>
+
+<p>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.<a name="FNanchor_83_83" id="FNanchor_83_83"></a><a href="#Footnote_83_83" class="fnanchor">[83]</a></p>
+
+<p>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.<span class="pagenum"><a name="Page_333" id="Page_333">[Pg 333]</a></span></p>
+
+<div class="figcenter" style="width: 600px;"><a name="fig_25" id="fig_25"></a>
+<img src="images/i065.png" width="600" height="424" alt="Figure 25." title="Figure 25." />
+
+<p class="caption2">Figure 25.&mdash;<span class="smcap">Mendenhall&rsquo;s 1/4-meter (1/2-second) apparatus.</span> 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 &ldquo;coincidence&rdquo; 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.</p>
+</div>
+
+<p>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<span class="pagenum"><a name="Page_334" id="Page_334">[Pg 334]</a></span>
+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.</p>
+
+<div class="figcenter" style="width: 500px;"><a name="fig_26" id="fig_26"></a>
+<img src="images/i068.png" width="500" height="431" alt="Figure 26." title="Figure 26." />
+
+<p class="caption2">Figure 26.&mdash;<span class="smcap">Vacuum receiver</span> 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 <i>q</i> and is
+started to swing by the lever <i>r</i>. The arc of swing is only about 1°. The stationary mirror
+is shown at <i>g</i>. The pendulum shown in outline in the center, is only about 9.7 inches long.</p>
+</div>
+
+<p>Under Supt. Mendenhall&rsquo;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.</p>
+
+<p>In accordance with Peirce&rsquo;s theory of the flexure
+of the stand under oscillations of the pendulum,
+determinations of the displacement of the receiver
+of the Mendenhall apparatus <a name="corr_44_04" id="corr_44_04"></a><ins class="mycorr" title="Original: was">were</ins> 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.<a name="FNanchor_84_84" id="FNanchor_84_84"></a><a href="#Footnote_84_84" class="fnanchor">[84]</a> 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.<a name="FNanchor_85_85" id="FNanchor_85_85"></a><a href="#Footnote_85_85" class="fnanchor">[85]</a><span class="pagenum"><a name="Page_335" id="Page_335">[Pg 335]</a></span></p>
+
+<div class="figcenter" style="width: 600px;">
+<img src="images/i071.png" width="600" height="395" alt="Figure 27." title="Figure 27." />
+
+<p class="caption2"><a name="fig_27" id="fig_27"></a>Figure 27.&mdash;<span class="smcap">The Michelson interferometer.</span> 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.</p>
+
+<p class="caption2">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.</p>
+</div>
+
+<p>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,<a name="FNanchor_86_86" id="FNanchor_86_86"></a><a href="#Footnote_86_86" class="fnanchor">[86]</a> and Dr. Schumann of
+Potsdam subsequently described a two-pendulum
+apparatus.<a name="FNanchor_87_87" id="FNanchor_87_87"></a><a href="#Footnote_87_87" class="fnanchor">[87]</a><span class="pagenum"><a name="Page_336" id="Page_336">[Pg 336]</a></span></p>
+
+<div class="figcenter" style="width: 600px;"><a name="fig_28" id="fig_28"></a>
+<img src="images/i074.png" width="600" height="473" alt="Figure 28." title="Figure 28." />
+
+<p class="caption2">Figure 28.&mdash;<span class="smcap">Apparatus which was developed in 1929</span> 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 <span class="nowrap">-starting</span> 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.</p>
+</div>
+
+<p>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
+&ldquo;Wippverfahren&rdquo; 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,<a name="FNanchor_88_88" id="FNanchor_88_88"></a><a href="#Footnote_88_88" class="fnanchor">[88]</a> and in his paper cited the mathematical
+methods of Peirce and Cellérier for the theory of
+Faye&rsquo;s proposal at Stuttgart in 1877 to swing two
+similar pendulums on the same support with equal
+amplitudes and in opposite phases.<span class="pagenum"><a name="Page_337" id="Page_337">[Pg 337]</a></span></p>
+
+<div class="figcenter" style="width: 600px;"><a name="fig_29" id="fig_29"></a>
+<img src="images/i077.png" width="600" height="490" alt="Figure 29." title="Figure 29." />
+
+<p class="caption2">Figure 29.&mdash;<span class="smcap">The Gulf pendulum</span> 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&rsquo;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.</p>
+</div>
+
+<p>In 1902, Dr. P. Furtwängler<a name="FNanchor_89_89" id="FNanchor_89_89"></a><a href="#Footnote_89_89" class="fnanchor">[89]</a> presented the mathematical
+theory of coupled pendulums in a paper in
+which he referred to Faye&rsquo;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&rsquo;s
+proposed method of swinging two pendulums on the
+same support.<a name="FNanchor_90_90" id="FNanchor_90_90"></a><a href="#Footnote_90_90" class="fnanchor">[90]</a> 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&mdash;which had been proposed by
+Faye and had been demonstrated theoretically to be
+sound by Peirce, who also published a design for its<span class="pagenum"><a name="Page_338" id="Page_338">[Pg 338]</a></span>
+application&mdash;was rapidly followed for pendulum apparatus
+for relative determinations by Potsdam,<a name="FNanchor_91_91" id="FNanchor_91_91"></a><a href="#Footnote_91_91" class="fnanchor">[91]</a>
+Cambridge (England),<a name="FNanchor_92_92" id="FNanchor_92_92"></a><a href="#Footnote_92_92" class="fnanchor">[92]</a> Gulf Oil and Development
+Company,<a name="FNanchor_93_93" id="FNanchor_93_93"></a><a href="#Footnote_93_93" class="fnanchor">[93]</a> and the Dominion Observatory at
+Ottawa.<a name="FNanchor_94_94" id="FNanchor_94_94"></a><a href="#Footnote_94_94" class="fnanchor">[94]</a> Heiskanen and Vening Meinesz state:</p>
+
+<blockquote><p>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.<a name="FNanchor_95_95" id="FNanchor_95_95"></a><a href="#Footnote_95_95" class="fnanchor">[95]</a></p></blockquote>
+
+<p>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&rsquo;s invention of the
+short, 1/4-meter pendulum.</p>
+
+<div class="figcenter" style="width: 600px;">
+<img src="images/i080.png" width="600" height="202" alt="Figure 30." title="Figure 30." />
+
+<p class="caption2"><a name="fig_30" id="fig_30"></a>Figure 30.&mdash;<span class="smcap">The accumulated data of gravity</span> observations over the earth&rsquo;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&rsquo;s crust produce a kind of hydrostatic
+equilibrium between its higher and lower parts, as they &ldquo;float&rdquo; on the earth&rsquo;s
+fluid core. The metals of different density floating in mercury in this
+diagram illustrate isostasy according to the theory of Pratt and Hayford.</p>
+</div>
+
+<hr style="width: 65%;" />
+<h3>Absolute Value of Gravity at Potsdam</h3>
+
+
+<p>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.<a name="FNanchor_96_96" id="FNanchor_96_96"></a><a href="#Footnote_96_96" class="fnanchor">[96]</a></p>
+
+<p>We have previously seen that in 1869 the Geodetic
+Institute&mdash;founded by Lt. Gen. Baeyer&mdash;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<span class="pagenum"><a name="Page_339" id="Page_339">[Pg 339]</a></span>
+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.<a name="FNanchor_97_97" id="FNanchor_97_97"></a><a href="#Footnote_97_97" class="fnanchor">[97]</a></p>
+
+<p>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.<a name="FNanchor_98_98" id="FNanchor_98_98"></a><a href="#Footnote_98_98" class="fnanchor">[98]</a> 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.</p>
+
+<p>In a memoir titled &ldquo;Effect of the flexure of a pendulum
+upon its period of oscillation,&rdquo;<a name="FNanchor_99_99" id="FNanchor_99_99"></a><a href="#Footnote_99_99" class="fnanchor">[99]</a> 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.<a name="FNanchor_100_100" id="FNanchor_100_100"></a><a href="#Footnote_100_100" class="fnanchor">[100]</a> 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:</p>
+
+<blockquote><p>1. The seconds pendulum of the Geodetic Institute
+procured in 1869.</p>
+
+<p>2. A seconds pendulum from the Astronomical Observatory,
+Padua.</p>
+
+<p>3. A heavy, seconds pendulum from the Imperial and
+Royal Military-Geographical Institute, Vienna.</p>
+
+<p>4. A light, seconds pendulum from the Imperial and
+Royal Military-Geographical Institute.</p>
+
+<p>5. A 1/2-second, reversible pendulum of the Geodetic
+Institute procured in 1892.</p></blockquote>
+
+<p>Work was begun in 1898, and in 1906 Kühnen and
+Furtwängler published their monumental memoir,
+&ldquo;Bestimmung der Absoluten Grösze der Schwerkraft
+zu Potsdam mit Reversionspendeln.&rdquo;</p>
+
+<p>The acceleration of gravity in the pendulum room
+of the Geodetic Institute was determined to be 981.274
+± 0.003 cm/sec<sup>2</sup>. 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 <i>T</i><sub>1</sub><sup>2</sup>/<i>T</i><sub>2</sub><sup>2</sup> = <i>g</i><sub>2</sub>/<i>g</i><sub>1</sub>.
+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<sup>2</sup>.<a name="FNanchor_101_101" id="FNanchor_101_101"></a><a href="#Footnote_101_101" class="fnanchor">[101]</a>
+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<sup>2</sup>.<a name="FNanchor_102_102" id="FNanchor_102_102"></a><a href="#Footnote_102_102" class="fnanchor">[102]</a></p>
+
+<p>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.<a name="FNanchor_103_103" id="FNanchor_103_103"></a><a href="#Footnote_103_103" class="fnanchor">[103]</a>
+This estimate was lowered slightly by Sir Harold
+Jeffreys of Cambridge, England, who recomputed
+the results of Heyl and Cook by different methods.<a name="FNanchor_104_104" id="FNanchor_104_104"></a><a href="#Footnote_104_104" class="fnanchor">[104]</a><span class="pagenum"><a name="Page_340" id="Page_340">[Pg 340]</a></span></p>
+
+<div class="figcenter" style="width: 700px;"><a name="fig_31" id="fig_31"></a>
+<img src="images/i085.png" width="700" height="437" alt="Figure 31." title="Figure 31." />
+<p class="caption">Figure 31.&mdash;<span class="smcap">Map showing the distribution</span> of gravity stations throughout the United States as of December 1908.</p>
+</div>
+<p><span class="pagenum"><a name="Page_341" id="Page_341">[Pg 341]</a></span></p>
+<div class="figcenter" style="width: 700px;"><a name="fig_32" id="fig_32"></a>
+<img src="images/i086.png" width="700" height="447" alt="Figure 32." title="Figure 32." />
+<p class="caption">Figure 32.&mdash;<span class="smcap">Map showing the distribution</span> of gravity stations throughout the United States in 1923.</p>
+</div>
+
+<p><span class="pagenum"><a name="Page_342" id="Page_342">[Pg 342]</a></span>
+In 1939, J. S. Clark published the results of a
+determination of gravity with pendulums of a non-ferrous
+Y-alloy<a name="FNanchor_105_105" id="FNanchor_105_105"></a><a href="#Footnote_105_105" class="fnanchor">[105]</a> 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 <a name="corr_44_05" id="corr_44_05"></a><ins class="mycorr" title="Original: Postdam">Potsdam</ins>, 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.<a name="FNanchor_106_106" id="FNanchor_106_106"></a><a href="#Footnote_106_106" class="fnanchor">[106]</a> 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<sup>2</sup> 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<sup>2</sup>. 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&rsquo;s and reported in the <i>Annual
+Report of the Superintendent of the Coast and Geodetic Survey
+for the year 1890-1891</i>, was 980.1017 cm/sec<sup>2</sup>.<a name="FNanchor_107_107" id="FNanchor_107_107"></a><a href="#Footnote_107_107" class="fnanchor">[107]</a> This
+value would appear to indicate that Peirce&rsquo;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.</p>
+
+<p>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:</p>
+
+<blockquote><p>At present about a dozen new absolute determinations
+are in progress or are being planned. Heyl and Cook&rsquo;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 <span class="nowrap">(V N I I M)</span> 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.<a name="FNanchor_108_108" id="FNanchor_108_108"></a><a href="#Footnote_108_108" class="fnanchor">[108]</a></p></blockquote>
+
+
+
+<hr style="width: 65%;" />
+<h3>Application of Gravity Surveys</h3>
+
+
+<p>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.</p>
+
+<div class="figcenter" style="width: 600px;"><a name="fig_33" id="fig_33"></a>
+<img src="images/i089.png" width="600" height="597" alt="Figure 33." title="Figure 33." />
+
+<p class="caption2">Figure 33.&mdash;<span class="smcap">Gravity characteristics of the globe.</span>
+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.</p>
+</div>
+
+<p>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,<span class="pagenum"><a name="Page_343" id="Page_343">[Pg 343]</a></span>
+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
+(<i>a</i>-<i>b</i>)/<i>a</i>, where <i>a</i> is the equatorial radius and <i>b</i> 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.<span class="pagenum"><a name="Page_344" id="Page_344">[Pg 344]</a></span></p>
+
+<div class="figcenter" style="width: 600px;"><a name="fig_34" id="fig_34"></a>
+<img src="images/i092.png" width="500" height="720" alt="Figure 34." title="Figure 34." />
+
+<p class="caption2">Figure 34.&mdash;<span class="smcap">An exhibit of gravity apparatus</span> 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. <a href="#fig_15">15</a>, p. <a href="#Page_319">319</a>), 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.</p>
+</div>
+
+<p><span class="pagenum"><a name="Page_345" id="Page_345">[Pg 345]</a></span>
+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.<a name="FNanchor_109_109" id="FNanchor_109_109"></a><a href="#Footnote_109_109" class="fnanchor">[109]</a></p>
+
+<p>The extensive sets of observations of gravity provided
+the basis of calculations of the ellipticity of the
+earth. Col. A. R. Clarke in his <i>Geodesy</i> (London,
+1880) calculated the ellipticity from the results of
+gravity surveys to be <span class="nowrap">1/(292.2 ± 1.5)</span>. 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.<a name="FNanchor_110_110" id="FNanchor_110_110"></a><a href="#Footnote_110_110" class="fnanchor">[110]</a>
+He calculated the ellipticity of the earth
+to be <span class="nowrap">1/(291.5 ± 0.9)</span>.</p>
+
+<p>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 <i>Die Internationale Erdmessung</i>, adopted
+in 1924 an international reference ellipsoid,<a name="FNanchor_111_111" id="FNanchor_111_111"></a><a href="#Footnote_111_111" class="fnanchor">[111]</a> of
+which the ellipticity, or flattening, is Hayford&rsquo;s
+value 1/297. In 1930, the general assembly adopted
+a correlated International Gravity Formula of the form
+&#947; = &#947;<sub><i>E</i></sub><span class="nowrap">(1 + &#946;(sin<sup>2</sup> &#966;) + &#949;(sin<sup>2</sup> 2&#966;))</span>
+where &#947; is normal gravity at latitude &#966;, &#947;<sub><i>E</i></sub> is the
+value of gravity at sea level at the equator, &#946; is a
+parameter which is computed on the basis of Clairaut&rsquo;s
+theorem from the flattening value of the meridian,
+and &#949; 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 <i>g</i><sub>0</sub> is observed
+gravity reduced to sea level and &#947; is normal gravity
+obtained from the International Gravity Formula,
+then &#916;<i>g</i> = <span class="nowrap"><i>g</i><sub>0</sub> - &#947;</span> is the gravity anomaly.<a name="FNanchor_112_112" id="FNanchor_112_112"></a><a href="#Footnote_112_112" class="fnanchor">[112]</a></p>
+
+<p>In 1849, Stokes derived a theorem whereby the
+distance <i>N</i> 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.</p>
+
+<p>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 <a name="corr_44_06" id="corr_44_06"></a><ins class="mycorr" title="Original: mtehods">methods</ins> 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<span class="pagenum"><a name="Page_346" id="Page_346">[Pg 346]</a></span>
+physical geodesy by stating<a name="FNanchor_113_113" id="FNanchor_113_113"></a><a href="#Footnote_113_113" class="fnanchor">[113]</a> that it alone can give:</p>
+
+<blockquote><p>1. The flattening of the reference ellipsoid.</p>
+
+<p>2. The undulations <i>N</i> of the geoid.</p>
+
+<p>3. The components of the deflection of the vertical &#950;
+and &#951; at any point, oceans and islands included.</p>
+
+<p>4. The conversion of existing geodetic systems to the
+same world geodetic system.</p>
+
+<p>5. The reduction of triangulation base lines from the
+geoid to the reference ellipsoid.</p>
+
+<p>6. The correction of errors in triangulation in mountainous
+regions due to the effect of the deflections of
+the vertical.</p>
+
+<p>7. Geophysical applications of gravity measurements,
+e.g., the isostatic study of the earth&rsquo;s interior and the
+exploration of oil fields and ore deposits.</p></blockquote>
+
+<p>With astronomical observations or with existing
+triangulations, the gravimetric method can accomplish
+further results. Heiskanen and Vening Meinesz state:</p>
+
+<blockquote><p>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.<a name="FNanchor_114_114" id="FNanchor_114_114"></a><a href="#Footnote_114_114" class="fnanchor">[114]</a></p></blockquote>
+
+
+
+<hr style="width: 65%;" />
+<h3>Summary</h3>
+
+<p>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.</p>
+
+<p>The history of gravity pendulums begins with the
+ball or &ldquo;simple&rdquo; 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.</p>
+
+<p>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.</p>
+
+<p>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 <i>Die Europäische Gradmessung</i>,
+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 &ldquo;double pendulum&rdquo; 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&rsquo; 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<span class="pagenum"><a name="Page_347" id="Page_347">[Pg 347]</a></span>
+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.</p>
+
+<p>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&rsquo;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.<a name="FNanchor_115_115" id="FNanchor_115_115"></a><a href="#Footnote_115_115" class="fnanchor">[115]</a> 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
+&ldquo;I&rdquo; 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.</p>
+
+
+
+<div class="footnotes">
+<h3>FOOTNOTES:</h3>
+
+<div class="footnote"><p><a name="Footnote_1_1" id="Footnote_1_1"></a><a href="#FNanchor_1_1"><span class="label">[1]</span></a> The basic historical documents have been collected, with a
+bibliography of works and memoirs published from 1629 to
+the end of 1885, in <i>Collection de mémoires relatifs a la physique,
+publiés par la Société <a name="corr_44_07" id="corr_44_07"></a><ins class="mycorr" title="Original: Française">française</ins> de Physique</i> [hereinafter referred
+to as <i>Collection de mémoires</i>]: vol. 4, <i>Mémoires sur le pendule, précédés
+d&rsquo;une bibliographie</i> (Paris: Gauthier-Villars, 1889); and vol. 5,
+<i>Mémoires sur le pendule</i>, part 2 (Paris: Gauthier-Villars, 1891).
+Important secondary sources are: <span class="smcap">C. Wolf</span>, &ldquo;Introduction
+historique,&rdquo; pp. 1-42 in vol. 4, above; and <span class="smcap">George Biddell
+Airy</span>, &ldquo;Figure of the Earth,&rdquo; pp. 165-240 in vol. 5 of <i>Encyclopaedia
+metropolitana</i> (London, 1845).</p></div>
+
+<div class="footnote"><p><a name="Footnote_2_2" id="Footnote_2_2"></a><a href="#FNanchor_2_2"><span class="label">[2]</span></a> Galileo Galilei&rsquo;s principal statements concerning the pendulum
+occur in his <i>Discourses Concerning Two New Sciences</i>,
+transl. from Italian and Latin into English by Henry Crew and
+Alfonso de Salvio (Evanston: Northwestern University Press,
+1939), pp. 95-97, 170-172.</p></div>
+
+<div class="footnote"><p><a name="Footnote_3_3" id="Footnote_3_3"></a><a href="#FNanchor_3_3"><span class="label">[3]</span></a> <span class="smcap">P. Marin Mersenne</span>, <i>Cogitata <a name="corr_44_17" id="corr_44_17"></a><ins class="mycorr" title="Original: physica">physico</ins>-mathematica</i> (Paris,
+1644), p. 44.</p></div>
+
+<div class="footnote"><p><a name="Footnote_4_4" id="Footnote_4_4"></a><a href="#FNanchor_4_4"><span class="label">[4]</span></a> <span class="smcap">Christiaan Huygens</span>, <i>Horologium oscillatorium, sive de motu
+pendulorum ad horologia adaptato demonstrationes geometricae</i> (Paris,
+1673), proposition 20.</p></div>
+
+<div class="footnote"><p><a name="Footnote_5_5" id="Footnote_5_5"></a><a href="#FNanchor_5_5"><span class="label">[5]</span></a> The historical events reported in the present section are
+from <span class="smcap">Airy</span>, &ldquo;Figure of the Earth.&rdquo;</p></div>
+
+<div class="footnote"><p><a name="Footnote_6_6" id="Footnote_6_6"></a><a href="#FNanchor_6_6"><span class="label">[6]</span></a> <span class="smcap">Abbé Jean Picard</span>, <i>La Mesure de la terre</i> (Paris, 1671).
+<span class="smcap">John W. Olmsted</span>, &ldquo;The &lsquo;Application&rsquo; of Telescopes to Astronomical
+Instruments, 1667-1669,&rdquo; <i>Isis</i> (1949), vol. 40, p. 213.</p></div>
+
+<div class="footnote"><p><a name="Footnote_7_7" id="Footnote_7_7"></a><a href="#FNanchor_7_7"><span class="label">[7]</span></a> The toise as a unit of length was 6 Paris feet or about 1,949
+millimeters.</p></div>
+
+<div class="footnote"><p><a name="Footnote_8_8" id="Footnote_8_8"></a><a href="#FNanchor_8_8"><span class="label">[8]</span></a> <span class="smcap">Jean Richer</span>, <i>Observations astronomiques et physiques faites
+en l&rsquo;isle de Caïenne</i> (Paris, 1679). <span class="smcap">John W. Olmsted</span>, &ldquo;The
+Expedition of Jean Richer to Cayenne 1672-1673,&rdquo; <i>Isis</i>
+(1942), vol. 34, pp. 117-128.</p></div>
+
+<div class="footnote"><p><a name="Footnote_9_9" id="Footnote_9_9"></a><a href="#FNanchor_9_9"><span class="label">[9]</span></a> The Paris foot was 1.066 English feet, and there were 12
+lines to the inch.</p></div>
+
+<div class="footnote"><p><a name="Footnote_10_10" id="Footnote_10_10"></a><a href="#FNanchor_10_10"><span class="label">[10]</span></a> <span class="smcap">Christiaan Huygens</span>, &ldquo;De la cause de la pesanteur,&rdquo;
+<i>Divers ouvrages de mathematiques et de physique par MM. de
+l&rsquo;Académie <a name="corr_44_18" id="corr_44_18"></a><ins class="mycorr" title="Original: Royal">Royale</ins> des Sciences</i> (Paris, 1693), p. 305.</p></div>
+
+<div class="footnote"><p><a name="Footnote_11_11" id="Footnote_11_11"></a><a href="#FNanchor_11_11"><span class="label">[11]</span></a> <span class="smcap">Isaac Newton</span>, <i>Philosophiae naturalis principia mathematica</i>
+(London, 1687), vol. 3, propositions 18-20.</p></div>
+
+<div class="footnote"><p><a name="Footnote_12_12" id="Footnote_12_12"></a><a href="#FNanchor_12_12"><span class="label">[12]</span></a> <span class="smcap">Pierre Bouguer</span>, <i>La figure de la terre, déterminée par les
+observations de Messieurs Bouguer et de La Condamine, envoyés
+par ordre du Roy au <a name="corr_44_08" id="corr_44_08"></a><ins class="mycorr" title="Original: Perou, pour observir">Pérou, pour observer</ins> aux environs de l&rsquo;equateur</i>
+(Paris, 1749).</p></div>
+
+<div class="footnote"><p><a name="Footnote_13_13" id="Footnote_13_13"></a><a href="#FNanchor_13_13"><span class="label">[13]</span></a> <span class="smcap">P. L. Moreau de Maupertuis</span>, <i>La figure de la terre déterminée
+par les observations de Messieurs de Maupertuis, Clairaut, Camus, Le
+Monnier, l&rsquo;Abbé Outhier et Celsius, faites par ordre du Roy au cercle
+polaire</i> (Paris, 1738).</p></div>
+
+<div class="footnote"><p><a name="Footnote_14_14" id="Footnote_14_14"></a><a href="#FNanchor_14_14"><span class="label">[14]</span></a> Paris, 1743.</p></div>
+
+<div class="footnote"><p><a name="Footnote_15_15" id="Footnote_15_15"></a><a href="#FNanchor_15_15"><span class="label">[15]</span></a> <span class="smcap">George Gabriel Stokes</span>, &ldquo;On Attraction and on Clairaut&rsquo;s
+Theorem,&rdquo; <i>Cambridge and Dublin Mathematical Journal</i>
+(1849), vol. 4, p. 194.</p></div>
+
+<div class="footnote"><p><a name="Footnote_16_16" id="Footnote_16_16"></a><a href="#FNanchor_16_16"><span class="label">[16]</span></a> See <i>Collection de mémoires</i>, vol. 4, p. B-34, and <span class="smcap">J. H. Poynting</span>
+and <span class="smcap">Sir J. J. Thomson</span>, <i>Properties of Matter</i> (London, 1927),
+p. 24.</p></div>
+
+<div class="footnote"><p><a name="Footnote_17_17" id="Footnote_17_17"></a><a href="#FNanchor_17_17"><span class="label">[17]</span></a> <span class="smcap">Poynting</span> and <span class="smcap">Thomson</span>, ibid., p. 22.</p></div>
+
+<div class="footnote"><p><a name="Footnote_18_18" id="Footnote_18_18"></a><a href="#FNanchor_18_18"><span class="label">[18]</span></a> <span class="smcap">Charles M. de la Condamine</span>, &ldquo;De la mesure du pendule
+à Saint Domingue,&rdquo; <i>Collection de mémoires</i>, vol. 4, pp. 3-16.</p></div>
+
+<div class="footnote"><p><a name="Footnote_19_19" id="Footnote_19_19"></a><a href="#FNanchor_19_19"><span class="label">[19]</span></a> <span class="smcap">Père R. J. Boscovich</span>, <i>Opera pertinentia ad Opticam et
+<a name="corr_44_19" id="corr_44_19"></a><ins class="mycorr" title="Original: Astronomian">Astronomiam</ins></i> (Bassani, 1785), vol. 5, no. 3.</p></div>
+
+<div class="footnote"><p><a name="Footnote_20_20" id="Footnote_20_20"></a><a href="#FNanchor_20_20"><span class="label">[20]</span></a> <span class="smcap">J. C. Borda</span> and <span class="smcap">J. D. Cassini de Thury</span>, &ldquo;Expériences
+pour <a name="corr_44_09" id="corr_44_09"></a><ins class="mycorr" title="Original: connaitre la longuer">connaître la longueur</ins> du pendule qui bat les secondes à
+Paris,&rdquo; <i>Collection de mémoires</i>, vol. 4, pp. 17-64.</p></div>
+
+<div class="footnote"><p><a name="Footnote_21_21" id="Footnote_21_21"></a><a href="#FNanchor_21_21"><span class="label">[21]</span></a> <span class="smcap">F. W. Bessel</span>, &ldquo;Untersuchungen über die Länge des
+einfachen Secundenpendels,&rdquo; <i>Abhandlungen der <a name="corr_44_20" id="corr_44_20"></a><ins class="mycorr" title="Original: Königliche">Königlichen</ins> Akademie
+der Wissenschaften zu Berlin, 1826</i> (Berlin, 1828).</p></div>
+
+<div class="footnote"><p><a name="Footnote_22_22" id="Footnote_22_22"></a><a href="#FNanchor_22_22"><span class="label">[22]</span></a> 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 &ldquo;toise de Peru&rdquo; by Arago.</p></div>
+
+<div class="footnote"><p><a name="Footnote_23_23" id="Footnote_23_23"></a><a href="#FNanchor_23_23"><span class="label">[23]</span></a> <span class="smcap">L. G. du Buat</span>, <i>Principes d&rsquo;hydraulique</i> (Paris, 1786). See
+excerpts in <i>Collection de mémoires</i>, pp. B-64 to B-67.</p></div>
+
+<div class="footnote"><p><a name="Footnote_24_24" id="Footnote_24_24"></a><a href="#FNanchor_24_24"><span class="label">[24]</span></a> <span class="smcap">Capt. Henry Kater</span>, &ldquo;An Account of Experiments for
+Determining the Length of the Pendulum Vibrating Seconds
+in the Latitude of London,&rdquo; <i>Philosophical Transactions of the
+Royal Society of London</i> (1818), vol. 108, p. 33. [Hereinafter
+abbreviated <i>Phil. Trans.</i>]</p></div>
+
+<div class="footnote"><p><a name="Footnote_25_25" id="Footnote_25_25"></a><a href="#FNanchor_25_25"><span class="label">[25]</span></a> <span class="smcap">M. G. de Prony</span>, &ldquo;Méthode pour déterminer la <a name="corr_44_21" id="corr_44_21"></a><ins class="mycorr" title="Original: longeur">longueur</ins>
+du pendule simple qui bat les secondes,&rdquo; <i>Collection de mémoires</i>,
+vol. 4, pp. 65-76.</p></div>
+
+<div class="footnote"><p><a name="Footnote_26_26" id="Footnote_26_26"></a><a href="#FNanchor_26_26"><span class="label">[26]</span></a> <i>Collection de mémoires</i>, vol. 4, p. B-74.</p></div>
+
+<div class="footnote"><p><a name="Footnote_27_27" id="Footnote_27_27"></a><a href="#FNanchor_27_27"><span class="label">[27]</span></a> <i>Phil. Trans.</i> (1819), vol. 109, p. 337.</p></div>
+
+<div class="footnote"><p><a name="Footnote_28_28" id="Footnote_28_28"></a><a href="#FNanchor_28_28"><span class="label">[28]</span></a> <span class="smcap">John Herschel</span>, &ldquo;Notes for a History of the Use of
+Invariable Pendulums,&rdquo; <i>The Great Trigonometrical Survey of India</i>
+(Calcutta, 1879), vol. 5.</p></div>
+
+<div class="footnote"><p><a name="Footnote_29_29" id="Footnote_29_29"></a><a href="#FNanchor_29_29"><span class="label">[29]</span></a> <span class="smcap">Capt. Edward Sabine</span>, &ldquo;An Account of Experiments to
+Determine the Figure of the Earth,&rdquo; <i>Phil. Trans.</i> (1828), vol. 118,
+p. 76.</p></div>
+
+<div class="footnote"><p><a name="Footnote_30_30" id="Footnote_30_30"></a><a href="#FNanchor_30_30"><span class="label">[30]</span></a> <span class="smcap">John Goldingham</span>, &ldquo;Observations for Ascertaining the
+Length of the Pendulum at Madras in the East Indies,&rdquo; <i>Phil.
+Trans.</i> (1822), vol. 112, p. 127.</p></div>
+
+<div class="footnote"><p><a name="Footnote_31_31" id="Footnote_31_31"></a><a href="#FNanchor_31_31"><span class="label">[31]</span></a> <span class="smcap">Basil Hall</span>, &ldquo;Letter to Captain Kater Communicating
+the Details of Experiments made by him and Mr. Henry
+Foster with an Invariable Pendulum,&rdquo; <i>Phil. Trans.</i> (1823),
+vol. 113, p. 211.</p></div>
+
+<div class="footnote"><p><a name="Footnote_32_32" id="Footnote_32_32"></a><a href="#FNanchor_32_32"><span class="label">[32]</span></a> See <i>Collection de mémoires</i>, vol. 4, p. B-103.</p></div>
+
+<div class="footnote"><p><a name="Footnote_33_33" id="Footnote_33_33"></a><a href="#FNanchor_33_33"><span class="label">[33]</span></a> Ibid., p. B-88.</p></div>
+
+<div class="footnote"><p><a name="Footnote_34_34" id="Footnote_34_34"></a><a href="#FNanchor_34_34"><span class="label">[34]</span></a> Ibid., p. B-94.</p></div>
+
+<div class="footnote"><p><a name="Footnote_35_35" id="Footnote_35_35"></a><a href="#FNanchor_35_35"><span class="label">[35]</span></a> <span class="smcap">Francis Baily</span>, &ldquo;On the Correction of a Pendulum for
+the Reduction to a Vacuum, Together with Remarks on Some
+Anomalies Observed in Pendulum Experiments,&rdquo; <i>Phil. Trans.</i>
+(1832), vol. 122, pp. 399-492. See also <i>Collection de mémoires</i>,
+vol. 4, pp. B-105, B-112, B-115, B-116, and B-117.</p></div>
+
+<div class="footnote"><p><a name="Footnote_36_36" id="Footnote_36_36"></a><a href="#FNanchor_36_36"><span class="label">[36]</span></a> 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. <a href="#fig_15">15</a>) 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.</p></div>
+
+<div class="footnote"><p><a name="Footnote_37_37" id="Footnote_37_37"></a><a href="#FNanchor_37_37"><span class="label">[37]</span></a> See Baily&rsquo;s remarks in the <i>Monthly Notices of the Royal
+Astronomical Society</i> (1839), vol. 4, pp. 141-143. See also letters
+mentioned in footnote <a href="#Footnote_38_38">38</a>.</p></div>
+
+<div class="footnote"><p><a name="Footnote_38_38" id="Footnote_38_38"></a><a href="#FNanchor_38_38"><span class="label">[38]</span></a> 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.</p></div>
+
+<div class="footnote"><p><a name="Footnote_39_39" id="Footnote_39_39"></a><a href="#FNanchor_39_39"><span class="label">[39]</span></a> <span class="smcap">G. B. Airy</span>, &ldquo;Account of Experiments Undertaken in the
+Harton Colliery, for the Purpose of Determining the Mean
+Density of the Earth,&rdquo; <i>Phil. Trans.</i> (1856), vol. 146, p. 297.</p></div>
+
+<div class="footnote"><p><a name="Footnote_40_40" id="Footnote_40_40"></a><a href="#FNanchor_40_40"><span class="label">[40]</span></a> <span class="smcap">T. C. Mendenhall</span>, &ldquo;Measurements of the Force of Gravity
+at Tokyo, and on the Summit of Fujiyama,&rdquo; <i>Memoirs of the
+Science Department, University of Tokyo</i> (1881), no. 5.</p></div>
+
+<div class="footnote"><p><a name="Footnote_41_41" id="Footnote_41_41"></a><a href="#FNanchor_41_41"><span class="label">[41]</span></a> <span class="smcap">J. T. Walker</span>, <i>Account of Operations of The Great Trigonometrical
+<a name="corr_44_10" id="corr_44_10"></a><ins class="mycorr" title="Original: Surey">Survey</ins> of India</i> (Calcutta, 1879), vol. 5, app. no. 2.</p></div>
+
+<div class="footnote"><p><a name="Footnote_42_42" id="Footnote_42_42"></a><a href="#FNanchor_42_42"><span class="label">[42]</span></a> <span class="smcap">Bessel</span>, op. cit. (footnote <a href="#Footnote_21_21">21</a>), article 31.</p></div>
+
+<div class="footnote"><p><a name="Footnote_43_43" id="Footnote_43_43"></a><a href="#FNanchor_43_43"><span class="label">[43]</span></a> <span class="smcap">C. A. F. Peters</span>, <i>Briefwechsel zwischen C. F. Gauss und H. C.
+Schumacher</i> (Altona, Germany, 1860), <i>Band</i> 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.</p></div>
+
+<div class="footnote"><p><a name="Footnote_44_44" id="Footnote_44_44"></a><a href="#FNanchor_44_44"><span class="label">[44]</span></a> <span class="smcap">F. W. Bessel</span>, &ldquo;Construction eines symmetrisch geformten
+Pendels mit reciproken Axen, von Bessel,&rdquo; <i>Astronomische
+Nachrichten</i> (1849), vol. 30, p. 1.</p></div>
+
+<div class="footnote"><p><a name="Footnote_45_45" id="Footnote_45_45"></a><a href="#FNanchor_45_45"><span class="label">[45]</span></a> <span class="smcap">E. Plantamour</span>, &ldquo;Expériences faites à Genève avec le
+pendule à réversion,&rdquo; <i>Mémoires de la Société de Physique et <a name="corr_44_11" id="corr_44_11"></a><ins class="mycorr" title="Original: d&rsquo;historire">d&rsquo;histoire</ins>
+naturelle de Genève, 1865</i> (Geneva, 1866), vol. 18, p. 309.</p></div>
+
+<div class="footnote"><p><a name="Footnote_46_46" id="Footnote_46_46"></a><a href="#FNanchor_46_46"><span class="label">[46]</span></a> Ibid., pp. 309-416.</p></div>
+
+<div class="footnote"><p><a name="Footnote_47_47" id="Footnote_47_47"></a><a href="#FNanchor_47_47"><span class="label">[47]</span></a> <span class="smcap">C. Cellérier</span>, &ldquo;Note sur la Mesure de la Pesanteur par
+le Pendule,&rdquo; <i>Mémoires de la Société de Physique et <a name="corr_44_12" id="corr_44_12"></a><ins class="mycorr" title="Original: d&rsquo;historire">d&rsquo;histoire</ins>
+naturelle de Genève, 1865</i> (Geneva, 1866), vol. 18, pp. 197-218.</p></div>
+
+<div class="footnote"><p><a name="Footnote_48_48" id="Footnote_48_48"></a><a href="#FNanchor_48_48"><span class="label">[48]</span></a> <span class="smcap">A. Sawitsch</span>, &ldquo;Les variations de la pesanteur dans les
+provinces occidentales de l&rsquo;Empire russe,&rdquo; <i>Memoirs of the
+Royal Astronomical Society</i> (1872), vol. 39, p. 19.</p></div>
+
+<div class="footnote"><p><a name="Footnote_49_49" id="Footnote_49_49"></a><a href="#FNanchor_49_49"><span class="label">[49]</span></a> <span class="smcap">J. J. Baeyer</span>, <i>Über die <a name="corr_44_22" id="corr_44_22"></a><ins class="mycorr" title="Original: Grosse">Grösse</ins> und Figur der Erde</i> (Berlin,
+1861).</p></div>
+
+<div class="footnote"><p><a name="Footnote_50_50" id="Footnote_50_50"></a><a href="#FNanchor_50_50"><span class="label">[50]</span></a> <i>Comptes-rendus de la Conférence Géodésique Internationale
+réunie à Berlin du 15-22 Octobre 1864</i> (Neuchâtel, 1865).</p></div>
+
+<div class="footnote"><p><a name="Footnote_51_51" id="Footnote_51_51"></a><a href="#FNanchor_51_51"><span class="label">[51]</span></a> Ibid., part III, subpart E.</p></div>
+
+<div class="footnote"><p><a name="Footnote_52_52" id="Footnote_52_52"></a><a href="#FNanchor_52_52"><span class="label">[52]</span></a> <i>Bericht über die Verhandlungen der vom 30 September bis 7
+October 1867 zu Berlin abgehaltenen allgemeinen Conferenz der
+Europäischen Gradmessung</i> (Berlin, 1868). See report of fourth
+session, October 3, 1867.</p></div>
+
+<div class="footnote"><p><a name="Footnote_53_53" id="Footnote_53_53"></a><a href="#FNanchor_53_53"><span class="label">[53]</span></a> <span class="smcap">C. Bruhns</span> and <span class="smcap">Albrecht</span>, &ldquo;Bestimmung der <a name="corr_44_13" id="corr_44_13"></a><ins class="mycorr" title="Original: Lange">Länge</ins>
+des Secundenpendels in Bonn, Leiden und Mannheim,&rdquo;
+<i><ins class="mycorr" title="Original: Astronomische">Astronomisch</ins>-Geodätische Arbeiten im Jahre 1870</i> (Leipzig: Veröffentlichungen
+des <ins class="mycorr" title="Original: Königliche">Königlichen</ins> Preussischen Geodätischen
+Instituts, 1871).</p></div>
+
+<div class="footnote"><p><a name="Footnote_54_54" id="Footnote_54_54"></a><a href="#FNanchor_54_54"><span class="label">[54]</span></a> <i>Bericht über die Verhandlungen der vom 23 bis 28 September
+1874 in Dresden abgehaltenen vierten allgemeinen Conferenz der
+Europäischen Gradmessung</i> (Berlin, 1875). See report of second
+session, September 24, 1874.</p></div>
+
+<div class="footnote"><p><a name="Footnote_55_55" id="Footnote_55_55"></a><a href="#FNanchor_55_55"><span class="label">[55]</span></a> <span class="smcap">Carolyn Eisele</span>, &ldquo;Charles S. Peirce&mdash;Nineteenth-Century
+Man of Science,&rdquo; <i>Scripta Mathematica</i> (1959), vol 24, p. 305.
+For the account of the work of Peirce, the authors are greatly
+indebted to this pioneer paper on Peirce&rsquo;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 &ldquo;the going rate of a clock&rdquo; at &ldquo;the forks
+of the river Brandiwine in Pennsylvania,&rdquo; in 1766-67. These
+observations were published in <i>Phil. Trans.</i> (1768), vol. 58,
+pp. <a name="corr_44_14" id="corr_44_14"></a><ins class="mycorr" title="Original: 329-235.">329-335.</ins></p></div>
+
+<div class="footnote"><p><a name="Footnote_56_56" id="Footnote_56_56"></a><a href="#FNanchor_56_56"><span class="label">[56]</span></a> The pendulums with conical bobs are described and illustrated
+in <span class="smcap">E. D. Preston</span>, &ldquo;Determinations of Gravity and the
+Magnetic Elements in Connection with the United States
+Scientific Expedition to the West Coast of Africa, 1889-90,&rdquo;
+<i>Report of the Superintendent of the Coast and Geodetic Survey for 1889-90</i>
+(Washington, 1891), app. no. 12.</p></div>
+
+<div class="footnote"><p><a name="Footnote_57_57" id="Footnote_57_57"></a><a href="#FNanchor_57_57"><span class="label">[57]</span></a> <span class="smcap">Eisele</span>, op. cit. (footnote <a href="#Footnote_55_55">55</a>), p. 311.</p></div>
+
+<div class="footnote"><p><a name="Footnote_58_58" id="Footnote_58_58"></a><a href="#FNanchor_58_58"><span class="label">[58]</span></a> The record of Peirce&rsquo;s observations in Europe during 1875-76
+is given in <span class="smcap">C. S. Peirce</span>, &ldquo;Measurements of Gravity at
+Initial Stations in America and Europe,&rdquo; <i>Report of the Superintendent
+of the Coast Survey for 1875-76</i> (Washington, 1879), pp.
+202-337 and 410-416. Peirce&rsquo;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.</p></div>
+
+<div class="footnote"><p><a name="Footnote_59_59" id="Footnote_59_59"></a><a href="#FNanchor_59_59"><span class="label">[59]</span></a> <i>Verhandlungen der vom 20 bis 29 September 1875 in Paris Vereinigten
+Permanenten Commission der Europäischen Gradmessung</i> (Berlin,
+1876).</p></div>
+
+<div class="footnote"><p><a name="Footnote_60_60" id="Footnote_60_60"></a><a href="#FNanchor_60_60"><span class="label">[60]</span></a> Ibid. See report for fifth session, September 25, 1875.</p></div>
+
+<div class="footnote"><p><a name="Footnote_61_61" id="Footnote_61_61"></a><a href="#FNanchor_61_61"><span class="label">[61]</span></a> 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 <i>Bericht</i> for 1878 in the <i>Verhandlungen
+der vom 4 bis 8 September 1878 in Hamburg Vereinigten Permanenten
+Commission der Europäischen Gradmessung</i> (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
+<span class="smcap">C. S. Peirce</span>, &ldquo;On the Flexure of Pendulum Supports,&rdquo; <i>Report
+of the Superintendent of the U.S. Coast and Geodetic Survey for 1880-81</i>
+(Washington, 1883), app. no. 14, pp. 359-441.</p></div>
+
+<div class="footnote"><p><a name="Footnote_62_62" id="Footnote_62_62"></a><a href="#FNanchor_62_62"><span class="label">[62]</span></a> <i>Verhandlungen der vom 5 bis 10 Oktober 1876 in Brussels Vereinigten
+Permanenten Commission der Europäischen Gradmessung</i>
+(Berlin, 1877). See report of third session, October 7, 1876.</p></div>
+
+<div class="footnote"><p><a name="Footnote_63_63" id="Footnote_63_63"></a><a href="#FNanchor_63_63"><span class="label">[63]</span></a> <i>Verhandlungen der vom 27 September bis 2 Oktober 1877 zu
+Stuttgart abgehaltenen fünften allgemeinen Conferenz der Europäischen
+Gradmessung</i> (Berlin, 1878).</p></div>
+
+<div class="footnote"><p><a name="Footnote_64_64" id="Footnote_64_64"></a><a href="#FNanchor_64_64"><span class="label">[64]</span></a> <i>Verhandlung der vom 16 bis 20 September 1879 in Genf Vereinigten
+Permanenten Commission der Europäischen Gradmessung</i>
+(Berlin, 1880).</p></div>
+
+<div class="footnote"><p><a name="Footnote_65_65" id="Footnote_65_65"></a><a href="#FNanchor_65_65"><span class="label">[65]</span></a> <i>Assistants&rsquo; Reports, U.S. Coast and Geodetic Survey, 1879-80.</i>
+Peirce&rsquo;s paper was published in the <i>American Journal of Science</i>
+(1879), vol. 18, p. 112.</p></div>
+
+<div class="footnote"><p><a name="Footnote_66_66" id="Footnote_66_66"></a><a href="#FNanchor_66_66"><span class="label">[66]</span></a> <i>Comptes-rendus de <a name="corr_44_15" id="corr_44_15"></a><ins class="mycorr" title="Original: L&rsquo;Académie">l&rsquo;Académie</ins> des Sciences</i> (Paris, 1879),
+vol. 89, p. 462.</p></div>
+
+<div class="footnote"><p><a name="Footnote_67_67" id="Footnote_67_67"></a><a href="#FNanchor_67_67"><span class="label">[67]</span></a> <i>Verhandlungen der vom 13 bis 16 September 1880 zu München
+abgehaltenen sechsten allgemeinen Conferenz der Europäischen Gradmessung</i>
+(Berlin, 1881).</p></div>
+
+<div class="footnote"><p><a name="Footnote_68_68" id="Footnote_68_68"></a><a href="#FNanchor_68_68"><span class="label">[68]</span></a> Ibid., app. 2.</p></div>
+
+<div class="footnote"><p><a name="Footnote_69_69" id="Footnote_69_69"></a><a href="#FNanchor_69_69"><span class="label">[69]</span></a> Ibid., app. 2a.</p></div>
+
+<div class="footnote"><p><a name="Footnote_70_70" id="Footnote_70_70"></a><a href="#FNanchor_70_70"><span class="label">[70]</span></a> <i>Verhandlungen der vom 11 bis zum 15 September 1882 im Haag
+Vereinigten Permanenten Commission der Europäischen Gradmessung</i>
+(Berlin, 1883).</p></div>
+
+<div class="footnote"><p><a name="Footnote_71_71" id="Footnote_71_71"></a><a href="#FNanchor_71_71"><span class="label">[71]</span></a> <i>Verhandlungen der vom 15 bis 24 Oktober 1883 zu Rom abgehaltenen
+siebenten allgemeinen Conferenz der Europäischen Gradmessung</i>
+(Berlin, 1884). Gen. Cutts attended for the U.S. Coast and
+Geodetic Survey.</p></div>
+
+<div class="footnote"><p><a name="Footnote_72_72" id="Footnote_72_72"></a><a href="#FNanchor_72_72"><span class="label">[72]</span></a> Ibid., app. 6. See also, <i>Zeitschrift für Instrumentenkunde</i>
+(1884), vol. 4, pp. 303 and 379.</p></div>
+
+<div class="footnote"><p><a name="Footnote_73_73" id="Footnote_73_73"></a><a href="#FNanchor_73_73"><span class="label">[73]</span></a> Op. cit. (footnote <a href="#Footnote_67_67">67</a>).</p></div>
+
+<div class="footnote"><p><a name="Footnote_74_74" id="Footnote_74_74"></a><a href="#FNanchor_74_74"><span class="label">[74]</span></a> <i>Report of the Superintendent of the U.S. Coast and Geodetic
+Survey for 1880-81</i> (Washington, 1883), p. 26.</p></div>
+
+<div class="footnote"><p><a name="Footnote_75_75" id="Footnote_75_75"></a><a href="#FNanchor_75_75"><span class="label">[75]</span></a> <i>Report of the Superintendent of the U.S. Coast and Geodetic
+Survey for 1889-90</i> (Washington, 1891), app. no. 12.</p></div>
+
+<div class="footnote"><p><a name="Footnote_76_76" id="Footnote_76_76"></a><a href="#FNanchor_76_76"><span class="label">[76]</span></a> <i>Report of the Superintendent of the U.S. Coast and Geodetic
+Survey for 1881-82</i> (Washington, 1883).</p></div>
+
+<div class="footnote"><p><a name="Footnote_77_77" id="Footnote_77_77"></a><a href="#FNanchor_77_77"><span class="label">[77]</span></a> <i>Transactions of the Cambridge Philosophical Society</i> (1856),
+vol. 9, part 2, p. 8. Also published in <i>Mathematical and Physical
+Papers</i> (Cambridge, 1901), vol. 3, p. 1.</p></div>
+
+<div class="footnote"><p><a name="Footnote_78_78" id="Footnote_78_78"></a><a href="#FNanchor_78_78"><span class="label">[78]</span></a> Peirce&rsquo;s comparison of theory and experiment is discussed
+in a report on the Peirce memoir by <span class="smcap">William Ferrel</span>, dated
+October 19, 1890, Martinsburg, West Virginia. <i>U.S. Coast and
+Geodetic Survey, Special Reports, 1887-1891</i> (MS, National
+Archives, Washington).</p></div>
+
+<div class="footnote"><p><a name="Footnote_79_79" id="Footnote_79_79"></a><a href="#FNanchor_79_79"><span class="label">[79]</span></a> 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.</p></div>
+
+<div class="footnote"><p><a name="Footnote_80_80" id="Footnote_80_80"></a><a href="#FNanchor_80_80"><span class="label">[80]</span></a> <i>Comptes-rendus de l&rsquo;Académie des Sciences</i> (Paris, 1880),
+vol. 90, p. 1401. <span class="smcap">Hervé Faye</span>&rsquo;s report, dated June 21, 1880, is
+in the same <i>Comptes-rendus</i>, p. 1463.</p></div>
+
+<div class="footnote"><p><a name="Footnote_81_81" id="Footnote_81_81"></a><a href="#FNanchor_81_81"><span class="label">[81]</span></a> <span class="smcap">Commandant C. Defforges</span>, &ldquo;Sur <a name="corr_44_23" id="corr_44_23"></a><ins class="mycorr" title="Original: l&rsquo;Intensite">l&rsquo;Intensité</ins> absolue de
+la pesanteur,&rdquo; <i>Journal de Physique</i> (1888), vol. 17, pp. 239, 347,
+455. See also, <span class="smcap">Defforges</span>, &ldquo;Observations du pendule,&rdquo;
+<i>Mémorial du Dépôt général de la Guerre</i> (Paris, 1894), vol. 15.
+In the latter work, Defforges described a pendulum &ldquo;reversible
+inversable,&rdquo; 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.</p></div>
+
+<div class="footnote"><p><a name="Footnote_82_82" id="Footnote_82_82"></a><a href="#FNanchor_82_82"><span class="label">[82]</span></a> Papers by <span class="smcap">Maj. von Sterneck</span> in <i>Mitteilungen des K. u. K.
+Militär-geographischen Instituts, Wien</i>, 1882-87; see, in particular,
+vol. 7 (1887).</p></div>
+
+<div class="footnote"><p><a name="Footnote_83_83" id="Footnote_83_83"></a><a href="#FNanchor_83_83"><span class="label">[83]</span></a> <span class="smcap">T. C. Mendenhall</span>, &ldquo;Determinations of Gravity with
+the New Half-Second Pendulum ...,&rdquo; <i>Report of the Superintendent
+of the U.S. Coast and Geodetic Survey for 1890-91</i> (Washington,
+1892), part 2, pp. 503-564.</p></div>
+
+<div class="footnote"><p><a name="Footnote_84_84" id="Footnote_84_84"></a><a href="#FNanchor_84_84"><span class="label">[84]</span></a> <span class="smcap">W. H. Burger</span>, &ldquo;The Measurement of the Flexure of
+Pendulum Supports with the Interferometer,&rdquo; <i>Report of the
+Superintendent of the U.S. Coast and Geodetic Survey for 1909-10</i>
+(Washington, 1911), app. no. 6.</p></div>
+
+<div class="footnote"><p><a name="Footnote_85_85" id="Footnote_85_85"></a><a href="#FNanchor_85_85"><span class="label">[85]</span></a> <span class="smcap">E. J. Brown</span>, <i>A Determination of the Relative Values of
+Gravity at Potsdam and Washington</i> (Special Publication No.
+204, U.S. Coast and Geodetic Survey; Washington, 1936).</p></div>
+
+<div class="footnote"><p><a name="Footnote_86_86" id="Footnote_86_86"></a><a href="#FNanchor_86_86"><span class="label">[86]</span></a> <span class="smcap">M. Haid</span>, &ldquo;Neues Pendelstativ,&rdquo; <i>Zeitschrift für Instrumentenkunde</i>
+(July 1896), vol. 16, p. 193.</p></div>
+
+<div class="footnote"><p><a name="Footnote_87_87" id="Footnote_87_87"></a><a href="#FNanchor_87_87"><span class="label">[87]</span></a> <span class="smcap">Dr. R. Schumann</span>, &ldquo;Über eine Methode, das Mitschwingen
+bei relativen Schweremessungen zu bestimmen,&rdquo; <i>Zeitschrift
+für Instrumentenkunde</i> (January 1897), vol. 17, p. 7. The design
+for the stand is similar to that of Peirce&rsquo;s of 1879.</p></div>
+
+<div class="footnote"><p><a name="Footnote_88_88" id="Footnote_88_88"></a><a href="#FNanchor_88_88"><span class="label">[88]</span></a> <span class="smcap">Dr. R. Schumann</span>, &ldquo;Über die Verwendung zweier Pendel
+auf gemeinsamer Unterlage zur Bestimmung der Mitschwingung,&rdquo;
+<i>Zeitschrift für Mathematik und Physik</i> (1899), vol. 44, p. 44.</p></div>
+
+<div class="footnote"><p><a name="Footnote_89_89" id="Footnote_89_89"></a><a href="#FNanchor_89_89"><span class="label">[89]</span></a> <span class="smcap">P. Furtwängler</span>, &ldquo;Über die Schwingungen zweier
+Pendel mit annähernd gleicher Schwingungsdauer auf gemeinsamer
+Unterlage,&rdquo; <i>Sitzungsberichte der <a name="corr_44_24" id="corr_44_24"></a><ins class="mycorr" title="Original: Königliche">Königlicher</ins> Preussischen
+Akademie der Wissenschaften zu Berlin</i> (Berlin, 1902) pp. 245-253.
+Peirce investigated the plan of swinging two pendulums on the
+same stand (<i>Report of the Superintendent of the U.S. Coast and
+Geodetic Survey for 1880-81</i>, Washington, 1883, p. 26; also in
+<span class="smcap">Charles Sanders Peirce</span>, <i>Collected Papers</i>, 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 (&ldquo;Report of a conference on gravity determinations
+held in Washington, D.C., in May, 1882,&rdquo; <i>Report
+of the Superintendent of the U.S. Coast and Geodetic Survey for 1881-82</i>,
+Washington, 1883, app. no. 22, pp. 503-516).</p></div>
+
+<div class="footnote"><p><a name="Footnote_90_90" id="Footnote_90_90"></a><a href="#FNanchor_90_90"><span class="label">[90]</span></a> <span class="smcap">F. A. Vening Meinesz</span>, <i>Observations de pendule dans les
+Pays-Bas</i> (Delft, 1923).</p></div>
+
+<div class="footnote"><p><a name="Footnote_91_91" id="Footnote_91_91"></a><a href="#FNanchor_91_91"><span class="label">[91]</span></a> <span class="smcap">A. Berroth</span>, &ldquo;Schweremessungen mit zwei und vier
+gleichzeitig auf demselben Stativ schwingenden Pendeln,&rdquo;
+<i>Zeitschrift für Geophysik</i>, vol. 1 (1924-25), no. 3, p. 93.</p></div>
+
+<div class="footnote"><p><a name="Footnote_92_92" id="Footnote_92_92"></a><a href="#FNanchor_92_92"><span class="label">[92]</span></a> &ldquo;Pendulum Apparatus for Gravity Determinations,&rdquo;
+<i>Engineering</i> (1926), vol. 122, pp. 271-272.</p></div>
+
+<div class="footnote"><p><a name="Footnote_93_93" id="Footnote_93_93"></a><a href="#FNanchor_93_93"><span class="label">[93]</span></a> <span class="smcap">Malcolm W. Gay</span>, &ldquo;Relative Gravity Measurements
+Using Precision Pendulum Equipment,&rdquo; <i>Geophysics</i> (1940), vol.
+5, pp. 176-191.</p></div>
+
+<div class="footnote"><p><a name="Footnote_94_94" id="Footnote_94_94"></a><a href="#FNanchor_94_94"><span class="label">[94]</span></a> <span class="smcap">L. G. D. Thompson</span>, &ldquo;An Improved Bronze Pendulum
+Apparatus for Relative Gravity Determinations,&rdquo; [published
+by] <i>Dominion Observatory</i> (Ottawa, 1959), vol. 21, no. 3, pp.
+145-176.</p></div>
+
+<div class="footnote"><p><a name="Footnote_95_95" id="Footnote_95_95"></a><a href="#FNanchor_95_95"><span class="label">[95]</span></a> <span class="smcap">W. A. Heiskanen</span> and <span class="smcap">F. A. Vening Meinesz</span>, <i>The Earth
+and its Gravity Field</i> (McGraw: New York, 1958).</p></div>
+
+<div class="footnote"><p><a name="Footnote_96_96" id="Footnote_96_96"></a><a href="#FNanchor_96_96"><span class="label">[96]</span></a> <span class="smcap">F. Kühnen</span> and <span class="smcap">P. Furtwängler</span>, <i>Bestimmung der Absoluten
+Grösze der Schwerkraft zu Potsdam mit Reversionspendeln</i> (Berlin:
+Veröffentlichungen des Königlichen Preussischen Geodätischen
+Instituts, 1906), new ser., no. 27.</p></div>
+
+<div class="footnote"><p><a name="Footnote_97_97" id="Footnote_97_97"></a><a href="#FNanchor_97_97"><span class="label">[97]</span></a> Reported by Dr. F. Kühnen to the fifth session, October 9,
+1895, of the Eleventh General Conference, <i>Die Internationale
+Erdmessung</i>, 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&rsquo;s
+prior discovery of the influence of the flexure of the pendulum
+itself upon the period (<i>Report of the Superintendent of the U.S.
+Coast and Geodetic Survey for 1883-84</i>, Washington, 1885, app.
+16, pp. 483-485).</p></div>
+
+<div class="footnote"><p><a name="Footnote_98_98" id="Footnote_98_98"></a><a href="#FNanchor_98_98"><span class="label">[98]</span></a> <i>Assistants&rsquo; Reports, U.S. Coast and Geodetic Survey, 1883-84</i>
+(MS, National Archives, Washington).</p></div>
+
+<div class="footnote"><p><a name="Footnote_99_99" id="Footnote_99_99"></a><a href="#FNanchor_99_99"><span class="label">[99]</span></a> <span class="smcap">C. S. Peirce</span>, &ldquo;Effect of the Flexure of a Pendulum Upon
+its Period of Oscillation,&rdquo; <i>Report of the Superintendent of the U.S.
+Coast and Geodetic Survey for 1883-84</i> (Washington, 1885), app.
+no. 16.</p></div>
+
+<div class="footnote"><p><a name="Footnote_100_100" id="Footnote_100_100"></a><a href="#FNanchor_100_100"><span class="label">[100]</span></a> <span class="smcap">F. R. Helmert</span>, <i>Beiträge zur Theorie des Reversionspendels</i>
+(Potsdam: Veröffentlichungen <a name="corr_44_25" id="corr_44_25"></a><ins class="mycorr" title="Original: Königliche">des Königlichen</ins> Preussischen Geodätischen
+Instituts, 1898).</p></div>
+
+<div class="footnote"><p><a name="Footnote_101_101" id="Footnote_101_101"></a><a href="#FNanchor_101_101"><span class="label">[101]</span></a> <span class="smcap">J. A. Duerksen</span>, <i>Pendulum Gravity Data in the United States</i>
+(Special Publication No. 244, U.S. Coast and Geodetic Survey;
+Washington, 1949).</p></div>
+
+<div class="footnote"><p><a name="Footnote_102_102" id="Footnote_102_102"></a><a href="#FNanchor_102_102"><span class="label">[102]</span></a> Ibid., p. 2. See also, <span class="smcap">E. J. Brown</span>, loc. cit. (footnote <a href="#Footnote_85_85">85</a>).</p></div>
+
+<div class="footnote"><p><a name="Footnote_103_103" id="Footnote_103_103"></a><a href="#FNanchor_103_103"><span class="label">[103]</span></a> <span class="smcap">Paul R. Heyl</span> and <span class="smcap">Guy S. Cook</span>, &ldquo;The Value of Gravity
+at Washington,&rdquo; <i>Journal of Research, National Bureau of Standards</i>
+(1936), vol. 17, p. 805.</p></div>
+
+<div class="footnote"><p><a name="Footnote_104_104" id="Footnote_104_104"></a><a href="#FNanchor_104_104"><span class="label">[104]</span></a> <span class="smcap">Sir Harold Jeffreys</span>, &ldquo;The Absolute Value of Gravity,&rdquo;
+<i>Monthly Notices of the Royal Astronomical Society, Geophysical Supplement</i>
+(London, 1949), vol. 5, p. 398.</p></div>
+
+<div class="footnote"><p><a name="Footnote_105_105" id="Footnote_105_105"></a><a href="#FNanchor_105_105"><span class="label">[105]</span></a> <span class="smcap">J. S. Clark</span>, &ldquo;The Acceleration Due to Gravity,&rdquo; <i>Phil.
+Trans.</i> (1939), vol. 238, p. 65.</p></div>
+
+<div class="footnote"><p><a name="Footnote_106_106" id="Footnote_106_106"></a><a href="#FNanchor_106_106"><span class="label">[106]</span></a> <span class="smcap">Hugh L. Dryden</span>, &ldquo;A Reexamination of the Potsdam
+Absolute Determination of Gravity,&rdquo; <i>Journal of Research,
+National Bureau of Standards</i> (1942), vol. 29, p. 303; and <span class="smcap">A.
+Berroth</span>, &ldquo;Das Fundamentalsystem der Schwere im Lichte
+neuer Reversionspendelmessungen,&rdquo; <i>Bulletin Géodésique</i> (1949),
+no. 12, pp. 183-204.</p></div>
+
+<div class="footnote"><p><a name="Footnote_107_107" id="Footnote_107_107"></a><a href="#FNanchor_107_107"><span class="label">[107]</span></a> <span class="smcap">T. C. Mendenhall</span>, op. cit. (footnote <a href="#Footnote_83_83">83</a>), p. 522.</p></div>
+
+<div class="footnote"><p><a name="Footnote_108_108" id="Footnote_108_108"></a><a href="#FNanchor_108_108"><span class="label">[108]</span></a> <span class="smcap">A. H. Cook</span>, &ldquo;Recent Developments in the Absolute
+Measurement of Gravity,&rdquo; <i>Bulletin Géodésique</i> (June 1, 1957),
+no. 44, pp. 34-59.</p></div>
+
+<div class="footnote"><p><a name="Footnote_109_109" id="Footnote_109_109"></a><a href="#FNanchor_109_109"><span class="label">[109]</span></a> See footnote <a href="#Footnote_89_89">89</a>.</p></div>
+
+<div class="footnote"><p><a name="Footnote_110_110" id="Footnote_110_110"></a><a href="#FNanchor_110_110"><span class="label">[110]</span></a> <span class="smcap">C. S. Peirce</span>, &ldquo;On the Deduction of the Ellipticity of the
+Earth, from Pendulum Experiments,&rdquo; <i>Report of the Superintendent
+of the U.S. Coast and Geodetic Survey for 1880-81</i> (Washington,
+1883), app. no. 15, pp. 442-456.</p></div>
+
+<div class="footnote"><p><a name="Footnote_111_111" id="Footnote_111_111"></a><a href="#FNanchor_111_111"><span class="label">[111]</span></a> <span class="smcap">Heiskanen</span> and <span class="smcap">Vening Meinesz</span>, op. cit. (footnote <a href="#Footnote_95_95">95</a>), p.
+74.</p></div>
+
+<div class="footnote"><p><a name="Footnote_112_112" id="Footnote_112_112"></a><a href="#FNanchor_112_112"><span class="label">[112]</span></a> Ibid., p. 76.</p></div>
+
+<div class="footnote"><p><a name="Footnote_113_113" id="Footnote_113_113"></a><a href="#FNanchor_113_113"><span class="label">[113]</span></a> Ibid., p. 309.</p></div>
+
+<div class="footnote"><p><a name="Footnote_114_114" id="Footnote_114_114"></a><a href="#FNanchor_114_114"><span class="label">[114]</span></a> Ibid., p. 310.</p></div>
+
+<div class="footnote"><p><a name="Footnote_115_115" id="Footnote_115_115"></a><a href="#FNanchor_115_115"><span class="label">[115]</span></a> <span class="smcap">K. Reicheneder</span>, &ldquo;Method of the New Measurements at Potsdam by
+Means of the Reversible Pendulum,&rdquo; <i>Bulletin Géodésique</i> (March 1, 1959),
+no. 51, p.72.</p></div>
+
+</div>
+<p class="center"><br /><br />
+U.S. GOVERNMENT PRINTING OFFICE: 1965<br />
+<br />
+For sale by the Superintendent of Documents, U.S. Government Printing Office<br />
+Washington, D.C., 20402&mdash;Price 70 cents.<br />
+</p>
+
+
+
+<hr style="width: 65%;" />
+<h3>INDEX</h3>
+
+
+<p>Airy, G. B., <a href="#Page_319">319</a>, <a href="#Page_324">324</a>, <a href="#Page_332">332</a></p>
+
+<p>Albrecht, Karl Theodore, <a href="#Page_322">322</a>, <a href="#Page_338">338</a></p>
+
+<p>Al-Mamun, seventh calif of Bagdad, <a href="#Page_306">306</a></p>
+
+<p>Almansi, Emilio, <a href="#Page_339">339</a></p>
+
+<p>Aristotle, <a href="#Page_306">306</a></p>
+
+
+<p>Baeyer, J. J., <a href="#Page_321">321</a>, <a href="#Page_322">322</a>, <a href="#Page_324">324</a>, <a href="#Page_338">338</a>, <a href="#Page_346">346</a></p>
+
+<p>Baily, Francis, <a href="#Page_317">317</a></p>
+
+<p>Basevi, James Palladio, <a href="#Page_345">345</a></p>
+
+<p>Berroth, A., <a href="#Page_342">342</a></p>
+
+<p>Bessel, Friedrich Wilhelm, <a href="#Page_313">313</a>, <a href="#Page_314">314</a>, <a href="#Page_319">319</a>, <a href="#Page_320">320</a>, <a href="#Page_324">324</a>, <a href="#Page_325">325</a>, <a href="#Page_338">338</a>, <a href="#Page_346">346</a></p>
+
+<p>Biot, Jean Baptiste, <a href="#Page_325">325</a>, <a href="#Page_329">329</a></p>
+
+<p>Bohnenberger, Johann Gottlieb Friedrich, <a href="#Page_315">315</a></p>
+
+<p>Borda, J. C., <a href="#Page_311">311</a>, <a href="#Page_312">312</a>, <a href="#Page_315">315</a>, <a href="#Page_325">325</a>, <a href="#Page_329">329</a>, <a href="#Page_346">346</a></p>
+
+<p>Boscovitch, Père R. J., <a href="#Page_310">310</a>, <a href="#Page_311">311</a></p>
+
+<p>Bouguer, Pierre, <a href="#Page_307">307</a>, <a href="#Page_309">309</a>, <a href="#Page_327">327</a>, <a href="#Page_343">343</a>, <a href="#Page_345">345</a></p>
+
+<p>Brahe, Tycho, <a href="#Page_306">306</a></p>
+
+<p>Brown, E. J., <a href="#Page_334">334</a>, <a href="#Page_339">339</a></p>
+
+<p>Browne, Henry, <a href="#Page_304">304</a>, <a href="#Page_314">314</a></p>
+
+<p>Bruhns, C., <a href="#Page_322">322</a>, <a href="#Page_324">324</a>, <a href="#Page_338">338</a></p>
+
+<p>Brunner Brothers (Paris), <a href="#Page_329">329</a></p>
+
+
+<p>Cassini, Giovanni-Domenico, <a href="#Page_306">306</a>, <a href="#Page_307">307</a></p>
+
+<p>Cassini, Jacques, <a href="#Page_306">306</a></p>
+
+<p>Cassini de Thury, J. D., <a href="#Page_311">311</a>, <a href="#Page_312">312</a>, <a href="#Page_315">315</a>, <a href="#Page_325">325</a>, <a href="#Page_329">329</a>, <a href="#Page_346">346</a></p>
+
+<p>Cellérier, Charles, <a href="#Page_320">320</a>, <a href="#Page_321">321</a>, <a href="#Page_325">325</a>, <a href="#Page_326">326</a>, <a href="#Page_329">329</a>, <a href="#Page_336">336</a></p>
+
+<p>Clairaut, Alexis Claude, <a href="#Page_308">308</a>, <a href="#Page_309">309</a>, <a href="#Page_343">343</a>, <a href="#Page_345">345</a></p>
+
+<p>Clark, J. S., <a href="#Page_342">342</a></p>
+
+<p>Clarke, A. R., <a href="#Page_345">345</a></p>
+
+<p>Colbert, Jean Baptiste, <a href="#Page_306">306</a></p>
+
+<p>Cook, A. H., <a href="#Page_342">342</a></p>
+
+<p>Cook, Guy S., <a href="#Page_339">339</a>, <a href="#Page_342">342</a></p>
+
+
+<p>Defforges, C., <a href="#Page_314">314</a>, <a href="#Page_329">329</a>, <a href="#Page_346">346</a></p>
+
+<p>De Freycinet, Louis Claude de Saulses, <a href="#Page_317">317</a></p>
+
+<p>De la Hire, Gabriel Philippe, <a href="#Page_306">306</a></p>
+
+<p>De Prony, M. G., <a href="#Page_314">314</a></p>
+
+<p>Dryden, Hugh L., <a href="#Page_342">342</a></p>
+
+<p>Du Buat, L. G., <a href="#Page_314">314</a></p>
+
+<p>Duperry, Capt. Louis Isidore, <a href="#Page_317">317</a></p>
+
+
+<p>Eratosthenes, <a href="#Page_306">306</a>, <a href="#Page_308">308</a>, <a href="#Page_342">342</a></p>
+
+<p>Eudoxus of Cnidus, <a href="#Page_306">306</a></p>
+
+
+<p>Faye, Hervé, <a href="#Page_325">325</a>, <a href="#Page_336">336</a>, <a href="#Page_346">346</a>, <a href="#Page_347">347</a></p>
+
+<p>Fernel, Jean, <a href="#Page_306">306</a></p>
+
+<p>Furtwängler, P., <a href="#Page_337">337</a></p>
+
+
+<p>Galilei, Galileo, <a href="#Page_304">304</a>, <a href="#Page_305">305</a>, <a href="#Page_346">346</a></p>
+
+<p>Gauss, C. F., <a href="#Page_320">320</a></p>
+
+<p>Gautier, P., <a href="#Page_339">339</a></p>
+
+<p>Godin, Louis, <a href="#Page_307">307</a></p>
+
+<p>Goldingham, John, <a href="#Page_316">316</a>, <a href="#Page_345">345</a></p>
+
+<p>Greely, A. W., <a href="#Page_329">329</a></p>
+
+<p>Gulf Oil and Development Company, <a href="#Page_338">338</a></p>
+
+
+<p>Haid, M., <a href="#Page_335">335</a></p>
+
+<p>Hall, Basil, <a href="#Page_316">316</a></p>
+
+<p>Heaviside, W. J., <a href="#Page_321">321</a>, <a href="#Page_345">345</a></p>
+
+<p>Heiskanen, W. A., <a href="#Page_338">338</a>, <a href="#Page_345">345</a>, <a href="#Page_346">346</a></p>
+
+<p>Helmert, F. R., <a href="#Page_338">338</a>, <a href="#Page_339">339</a></p>
+
+<p>Helmholtz, Hermann von, <a href="#Page_326">326</a></p>
+
+<p>Herschel, John, <a href="#Page_319">319</a>, <a href="#Page_328">328</a>, <a href="#Page_345">345</a></p>
+
+<p>Heyl, Paul R., <a href="#Page_339">339</a>, <a href="#Page_342">342</a></p>
+
+<p>Hirsch, Adolph, <a href="#Page_322">322</a>, <a href="#Page_324">324</a></p>
+
+<p>Huygens, Christiaan, <a href="#Page_304">304</a>, <a href="#Page_305">305</a>, <a href="#Page_307">307</a>, <a href="#Page_314">314</a>, <a href="#Page_342">342</a>, <a href="#Page_346">346</a></p>
+
+
+<p>Ibañez, Carlos, <a href="#Page_325">325</a></p>
+
+
+<p>Jeffreys, Sir Harold, <a href="#Page_342">342</a></p>
+
+<p>Jones, Thomas, <a href="#Page_318">318</a></p>
+
+
+<p>Kater, Henry, <a href="#Page_304">304</a>, <a href="#Page_314">314</a>, <a href="#Page_325">325</a>, <a href="#Page_327">327</a>, <a href="#Page_329">329</a>, <a href="#Page_345">345</a>, <a href="#Page_346">346</a></p>
+
+<p>Kühnen, F., <a href="#Page_338">338</a>, <a href="#Page_339">339</a></p>
+
+
+<p>La Condamine, Charles Marie de, <a href="#Page_307">307</a>, <a href="#Page_310">310</a>, <a href="#Page_311">311</a>, <a href="#Page_343">343</a></p>
+
+<p>Laplace, Marquis Pierre Simon de, <a href="#Page_309">309</a>, <a href="#Page_313">313</a>, <a href="#Page_320">320</a></p>
+
+<p>Lorenzoni, Giuseppe, <a href="#Page_336">336</a>, <a href="#Page_339">339</a></p>
+
+<p>Lütke, Count Feodor Petrovich, <a href="#Page_316">316</a>, <a href="#Page_345">345</a></p>
+
+
+<p>Maupertius, P. L. Moreau de, <a href="#Page_308">308</a>, <a href="#Page_343">343</a></p>
+
+<p>Maxwell, James Clerk, <a href="#Page_324">324</a></p>
+
+<p>Medi, Enrico, <a href="#Page_342">342</a></p>
+
+<p>Mendenhall, Thomas Corwin, <a href="#Page_319">319</a>, <a href="#Page_331">331</a>, <a href="#Page_332">332</a>, <a href="#Page_334">334</a>, <a href="#Page_347">347</a></p>
+
+<p>Mersenne, P. Marin, <a href="#Page_305">305</a></p>
+
+
+<p>Newton, Sir Isaac, <a href="#Page_303">303</a>, <a href="#Page_307">307</a>, <a href="#Page_308">308</a>, <a href="#Page_342">342</a>, <a href="#Page_343">343</a></p>
+
+<p>Norwood, Richard, <a href="#Page_306">306</a></p>
+
+
+<p>Oppolzer, Theodor von, <a href="#Page_322">322</a>, <a href="#Page_324">324</a></p>
+
+
+<p>Patterson, Carlile Pollock, <a href="#Page_325">325</a>, <a href="#Page_326">326</a></p>
+
+<p>Peirce, Charles Sanders, <a href="#Page_314">314</a>, <a href="#Page_322">322</a>, <a href="#Page_332">332</a>, <a href="#Page_336">336</a>, <a href="#Page_342">342</a>, <a href="#Page_345">345</a></p>
+
+<p>Peters, C. A. F., <a href="#Page_322">322</a>, <a href="#Page_324">324</a></p>
+
+<p>Picard, Abbé Jean, <a href="#Page_306">306</a>, <a href="#Page_308">308</a>, <a href="#Page_342">342</a></p>
+
+<p>Plantamour, E., <a href="#Page_319">319</a>, <a href="#Page_324">324</a></p>
+
+<p>Posidonius, <a href="#Page_306">306</a></p>
+
+<p>Preston, E. D., <a href="#Page_328">328</a>, <a href="#Page_329">329</a></p>
+
+<p>Putnam, G. R., <a href="#Page_339">339</a></p>
+
+<p>Pythagoras, <a href="#Page_306">306</a></p>
+
+
+<p>Repsold, A., and Sons (Hamburg), <a href="#Page_320">320</a>, <a href="#Page_322">322</a>, <a href="#Page_338">338</a>, <a href="#Page_339">339</a>, <a href="#Page_346">346</a></p>
+
+<p>Richer, Jean, <a href="#Page_307">307</a>, <a href="#Page_342">342</a></p>
+
+
+<p>Sabine, Capt. Edward, <a href="#Page_315">315</a>, <a href="#Page_325">325</a>, <a href="#Page_329">329</a>, <a href="#Page_345">345</a></p>
+
+<p>Sawitsch, A., <a href="#Page_321">321</a>, <a href="#Page_322">322</a></p>
+
+<p>Schumacher, H. C., <a href="#Page_320">320</a></p>
+
+<p>Schumann, R., <a href="#Page_335">335</a>, <a href="#Page_336">336</a></p>
+
+<p>Snell, Willebrord, <a href="#Page_306">306</a></p>
+
+<p>Sterneck, Robert von, <a href="#Page_331">331</a>, <a href="#Page_332">332</a>, <a href="#Page_335">335</a>, <a href="#Page_338">338</a>, <a href="#Page_346">346</a></p>
+
+<p>Stokes, George Gabriel, <a href="#Page_324">324</a>, <a href="#Page_328">328</a>, <a href="#Page_329">329</a>, <a href="#Page_345">345</a>, <a href="#Page_346">346</a></p>
+
+
+<p>Ulloa, Antonio de, <a href="#Page_308">308</a></p>
+
+
+<p>Vening Meinesz, F. A., <a href="#Page_337">337</a>, <a href="#Page_338">338</a>, <a href="#Page_345">345</a></p>
+
+<p>Volet, Charles, <a href="#Page_342">342</a></p>
+
+
+<p>Wilkes, Charles, <a href="#Page_317">317</a>, <a href="#Page_318">318</a><br /><br /></p>
+
+<div class="tnote"><a name="corrections_44" id="corrections_44"></a>
+<h3>Transcriber&rsquo;s Corrections</h3>
+
+<p>Formatting of equations has been altered from the original to display them
+&lsquo;in line,&rsquo; and brackets have been added to clarify expressions where necessary.</p>
+
+<p>Footnotes have been moved to the end of the paper. Illustrations and the
+<span class="smcap">Glossary of Gravity Terminology</span> 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>
+<div style="margin-left: 2em;">
+  <p class="hang">P. <a href="#corr_44_01">320</a> &lsquo;difference <i>T</i><sub>1</sub> - <i>T</i><sub>2</sub> is sufficiently&rsquo;&mdash;had &lsquo;sufficlently.&rsquo;</p>
+  <p class="hang">P. <a href="#corr_44_02">321</a> &lsquo;faites à Genève avec le pendule à réversion&rsquo;&mdash;had &lsquo;reversion.&rsquo;</p>
+  <p class="hang">P. <a href="#corr_44_16">326</a> &lsquo;Schwere mit Hilfe verschiedener Apparate&rsquo;&mdash;had &lsquo;verschiedene.&rsquo;</p>
+  <p class="hang">P. <a href="#corr_44_03">328</a> &lsquo;between the yard and the meter.&rsquo;&mdash;closing quote mark deleted.</p>
+  <p class="hang">P. <a href="#corr_44_04">334</a> &lsquo;Mendenhall apparatus were part of&rsquo;&mdash;&lsquo;was&rsquo; changed to &lsquo;were.&rsquo;</p>
+  <p class="hang">P. <a href="#corr_44_05">342</a> &lsquo;of the Geodetic Institute at Potsdam&rsquo;&mdash;had &lsquo;Postdam.&rsquo;</p>
+  <p class="hang">P. <a href="#corr_44_06">345</a> &lsquo;The gravimetric methods of physical&rsquo;&mdash;had &lsquo;mtehods.&rsquo;</p>
+  <p class="hang">Footnote <a href="#corr_44_07">1</a> &lsquo;Société française de Physique&rsquo;&mdash;had &lsquo;Française.&rsquo;</p>
+  <p class="hang">Footnote <a href="#corr_44_17">3</a> &lsquo;Cogitata physico-mathematica&rsquo;&mdash;had &lsquo;physica.&rsquo;</p>
+  <p class="hang">Footnote <a href="#corr_44_18">10</a> &lsquo;mathématiques et de physique par MM. de l&rsquo;Académie Royale&rsquo;&mdash;had &lsquo;mathematiques,&rsquo; &lsquo;Royal.&rsquo;</p>
+  <p class="hang">Footnote <a href="#corr_44_08">12</a> &lsquo;par ordre du Roy au Pérou, pour observer&rsquo;&mdash;had &lsquo;Perou, pour observir.&rsquo;</p>
+  <p class="hang">Footnote <a href="#corr_44_19">19</a> &lsquo;Opticam et Astronomiam&rsquo;&mdash;had &lsquo;Astronomian.&rsquo;</p>
+  <p class="hang">Footnote <a href="#corr_44_09">20</a> &lsquo;connaître la longueur du pendule qui&rsquo;&mdash;had &lsquo;connaitre la longuer.&rsquo;</p>
+  <p class="hang">Footnote <a href="#corr_44_20">21</a> &lsquo;Abhandlungen der Königlichen Akademie&rsquo;&mdash;had &lsquo;Königliche.&rsquo;</p>
+  <p class="hang">Footnote <a href="#corr_44_21">25</a> &lsquo;pour déterminer la longueur du pendule&rsquo;&mdash;had &lsquo;longeur.&rsquo;</p>
+  <p class="hang">Footnote <a href="#corr_44_10">41</a> &lsquo;Survey of India (Calcutta, 1879)&rsquo;&mdash; had &lsquo;Surey.&rsquo;</p>
+  <p class="hang">Footnotes <a href="#corr_44_11">45</a> and <a href="#corr_44_12">47</a> &lsquo;Société de Physique et d&rsquo;histoire&rsquo;&mdash;had
+      &lsquo;d&rsquo;historire.&rsquo;</p>
+  <p class="hang">Footnote <a href="#corr_44_22">49</a> &lsquo;Über die Grösse und Figur der Erde&rsquo;&mdash;had &lsquo;Grosse.&rsquo;</p>
+  <p class="hang">Footnote <a href="#corr_44_13">53</a> &lsquo;Bestimmung der Länge&rsquo;&mdash;had &lsquo;Lange&rsquo;; &lsquo;Astronomisch-Geodätische Arbeiten&rsquo;&mdash;had
+&lsquo;Astronomische&rsquo;; &lsquo;Veröffentlichungen des Königlichen&rsquo;&mdash;had &lsquo;Königliche.&rsquo;</p>
+  <p class="hang">Footnote <a href="#corr_44_14">55</a> &lsquo;(1768), vol. 58, pp. 329-335.&rsquo;&mdash;had &lsquo;329-235.&rsquo;</p>
+  <p class="hang">Footnote <a href="#corr_44_15">66</a> &lsquo;Comptes-rendus de l&rsquo;Académie&rsquo;&mdash;had &lsquo;L&rsquo;Académie.&rsquo;</p>
+  <p class="hang">Footnote <a href="#corr_44_23">81</a> &lsquo;Sur l&rsquo;Intensité absolue&rsquo;&mdash;had &lsquo;l&rsquo;Intensite.&rsquo;</p>
+  <p class="hang">Footnote <a href="#corr_44_24">89</a> &lsquo;Sitzungsberichte der Königlicher&rsquo;&mdash;had &lsquo;Königliche.&rsquo;</p>
+  <p class="hang">Footnote <a href="#corr_44_25">100</a> &lsquo;Veröffentlichungen des Königlichen&rsquo; had &lsquo;Veröffentlichungen Königliche.&rsquo;</p>
+</div>
+<p>Capitalisation of &lsquo;Von&rsquo;/&lsquo;von&rsquo; has been regulaized to &lsquo;von&rsquo; for all
+personal names, except at the beginning of a sentence, and when
+referring to the Von Sterneck pendulum.</p>
+</div>
+
+<p>&nbsp;</p>
+<p>&nbsp;</p>
+<hr class="full" />
+<p>***END OF THE PROJECT GUTENBERG EBOOK DEVELOPMENT OF GRAVITY PENDULUMS IN THE 19TH CENTURY***</p>
+<p>******* This file should be named 35024-h.txt or 35024-h.zip *******</p>
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+The Project Gutenberg eBook, Development of Gravity Pendulums in the 19th
+Century, by Victor Fritz Lenzen and Robert P. Multhauf
+
+
+This eBook is for the use of anyone anywhere at no cost and with
+almost no restrictions whatsoever.  You may copy it, give it away or
+re-use it under the terms of the Project Gutenberg License included
+with this eBook or online at www.gutenberg.org
+
+
+
+
+
+Title: Development of Gravity Pendulums in the 19th Century
+       Contributions from the Museum of History and Technology, Papers 34-44 On Science and Technology, Smithsonian Institution, 1966
+
+
+Author: Victor Fritz Lenzen and Robert P. Multhauf
+
+
+
+Release Date: January 21, 2011  [eBook #35024]
+
+Language: English
+
+Character set encoding: ISO-646-US (US-ASCII)
+
+
+***START OF THE PROJECT GUTENBERG EBOOK DEVELOPMENT OF GRAVITY PENDULUMS
+IN THE 19TH CENTURY***
+
+
+E-text prepared by Chris Curnow, Joseph Cooper, Louise Pattison, and the
+Online Distributed Proofreading Team (http://www.pgdp.net)
+
+
+
+Note: Project Gutenberg also has an HTML version of this
+      file which includes the original illustrations.
+      See 35024-h.htm or 35024-h.zip:
+      (http://www.gutenberg.org/files/35024/35024-h/35024-h.htm)
+      or
+      (http://www.gutenberg.org/files/35024/35024-h.zip)
+
+
+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.
+
+      The front material, introduction and relevant index entries
+      from the _Bulletin_ are included in each single-paper e-book.
+
+      Mathematical notation used in this e-text:
+
+        1. Greek letters are represented by the name of the letter
+           in square brackets; _e.g._, [pi].
+
+        2. Subscripts are denoted by underscore followed by the
+           subscript in curly braces; _e.g._, T_{n}. Superscripts
+           are denoted by a caret followed by the superscript in
+           curly braces; _e.g._, T^{n}. To avoid possible confusion,
+           subscripted variables which are raised to a power are
+           enclosed in brackets, thus (T_{1})^{2} represents
+           'T one squared'.
+
+        3. Square root is denoted by [sqrt].
+
+        4. To improve readability, italic markup (underscores
+           enclosing text) has been omitted from letters used in
+           mathematical formulae, and some equations have been set
+           'out of line'.
+
+      Please see the end of the book for a list of corrections.
+
+
+
+
+
+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 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.
+
+[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 Kuebel, Washington,
+D.C., which is no. 316,876 in the collection of the U.S. National
+Museum. Among a large number of pendulums tried by Baily in London
+(1827-1840), was one which resembles the reversible pendulum
+superficially, but which is actually an invariable pendulum having knife
+edges at both ends. The purpose was apparently economy, since it is
+equivalent to two separate invariable pendulums. This is the type of
+pendulum used on the U.S. Exploring Expedition of 1838-1842. It is not
+known what use was made of the Kuebel pendulum.]
+
+
+
+
+Repsold-Bessel Reversible Pendulum
+
+
+As we have noted, Bessel made determinations of gravity with a ball
+("simple") pendulum in the period 1825-1827 and in 1835 at Koenigsberg
+and Berlin, respectively. In the memoir on his observations at
+Koenigsberg, he set forth the theory of the symmetrical compound pendulum
+with interchangeable knife edges.[42] Bessel demonstrated theoretically
+that if the pendulum were symmetrical with respect to its geometrical
+center, if the times of swing about each axis were the same, the effects
+of buoyancy and of air set in motion would be eliminated. Laplace had
+already shown that the knife edge must be regarded as a cylinder and not
+as a mere line of support. Bessel then showed that if the knife edges
+were equal cylinders, their effects were eliminated by inverting the
+pendulum; and if the knife edges were not equal cylinders, the
+difference in their effects was canceled by interchanging the knives and
+again determining the times of swing in the so-called erect and inverted
+positions. Bessel further showed that it is unnecessary to make the
+times of swing exactly equal for the two knife edges.
+
+The simplified discussion for infinitely small oscillations in a vacuum
+is as follows: If T_{1} and T_{2} are the times of swing about the knife
+edges, and if h_{1} and h_{2} are distances of the knife edges from the
+center of gravity, and if k is the radius of gyration about an axis
+through the center of gravity, then from the equation of motion of a
+rigid body oscillating about a fixed axis under gravity
+
+  (T_{1})^{2} = [pi]^{2}(k^{2} + (h_{1})^{2})/gh_{1},
+
+  (T_{2})^{2} = [pi]^{2}(k^{2} + (h_{2})^{2})/gh_{2}.
+
+Then
+
+  (h_{1}(T_{1})^{2} - h_{2}(T_{2})^{2})/(h_{1} - h_{2})
+
+  = ([pi]^{2}/g)(h_{1} + h_{2})
+
+  = [tau]^{2}.
+
+[tau] is then the time of swing of a simple pendulum of length h_{1} +
+h_{2}. If the difference T_{1} - T_{2} is sufficiently small,
+
+  [tau] = (h_{1}T_{1} - h_{2}T_{2})/(h_{1} - h_{2}).
+
+Prior to its publication by Bessel in 1828, the formula for the time of
+swing of a simple pendulum of length h_{1} + h_{2} in terms of T_{1},
+T_{2} had been given by C. F. Gauss in a letter to H. C. Schumacher
+dated November 28, 1824.[43]
+
+The symmetrical compound pendulum with interchangeable knives, for which
+Bessel gave a posthumously published design and specifications,[44] has
+been called a reversible pendulum; it may thereby be distinguished from
+Kater's unsymmetrical convertible pendulum. In 1861, the Swiss Geodetic
+Commission was formed, and in one of its first sessions in 1862 it was
+decided to add determinations of gravity to the operations connected
+with the measurement--at different points in Switzerland--of the arc of
+the meridian traversing central Europe.[45] It was decided further to
+employ a reversible pendulum of Bessel's design and to have it
+constructed by the firm of A. Repsold and Sons, Hamburg. It was also
+decided to make the first observations with the pendulum in Geneva;
+accordingly, the Repsold-Bessel pendulum (fig. 16) was sent to Prof. E.
+Plantamour, director of the observatory at Geneva, in the autumn of
+1864.[46]
+
+The Swiss reversible pendulum was about 560 mm. in length (distance
+between the knife edges) and the time of swing was approximately 3/4
+second. At the extremities of the stem of the pendulum were movable
+cylindrical disks, one of which was solid and heavy, the other hollow
+and light. It was intended by the mechanicians that equality of times of
+oscillation about the knife edges would be achieved by adjusting the
+position of a movable disk. The pendulum was hung by a knife edge on a
+plate supported by a tripod and having an attachment from which a
+measuring rod could be suspended so that the distance between the knife
+edges could be measured by a comparator. Plantamour found it
+impracticable to adjust a disk until the times of swing about each knife
+edge were equal. His colleague, Charles Cellerier,[47] then showed that
+if (T_{1} - T_{2})/T_{1} is sufficiently small so that one can neglect
+its square, one can determine the length of the seconds pendulum from
+the times of swing about the knife edges by a theory which uses the
+distances of the center of gravity from the respective knife edges.
+Thus, a role for the position of the center of gravity in the theory of
+the reversible pendulum, which had been set forth earlier by Bessel, was
+discovered independently by Cellerier for the Swiss observers of
+pendulums.
+
+In 1866, Plantamour published an extensive memoir "Experiences faites a
+Geneve avec le pendule a reversion." Another memoir, published in 1872,
+presented further results of determinations of gravity in Switzerland.
+Plantamour was the first scientist in western Europe to use a
+Repsold-Bessel reversible pendulum and to work out methods for its
+employment.
+
+The Russian Imperial Academy of Sciences acquired two Repsold-Bessel
+pendulums, and observations with them were begun in 1864 by Prof.
+Sawitsch, University of St. Petersburg, and others.[48] In 1869, the
+Russian pendulums were loaned to the India Survey in order to enable
+members of the Survey to supplement observations with the Kater
+invariable pendulums nos. 4 and 6 (1821). During the transport of the
+Russian apparatus to India, the knives became rusted and the apparatus
+had to be reconditioned. Capt. Heaviside of the India Survey observed
+with both pendulums at Kew Observatory, near London, in the spring of
+1874, after which the Russian pendulums were sent to Pulkowa (Russia)
+and were used for observations there and in the Caucasus.
+
+The introduction of the Repsold-Bessel reversible pendulum for the
+determination of gravity was accompanied by the creation of the first
+international scientific association, one for geodesy. In 1861, Lt. Gen.
+J. J. Baeyer, director of the Prussian Geodetic Survey, sent a
+memorandum to the Prussian minister of war in which he proposed that the
+independent geodetic surveys of the states of central Europe be
+coordinated by the creation of an international organization.[49] In
+1862, invitations were sent to the various German states and to other
+states of central Europe. The first General Conference of the
+association, initially called _Die Mittel-Europaeische Gradmessung_, also
+_L'Association Geodesique Internationale_, was held from the 15th to
+the 22d of October 1864 in Berlin.[50] The Conference decided upon
+questions of organization: a general conference was to be held
+ordinarily every three years; a permanent commission initially
+consisting of seven members was to be the scientific organ of the
+association and to meet annually; a central bureau was to be established
+for the reception, publication, and distribution of reports from the
+member states.
+
+[Illustration: Figure 16.--FROM A DESIGN LEFT BY BESSEL, this portable
+apparatus was developed in 1862 by the firm of Repsold in Hamburg, whose
+founder had assisted Bessel in the construction of his pendulum
+apparatus of 1826. The pendulum is convertible, but differs from Kater's
+in being geometrically symmetrical and, for this reason, Repsold's is
+usually called "reversible." Just to the right of the pendulum is a
+standard scale. To the left is a "vertical comparator" designed by
+Repsold to measure the distance between the knife edges of the pendulum.
+To make this measurement, two micrometer microscopes which project
+horizontally through the comparator are alternately focused on the knife
+edges and on the standard scale.]
+
+Under the topic "Astronomical Questions," the General Conference of 1864
+resolved that there should be determinations of the intensity of gravity
+at the greatest possible number of points of the geodetic network, and
+recommended the reversible pendulum as the instrument of
+observation.[51] At the second General Conference, in Berlin in 1867, on
+the basis of favorable reports by Dr. Hirsch, director of the
+observatory at Neuchatel, of Swiss practice with the Repsold-Bessel
+reversible pendulum, this instrument was specifically recommended for
+determinations of gravity.[52] The title of the association was changed
+to _Die Europaeische Gradmessung_; in 1886, it became _Die Internationale
+Erdmessung_, under which title it continued until World War I.
+
+On April 1, 1866, the Central Bureau of _Die Europaeische Gradmessung_
+was opened in Berlin under the presidency of Baeyer, and in 1868 there
+was founded at Berlin, also under his presidency, the Royal Prussian
+Geodetic Institute, which obtained regular budgetary status on January
+1, 1870. A reversible pendulum for the Institute was ordered from A.
+Repsold and Sons, and it was delivered in the spring of 1869. The
+Prussian instrument was symmetrical geometrically, as specified by
+Bessel, but different in form from the Swiss and Russian pendulums. The
+distance between the knife edges was 1 meter, and the time of swing
+approximately 1 second. The Prussian Repsold-Bessel pendulum was swung
+at Leipzig and other stations in central Europe during the years
+1869-1870 by Dr. Albrecht under the direction of Dr. Bruhns, director of
+the observatory at Leipzig and chief of the astronomical section of the
+Geodetic Institute. The results of these first observations appeared in
+a publication of the Royal Prussian Geodetic Institute in 1871.[53]
+
+Results of observations with the Russian Repsold-Bessel pendulums were
+published by the Imperial Academy of Sciences. In 1872, Prof. Sawitsch
+reported the work for western Europeans in "Les variations de la
+pesanteur dans les provinces occidentales de l'Empire russe."[48] In
+November 1873, the Austrian Geodetic Commission received a
+Repsold-Bessel reversible pendulum and on September 24, 1874, Prof.
+Theodor von Oppolzer reported on observations at Vienna and other
+stations to the Fourth General Conference of _Die Europaeische
+Gradmessung_ in Dresden.[54] At the fourth session of the Conference, on
+September 28, 1874, a Special Commission, consisting of Baeyer, as
+chairman, and Bruhns, Hirsch, von Oppolzer, Peters, and Albrecht, was
+appointed to consider (under Topic 3 of the program): "Observations for
+the determination of the intensity of gravity," the question, "Which
+Pendulum-apparatuses are preferable for the determination of many
+points?"
+
+After the adoption of the Repsold-Bessel reversible pendulum for gravity
+determinations in Europe, work in the field was begun by the U.S. Coast
+Survey under the superintendency of Prof. Benjamin Peirce. There is
+mention in reports of observations with pendulums prior to Peirce's
+direction to his son Charles on November 30, 1872, "to take charge of
+the Pendulum Experiments of the Coast Survey and to direct and inspect
+all parties engaged in such experiments and as often as circumstances
+will permit, to take the field with a party...."[55] Systematic and
+important gravity work by the Survey was begun by Charles Sanders
+Peirce. Upon receiving notice of his appointment, the latter promptly
+ordered from the Repsolds a pendulum similar to the Prussian instrument.
+Since the firm of mechanicians was engaged in making instruments for
+observations of the transit of Venus in 1874, the pendulum for the
+Coast Survey could not be constructed immediately. Meanwhile, during the
+years 1873-1874, Charles Peirce conducted a party which made
+observations of gravity in the Hoosac Tunnel near North Adams, and at
+Northampton and Cambridge, Massachusetts. The pendulums used were
+nonreversible, invariable pendulums with conical bobs. Among them was a
+silver pendulum, but similar pendulums of brass were used also.[56]
+
+[Illustration: Figure 17.--REPSOLD-BESSEL REVERSIBLE PENDULUM apparatus
+as made in 1875, and used in the gravity work of the U.S. Coast and
+Geodetic Survey. Continental geodesists continued to favor the general
+use of convertible pendulums and absolute determinations of gravity,
+while their English colleagues had turned to invariable pendulums and
+relative determinations, except for base stations. Perhaps the first
+important American contribution to gravity work was C. S. Peirce's
+demonstration of the error inherent in the Repsold apparatus through
+flexure of the stand.]
+
+[Illustration: Figure 18.--CHARLES SANDERS PEIRCE (1839-1914), son of
+Benjamin Peirce, Perkins Professor of Astronomy and Mathematics at
+Harvard College. C. S. Peirce graduated from Harvard in 1859. From 1873
+to 1891, as an assistant at the U.S. Coast and Geodetic Survey, he
+accomplished the important gravimetric work described in this article.
+Peirce was also interested in many other fields, but above all in the
+logic, philosophy, and history of science, in which he wrote
+extensively. His greatest fame is in philosophy, where he is regarded as
+the founder of pragmatism.]
+
+In 1874, Charles Peirce expressed the desire to be sent to Europe for at
+least a year, beginning about March 1, 1875, "to learn the use of the
+new convertible pendulum and to compare it with those of the European
+measure of a Degree and the Swiss and to compare" his "invariable
+pendulums in the manner which has been used by swinging them in London
+and Paris."[57]
+
+Charles S. Peirce, assistant, U.S. Coast Survey, sailed for Europe on
+April 3, 1875, on his mission to obtain the Repsold-Bessel reversible
+pendulum ordered for the Survey and to learn the methods of using it for
+the determination of gravity. In England, he conferred with Maxwell,
+Stokes, and Airy concerning the theory and practice of research with
+pendulums. In May, he continued on to Hamburg and obtained delivery from
+the Repsolds of the pendulum for the Coast Survey (fig. 17). Peirce then
+went to Berlin and conferred with Gen. Baeyer, who expressed doubts of
+the stability of the Repsold stand for the pendulum. Peirce next went to
+Geneva, where, under arrangements with Prof. Plantamour, he swung the
+newly acquired pendulum at the observatory.[58]
+
+In view of Baeyer's expressed doubts of the rigidity of the Repsold
+stand, Peirce performed experiments to measure the flexure of the stand
+caused by the oscillations of the pendulum. His method was to set up a
+micrometer in front of the pendulum stand and, with a microscope, to
+measure the displacement caused by a weight passing over a pulley, the
+friction of which had been determined. Peirce calculated the correction
+to be applied to the length of the seconds pendulum--on account of the
+swaying of the stand during the swings of the pendulum--to amount to
+over 0.2 mm. Although Peirce's measurements of flexure in Geneva were
+not as precise as his later measurements, he believed that failure to
+correct for flexure of the stand in determinations previously made with
+Repsold pendulums was responsible for appreciable errors in reported
+values of the length of the seconds pendulum.
+
+The Permanent Commission of _Die Europaeische Gradmessung_ met in Paris,
+September 20-29, 1875. In conjunction with this meeting, there was held
+on September 21 a meeting of the Special Commission on the Pendulum. The
+basis of the discussion by the Special Commission was provided by
+reports which had been submitted in response to a circular sent out by
+the Central Bureau to the members on February 26, 1874.[59]
+
+Gen. Baeyer stated that the distance of 1 meter between the knife edges
+of the Prussian Repsold-Bessel pendulum made it unwieldy and unsuited
+for transport. He declared that the instability of the stand also was a
+source of error. Accordingly, Gen. Baeyer expressed the opinion that
+absolute determinations of gravity should be made at a control station
+by a reversible pendulum hung on a permanent, and therefore stable
+stand, and he said that relative values of gravity with respect to the
+control station should be obtained in the field by means of a Bouguer
+invariable pendulum. Dr. Bruhns and Dr. Peters agreed with Gen. Baeyer;
+however, the Swiss investigators, Prof. Plantamour and Dr. Hirsch
+reported in defense of the reversible pendulum as a field instrument, as
+did Prof. von Oppolzer of Vienna. The circumstance that an invariable
+pendulum is subject to changes in length was offered as an argument in
+favor of the reversible pendulum as a field instrument.
+
+Peirce was present during these discussions by the members of the
+Special Commission, and he reported that his experiments at Geneva
+demonstrated that the oscillations of the pendulum called forth a
+flexure of the support which hitherto had been neglected. The observers
+who used the Swiss and Austrian Repsold pendulums contended, in
+opposition to Peirce, that the Repsold stand was stable.
+
+The outcome of these discussions was that the Special Commission
+reported to the Permanent Commission that the Repsold-Bessel reversible
+pendulum, except for some small changes, satisfied all requirements for
+the determination of gravity. The Special Commission proposed that the
+Repsold pendulums of the several states be swung at the Prussian
+Eichungsamt in Berlin where, as Peirce pointed out, Bessel had made his
+determination of the intensity of gravity with a ball pendulum in 1835.
+Peirce was encouraged to swing the Coast Survey reversible pendulum at
+the stations in France, England, and Germany where Borda and Cassini,
+Kater, and Bessel, respectively, had made historic determinations. The
+Permanent Commission, in whose sessions Peirce also participated, by
+resolutions adopted the report of the Special Commission on the
+Pendulum.[60]
+
+During the months of January and February 1876, Peirce conducted
+observations in the Grande Salle du Meridien at the observatory in Paris
+where Borda, Biot, and Capt. Edward Sabine had swung pendulums early in
+the 19th century. He conducted observations in Berlin from April to June
+1876 and, by experiment, determined the correction for flexure to be
+applied to the value of gravity previously obtained with the Prussian
+instrument. Subsequent observations were made at Kew. After his return
+to the United States on August 26, 1876, Peirce conducted experiments at
+the Stevens Institute in Hoboken, New Jersey, where he made careful
+measurements of the flexure of the stand by statical and dynamical
+methods. In Geneva, he had secured the construction of a vacuum chamber
+in which the pendulum could be swung on a support which he called the
+Geneva support. At the Stevens Institute, Peirce swung the
+Repsold-Bessel pendulum on the Geneva support and determined the effect
+of different pressures and temperatures on the period of oscillation of
+the pendulum. These experiments continued into 1878.[61]
+
+Meanwhile, the Permanent Commission met October 5-10, 1876, in Brussels
+and continued the discussion of the pendulum.[62] Gen. Baeyer reported
+on Peirce's experiments in Berlin to determine the flexure of the stand.
+The difference of 0.18 mm. in the lengths of the seconds pendulum as
+determined by Bessel and as determined by the Repsold instrument agreed
+with Peirce's estimate of error caused by neglect of flexure of the
+Repsold stand. Dr. Hirsch, speaking for the Swiss survey, and Prof. von
+Oppolzer, speaking for the Austrian survey, contended, however, that
+their stands possessed sufficient stability and that the results found
+by Peirce applied only to the stands and bases investigated by him. The
+Permanent Commission proposed further study of the pendulum.
+
+The Fifth General Conference of _Die Europaeische 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.
+Ibanez, president of the Permanent Commission, Peirce had sent on July
+13, 1877, from New York, the manuscript of a memoir titled "De
+l'Influence de la flexibilite du trepied sur l'oscillation du pendule a
+reversion." This memoir and others by Cellerier and Plantamour
+confirming Peirce's work were published as appendices to the proceedings
+of the conference. As appendices to Peirce's contribution were published
+also two notes by Prof. von Oppolzer. At the second session on September
+29, 1877, when Plantamour reported that the work of Hirsch and himself
+had confirmed experimentally the independent theoretical work of
+Cellerier and the theoretical and experimental work of Peirce on
+flexure, Peirce described his Hoboken experiments.
+
+During the discussions at Stuttgart on the flexure of the Repsold stand,
+Herve Faye, president of the Bureau of Longitudes, Paris, suggested that
+the swaying of the stand during oscillations of the pendulum could be
+overcome by the suspension from one support of two similar pendulums
+which oscillated with equal amplitudes and in opposite phases. This
+proposal was criticized by Dr. Hirsch, who declared that exact
+observation of passages of a "double pendulum" would be difficult and
+that two pendulums swinging so close together would interfere with each
+other. The proposal of the double pendulum came up again at the meeting
+of the Permanent Commission at Geneva in 1879.[64] On February 17, 1879,
+Peirce had completed a paper "On a Method of Swinging Pendulums for the
+Determination of Gravity, Proposed by M. Faye." In this paper, Peirce
+presented the results of an analytical mechanical investigation of
+Faye's proposal. Peirce set up the differential equations, found the
+solutions, interpreted them physically, and arrived at the conclusion
+"that the suggestion of M. Faye ... is as sound as it is brilliant and
+offers some peculiar advantages over the existing method of swinging
+pendulums."
+
+In a report to Supt. Patterson, dated July 1879, Peirce stated: "I think
+it is important before making a new pendulum apparatus to experiment
+with Faye's proposed method."[65] He wrote further: "The method proves
+to be perfectly sound in theory, and as it would greatly facilitate the
+work it is probably destined eventually to prevail. We must
+unfortunately leave to other surveys the merit of practically testing
+and introducing the new method, as our appropriations are insufficient
+for us to maintain the leading position in this matter, which we
+otherwise might take." Copies of the published version of Peirce's
+remarks were sent to Europe. At a meeting of the Academy of Sciences in
+Paris on September 1, 1879, Faye presented a report on Peirce's
+findings.[66] The Permanent Commission met September 16-20, 1879, in
+Geneva. At the third session on September 19, by action of Gen. Baeyer,
+copies of Peirce's paper on Faye's proposed method of swinging pendulums
+were distributed. Dr. Hirsch again commented adversely on the proposal,
+but moved that the question be investigated and reported on at the
+coming General Conference. The Permanent Commission accepted the
+proposal of Dr. Hirsch, and Prof. Plantamour was named to report on the
+matter at the General Conference. At Plantamour's request, Charles
+Cellerier was appointed to join him, since the problem essentially was a
+theoretical one.
+
+The Sixth General Conference of _Die Europaeische Gradmessung_ met
+September 13-16, 1880, in Munich.[67] Topic III, part 7 of the program
+was entitled "On Determinations of Gravity through pendulum
+observations. Which construction of a pendulum apparatus corresponds
+completely to all requirements of science? Special report on the
+pendulum."
+
+The conference received a memoir by Cellerier[68] on the theory of the
+double pendulum and a report by Plantamour and Cellerier.[69]
+Cellerier's mathematical analysis began with the equations of Peirce and
+used the latter's notation as far as possible. His general discussion
+included the results of Peirce, but he stated that the difficulties to
+be overcome did not justify the employment of the "double pendulum." He
+presented an alternative method of correcting for flexure based upon a
+theory by which the flexure caused by the oscillation of a given
+reversible pendulum could be determined from the behavior of an
+auxiliary pendulum of the same length but of different weight. This
+method of correcting for flexure was recommended to the General
+Conference by Plantamour and Cellerier in their joint report. At the
+fourth session of the conference on September 16, 1880, the problem of
+the pendulum was discussed and, in consequence, a commission consisting
+of Faye, Helmholtz, Plantamour (replaced in 1882 by Hirsch), and von
+Oppolzer was appointed to study apparatus suitable for relative
+determinations of gravity.
+
+The Permanent Commission met September 11-15, 1882, at The Hague,[70]
+and at its last session appointed Prof. von Oppolzer to report to the
+Seventh General Conference on different forms of apparatus for the
+determination of gravity. The Seventh Conference met October 15-24,
+1883, in Rome,[71] and, at its eighth session, on October 22, received a
+comprehensive, critical review from Prof. von Oppolzer entitled "Ueber
+die Bestimmung der Schwere mit Hilfe verschiedener Apparate."[72] Von
+Oppolzer especially expounded the advantages of the Bessel reversible
+pendulum, which compensated for air effects by symmetry of form if the
+times of swing for both positions were maintained between the same
+amplitudes, and compensated for irregular knife edges by making them
+interchangeable. Prof. von Oppolzer reviewed the problem of flexure of
+the Repsold stand and stated that a solution in the right direction
+was the proposal--made by Faye and theoretically pursued by Peirce--to
+swing two pendulums from the same stand with equal amplitudes and in
+opposite phases, but that the proposal was not practicable. He concluded
+that for absolute determinations of gravity, the Bessel reversible
+pendulum was highly appropriate if one swung two exemplars of different
+weight from the same stand for the elimination of flexure. Prof. von
+Oppolzer's important report recognized that absolute determinations were
+less accurate than relative ones, and should be conducted only at
+special places.
+
+The discussions initiated by Peirce's demonstration of the flexure of
+the Repsold stand resulted, finally, in the abandonment of the plan to
+make absolute determinations of gravity at all stations with the
+reversible pendulum.
+
+[Illustration: Figure 19.--THREE PENDULUMS USED IN EARLY WORK at the
+U.S. Coast and Geodetic Survey. Shown on the left is the Peirce
+invariable; center, the Peirce reversible; and, right, the Repsold
+reversible. Peirce designed the cylindrical pendulum in 1881-1882 to
+study the effect of air resistance according to the theory of G. G.
+Stokes on the motion of a pendulum in a viscous field. Three examples of
+the Peirce pendulums are in the U.S. National Museum.]
+
+
+
+
+Peirce and Defforges Invariable, Reversible Pendulums
+
+
+The Repsold-Bessel reversible pendulum was designed and initially used
+to make absolute determinations of gravity not only at initial stations
+such as Kew, the observatory in Paris, and the Smithsonian Institution
+in Washington, D.C., but also at stations in the field. An invariable
+pendulum with a single knife edge, however, is adequate for relative
+determinations. As we have seen, such invariable pendulums had been used
+by Bouguer and Kater, and after the experiences with the Repsold
+apparatus had been recommended again by Baeyer for relative
+determinations. But an invariable pendulum is subject to uncontrollable
+changes of length. Peirce proposed to detect such changes in an
+invariable pendulum in the field by combining the invariable and
+reversible principles. He explained his proposal to Faye in a letter
+dated July 23, 1880, and he presented it on September 16, 1880, at the
+fourth session of the sixth General Conference of _Die Europaeische
+Gradmessung_, in Munich.[73]
+
+As recorded in the Proceedings of the Conference, Peirce wrote:
+
+    But I obviate it in making my pendulum both invariable and
+    reversible. Every alteration of the pendulum will be revealed
+    immediately by the change in the difference of the two periods
+    of oscillation in the two positions. Once discovered, it will be
+    taken account of by means of new measures of the distance
+    between the two supports.
+
+Peirce added that it seemed to him that if the reversible pendulum
+perhaps is not the best instrument to determine absolute gravity, it is,
+on condition that it be truly invariable, the best to determine relative
+gravity. Peirce further stated that he would wish that the pendulum be
+formed of a tube of drawn brass with heavy plugs of brass equally drawn.
+The cylinder would be terminated by two hemispheres; the knives would be
+attached to tongues fixed near the ends of the cylinder.
+
+During the years 1881 and 1882, four invariable, reversible pendulums
+were made after the design of Peirce at the office of the U.S. Coast and
+Geodetic Survey in Washington, D.C. The report of the superintendent for
+the year 1880-1881 states:
+
+    A new pattern of the reversible pendulum has been invented,
+    having its surface as nearly as convenient in the form of an
+    elongated ellipsoid. Three of these instruments have been
+    constructed, two having a distance of one meter between the
+    knife edges and the third a distance of one yard. It is proposed
+    to swing one of the meter pendulums at a temperature near 32 deg.
+    F. at the same time that the yard is swung at 60 deg. F., in order
+    to determine anew the relation between the yard and the meter.[74]
+
+The report for 1881-1882 mentions four of these Peirce pendulums.
+
+A description of the Peirce invariable, reversible pendulums was given
+by Assistant E. D. Preston in "Determinations of Gravity and the
+Magnetic Elements in Connection with the United States Scientific
+Expedition to the West Coast of Africa, 1889-90."[75] The invariable,
+reversible pendulum, Peirce no. 4, now preserved in the Smithsonian
+Institution's Museum of History and Technology (fig. 34), may be taken
+as typical of the meter pendulums: In the same memoir, Preston gives the
+diameter of the tube as 63.7 mm., thickness of tube 1.5 mm., weight
+10.680 kilograms, and distance between the knives 1.000 meter.
+
+The combination of invariability and reversibility in the Peirce
+pendulums was an innovation for relative determinations. Indeed, the
+combination was criticized by Maj. J. Herschel, R.E., of the Indian
+Survey, at a conference on gravity held in Washington in May 1882 on the
+occasion of his visit to the United States for the purpose of
+connecting English and American stations by relative determinations with
+three Kater invariable pendulums. These three pendulums have been
+designated as nos. 4, 6 (1821), and 11.[76]
+
+[Illustration: Figure 20.--SUPPORT FOR THE PEIRCE PENDULUM, 1889. Much
+of the work of C. S. Peirce was concerned with the determination of the
+error introduced into observations made with the portable apparatus by
+the vibration of the stand with the pendulum. He showed that the popular
+Bessel-Repsold apparatus was subject to such an error. His own pendulums
+were swung from a simple but rugged wooden frame to which a hardened
+steel bearing was fixed.]
+
+Another novel characteristic of the Peirce pendulums was the mainly
+cylindrical form. Prof. George Gabriel Stokes, in a paper "On the Effect
+of the Internal Friction of Fluids on the Motion of Pendulums"[77] that
+was read to the Cambridge Philosophical Society on December 9, 1850, had
+solved the hydrodynamical equations to obtain the resistance to the
+motions of a sphere and a cylinder in a viscous fluid. Peirce had
+studied the effect of viscous resistance on the motion of his
+Repsold-Bessel pendulum, which was symmetrical in form but not
+cylindrical. The mainly cylindrical form of his pendulums (fig. 19)
+permitted Peirce to predict from Stokes' theory the effect of viscosity
+and to compare the results with experiment. His report of November 20,
+1889, in which he presented the comparison of experimental results with
+the theory of Stokes, was not published.[78]
+
+Peirce used his pendulums in 1883 to establish a station at the
+Smithsonian Institution that was to serve as the base station for the
+Coast and Geodetic Survey for some years. Pendulum Peirce no. 1 was
+swung at Washington in 1881 and was then taken by the party of
+Lieutenant Greely, U.S.A., on an expedition to Lady Franklin Bay where
+it was swung in 1882 at Fort Conger, Grinnell Land, Canada. Peirce nos.
+2 and 3 were swung by Peirce in 1882 at Washington, D.C.; Hoboken, New
+Jersey; Montreal, Canada; and Albany, New York. Assistant Preston took
+Peirce no. 3 on a U.S. eclipse expedition to the Caroline Islands in
+1883. Peirce in 1885 swung pendulums nos. 2 and 3 at Ann Arbor,
+Michigan; Madison, Wisconsin; and Ithaca, New York. Assistant Preston in
+1887 swung Peirce nos. 3 and 4 at stations in the Hawaiian Islands, and
+in 1890 he swung Peirce nos. 3 and 4 at stations on the west coast of
+Africa.[79]
+
+The new pattern of pendulum designed by Peirce was also adopted in
+France, after some years of experience with a Repsold-Bessel pendulum.
+Peirce in 1875 had swung his Repsold-Bessel pendulum at the observatory
+in Paris, where Borda and Cassini, and Biot, had made historic
+observations and where Sabine also had determined gravity by comparison
+with Kater's value at London. During the spring of 1880, Peirce made
+studies of the supports for the pendulums of these earlier
+determinations and calculated corrections to those results for
+hydrodynamic effects, viscosity, and flexure. On June 14, 1880, Peirce
+addressed the Academy of Sciences, Paris, on the value of gravity at
+Paris, and compared his results with the corrected results of Borda and
+Biot and with the transferred value of Kater.[80]
+
+In the same year the French Geographic Service of the Army acquired a
+Repsold-Bessel reversible pendulum of the smaller type, and Defforges
+conducted experiments with it.[81] He introduced the method of measuring
+flexure from the movement of interference fringes during motion of the
+pendulum. He found an appreciable difference between dynamical and
+statical coefficients of flexure and concluded that the "correction
+formula of Peirce and Cellerier is suited perfectly to practice and
+represents exactly the variation of period caused by swaying of the
+support, on the condition that one uses the statical coefficient."
+Defforges developed a theory for the employment of two similar pendulums
+of the same weight, but of different length, and hung by the same
+knives. This theory eliminated the flexure of the support and the
+curvature of the knives from the reduction of observations.
+
+Pendulums of 1-meter and of 1/2-meter distance between the knife edges
+were constructed from Defforges' design by Brunner Brothers in Paris
+(fig. 21). These Defforges pendulums were cylindrical in form with
+hemispherical ends like the Peirce pendulums, and were hung on knives
+that projected from the sides of the pendulum, as in some unfinished
+Gautier pendulums designed by Peirce in 1883 in Paris.
+
+[Illustration: Figure 21.--REVERSIBLE PENDULUM APPARATUS of Defforges,
+as constructed by Brunner, Paris, about 1887. The clock and telescope
+used to observe coincidences are not shown. The telescope shown is part
+of an interferometer used to measure flexure of the support. One mirror
+of the interferometer is attached to the pendulum support; the other to
+the separate masonry pillar at the left.]
+
+[Illustration: Figure 22.--BECAUSE OF THE GREATER SIMPLICITY of its use,
+the invariable pendulum superseded the convertible pendulum towards the
+end of the 19th century, except at various national base stations (Kew,
+Paris, Potsdam, Washington, D.C., etc.). Shown here are, right to left,
+a pendulum of the type used by Peirce at the Hoosac Tunnel in 1873-74,
+the Mendenhall 1/2-second pendulum of 1890, and the pendulum designed by
+Peirce in 1881-1882.]
+
+[Illustration: Figure 23.--THE OVERALL SIZE of portable pendulum
+apparatus was greatly reduced with the introduction of this 1/2-second
+apparatus in 1887, by the Austrian military officer, Robert von
+Sterneck. Used with a vacuum chamber not shown here, the apparatus is
+only about 2 feet high. Coincidences are observed by the reflection of a
+periodic electric spark in two mirrors, one on the support and the other
+on the pendulum itself.]
+
+[Illustration: Figure 24.--THOMAS C. MENDENHALL (1841-1924). Although
+largely self-educated, he became the first professor of physics and
+mechanics at the Ohio Agricultural and Mechanical College (later Ohio
+State University), and was subsequently connected with several other
+universities. In 1878, while teaching at the Tokyo Imperial University
+in Japan, he made gravity measurements between Tokyo and Fujiyama from
+which he calculated the mean density of the earth. While superintendent
+of the U.S. Coast and Geodetic Survey, 1889-94, he developed the
+pendulum apparatus which bears his name.]
+
+
+
+
+Von Sterneck and Mendenhall Pendulums
+
+
+While scientists who had used the Repsold-Bessel pendulum apparatus
+discussed its defects and limitations for gravity surveys, Maj. Robert
+von Sterneck of Austria-Hungary began to develop an excellent apparatus
+for the rapid determination of relative values of gravity.[82] Maj. von
+Sterneck's apparatus contained a nonreversible pendulum 1/4-meter in
+length, and 1/2-second time of swing. The pendulum was hung by a single
+knife edge, which rested on a plate that was supported by a tripod. The
+pendulum was swung in a chamber from which air was exhausted and which
+could be maintained at any desired temperature. Times of swing were
+determined by the observation of coincidences of the pendulum with
+chronometer signals. In the final form a small mirror was attached to
+the knife edge perpendicular to the plane of vibration of the pendulum
+and a second fixed mirror was placed close to it so that the two mirrors
+were parallel when the pendulum was at rest. The chronometer signals
+worked a relay that gave a horizontal spark which was reflected into the
+telescope from the mirrors. When the pendulum was at rest, the image of
+the spark in both mirrors appeared on the horizontal cross wire in the
+telescope, and during oscillation of the pendulum the two images
+appeared in that position upon coincidence. In view of the reduced size
+of the pendulum, the chamber in which it was swung was readily portable,
+and with an improved method of observing coincidences, relative
+determinations of gravity could be made with rapidity and accuracy.
+
+By 1887 Maj. von Sterneck had perfected his apparatus, and it was widely
+adopted in Europe for relative determinations of gravity. He used his
+apparatus in extensive gravity surveys and also applied it in the silver
+mines in Saxony and Bohemia, by the previously described methods of
+Airy, for investigations into the internal constitution of the earth.
+
+On July 1, 1889, Thomas Corwin Mendenhall became superintendent of the
+U.S. Coast and Geodetic Survey. Earlier, he had been professor of
+physics at the University of Tokyo and had directed observations of
+pendulums for the determination of gravity on Fujiyama and at Tokyo.
+Supt. Mendenhall, with the cooperation of members of his staff in
+Washington, designed a new pendulum apparatus of the Von Sterneck type,
+and in October 1890 he ordered construction of the first model.[83]
+
+Like the Von Sterneck apparatus, the Mendenhall pendulum apparatus
+employed a nonreversible, invariable pendulum 1/4-meter in length and of
+slightly more than 1/2-second in time of swing. Initially, the knife
+edge was placed in the head of the pendulum and hung on a fixed plane
+support, but after some experimentation Mendenhall attached the plane
+surface to the pendulum and hung it on a fixed knife edge. An apparatus
+was provided with a set of three pendulums, so that if discrepancies
+appeared in the results, the pendulum at fault could be detected. There
+was also a dummy pendulum which carried a thermometer. A pendulum was
+swung in a receiver in which the pressure and temperature of the air
+were controlled. The time of swing was measured by coincidences with the
+beat of a chronometer. The coincidences were determined by an optical
+method with the aid of a flash apparatus.
+
+[Illustration: Figure 25.--MENDENHALL'S 1/4-METER (1/2-SECOND)
+APPARATUS. Shown on the left is the flash apparatus and, on the right,
+the vacuum chamber within which the pendulum is swung. The flash
+apparatus consists of a kerosene lantern and a telescope, mounted on a
+box containing an electromagnetically operated shutter. The operation of
+the shutter is controlled by a chronograph (not shown), so that it emits
+a slit of light at regular intervals. The telescope is focused on two
+mirrors within the apparatus, one fixed, the other attached to the top
+of the pendulum. It is used to observe the reflection of the flashes
+from these mirrors. When the two reflections are aligned, a
+"coincidence" is marked on the chronograph tape. The second telescope
+attached to the bottom of the vacuum chamber is for observing the
+amplitude of the pendulum swing.]
+
+The flash apparatus was contained in a light metal box which supported
+an observing telescope and which was mounted on a stand. Within the box
+was an electromagnet whose coils were connected with a chronometer
+circuit and whose armature carried a long arm that moved two shutters,
+in both of which were horizontal slits of the same size. The shutters
+were behind the front face of the box, which also had a horizontal slit.
+A flash of light from an oil lamp or an electric spark was emitted from
+the box when the circuit was broken, but not when it was closed. When
+the circuit was broken a spring caused the arm to rise, and the shutters
+were actuated so that the three slits came into line and a flash of
+light was emitted. A small circular mirror was set in each side of the
+pendulum head, so that from either face of the pendulum the image of the
+illuminated slit could be reflected into the field of the observing
+telescope. A similar mirror was placed parallel to these two mirrors and
+rigidly attached to the support. The chronometer signals broke the
+circuit, causing the three slits momentarily to be in line, and when the
+images of the slit in the two mirrors coincided, a coincidence was
+observed. A coincidence occurred whenever the pendulum gained or lost
+one oscillation on the beat of the chronometer. The relative intensity
+of gravity was determined by observations with the first Mendenhall
+apparatus at Washington, D.C., at stations on the Pacific Coast and in
+Alaska, and at the Stevens Institute, Hoboken, New Jersey, between March
+and October 1891.
+
+[Illustration: Figure 26.--VACUUM RECEIVER within which the Mendenhall
+pendulum is swung. The pressure is reduced to about 50 mm. to reduce the
+disturbing effect of air resistance. When the apparatus is sealed, the
+pendulum is lifted on the knife edge by the lever _q_ and is started to
+swing by the lever _r_. The arc of swing is only about 1 deg. The
+stationary mirror is shown at _g_. The pendulum shown in outline in the
+center, is only about 9.7 inches long.]
+
+Under Supt. Mendenhall's direction a smaller, 1/4-second, pendulum
+apparatus was also constructed and tested, but did not offer advantages
+over the 1/2-second apparatus, which therefore continued in use.
+
+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 deg. 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 Cellerier 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. Furtwaengler[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 Stueckrath 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 Kuehnen and Furtwaengler 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. Kuehnen 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, Kuehnen and
+Furtwaengler 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 Kuehnen and Furtwaengler published
+their monumental memoir, "Bestimmung der Absoluten Groesze 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 Kuebel (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.
+Luetke 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
+Koenigsberg 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 Europaeische 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, Herve 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 Stueckrath
+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 memoires relatifs a la physique, publies par la
+Societe francaise de Physique_ [hereinafter referred to as _Collection
+de memoires_]: vol. 4, _Memoires sur le pendule, precedes d'une
+bibliographie_ (Paris: Gauthier-Villars, 1889); and vol. 5, _Memoires
+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] ABBE 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 Caienne_ (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 mathematiques[mathematiques] et de physique par MM. de l'Academie
+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, determinee par les
+observations de Messieurs Bouguer et de La Condamine, envoyes par ordre
+du Roy au Perou, pour observer aux environs de l'equateur_ (Paris,
+1749).
+
+[13] P. L. MOREAU DE MAUPERTUIS, _La figure de la terre determinee par
+les observations de Messieurs de Maupertuis, Clairaut, Camus, Le
+Monnier, l'Abbe 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 memoires_, 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 a Saint
+Domingue," _Collection de memoires_, vol. 4, pp. 3-16.
+
+[19] PERE 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, "Experiences pour connaitre
+la longueur du pendule qui bat les secondes a Paris," _Collection de
+memoires_, vol. 4, pp. 17-64.
+
+[21] F. W. BESSEL, "Untersuchungen ueber die Laenge des einfachen
+Secundenpendels," _Abhandlungen der Koeniglichen 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 memoires_, 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, "Methode pour determiner la longueur du pendule
+simple qui bat les secondes," _Collection de memoires_, vol. 4, pp.
+65-76.
+
+[26] _Collection de memoires_, 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 memoires_, 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 memoires_, 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 Kuebel
+(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, "Experiences faites a Geneve avec le pendule a
+reversion," _Memoires de la Societe de Physique et d'histoire naturelle
+de Geneve, 1865_ (Geneva, 1866), vol. 18, p. 309.
+
+[46] Ibid., pp. 309-416.
+
+[47] C. CELLERIER, "Note sur la Mesure de la Pesanteur par le Pendule,"
+_Memoires de la Societe de Physique et d'histoire naturelle de Geneve,
+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, _Ueber die Groesse und Figur der Erde_ (Berlin, 1861).
+
+[50] _Comptes-rendus de la Conference Geodesique Internationale reunie a
+Berlin du 15-22 Octobre 1864_ (Neuchatel, 1865).
+
+[51] Ibid., part III, subpart E.
+
+[52] _Bericht ueber die Verhandlungen der vom 30 September bis 7 October
+1867 zu Berlin abgehaltenen allgemeinen Conferenz der Europaeischen
+Gradmessung_ (Berlin, 1868). See report of fourth session, October 3,
+1867.
+
+[53] C. BRUHNS and ALBRECHT, "Bestimmung der Laenge des Secundenpendels
+in Bonn, Leiden und Mannheim," _Astronomisch-Geodaetische Arbeiten im
+Jahre 1870_ (Leipzig: Veroeffentlichungen des Koeniglichen Preussischen
+Geodaetischen Instituts, 1871).
+
+[54] _Bericht ueber die Verhandlungen der vom 23 bis 28 September 1874 in
+Dresden abgehaltenen vierten allgemeinen Conferenz der Europaeischen
+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 Europaeischen 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 Europaeischen 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 Europaeischen Gradmessung_
+(Berlin, 1877). See report of third session, October 7, 1876.
+
+[63] _Verhandlungen der vom 27 September bis 2 Oktober 1877 zu Stuttgart
+abgehaltenen fuenften allgemeinen Conferenz der Europaeischen Gradmessung_
+(Berlin, 1878).
+
+[64] _Verhandlung der vom 16 bis 20 September 1879 in Genf Vereinigten
+Permanenten Commission der Europaeischen 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'Academie des Sciences_ (Paris, 1879), vol. 89,
+p. 462.
+
+[67] _Verhandlungen der vom 13 bis 16 September 1880 zu Muenchen
+abgehaltenen sechsten allgemeinen Conferenz der Europaeischen
+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 Europaeischen Gradmessung_
+(Berlin, 1883).
+
+[71] _Verhandlungen der vom 15 bis 24 Oktober 1883 zu Rom abgehaltenen
+siebenten allgemeinen Conferenz der Europaeischen Gradmessung_ (Berlin,
+1884). Gen. Cutts attended for the U.S. Coast and Geodetic Survey.
+
+[72] Ibid., app. 6. See also, _Zeitschrift fuer 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'Academie des Sciences_ (Paris, 1880), vol. 90,
+p. 1401. HERVE FAYE's report, dated June 21, 1880, is in the same
+_Comptes-rendus_, p. 1463.
+
+[81] COMMANDANT C. DEFFORGES, "Sur l'Intensite absolue de la pesanteur,"
+_Journal de Physique_ (1888), vol. 17, pp. 239, 347, 455. See also,
+DEFFORGES, "Observations du pendule," _Memorial du Depot general 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.
+Militaer-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 fuer Instrumentenkunde_
+(July 1896), vol. 16, p. 193.
+
+[87] DR. R. SCHUMANN, "Ueber eine Methode, das Mitschwingen bei relativen
+Schweremessungen zu bestimmen," _Zeitschrift fuer 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, "Ueber die Verwendung zweier Pendel auf gemeinsamer
+Unterlage zur Bestimmung der Mitschwingung," _Zeitschrift fuer Mathematik
+und Physik_ (1899), vol. 44, p. 44.
+
+[89] P. FURTWAENGLER, "Ueber die Schwingungen zweier Pendel mit annaehernd
+gleicher Schwingungsdauer auf gemeinsamer Unterlage," _Sitzungsberichte
+der Koeniglicher 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 fuer 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. KUEHNEN and P. FURTWAENGLER, _Bestimmung der Absoluten
+Groesze der Schwerkraft zu Potsdam mit Reversionspendeln_ (Berlin:
+Veroeffentlichungen des Koeniglichen Preussischen Geodaetischen Instituts,
+1906), new ser., no. 27.
+
+[97] Reported by Dr. F. Kuehnen 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, _Beitraege zur Theorie des Reversionspendels_
+(Potsdam: Veroeffentlichungen des Koeniglichen Preussischen Geodaetischen
+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 Geodesique_ (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 Geodesique_ (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 Geodesique_ (March 1, 1959),
+no. 51, p.72.
+
+
+    U.S. GOVERNMENT PRINTING OFFICE: 1965
+
+    For sale by the Superintendent of Documents, U.S. Government Printing
+    Office Washington, D.C., 20402--Price 70 cents.
+
+
+
+
+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, Pere 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
+
+    Cellerier, 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, Herve, 325, 336, 346, 347
+
+    Fernel, Jean, 306
+
+    Furtwaengler, 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
+
+
+    Ibanez, Carlos, 325
+
+
+    Jeffreys, Sir Harold, 342
+
+    Jones, Thomas, 318
+
+
+    Kater, Henry, 304, 314, 325, 327, 329, 345, 346
+
+    Kuehnen, F., 338, 339
+
+
+    La Condamine, Charles Marie de, 307, 310, 311, 343
+
+    Laplace, Marquis Pierre Simon de, 309, 313, 320
+
+    Lorenzoni, Giuseppe, 336, 339
+
+    Luetke, 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, Abbe 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 a Geneve avec le pendule a reversion'--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 'Societe francaise de Physique'--had 'Francaise.'
+  Footnote 3 'Cogitata physico-mathematica'--had 'physica.'
+  Footnote 10 'mathematiques et de physique par MM. de l'Academie
+      Royale'--had 'mathematiques,' 'Royal.'
+  Footnote 12 'par ordre du Roy au Perou, pour observer'--had 'Perou,
+      pour observir.'
+  Footnote 19 'Opticam et Astronomiam'--had 'Astronomian.'
+  Footnote 20 'connaitre la longueur du pendule qui'--had 'connaitre la
+      longuer.'
+  Footnote 21 'Abhandlungen der Koeniglichen Akademie'--had 'Koenigliche.'
+  Footnote 25 'pour determiner la longueur du pendule'--had 'longeur.'
+  Footnote 41 'Survey of India (Calcutta, 1879)'-- had 'Surey.'
+  Footnotes 45 and 47 'Societe de Physique et d'histoire'--had
+      'd'historire.'
+  Footnote 49 'Ueber die Groesse und Figur der Erde'--had 'Grosse.'
+  Footnote 53 'Bestimmung der Laenge'--had 'Lange';
+      'Astronomisch-Geodaetische Arbeiten'--had 'Astronomische';
+      'Veroeffentlichungen des Koeniglichen'--had 'Koenigliche.'
+  Footnote 55 '(1768), vol. 58, pp. 329-335.'--had '329-235.'
+  Footnote 66 'Comptes-rendus de l'Academie'--had 'L'Academie.'
+  Footnote 81 'Sur l'Intensite absolue'--had 'l'Intensite.'
+  Footnote 89 'Sitzungsberichte der Koeniglicher'--had 'Koenigliche.'
+  Footnote 100 'Veroeffentlichungen des Koeniglichen' had
+      'Veroeffentlichungen Koenigliche.'
+
+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.
+
+
+
+***END OF THE PROJECT GUTENBERG EBOOK DEVELOPMENT OF GRAVITY PENDULUMS IN
+THE 19TH CENTURY***
+
+
+******* This file should be named 35024.txt or 35024.zip *******
+
+
+This and all associated files of various formats will be found in:
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