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diff --git a/35024.txt b/35024.txt new file mode 100644 index 0000000..ceeb837 --- /dev/null +++ b/35024.txt @@ -0,0 +1,3349 @@ +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: +http://www.gutenberg.org/dirs/3/5/0/2/35024 + + + +Updated editions will replace the previous one--the old editions +will be renamed. + +Creating the works from public domain print editions means that no +one owns a United States copyright in these works, so the Foundation +(and you!) can copy and distribute it in the United States without +permission and without paying copyright royalties. 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