Chapter I: Part 1

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States National Museum Bulletin 240_, comprising Papers 34-44,
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Smithsonian Institution United States National Museum Bulletin 240

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

_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.

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).

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.

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]

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.

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.

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.)]

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.

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]

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.

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.

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.

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]

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]