file which includes the original illustrations4 ,see #cejbd-;h4htm or #cejbd-;h4zip3 7https3_/_/www4gutenberg4org_/cache_/epu b_/#cejbd_/pg#cejbd-;images4html7 or 7https3_/_/www4gutenberg4org_/files_/#ce jbd_/cejbd-;h4zip7 ,transcriber's note3 ,this is ,paper #dd from the ,smithsonian ,institution ,united ,states ,national ,museum ,bulletin #bdj1 comprising ,papers #cd-dd1 which will also be available as a complete e-book4 ,the front material1 introduction and relevant index entries from the ,bulletin are included in each single-paper e-book4 ,mathematical notation used in this e-text3 #a4 ,greek letters are represented by the name of the letter in square brackets2 e4g41 pi4 #b4 ,subscripts are denoted by underscore followed by the subscript in curly braces2 e4g41 ,tn4 ,superscripts are denoted by a caret followed by the superscript in curly braces2 e4g41 ,tn4 ,to avoid possible confusion1 subscripted variables which are raised to a power are enclosed in brackets1 thus 7,t#a7b represents ',t one squared'4 #c4 ,square root is denoted by sqrt4 #d4 ,to improve readability1 italic markup 7underscores enclosing text7 has been omitted from letters used in mathematical formulae1 and some equations have been set 'out of line'4 ,please see the end of the book for a list of corrections4 ,smithsonian ,institution ,united ,states ,national ,museum ,bulletin #bdj ,illustration ,smithsonian ,press ,museum of ,history and ,technology ,contributions from the ,museum of ,history and ,technology ,papers #cd-dd ,on ,science and ,technology ,smithsonian ,institution 4 ,washington1 ,d4,c4 #aiff 99 99 99 99 99 ,publications of the ,united ,states ,national ,museum ,the scholarly and scientific publications of the ,united ,states ,national ,museum include two series1 ,proceedings of the ,united ,states ,national ,museum and ,united ,states ,national ,museum ,bulletin4 ,in these series1 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 anthropology1 biology1 history1 geology1 and technology4 ,copies of each publication are distributed to libraries1 to cultural and scientific organizations1 and to specialists and others interested in the different subjects4 ,the ,proceedings1 begun in #ahgh1 are intended for the publication1 in separate form1 of shorter papers from the ,museum of ,natural ,history4 ,these are gathered in volumes1 octavo in size1 with the publication date of each paper recorded in the table of contents of the volume4 ,in the ,bulletin series1 the first of which was issued in #ahge1 appear longer1 separate publications consisting of monographs 7occasionally in several parts7 and volumes in which are collected works on related subjects4 ,bulletins are either octavo or quarto in size1 depending on the needs of the presentation4 ,since #aijb 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 ,herbarium1 and since #aiei1 in ,bulletins titled ,contributions from the ,museum of ,history and ,technology1 have been gathered shorter papers relating to the collections and research of that ,museum4 ,the present collection of ,contributions1 ,papers #cd-dd1 comprises ,bulletin #bdj4 ,each of these papers has been previously published in separate form4 ,the year of publication is shown on the last page of each paper4 ,,frank ,a4 ,,taylor ,director1 ,united ,states ,national ,museum 99 99 99 99 99 ,contributions from the ,museum of ,history and ,technology3 ,paper #dd ,,development ,,of ,,gravity ,,pendulums ,,in ,,the #ai,,th ,,century by ,victor ,f4 ,lenzen and ,robert ,p4 ,multhauf ,,galileo1 ,,huygens1 ,,and ,,newton #cjd ,,figure ,,of ,,the ,,earth #cjf ,,early ,,types ,,of ,,pendulums #cji ,,kater',s ,,convertible ,,and ,,invariable ,,pendulums #cad ,,repsold-,,bessel ,,reversible ,,pendulum #cbj ,,peirce ,,and ,,defforges ,,invariable1 ,,reversible ,,pendulums #cbg ,,von ,,sterneck ,,and ,,mendenhall ,,pendulums #cca ,,absolute ,,value ,,of ,,gravity ,,at ,,potsdam #cch ,,application ,,of ,,gravity ,,surveys #cdb ,,summary #cdf ,,victor ,f4 ,,lenzen ,,and ,,robert ,p4 ,,multhauf ,,development ,,of ,,gravity ,,pendulums ,,in ,,the #aith ,,century ,illustration3 ,figure #a4--;,a ,,study ,,of ,,the ,,figure ,,of ,,the ,,earth ,,was one of the earliest projects of the ,french ,academy of ,sciences4 ,in order to test the effect of the earth's rotation on its gravitational force1 the ,academy in #afgb 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 ,paris4 ,richer found that the clock lost #b minutes and #bh seconds at ,cayenne1 indicating a substantial decrease in the force of gravity on the pendulum4 ,subsequent pendulum experiments revealed that the period of a pendulum varied not only with the latitude but also regionally1 under the influence of topographical features such as mountains4 ,it became clear that the measurement of gravity should be made a part of the work of the geodetic surveyor4 ,the history of gravity pendulums dates back to the time of ,galileo4 ,after the discovery of the variation of the force of gravity over the surface of the earth1 gravity measurement became a major concern of physics and geodesy4 ,this article traces the history of the development of instruments for this purpose4 ,,the ,,authors3 ,victor ,f4 ,lenzen is ,professor of ,physics1 ,emeritus1 at the ,university of ,california at ,berkeley and ,robert ,p4 ,multhauf is ,chairman of the ,department of ,science and ,technology in the ,smithsonian ,institution's ,museum of ,history and ,technology4 ,the intensity of gravity1 or the acceleration of a freely falling body1 is an important physical quantity for the several physical sciences4 ,the intensity of gravity determines the weight of a standard pound or kilogram as a standard or unit of force4 ,in physical experiments1 the force on a body may be measured by determining the weight of a known mass which serves to establish equilibrium against it4 ,thus1 in the absolute determination of the ampere with a current balance1 the force between two coils carrying current is balanced by the earth's gravitational force upon a body of determinable mass4 ,the intensity of gravity enters into determinations of the size of the earth from the angular velocity of the moon1 its distance from the earth1 and ,newton's inverse square law of gravitation and the laws of motion4 ,prediction of the motion of an artificial satellite requires an accurate knowledge of gravity for this astronomical problem4 ,the gravity field of the earth also provides data for a determination of the figure of the earth1 or geoid1 but for this problem of geodesy relative values of gravity are sufficient4 ,if g is the intensity of gravity at some reference station1 and ,deltag is the difference between intensities at two stations1 the values of gravity in geodetic calculations enter as ratios 7,deltag7_/g over the surface of the earth4 ,gravimetric investigations in conjunction with other forms of geophysical investigation1 such as seismology1 furnish data to test hypotheses concerning the internal structure of the earth4 ,whether the intensity of gravity is sought in absolute or relative measure1 the most widely used instrument for its determination since the creation of classical mechanics has been the pendulum4 ,in recent decades1 there have been invented gravity meters based upon the principle of the spring1 and these instruments have made possible the rapid determination of relative values of gravity to a high degree of accuracy4 ,the gravity meter1 however1 must be calibrated at stations where the absolute value of gravity has been determined by other means if absolute values are sought4 ,for absolute determinations of gravity1 the pendulum historically has been the principal instrument employed4 ,although alternative methods of determining absolute values of gravity are now in use1 the pendulum retains its value for absolute determinations1 and even retains it for relative determinations1 as is exemplified by the ,cambridge ,pendulum ,apparatus and that of the ,dominion ,observatory at ,ottawa1 ,ontario4 ,the pendulums employed for absolute or relative determinations of gravity have been of two basic types4 ,the first form of pendulum used as a physical instrument consisted of a weight suspended by a fiber1 cord1 or fine wire1 the upper end of which was attached to a fixed support4 ,such a pendulum may be called a simple pendulum2 the enclosure of the word simple by quotation marks is to indicate that such a pendulum is an approximation to a simple1 or mathematical pendulum1 a conceptual object which consists of a mass-point suspended by a weightless inextensible cord4 ,if l is the length of the simple pendulum1 the time of swing 7half-period in the sense of physics7 for vibrations of infinitely small amplitude1 as derived from ,newton's laws of motion and the hypothesis that weight is proportional to mass1 is ,t pisqrt7l_/g74 ,the second form of pendulum is the compound1 or physical1 pendulum4 ,it consists of an extended solid body which vibrates about a fixed axis under the action of the weight of the body4 ,a compound pendulum may be constituted to oscillate about one axis only1 in which case it is nonreversible and applicable only for relative measurements4 ,or a compound pendulum may be constituted to oscillate about two axes1 in which case it is reversible 7or convertible7 and may be used to determine absolute values of gravity4 ,capt4 ,henry ,kater1 ,f4,r4,s41 during the years #ahag-ahah was the first to design1 construct1 and use a compound pendulum for the absolute determination of gravity4 ,he constructed a convertible pendulum with two knife edges and with it determined the absolute value of gravity at the house of ,henry ,browne1 ,f4,r4,s41 in ,portland ,place1 ,london4 ,he then constructed a similar compound pendulum with only one knife edge1 and swung it to determine relative values of gravity at a number of stations in the ,british ,isles4 ,the #aith century witnessed the development of the theory and practice of observations with pendulums for the determination of absolute and relative values of gravity4 ,galileo1 ,huygens1 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 #agth century4#a ,it is tradition that the youthful ,galileo discovered that the period of oscillation of a pendulum is constant by observations of the swings of the great lamp suspended from the ceiling in the cathedral of ,pisa4#b ,the lamp was only a rough approximation to a simple pendulum1 but ,galileo later performed more accurate experiments with a simple pendulum which consisted of a heavy ball suspended by a cord4 ,in an experiment designed to confirm his laws of falling bodies1 ,galileo lifted the ball to the level of a given altitude and released it4 ,the ball ascended to the same level on the other side of the vertical equilibrium position and thereby confirmed a prediction from the laws4 ,galileo also discovered that the period of vibration of a simple pendulum varies as the square root of its length1 a result which is expressed by the formula for the time of swing of the ideal simple pendulum4 ,he also used a pendulum to measure lapse of time1 and he designed a pendulum clock4 ,galileo's experimental results are important historically1 but have required correction in the light of subsequent measurements of greater precision4 ,mersenne in #afdd made the first determination of the length of the seconds pendulum1#c that is1 the length of a simple pendulum that beats seconds 7half-period in the sense of physics74 ,subsequently1 he proposed the problem to determine the length of the simple pendulum equivalent in period to a given compound pendulum4 ,this problem was solved by ,huygens1 who in his famous work ,horologium oscillatorium 444 7#afgc7 set forth the theory of the compound pendulum4#d ,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 gravity4 ,the theorem is that a given compound pendulum possesses conjugate points on opposite sides of the center of gravity2 about these points1 the periods of oscillation are the same4 ,for each of these points as center of suspension the other point is the center of oscillation1 and the distance between them is the length of the equivalent simple pendulum4 ,earlier1 in #afeg1 ,huygens independently had invented and patented the pendulum clock1 which rapidly came into use for the measurement of time4 ,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 gravity4 ,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 7#afhg74 ,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 factors3 the principal factor is the gravitational attraction of the earth upon bodies1 and the subsidiary factor is the effect of the rotation of the earth4 ,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 axis4 ,if the rotating earth is used as a frame of reference1 the effect of the rotation is expressed as a centrifugal force which acts to diminish the observed intensity of gravity4 99 99 99 99 99 ,,glossary ,,of ,,gravity ,,terminology ,,absolute ,,gravity3 the value of the acceleration of gravity1 also expressed by the length of the seconds pendulum4 ,,relative ,,gravity3 the value of the acceleration of gravity relative to the value at some standard point4 ,,simple ,,pendulum3 see theoretical pendulum4 ,,theoretical ,,pendulum3 a heavy bob 7point-mass7 at the end of a weightless rod4 ,,seconds ,,pendulum3 a theoretical or simple pendulum of such length that its time of swing 7half-period7 is one second4 7,this length is about one meter47 ,,gravity ,,pendulum3 a precisely made pendulum used for the measurement of gravity4 ,,compound ,,pendulum3 a pendulum in which the supporting rod is not weightless2 in other words1 any actual pendulum4 ,,convertible ,,pendulum3 a compound pendulum having knife edges at different distances from the center of gravity4 ,huygens demonstrated 7#afgc7 that if such a pendulum were to swing with equal periods from either knife edge1 the distance between those knife edges would be equal to the length of a theoretical or simple pendulum of the same period4 ,,reversible ,,pendulum3 a convertible pendulum which is also symmetrical in form4 ,,invariable ,,pendulum3 a compound pendulum with only one knife edge1 used for relative measurement of gravity4 99 99 99 99 99 ,from ,newton's laws of motion and the hypothesis that weight is proportional to mass1 the formula for the half-period of a simple pendulum is given by ,t pisqrt7l_/g74 ,if a simple pendulum beats seconds1 #a pisqrt7lambda_/g71 where lambda is the length of the seconds pendulum4 ,from ,t pisqrt7l_/g7 and #a pisqrt7lambda_/g71 it follows that lambda l_/,t#b4 ,then g pi#blambda4 ,thus1 the intensity of gravity can be expressed in terms of the length of the seconds pendulum1 as well as by the acceleration of a freely falling body4 ,during the #aith century1 gravity usually was expressed in terms of the length of the seconds pendulum1 but present practice is to express gravity in terms of g1 for which the unit is the gal1 or one centimeter per second per second4 ,illustration3 ,figure #b4--,,this ,,drawing1 ,,from ,,richer',s ,observations astronomiques et physiques faites en l'isle de ,caienne 7,paris1 #afgi71 shows most of the astronomical instruments used by ,richer1 namely1 one of the two pendulum clocks made by ,thuret1 the #bj-;foot and the #e-;foot telescopes and the large quadrant4 ,the figure may be intended as a portrait of ,richer4 ,this drawing was done by ,sebastian ,le ,clerc1 a young illustrator who made many illustrations of the early work of the ,paris ,academy4 ,figure of the ,earth ,a principal contribution of the pendulum as a physical instrument has been the determination of the figure of the earth4#e ,that the earth is spherical in form was accepted doctrine among the ancient ,greeks4 ,pythagoras is said to have been the first to describe the earth as a sphere1 and this view was adopted by ,eudoxus and ,aristotle4 ,the ,alexandrian scientist ,eratosthenes made the first estimate of the diameter and circumference of a supposedly spherical earth by an astronomical-geodetic method4 ,he measured the angle between the directions of the rays of the sun at ,alexandria and ,syene 7,aswan71 ,egypt1 and estimated the distance between these places from the length of time required by a caravan of camels to travel between them4 ,from the central angle corresponding to the arc on the surface1 he calculated the radius and hence the circumference of the earth4 ,a second measurement was undertaken by ,posidonius1 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 other4 ,with the decline of classical antiquity1 the doctrine of the spherical shape of the earth was lost1 and only one investigation1 that by the ,arabs under ,calif ,al-,mamun in ,a4,d4 #hbg1 is recorded until the #afth century4 ,in #aebe1 the ,french mathematician ,fernel measured the length of a degree of latitude between ,paris and ,amiens by the revolutions of the wheels of his carriage1 the circumference of which he had determined4 ,in ,england1 ,norwood in #afce measured the length of an arc between ,london and ,york with a chain4 ,an important forward step in geodesy was the measurement of distance by triangulation1 first by ,tycho ,brahe1 in ,denmark1 and later1 in #afae1 by ,willebrord ,snell1 in ,holland4 ,of historic importance1 was the use of telescopes in the triangulation for the measurement of a degree of arc by the ,abbe ,jean ,picard in #affi4f ,he had been commissioned by the newly established ,academy of ,sciences to measure an arc corresponding to an angle of #a deg41 #bb'1 #ee of the meridian between ,amiens and ,malvoisine1 near ,paris4 ,picard proposed to the ,academy the measurement of the meridian of ,paris through all of ,france1 and this project was supported by ,colbert1 who obtained the approval of the ,king4 ,in #afhd1 ,giovanni-,domenico ,cassini and ,de la ,hire commenced a trigonometrical measure of an arc south of ,paris2 subsequently1 ,jacques ,cassini1 the son of ,giovanni-,domenico1 added the arc to the north of ,paris4 ,the project was completed in #agah4 ,the length of a degree of arc south of ,paris was found to be greater than the length north of ,paris4 ,from the difference1 #eg1jig toises#g minus #ef1ifj toises1 it was concluded that the polar diameter of the earth is larger than the equatorial diameter1 i4e41 that the earth is a prolate spheroid 7fig4 #c74 ,illustration3 ,figure #c4--,,measurements ,,of ,,the ,,length of a degree of latitude which were completed in different parts of ,france in #affi and #agah gave differing results which suggested that the shape of the earth is not a sphere but a prolate spheroid 7#a74 ,but ,richer's pendulum observation of #afgb1 as explained by ,huygens and ,newton1 indicated that its shape is that of an oblate spheroid 7#b74 ,the disagreement is reflected in this drawing4 ,in the #agcj'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 7,ecuador--then part of ,peru--and ,lapland74 ,meanwhile1 ,richer in #afgb had been sent to ,cayenne1 ,french ,guiana1 to make astronomical observations and to measure the length of the seconds pendulum4#h ,he took with him a pendulum clock which had been adjusted to keep accurate time in ,paris4 ,at ,cayenne1 however1 ,richer found that the clock was retarded by #b minutes and #bh seconds per day 7fig4 #a74 ,he also fitted up a simple pendulum to vibrate in seconds and measured the length of this seconds pendulum several times every week for #aj months4 ,upon his return to ,paris1 he found that the length of the simple pendulum which beat seconds at ,cayenne was #a-a_/d ,paris lines#i shorter than the length of the seconds pendulum at ,paris4 ,huygens explained the reduction in the length of the seconds pendulum--and1 therefore1 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 pendulum4#aj ,a more complete theory was given by ,newton in the ,principia4#aa ,newton showed that if the earth is assumed to be a homogeneous1 mutually gravitating fluid globe1 its rotation will result in a bulging at the equator4 ,the earth will then have the form of an oblate spheroid1 and the intensity of gravity as a form of universal gravitation will vary with position on the surface of the earth4 ,newton took into account gravitational attraction and centrifugal action1 and he calculated the ratio of the axes of the spheroid to be #bcj3bbi4 ,he calculated and prepared a table of the lengths of a degree of latitude and of the seconds pendulum for every #e deg4 of latitude from the equator to the pole4 ,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 equator4 ,newton's theory that the earth is an oblate spheroid was confirmed by the measurements of ,richer1 but was rejected by the ,paris ,academy of ,sciences1 for it contradicted the results of the ,cassinis1 father and son1 whose measurements of arcs to the south and north of ,paris had led to the conclusion that the earth is a prolate spheroid4 ,thus1 a controversy arose between the ,english scientists and the ,paris ,academy4 ,the conflict was finally resolved by the results of expeditions sent by the ,academy to ,peru and ,sweden4 ,the first expedition1 under ,bouguer1 ,la ,condamine1 and ,godin in #agce1 went to a region in ,peru1 and1 with the help of the ,spaniard ,ullo1 measured a meridian arc of about #c deg4 #g' near ,quito1 now in ,ecuador4#ab ,the second expedition1 with ,maupertuis and ,clairaut in #agcf1 went to ,lapland within the ,arctic ,circle and measured an arc of about #a deg4 in length4#ac ,the northern arc of #a deg4 was found to be longer than the ,peruvian arc of #a deg41 and thus it was confirmed that the earth is an oblate spheroid1 that is1 flattened at the poles1 as predicted by the theory of ,newton4 ,illustration3 ,figure #d4--,,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 clock1 and by the unavoidable difference between the characteristics of a clock pendulum and those of a theoretical 7usually called simple7 pendulum1 in which the mass is concentrated in the bob1 and the supporting rod is weightless4 ,after #agce1 the clock was used only to time the swing of a detached pendulum1 by the method of coincidences4 ,in this method1 invented by ,j4 ,j4 ,mairan1 the length of the detached pendulum is first accurately measured1 and the clock is corrected by astronomical observation4 ,the detached pendulum is then swung before the clock pendulum as shown here4 ,the two pendulums swing more or less out of phase1 coming into coincidence each time one has gained a vibration4 ,by counting the number of coincidences over several hours1 the period of the detached pendulum can be very accurately determined4 ,the length and period of the detached pendulum are the data required for the calculation of the force of gravity4 ,the period from ,eratosthenes to ,picard has been called the spherical era of geodesy2 the period from ,picard to the end of the #aith century has been called the ellipsoidal period4 ,during the latter period the earth was conceived to be an ellipsoid1 and the determination of its ellipticity1 that is1 the difference of equatorial radius and polar radius divided by the equatorial radius1 became an important geodetic problem4 ,a significant contribution to the solution of this problem was made by determinations of gravity by the pendulum4 ,an epoch-making work during the ellipsoidal era of geodesy was ,clairaut's treatise1 ,theorie de la figure de la terre4#ad ,on the hypothesis that the earth is a spheroid of equilibrium1 that is1 such that a layer of water would spread all over it1 and that the internal density varies so that layers of equal density are coaxial spheroids1 ,clairaut derived a historic theorem3 ,if gamma,e1 gamma,p are the values of gravity at the equator and pole1 respectively1 and c the centrifugal force at the equator divided by gamma,e1 then the ellipticity alpha 7#e_/b7;c - 7gamma,p - gamma,e7_/gamma,e4 ,laplace showed that the surfaces of equal density might have any nearly spherical form1 and ,stokes showed that it is unnecessary to assume any law of density as long as the external surface is a spheroid of equilibrium4#ae ,it follows from ,clairaut's theorem that if the earth is an oblate spheroid1 its ellipticity can be determined from relative values of gravity and the absolute value at the equator involved in c4 ,observations with nonreversible1 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 earth4 ,early ,types of ,pendulums ,the pendulum employed in observations of gravity prior to the #aith century usually consisted of a small weight suspended by a filament 7figs4 #d-f74 ,the pioneer experimenters with simple pendulums changed the length of the suspension until the pendulum beat seconds4 ,picard in #affi 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 7pite was a preparation of the leaf of a species of aloe and was not affected appreciably by moisture74 ,a celebrated set of experiments with a simple pendulum was conducted by ,bouguer#af in #agcg in the ,andes1 as part of the expedition to measure the ,peruvian arc4 ,the bob of the pendulum was a double truncated cone1 and the length was measured from the jaw suspension to the center of oscillation of the thread and bob4 ,bouguer allowed for change of length of his measuring rod with temperature and also for the buoyancy of the air4 ,he determined the time of swing by an elementary form of the method of coincidences4 ,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 clock4 ,for this purpose1 he noted the time when the beat of the clock was heard and1 simultaneously1 the thread moved past the center of the scale4 ,a historic aspect of ,bouguer's method was that he employed an invariable pendulum1 that is1 the length was maintained the same at the various stations of observation1 a procedure that has been described as having been invented by ,bouguer4 ,since ,t pisqrt7l_/g71 it follows that 7,t#a7b_/7,t#b7b g#b_/;g#a4 ,thus1 if the absolute value of gravity is known at one station1 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 stations4 ,from the above equation1 if ,t#a is the time of swing at a station where the intensity of gravity is g1 and ,t#b is the time at a station where the intensity is g ,deltag1 then ,deltag_/g 7,t#a7b_/7,t#b7b - #a4 ,bouguer's investigations with his invariable pendulum yielded methods for the determination of the internal structure of the earth4 ,on the ,peruvian expedition1 he determined the length of the seconds pendulum at three stations1 including one at ,quito1 at varying distances above sea level4 ,if values of gravity at stations of different elevation are to be compared1 they must be reduced to the same level1 usually to sea level4 ,since gravity decreases with height above sea level in accordance with the law of gravitation1 a free-air reduction must be applied to values of gravity determined above the level of the sea4 ,bouguer originated the additional reduction for the increase in gravity on a mountain or plateau caused by the attraction of the matter in a plate4 ,from the relative values of gravity at elevated stations in ,peru and at sea level1 ,bouguer calculated that the mean density of the earth was #d4g times greater than that of the cordilleras4#ag ,for greater accuracy in the study of the internal structure of the earth1 in the #aith century the ,bouguer plate reduction came to be supplemented by corrections for irregularities of terrain and by different types of isostatic reduction4 ,la ,condamine1 who like ,bouguer was a member of the ,peruvian expedition1 conducted his own pendulum experiments 7fig4 #d74 ,he experimented in #agce at ,santo ,domingo en route to ,south ,america1#ah then at various stations in ,south ,america1 and again at ,paris upon his return to ,france4 ,his pendulum consisted of a copper ball suspended by a thread of pite4 ,for experimentation the length initially was about #ab feet1 and the time of swing #b seconds1 but then the length was reduced to about #c feet with time of swing #a second4 ,earlier1 when it was believed that gravity was constant over the earth1 ,picard and others had proposed that the length of the seconds pendulum be chosen as the standard4 ,la ,condamine in #agdg revived the proposal in the form that the length of the seconds pendulum at the equator be adopted as the standard of length4 ,subsequently1 he investigated the expansion of a toise of iron from the variation in the period of his pendulum4 ,in #agee1 he observed the pendulum at ,rome with ,boscovich4 ,la ,condamine's pendulum was used by other observers and finally was lost at sea on an expedition around the world4 ,the knowledge of the pendulum acquired by the end of the #ahth century was summarized in #aghe in a memoir by ,boscovich4#ai ,illustration3 ,figure #e4--;,,an ,,apparatus ,,for ,,the ,,practice ,,measurement of the length of the pendulum devised on the basis of a series of preliminary experiments by ,c4 ,m4 de la ,condamine who1 in the course of the ,french geodetic expedition to ,peru in #agce1 devoted a #c-month sojourn on the island of ,santo ,domingo to pendulum observations by ,mairan's ,method4 ,in this arrangement1 shown here1 a vertical rod of ironwood is used both as the scale and as the support for the apparatus1 having at its top the brass pendulum support 7,a7 and1 below1 a horizontal mirror 7,o7 which serves to align the apparatus vertically through visual observation of the reflection of the pointer projecting from ,a4 ,the pendulum1 about #cg inches long1 consists of a thread of pite 7a humidity-resistant1 natural fiber7 and a copper ball of about #f ounces4 ,its exact length is determined by adjusting the micrometer 7,s7 so that the ball nearly touches the mirror4 ,it will be noted that the clock pendulum would be obscured by the scale4 ,la ,condamine seems to have determined the times of coincidence by visual observation of the occasions on which the pendulums swing parallel4 7,portion of plate #a1 ,memoires publies par la ,societe francaise de ,physique1 vol4 #d47 ,illustration3 ,figure #f4--,,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 pendulum4 ,in #agib1 ,j4 ,c4 ,borda and ,j4 ,d4 ,cassini determined the length of the seconds pendulum at ,paris with this apparatus4 ,the pendulum consists of a platinum ball about #a-a_/b inches in diameter1 suspended by a fine iron wire4 ,the length1 about #ab feet1 was such that its period would be nearly twice as long as that of the pendulum of the clock 7,a74 ,the interval between coincidences was determined by observing1 through the telescope at the left1 the times when the two pendulums emerge together from behind the screen 7,m74 ,the exact length of the pendulum was measured by a platinum scale 7not shown7 equipped with a vernier and an auxiliary copper scale for temperature correction4 ,when1 at the end of the #ahth century1 the ,french revolutionary government established the metric system of weights and measures1 the length of the seconds pendulum at ,paris was considered1 but not adopted1 as the unit of length4 7,plate #b1 ,memoires publies par la ,societe francaise de ,physique1 vol4 #d47 ,the practice with the simple pendulum on the part of ,picard1 ,bouguer1 ,la ,condamine and others in ,france culminated in the work of ,borda and ,cassini in #agib at the observatory in ,paris#bj 7fig4 #f74 ,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 ,france4 ,the bob consisted of a platinum ball #af-a_/f ,paris lines in diameter1 and #i1iaa grains 7slightly more than #ag ounces7 in weight4 ,the bob was held to a brass cup covering about one-fifth of its surface by the interposition of a small quantity of grease4 ,the cup with ball was hung by a fine iron wire about #ab ,paris feet long4 ,the upper end of the wire was attached to a cylinder which was part of a wedge-shaped knife edge1 on the upper surface of which was a stem on which a small adjustable weight was held by a screw thread4 ,the knife edge rested on a steel plate4 ,the weight on the knife-edge apparatus was adjusted so that the apparatus would vibrate with the same period as the pendulum4 ,thus1 the mass of the suspending apparatus could be neglected in the theory of motion of the pendulum about the knife edge4 ,illustration3 ,figure #g4--,,results ,,of ,,experiments in the determination of the length of the seconds pendulum at ,koenigsberg by a new method were reported by ,f4 ,w4 ,bessel in #ahbf and published in #ahbh4 ,with this apparatus1 he obtained two sets of data from the same pendulum1 by using two different points of suspension4 ,the pendulum was about #aj feet long4 ,the distance between the two points of suspension 7a and b7 was #a toise 7about six feet74 ,a micrometric balance 7c7 below the bob was used to determine the increase in length due to the weight of the bob4 ,he projected the image of the clock pendulum 7not shown7 onto the gravity pendulum by means of a lens1 thus placing the clock some distance away and eliminating the disturbing effect of its motion4 7,portion of plate #f1 ,memoires publies par la ,societe francaise de ,physique1 vol4 #d47 ,in the earlier suspension from jaws there was uncertainty as to the point about which the pendulum oscillated4 ,borda and ,cassini hung their pendulum in front of a seconds clock and determined the time of swing by the method of coincidences4 ,the times on the clock were observed when the clock gained or lost one complete vibration 7two swings7 on the pendulum4 ,suppose that the wire pendulum makes n swings while the clock makes #bn #b4 ,if the clock beats seconds exactly1 the time of one complete vibration is #b seconds1 and the time of swing of the wire pendulum is ,t 7#bn #b7_/n #b7a #a_/n74 ,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 #bn4 ,the whole apparatus was enclosed in a box1 in order to exclude disturbances from currents of air4 ,corrections were made for buoyancy1 for amplitude of swing and for variations in length of the wire with temperature4 ,the final result was that the length of the seconds pendulum at the observatory in ,paris was determined to be #ddj4eeic ,paris lines1 or #iic4ec mm41 reduced to sea level #iic4he mm4 ,some years later the methods of ,borda were used by other ,french investigators1 among whom was ,biot who used the platinum ball of ,borda suspended by a copper wire #fj cm4 long4 ,another historic simple pendulum was the one swung by ,bessel 7fig4 #g7 for the determination of gravity at ,koenigsberg #ahbe-ahbg4ba ,the pendulum consisted of a ball of brass1 copper1 or ivory that was suspended by a fine wire1 the upper end of which was wrapped and unwrapped on a horizontal cylinder as support4 ,the pendulum was swung first from one point and then from another1 exactly a toise de ,peru#bb higher up1 the bob being at the same level in each case 7fig4 #g74 ,bessel found the period of vibration of the pendulum by the method of coincidences2 and in order to avoid disturbances from the comparison clock1 it was placed at some distance from the pendulum under observation4 ,bessel's experiments were significant in view of the care with which he determined the corrections4 ,he corrected for the stiffness of the wire and for the lack of rigidity of connection between the bob and wire4 ,the necessity for the latter correction had been pointed out by ,laplace1 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 gravity1 the bob acquires angular momentum about its center of gravity1 which cannot be accounted for if the line of the wire1 and therefore the force that it exerts1 always passed through the center4 ,in addition to a correction for buoyancy of the air considered by his predecessors1 ,bessel also took account of the inertia of the air set in motion by the pendulum4 ,illustration3 ,figure #h4--,,mode ,,of ,,suspension of ,bessel's pendulum is shown here4 ,the iron wire is supported by the thumbscrew and clamp at the left1 but passes over a pin at the center1 which is actually the upper terminal of the pendulum4 ,bessel found this cylinder of unrolling superior to the clamps and knife edges of earlier pendulums4 ,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 weight4 ,with this apparatus1 ,bessel determined the ratio of the lengths of the two pendulums and their times of vibration4 ,from this the length of the seconds pendulum was calculated4 ,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 bob4 7,portion of plate #g1 ,memoires publies par la ,societe francaise de ,physique1 vol4 #d47 ,illustration3 ,figure #i4--;,,friedrich ,,wilhelm ,,bessel 7#aghd-ahdf71 ,german mathematician and astronomer4 ,he became the first superintendent of the ,prussian observatory established at ,koenigsberg in #ahaj1 and remained there during the remainder of his life4 ,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 science4 ,this characteristic also shows in his venture into geodesy1 #ahbf-ahcj1 one product of which was the pendulum experiment reported in this article4 ,the latter effect had been discovered by ,du ,buat in #aghf1bc but his work was unknown to ,bessel4 ,the length of the seconds pendulum at ,koenigsberg1 reduced to sea level1 was found by ,bessel to be #ddj4hagi lines4 ,in #ahce1 ,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 ,s4 ,peirce of the ,u4,s4 ,coast ,survey4 ,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 ,capt4 ,henry ,kater1 ,f4,r4,s4 ,in #ahag1 he designed1 constructed1 and applied a convertible compound pendulum for the absolute determination of gravity at the house of ,henry ,browne1 ,f4,r4,s41 in ,portland ,place1 ,london4#bd ,kater's convertible pendulum 7fig4 #aa7 consisted of a brass rod to which were attached a flat circular bob of brass and two adjustable weights1 the smaller of which was adjusted by a screw4 ,the convertibility of the pendulum was constituted by the provision of two knife edges turned inwards on opposite sides of the center of gravity4 ,the pendulum was swung on each knife edge1 and the adjustable weights were moved until the times of swing were the same about each knife edge4 ,when the times were judged to be the same1 the distance between the knife edges was inferred to be the length of the equivalent simple pendulum1 in accordance with ,huygens' theorem on conjugate points of a compound pendulum4 ,kater determined the time of swing by the method of coincidences 7fig4 #ab74 ,he corrected for the buoyancy of the air4 ,the final value of the length of the seconds pendulum at ,browne's house in ,london1 reduced to sea level1 was determined to be #ci4acibi inches4 ,the convertible compound pendulum had been conceived prior to its realization by ,kater4 ,in #agib1 on the occasion of the proposal in ,paris to establish the standard of length as the length of the seconds pendulum1 ,baron de ,prony had proposed the employment of a compound pendulum with three axes of oscillation4#be ,in #ahjj1 he proposed the convertible compound pendulum with knife edges about which the pendulum could complete swings in equal times4 ,de ,prony's proposals were not accepted and his papers remained unpublished until #ahhi1 at which time they were discovered by ,defforges4 ,the ,french decision was to experiment with the ball pendulum1 and the determination of the length of the seconds pendulum was carried out by ,borda and ,cassini by methods previously described4 ,bohnenberger in his ,astronomie 7#ahaa71bf made the proposal to employ a convertible pendulum for the absolute determination of gravity2 thus1 he has received credit for priority in publication4 ,capt4 ,kater independently conceived of the convertible pendulum and was the first to design1 construct1 and swing one4 ,illustration3 ,figure #aj4--;,,henry ,,kater 7#aggg-ahce71 ,english army officer and physicist4 ,his scientific career began during his military service in ,india1 where he assisted in the great trigonometrical survey4 ,returned to ,england because of bad health1 and retired in #ahad1 he pioneered 7#ahah7 in the development of the convertible pendulum as an alternative to the approximation of the simple pendulum for the measurement of the seconds pendulum4 ,kater's convertible pendulum and the invariable pendulum introduced by him in #ahai were the basis of ,english pendulum work4 7,photo courtesy ,national ,portrait ,gallery1 ,london47 ,after his observations with the convertible pendulum1 ,capt4 ,kater designed an invariable compound pendulum with a single knife edge but otherwise similar in external form to the convertible pendulum#bg 7fig4 #ac74 ,thirteen of these ,kater invariable pendulums have been reported as constructed and swung at stations throughout the world4#bh ,kater himself swung an invariable pendulum at a station in ,london and at various other stations in the ,british ,isles4 ,capt4 ,edward ,sabine1 between #ahbj and #ahbe1 made voyages and swung ,kater invariable pendulums at stations from the ,west ,indies to ,greenland and ,spitzbergen4#bi ,in #ahbj1 ,kater swung a ,kater invariable pendulum at ,london and then sent it to ,goldingham1 who swung it in #ahba at ,madras1 ,india4#cj ,also in #ahbj1 ,kater supplied an invariable pendulum to ,hall1 who swung it at ,london and then made observations near the equator and in the ,southern ,hemisphere1 and at ,london again in #ahbc4ca ,the same pendulum1 after its knives were reground1 was delivered to ,adm4 ,luetke of ,russia1 who observed gravity with it on a trip around the world between #ahbf and #ahbi4cb ,illustration3 ,figure #aa4--,,the ,,attempt ,,to ,,approximate the simple 7theoretical7 pendulum in gravity experiments ended in #ahag-ah when ,henry ,kater invented the compound convertible pendulum1 from which the equivalent simple pendulum could be obtained according to the method of ,huygens 7see text1 p4 #cad74 ,developed in connection with a project to fix the standard of ,english measure1 ,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 rod4 ,it was called convertible because it is alternately swung from the two knife edges 7a and b7 at opposite ends4 ,the weights 7f and g7 are adjusted so that the period of the pendulum is the same from either knife edge4 ,the distance between the two knife edges is then equal to the length of the equivalent simple pendulum4 ,illustration3 ,figure #ab4--,,the ,,kater ,,convertible ,,pendulum in use is placed before a clock1 whose pendulum bob is directly behind the extended tail of the ,kater pendulum4 ,a white spot is painted on the center of the bob of the clock pendulum4 ,the observing telescope1 left1 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 rest4 ,when the two pendulums are swinging1 the white spot on the clock pendulum can be seen on each swing except that in which the two pendulums are in coincidence2 thus1 the coincidences are determined4 7,portion of plate #e1 ,memoires publies par la ,societe francaise de ,physique1 vol4 #d47 ,illustration3 ,figure #ac4--,,this ,,drawing ,,accompanied ,john ,goldingham's report on the work done in ,india with ,kater's invariable pendulum4 ,the value of gravity obtained1 directly or indirectly1 in terms of the simple pendulum1 is called absolute4 ,once absolute values of gravity were established at a number of stations1 it became possible to use the much simpler relative method for the measurement of gravity at new stations4 ,because it has only one knife edge1 and does not involve the adjustments of the convertible pendulum1 this one is called invariable4 ,in use1 it is first swung at a station where the absolute value of gravity has been established1 and this period is then compared with its period at one or more new stations4 ,kater developed an invariable pendulum in #ahai1 which was used in ,england and in ,madras1 ,india1 in #ahba4 ,while the ,british were engaged in swinging the ,kater invariable pendulums to determine relative values of the length of the seconds pendulum1 or of gravity1 the ,french also sent out expeditions4 ,capt4 de ,freycinet made initial observations at ,paris with three invariable brass pendulums and one wooden one1 and then carried out observations at ,rio de ,janeiro1 ,cape of ,good ,hope1 ,ile de ,france1 ,rawak 7near ,new ,guinea71 ,guam1 ,maui1 and various other places4#cc ,a similar expedition was conducted in #ahbb-ahbe by ,captain ,duperry4#cd ,during the years from #ahbg to #ahdj1 various types of pendulum were constructed and swung by ,francis ,baily1 a member of the ,royal ,astronomical ,society1 who reported in #ahcb on experiments in which no less than #da different pendulums were swung in vacuo1 and their characteristics determined4#ce ,in #ahcf1 ,baily undertook to advise the ,american ,lt4 ,charles ,wilkes1 who was to head the ,united ,states ,exploring ,expedition of #ahch-ahdb1 on the procurement of pendulums for this voyage4 ,wilkes ordered from the ,london instrument maker1 ,thomas ,jones1 two unusual pendulums1 which ,wilkes described as those considered the best form by ,mr4 ,baily for traveling pendulums1 and which ,baily1 himself1 described as precisely the same as the two invariable pendulums belonging to this ,royal ,astronomical ,society1 except for the location of the knife edges4 ,illustration3 ,figure #ad4--,,vacuum ,,chamber ,,for ,,use with the ,kater pendulum4 ,of a number of extraneous effects which tend to disturb the accuracy of pendulum observations the most important is air resistance4 ,experiments reported by the ,greenwich 7,england7 observatory in #ahbi led to the development of a vacuum chamber within which the pendulum was swung4 ,the unusual feature of these pendulums was in their symmetry of mass as well as of form4 ,they were made of bars1 of iron in one case1 and of brass in the other1 and each had two knife edges at opposite ends equidistant from the center4 ,thus1 although they resembled reversible pendulums1 their symmetry of mass prevented their use as such1 and they were rather equivalent to four separate invariable pendulums4#cf ,wilkes was taught the use of the pendulum by ,baily1 and conducted experiments at ,baily's house1 where the latter had carried out the work reported on in #ahcb4 ,the subsequent experiments made on the ,u4,s4 ,exploring ,expedition were under the charge of ,wilkes1 himself1 who made observations on #aa separate occasions1 beginning with that in ,london 7#ahcf7 and followed by others in ,new ,york1 ,washington1 ,d4,c41 ,rio de ,janeiro1 ,sydney1 ,honolulu1 ,pendulum ,peak 7,mauna ,loa71 ,mount ,kanoha1 ,nesqually 7,oregon ,territory71 and1 finally1 two more times in ,washington1 ,d4,c4 7#ahda and #ahde74 ,wilkes' results were communicated to ,baily1 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 pendulums4#cg ,the results were also to have been included in the publications of the ,expedition1 but were part of the unpublished #bdth volume4 ,fortunately they still exist1 in what appears to be a printer's proof4#ch ,the ,kater invariable pendulums were used to investigate the internal constitution of the earth4 ,airy sought to determine the density of the earth by observing the times of swing of pendulums at the top and bottom of a mine4 ,the first experiments were made in #ahbf at the ,dolcoath copper mine in ,cornwall1 and failed when the pendulum fell to the bottom4 ,in #ahed1 the experiments were again undertaken in the ,harton coalpit1 near ,sunderland4#ci ,gravity at the surface was greater than below1 because of the attraction of a shell equal to the depth of the pit4 ,from the density of the shell as determined from specimens of rock1 ,airy found the density of the earth to be #f-a_/b times greater than that of water4 ,t4 ,c4 ,mendenhall1 in #ahhj1 used a ,kater convertible pendulum in an invariable manner to compare values of gravity on ,fujiyama and at ,tokyo1 ,japan4#dj ,he used a simple pendulum of the ,borda type to determine the absolute value of gravity at ,tokyo4 ,from the values of gravity on the mountain and at ,tokyo1 and an estimate of the volume of the mountain1 he estimated the mean density of the earth as #e4gg times greater than that of water4 ,in #ahgi1 ,maj4 ,j4 ,herschel1 ,r4,e41 stated3 ,the years from #ahdj to #ahfe are a complete blank1 if we except ,airy's relative density experiments in #ahed4 ,this pause was broken simultaneously in three different ways4 ,two pendulums of the ,kater pattern were sent to ,india2 two after ,bessel's design were set to work in ,russia2 and at ,geneva1 ,plantamour's zealous experiments with a pendulum of the same kind mark the commencement of an era of renewed activity on the ,european continent4#da ,with the statement that ,kater invariable pendulums nos4 #d and #f 7#ahba7 were used in ,india between #ahfe and #ahgc1 we now consider the other events mentioned by ,herschel4 ,illustration3 ,figure #ae4--,,one ,,of ,,francis ,,baily',s ,,pendulums 7#fb-a_/b inches long71 shown on the left1 is now in the possession of the ,science ,museum1 ,london1 and1 right1 two views of a similar pendulum 7#cg-e_/h inches long7 made in the late #aith century by ,edward ,kuebel1 ,washington1 ,d4,c41 which is no4 #caf1hgf in the collection of the ,u4,s4 ,national ,museum4 ,among a large number of pendulums tried by ,baily in ,london 7#ahbg-ahdj71 was one which resembles the reversible pendulum superficially1 but which is actually an invariable pendulum having knife edges at both ends4 ,the purpose was apparently economy1 since it is equivalent to two separate invariable pendulums4 ,this is the type of pendulum used on the ,u4,s4 ,exploring ,expedition of #ahch-ahdb4 ,it is not known what use was made of the ,kuebel pendulum4 ,repsold-,bessel ,reversible ,pendulum ,as we have noted1 ,bessel made determinations of gravity with a ball 7simple7 pendulum in the period #ahbe-ahbg and in #ahce at ,koenigsberg and ,berlin1 respectively4 ,in the memoir on his observations at ,koenigsberg1 he set forth the theory of the symmetrical compound pendulum with interchangeable knife edges4#db ,bessel demonstrated theoretically that if the pendulum were symmetrical with respect to its geometrical center1 if the times of swing about each axis were the same1 the effects of buoyancy and of air set in motion would be eliminated4 ,laplace had already shown that the knife edge must be regarded as a cylinder and not as a mere line of support4 ,bessel then showed that if the knife edges were equal cylinders1 their effects were eliminated by inverting the pendulum2 and if the knife edges were not equal cylinders1 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 positions4 ,bessel further showed that it is unnecessary to make the times of swing exactly equal for the two knife edges4 ,the simplified discussion for infinitely small oscillations in a vacuum is as follows3 ,if ,t#a and ,t#b are the times of swing about the knife edges1 and if h#a and h#b are distances of the knife edges from the center of gravity1 and if k is the radius of gyration about an axis through the center of gravity1 then from the equation of motion of a rigid body oscillating about a fixed axis under gravity 7,t#a7b pi#b7k#b 7h#a7b7_/;gh#a1 7,t#b7b pi#b7k#b 7h#b7b7_/;gh#b4 ,then 7h#a7,t#a7b - h#b7,t#b7b7_/7;h#a - h#b7 7pi#b_/;g77h#a h#b7 tau#b4 tau is then the time of swing of a simple pendulum of length h#a h#b4 ,if the difference ,t#a - ,t#b is sufficiently small1 tau 7h#a,t#a - h#b,t#b7_/7;h#a - h#b74 ,prior to its publication by ,bessel in #ahbh1 the formula for the time of swing of a simple pendulum of length h#a h#b in terms of ,t#a1 ,t#b had been given by ,c4 ,f4 ,gauss in a letter to ,h4 ,c4 ,schumacher dated ,november #bh1 #ahbd4dc ,the symmetrical compound pendulum with interchangeable knives1 for which ,bessel gave a posthumously published design and specifications1#dd has been called a reversible pendulum2 it may thereby be distinguished from ,kater's unsymmetrical convertible pendulum4 ,in #ahfa1 the ,swiss ,geodetic ,commission was formed1 and in one of its first sessions in #ahfb 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 ,europe4#de ,it was decided further to employ a reversible pendulum of ,bessel's design and to have it constructed by the firm of ,a4 ,repsold and ,sons1 ,hamburg4 ,it was also decided to make the first observations with the pendulum in ,geneva2 accordingly1 the ,repsold-,bessel pendulum 7fig4 #af7 was sent to ,prof4 ,e4 ,plantamour1 director of the observatory at ,geneva1 in the autumn of #ahfd4df ,the ,swiss reversible pendulum was about #efj mm4 in length 7distance between the knife edges7 and the time of swing was approximately #c_/d second4 ,at the extremities of the stem of the pendulum were movable cylindrical disks1 one of which was solid and heavy1 the other hollow and light4 ,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 disk4 ,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 comparator4 ,plantamour found it impracticable to adjust a disk until the times of swing about each knife edge were equal4 ,his colleague1 ,charles ,cellerier1#dg then showed that if 7,t#a - ,t#b7_/,t#a is sufficiently small so that one can neglect its square1 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 edges4 ,thus1 a role for the position of the center of gravity in the theory of the reversible pendulum1 which had been set forth earlier by ,bessel1 was discovered independently by ,cellerier for the ,swiss observers of pendulums4 ,in #ahff1 ,plantamour published an extensive memoir ,experiences faites a ,geneve avec le pendule a reversion4 ,another memoir1 published in #ahgb1 presented further results of determinations of gravity in ,switzerland4 ,plantamour was the first scientist in western ,europe to use a ,repsold-,bessel reversible pendulum and to work out methods for its employment4 ,the ,russian ,imperial ,academy of ,sciences acquired two ,repsold-,bessel pendulums1 and observations with them were begun in #ahfd by ,prof4 ,sawitsch1 ,university of ,st4 ,petersburg1 and others4#dh ,in #ahfi1 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 nos4 #d and #f 7#ahba74 ,during the transport of the ,russian apparatus to ,india1 the knives became rusted and the apparatus had to be reconditioned4 ,capt4 ,heaviside of the ,india ,survey observed with both pendulums at ,kew ,observatory1 near ,london1 in the spring of #ahgd1 after which the ,russian pendulums were sent to ,pulkowa 7,russia7 and were used for observations there and in the ,caucasus4 ,the introduction of the ,repsold-,bessel reversible pendulum for the determination of gravity was accompanied by the creation of the first international scientific association1 one for geodesy4 ,in #ahfa1 ,lt4 ,gen4 ,j4 ,j4 ,baeyer1 director of the ,prussian ,geodetic ,survey1 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 organization4#di ,in #ahfb1 invitations were sent to the various ,german states and to other states of central ,europe4 ,the first ,general ,conference of the association1 initially called ,die ,mittel-,europaeische ,gradmessung1 also ,l',association ,geodesique ,internationale1 was held from the #aeth to the #bb;d of ,october #ahfd in ,berlin4#ej ,the ,conference decided upon questions of organization3 a general conference was to be held ordinarily every three years2 a permanent commission initially consisting of seven members was to be the scientific organ of the association and to meet annually2 a central bureau was to be established for the reception1 publication1 and distribution of reports from the member states4 ,illustration3 ,figure #af4--;,,from ,a ,,design ,,left ,,by ,,bessel1 this portable apparatus was developed in #ahfb by the firm of ,repsold in ,hamburg1 whose founder had assisted ,bessel in the construction of his pendulum apparatus of #ahbf4 ,the pendulum is convertible1 but differs from ,kater's in being geometrically symmetrical and1 for this reason1 ,repsold's is usually called reversible4 ,just to the right of the pendulum is a standard scale4 ,to the left is a vertical comparator designed by ,repsold to measure the distance between the knife edges of the pendulum4 ,to make this measurement1 two micrometer microscopes which project horizontally through the comparator are alternately focused on the knife edges and on the standard scale4 ,under the topic ,astronomical ,questions1 the ,general ,conference of #ahfd resolved that there should be determinations of the intensity of gravity at the greatest possible number of points of the geodetic network1 and recommended the reversible pendulum as the instrument of observation4#ea ,at the second ,general ,conference1 in ,berlin in #ahfg1 on the basis of favorable reports by ,dr4 ,hirsch1 director of the observatory at ,neuchatel1 of ,swiss practice with the ,repsold-,bessel reversible pendulum1 this instrument was specifically recommended for determinations of gravity4#eb ,the title of the association was changed to ,die ,europaeische ,gradmessung2 in #ahhf1 it became ,die ,internationale ,erdmessung1 under which title it continued until ,world ,war ,i4 ,on ,april #a1 #ahff1 the ,central ,bureau of ,die ,europaeische ,gradmessung was opened in ,berlin under the presidency of ,baeyer1 and in #ahfh there was founded at ,berlin1 also under his presidency1 the ,royal ,prussian ,geodetic ,institute1 which obtained regular budgetary status on ,january #a1 #ahgj4 ,a reversible pendulum for the ,institute was ordered from ,a4 ,repsold and ,sons1 and it was delivered in the spring of #ahfi4 ,the ,prussian instrument was symmetrical geometrically1 as specified by ,bessel1 but different in form from the ,swiss and ,russian pendulums4 ,the distance between the knife edges was #a meter1 and the time of swing approximately #a second4 ,the ,prussian ,repsold-,bessel pendulum was swung at ,leipzig and other stations in central ,europe during the years #ahfi-ahgj by ,dr4 ,albrecht under the direction of ,dr4 ,bruhns1 director of the observatory at ,leipzig and chief of the astronomical section of the ,geodetic ,institute4 ,the results of these first observations appeared in a publication of the ,royal ,prussian ,geodetic ,institute in #ahga4ec ,results of observations with the ,russian ,repsold-,bessel pendulums were published by the ,imperial ,academy of ,sciences4 ,in #ahgb1 ,prof4 ,sawitsch reported the work for western ,europeans in ,les variations de la pesanteur dans les provinces occidentales de l',empire russe4#dh ,in ,november #ahgc1 the ,austrian ,geodetic ,commission received a ,repsold-,bessel reversible pendulum and on ,september #bd1 #ahgd1 ,prof4 ,theodor von ,oppolzer reported on observations at ,vienna and other stations to the ,fourth ,general ,conference of ,die ,europaeische ,gradmessung in ,dresden4#ed ,at the fourth session of the ,conference1 on ,september #bh1 #ahgd1 a ,special ,commission1 consisting of ,baeyer1 as chairman1 and ,bruhns1 ,hirsch1 von ,oppolzer1 ,peters1 and ,albrecht1 was appointed to consider 7under ,topic #c of the program73 ,observations for the determination of the intensity of gravity1 the question1 ,which ,pendulum-apparatuses are preferable for the determination of many points8 ,after the adoption of the ,repsold-,bessel reversible pendulum for gravity determinations in ,europe1 work in the field was begun by the ,u4,s4 ,coast ,survey under the superintendency of ,prof4 ,benjamin ,peirce4 ,there is mention in reports of observations with pendulums prior to ,peirce's direction to his son ,charles on ,november #cj1 #ahgb1 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 permit1 to take the field with a party4444#ee ,systematic and important gravity work by the ,survey was begun by ,charles ,sanders ,peirce4 ,upon receiving notice of his appointment1 the latter promptly ordered from the ,repsolds a pendulum similar to the ,prussian instrument4 ,since the firm of mechanicians was engaged in making instruments for observations of the transit of ,venus in #ahgd1 the pendulum for the ,coast ,survey could not be constructed immediately4 ,meanwhile1 during the years #ahgc-ahgd1 ,charles ,peirce conducted a party which made observations of gravity in the ,hoosac ,tunnel near ,north ,adams1 and at ,northampton and ,cambridge1 ,massachusetts4 ,the pendulums used were nonreversible1 invariable pendulums with conical bobs4 ,among them was a silver pendulum1 but similar pendulums of brass were used also4#ef ,illustration3 ,figure #ag4--,,repsold-,,bessel ,,reversible ,,pendulum apparatus as made in #ahge1 and used in the gravity work of the ,u4,s4 ,coast and ,geodetic ,survey4 ,continental geodesists continued to favor the general use of convertible pendulums and absolute determinations of gravity1 while their ,english colleagues had turned to invariable pendulums and relative determinations1 except for base stations4 ,perhaps the first important ,american contribution to gravity work was ,c4 ,s4 ,peirce's demonstration of the error inherent in the ,repsold apparatus through flexure of the stand4 ,illustration3 ,figure #ah4--;,,charles ,,sanders ,,peirce 7#ahci-aiad71 son of ,benjamin ,peirce1 ,perkins ,professor of ,astronomy and ,mathematics at ,harvard ,college4 ,c4 ,s4 ,peirce graduated from ,harvard in #ahei4 ,from #ahgc to #ahia1 as an assistant at the ,u4,s4 ,coast and ,geodetic ,survey1 he accomplished the important gravimetric work described in this article4 ,peirce was also interested in many other fields1 but above all in the logic1 philosophy1 and history of science1 in which he wrote extensively4 ,his greatest fame is in philosophy1 where he is regarded as the founder of pragmatism4 ,in #ahgd1 ,charles ,peirce expressed the desire to be sent to ,europe for at least a year1 beginning about ,march #a1 #ahge1 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 ,paris4#eg ,charles ,s4 ,peirce1 assistant1 ,u4,s4 ,coast ,survey1 sailed for ,europe on ,april #c1 #ahge1 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 gravity4 ,in ,england1 he conferred with ,maxwell1 ,stokes1 and ,airy concerning the theory and practice of research with pendulums4 ,in ,may1 he continued on to ,hamburg and obtained delivery from the ,repsolds of the pendulum for the ,coast ,survey 7fig4 #ag74 ,peirce then went to ,berlin and conferred with ,gen4 ,baeyer1 who expressed doubts of the stability of the ,repsold stand for the pendulum4 ,peirce next went to ,geneva1 where1 under arrangements with ,prof4 ,plantamour1 he swung the newly acquired pendulum at the observatory4#eh ,in view of ,baeyer's expressed doubts of the rigidity of the ,repsold stand1 ,peirce performed experiments to measure the flexure of the stand caused by the oscillations of the pendulum4 ,his method was to set up a micrometer in front of the pendulum stand and1 with a microscope1 to measure the displacement caused by a weight passing over a pulley1 the friction of which had been determined4 ,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 #j4b mm4 ,although ,peirce's measurements of flexure in ,geneva were not as precise as his later measurements1 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 pendulum4 ,the ,permanent ,commission of ,die ,europaeische ,gradmessung met in ,paris1 ,september #bj-bi1 #ahge4 ,in conjunction with this meeting1 there was held on ,september #ba a meeting of the ,special ,commission on the ,pendulum4 ,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 #bf1 #ahgd4ei ,gen4 ,baeyer stated that the distance of #a meter between the knife edges of the ,prussian ,repsold-,bessel pendulum made it unwieldy and unsuited for transport4 ,he declared that the instability of the stand also was a source of error4 ,accordingly1 ,gen4 ,baeyer expressed the opinion that absolute determinations of gravity should be made at a control station by a reversible pendulum hung on a permanent1 and therefore stable stand1 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 pendulum4 ,dr4 ,bruhns and ,dr4 ,peters agreed with ,gen4 ,baeyer2 however1 the ,swiss investigators1 ,prof4 ,plantamour and ,dr4 ,hirsch reported in defense of the reversible pendulum as a field instrument1 as did ,prof4 von ,oppolzer of ,vienna4 ,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 instrument4 ,peirce was present during these discussions by the members of the ,special ,commission1 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 neglected4 ,the observers who used the ,swiss and ,austrian ,repsold pendulums contended1 in opposition to ,peirce1 that the ,repsold stand was stable4 ,the outcome of these discussions was that the ,special ,commission reported to the ,permanent ,commission that the ,repsold-,bessel reversible pendulum1 except for some small changes1 satisfied all requirements for the determination of gravity4 ,the ,special ,commission proposed that the ,repsold pendulums of the several states be swung at the ,prussian ,eichungsamt in ,berlin where1 as ,peirce pointed out1 ,bessel had made his determination of the intensity of gravity with a ball pendulum in #ahce4 ,peirce was encouraged to swing the ,coast ,survey reversible pendulum at the stations in ,france1 ,england1 and ,germany where ,borda and ,cassini1 ,kater1 and ,bessel1 respectively1 had made historic determinations4 ,the ,permanent ,commission1 in whose sessions ,peirce also participated1 by resolutions adopted the report of the ,special ,commission on the ,pendulum4#fj ,during the months of ,january and ,february #ahgf1 ,peirce conducted observations in the ,grande ,salle du ,meridien at the observatory in ,paris where ,borda1 ,biot1 and ,capt4 ,edward ,sabine had swung pendulums early in the #aith century4 ,he conducted observations in ,berlin from ,april to ,june #ahgf and1 by experiment1 determined the correction for flexure to be applied to the value of gravity previously obtained with the ,prussian instrument4 ,subsequent observations were made at ,kew4 ,after his return to the ,united ,states on ,august #bf1 #ahgf1 ,peirce conducted experiments at the ,stevens ,institute in ,hoboken1 ,new ,jersey1 where he made careful measurements of the flexure of the stand by statical and dynamical methods4 ,in ,geneva1 he had secured the construction of a vacuum chamber in which the pendulum could be swung on a support which he called the ,geneva support4 ,at the ,stevens ,institute1 ,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 pendulum4 ,these experiments continued into #ahgh4fa ,meanwhile1 the ,permanent ,commission met ,october #e-aj1 #ahgf1 in ,brussels and continued the discussion of the pendulum4#fb ,gen4 ,baeyer reported on ,peirce's experiments in ,berlin to determine the flexure of the stand4 ,the difference of #j4ah mm4 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 stand4 ,dr4 ,hirsch1 speaking for the ,swiss survey1 and ,prof4 von ,oppolzer1 speaking for the ,austrian survey1 contended1 however1 that their stands possessed sufficient stability and that the results found by ,peirce applied only to the stands and bases investigated by him4 ,the ,permanent ,commission proposed further study of the pendulum4 ,the ,fifth ,general ,conference of ,die ,europaeische ,gradmessung was held from ,september #bg to ,october #b1 #ahgg1 in ,stuttgart4#fc ,peirce had instructions from ,supt4 ,patterson of the ,u4,s4 ,coast ,survey to attend this conference1 and on arrival presented a letter of introduction from ,patterson requesting that he1 ,peirce1 be permitted to participate in the sessions4 ,upon invitation from ,prof4 ,plantamour1 as approved by ,gen4 ,ibanez1 president of the ,permanent ,commission1 ,peirce had sent on ,july #ac1 #ahgg1 from ,new ,york1 the manuscript of a memoir titled ,de l',influence de la flexibilite du trepied sur l'oscillation du pendule a reversion4 ,this memoir and others by ,cellerier and ,plantamour confirming ,peirce's work were published as appendices to the proceedings of the conference4 ,as appendices to ,peirce's contribution were published also two notes by ,prof4 von ,oppolzer4 ,at the second session on ,september #bi1 #ahgg1 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 flexure1 ,peirce described his ,hoboken experiments4 ,during the discussions at ,stuttgart on the flexure of the ,repsold stand1 ,herve ,faye1 president of the ,bureau of ,longitudes1 ,paris1 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 phases4 ,this proposal was criticized by ,dr4 ,hirsch1 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 other4 ,the proposal of the double pendulum came up again at the meeting of the ,permanent ,commission at ,geneva in #ahgi4fd ,on ,february #ag1 #ahgi1 ,peirce had completed a paper ,on a ,method of ,swinging ,pendulums for the ,determination of ,gravity1 ,proposed by ,m4 ,faye4 ,in this paper1 ,peirce presented the results of an analytical mechanical investigation of ,faye's proposal4 ,peirce set up the differential equations1 found the solutions1 interpreted them physically1 and arrived at the conclusion that the suggestion of ,m4 ,faye 444 is as sound as it is brilliant and offers some peculiar advantages over the existing method of swinging pendulums4 ,in a report to ,supt4 ,patterson1 dated ,july #ahgi1 ,peirce stated3 ,i think it is important before making a new pendulum apparatus to experiment with ,faye's proposed method4#fe ,he wrote further3 ,the method proves to be perfectly sound in theory1 and as it would greatly facilitate the work it is probably destined eventually to prevail4 ,we must unfortunately leave to other surveys the merit of practically testing and introducing the new method1 as our appropriations are insufficient for us to maintain the leading position in this matter1 which we otherwise might take4 ,copies of the published version of ,peirce's remarks were sent to ,europe4 ,at a meeting of the ,academy of ,sciences in ,paris on ,september #a1 #ahgi1 ,faye presented a report on ,peirce's findings4#ff ,the ,permanent ,commission met ,september #af-bj1 #ahgi1 in ,geneva4 ,at the third session on ,september #ai1 by action of ,gen4 ,baeyer1 copies of ,peirce's paper on ,faye's proposed method of swinging pendulums were distributed4 ,dr4 ,hirsch again commented adversely on the proposal1 but moved that the question be investigated and reported on at the coming ,general ,conference4 ,the ,permanent ,commission accepted the proposal of ,dr4 ,hirsch1 and ,prof4 ,plantamour was named to report on the matter at the ,general ,conference4 ,at ,plantamour's request1 ,charles ,cellerier was appointed to join him1 since the problem essentially was a theoretical one4 ,the ,sixth ,general ,conference of ,die ,europaeische ,gradmessung met ,september #ac-af1 #ahhj1 in ,munich4#fg ,topic ,,iii1 part #g of the program was entitled ,on ,determinations of ,gravity through pendulum observations4 ,which construction of a pendulum apparatus corresponds completely to all requirements of science8 ,special report on the pendulum4 ,the conference received a memoir by ,cellerier#fh on the theory of the double pendulum and a report by ,plantamour and ,cellerier4#fi ,cellerier's mathematical analysis began with the equations of ,peirce and used the latter's notation as far as possible4 ,his general discussion included the results of ,peirce1 but he stated that the difficulties to be overcome did not justify the employment of the double pendulum4 ,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 weight4 ,this method of correcting for flexure was recommended to the ,general ,conference by ,plantamour and ,cellerier in their joint report4 ,at the fourth session of the conference on ,september #af1 #ahhj1 the problem of the pendulum was discussed and1 in consequence1 a commission consisting of ,faye1 ,helmholtz1 ,plantamour 7replaced in #ahhb by ,hirsch71 and von ,oppolzer was appointed to study apparatus suitable for relative determinations of gravity4 ,the ,permanent ,commission met ,september #aa-ae1 #ahhb1 at ,the ,hague1#gj and at its last session appointed ,prof4 von ,oppolzer to report to the ,seventh ,general ,conference on different forms of apparatus for the determination of gravity4 ,the ,seventh ,conference met ,october #ae-bd1 #ahhc1 in ,rome1#ga and1 at its eighth session1 on ,october #bb1 received a comprehensive1 critical review from ,prof4 von ,oppolzer entitled ,ueber die ,bestimmung der ,schwere mit ,hilfe verschiedener ,apparate4#gb ,von ,oppolzer especially expounded the advantages of the ,bessel reversible pendulum1 which compensated for air effects by symmetry of form if the times of swing for both positions were maintained between the same amplitudes1 and compensated for irregular knife edges by making them interchangeable4 ,prof4 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 phases1 but that the proposal was not practicable4 ,he concluded that for absolute determinations of gravity1 the ,bessel reversible pendulum was highly appropriate if one swung two exemplars of different weight from the same stand for the elimination of flexure4 ,prof4 von ,oppolzer's important report recognized that absolute determinations were less accurate than relative ones1 and should be conducted only at special places4 ,the discussions initiated by ,peirce's demonstration of the flexure of the ,repsold stand resulted1 finally1 in the abandonment of the plan to make absolute determinations of gravity at all stations with the reversible pendulum4 ,illustration3 ,figure #ai4--,,three ,,pendulums ,,used ,,in ,,early ,,work at the ,u4,s4 ,coast and ,geodetic ,survey4 ,shown on the left is the ,peirce invariable2 center1 the ,peirce reversible2 and1 right1 the ,repsold reversible4 ,peirce designed the cylindrical pendulum in #ahha-ahhb to study the effect of air resistance according to the theory of ,g4 ,g4 ,stokes on the motion of a pendulum in a viscous field4 ,three examples of the ,peirce pendulums are in the ,u4,s4 ,national ,museum4 ,peirce and ,defforges ,invariable1 ,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 ,kew1 the observatory in ,paris1 and the ,smithsonian ,institution in ,washington1 ,d4,c41 but also at stations in the field4 ,an invariable pendulum with a single knife edge1 however1 is adequate for relative determinations4 ,as we have seen1 such invariable pendulums had been used by ,bouguer and ,kater1 and after the experiences with the ,repsold apparatus had been recommended again by ,baeyer for relative determinations4 ,but an invariable pendulum is subject to uncontrollable changes of length4 ,peirce proposed to detect such changes in an invariable pendulum in the field by combining the invariable and reversible principles4 ,he explained his proposal to ,faye in a letter dated ,july #bc1 #ahhj1 and he presented it on ,september #af1 #ahhj1 at the fourth session of the sixth ,general ,conference of ,die ,europaeische ,gradmessung1 in ,munich4#gc ,as recorded in the ,proceedings of the ,conference1 ,peirce wrote3 ,but ,i obviate it in making my pendulum both invariable and reversible4 ,every alteration of the pendulum will be revealed immediately by the change in the difference of the two periods of oscillation in the two positions4 ,once discovered1 it will be taken account of by means of new measures of the distance between the two supports4 ,peirce added that it seemed to him that if the reversible pendulum perhaps is not the best instrument to determine absolute gravity1 it is1 on condition that it be truly invariable1 the best to determine relative gravity4 ,peirce further stated that he would wish that the pendulum be formed of a tube of drawn brass with heavy plugs of brass equally drawn4 ,the cylinder would be terminated by two hemispheres2 the knives would be attached to tongues fixed near the ends of the cylinder4 ,during the years #ahha and #ahhb1 four invariable1 reversible pendulums were made after the design of ,peirce at the office of the ,u4,s4 ,coast and ,geodetic ,survey in ,washington1 ,d4,c4 ,the report of the superintendent for the year #ahhj-ahha states3 ,a new pattern of the reversible pendulum has been invented1 having its surface as nearly as convenient in the form of an elongated ellipsoid4 ,three of these instruments have been constructed1 two having a distance of one meter between the knife edges and the third a distance of one yard4 ,it is proposed to swing one of the meter pendulums at a temperature near #cb deg4 ,f4 at the same time that the yard is swung at #fj deg4 ,f41 in order to determine anew the relation between the yard and the meter4#gd ,the report for #ahha-ahhb mentions four of these ,peirce pendulums4 ,a description of the ,peirce invariable1 reversible pendulums was given by ,assistant ,e4 ,d4 ,preston in ,determinations of ,gravity and the ,magnetic ,elements in ,connection with the ,united ,states ,scientific ,expedition to the ,west ,coast of ,africa1 #ahhi-ij4ge ,the invariable1 reversible pendulum1 ,peirce no4 #d1 now preserved in the ,smithsonian ,institution's ,museum of ,history and ,technology 7fig4 #cd71 may be taken as typical of the meter pendulums3 ,in the same memoir1 ,preston gives the diameter of the tube as #fc4g mm41 thickness of tube #a4e mm41 weight #aj4fhj kilograms1 and distance between the knives #a4jjj meter4 ,the combination of invariability and reversibility in the ,peirce pendulums was an innovation for relative determinations4 ,indeed1 the combination was criticized by ,maj4 ,j4 ,herschel1 ,r4,e41 of the ,indian ,survey1 at a conference on gravity held in ,washington in ,may #ahhb 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 pendulums4 ,these three pendulums have been designated as nos4 #d1 #f 7#ahba71 and #aa4gf ,illustration3 ,figure #bj4--,,support ,,for ,,the ,,peirce ,,pendulum1 #ahhi4 ,much of the work of ,c4 ,s4 ,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 pendulum4 ,he showed that the popular ,bessel-,repsold apparatus was subject to such an error4 ,his own pendulums were swung from a simple but rugged wooden frame to which a hardened steel bearing was fixed4 ,another novel characteristic of the ,peirce pendulums was the mainly cylindrical form4 ,prof4 ,george ,gabriel ,stokes1 in a paper ,on the ,effect of the ,internal ,friction of ,fluids on the ,motion of ,pendulums#gg that was read to the ,cambridge ,philosophical ,society on ,december #i1 #ahej1 had solved the hydrodynamical equations to obtain the resistance to the motions of a sphere and a cylinder in a viscous fluid4 ,peirce had studied the effect of viscous resistance on the motion of his ,repsold-,bessel pendulum1 which was symmetrical in form but not cylindrical4 ,the mainly cylindrical form of his pendulums 7fig4 #ai7 permitted ,peirce to predict from ,stokes' theory the effect of viscosity and to compare the results with experiment4 ,his report of ,november #bj1 #ahhi1 in which he presented the comparison of experimental results with the theory of ,stokes1 was not published4#gh ,peirce used his pendulums in #ahhc to establish a station at the ,smithsonian ,institution that was to serve as the base station for the ,coast and ,geodetic ,survey for some years4 ,pendulum ,peirce no4 #a was swung at ,washington in #ahha and was then taken by the party of ,lieutenant ,greely1 ,u4,s4,a41 on an expedition to ,lady ,franklin ,bay where it was swung in #ahhb at ,fort ,conger1 ,grinnell ,land1 ,canada4 ,peirce nos4 #b and #c were swung by ,peirce in #ahhb at ,washington1 ,d4,c42 ,hoboken1 ,new ,jersey2 ,montreal1 ,canada2 and ,albany1 ,new ,york4 ,assistant ,preston took ,peirce no4 #c on a ,u4,s4 eclipse expedition to the ,caroline ,islands in #ahhc4 ,peirce in #ahhe swung pendulums nos4 #b and #c at ,ann ,arbor1 ,michigan2 ,madison1 ,wisconsin2 and ,ithaca1 ,new ,york4 ,assistant ,preston in #ahhg swung ,peirce nos4 #c and #d at stations in the ,hawaiian ,islands1 and in #ahij he swung ,peirce nos4 #c and #d at stations on the west coast of ,africa4#gi ,the new pattern of pendulum designed by ,peirce was also adopted in ,france1 after some years of experience with a ,repsold-,bessel pendulum4 ,peirce in #ahge had swung his ,repsold-,bessel pendulum at the observatory in ,paris1 where ,borda and ,cassini1 and ,biot1 had made historic observations and where ,sabine also had determined gravity by comparison with ,kater's value at ,london4 ,during the spring of #ahhj1 ,peirce made studies of the supports for the pendulums of these earlier determinations and calculated corrections to those results for hydrodynamic effects1 viscosity1 and flexure4 ,on ,june #ad1 #ahhj1 ,peirce addressed the ,academy of ,sciences1 ,paris1 on the value of gravity at ,paris1 and compared his results with the corrected results of ,borda and ,biot and with the transferred value of ,kater4#hj ,in the same year the ,french ,geographic ,service of the ,army acquired a ,repsold-,bessel reversible pendulum of the smaller type1 and ,defforges conducted experiments with it4#ha ,he introduced the method of measuring flexure from the movement of interference fringes during motion of the pendulum4 ,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 support1 on the condition that one uses the statical coefficient4 ,defforges developed a theory for the employment of two similar pendulums of the same weight1 but of different length1 and hung by the same knives4 ,this theory eliminated the flexure of the support and the curvature of the knives from the reduction of observations4 ,pendulums of #a-meter and of #a_/b-meter distance between the knife edges were constructed from ,defforges' design by ,brunner ,brothers in ,paris 7fig4 #ba74 ,these ,defforges pendulums were cylindrical in form with hemispherical ends like the ,peirce pendulums1 and were hung on knives that projected from the sides of the pendulum1 as in some unfinished ,gautier pendulums designed by ,peirce in #ahhc in ,paris4 ,illustration3 ,figure #ba4--,,reversible ,,pendulum ,,apparatus of ,defforges1 as constructed by ,brunner1 ,paris1 about #ahhg4 ,the clock and telescope used to observe coincidences are not shown4 ,the telescope shown is part of an interferometer used to measure flexure of the support4 ,one mirror of the interferometer is attached to the pendulum support2 the other to the separate masonry pillar at the left4 ,illustration3 ,figure #bb4--;,,because ,,of ,,the ,,greater ,,simplicity of its use1 the invariable pendulum superseded the convertible pendulum towards the end of the #aith century1 except at various national base stations 7,kew1 ,paris1 ,potsdam1 ,washington1 ,d4,c41 etc474 ,shown here are1 right to left1 a pendulum of the type used by ,peirce at the ,hoosac ,tunnel in #ahgc-gd1 the ,mendenhall #a_/b-second pendulum of #ahij1 and the pendulum designed by ,peirce in #ahha-ahhb4 ,illustration3 ,figure #bc4--,,the ,,overall ,,size of portable pendulum apparatus was greatly reduced with the introduction of this #a_/b-second apparatus in #ahhg1 by the ,austrian military officer1 ,robert von ,sterneck4 ,used with a vacuum chamber not shown here1 the apparatus is only about #b feet high4 ,coincidences are observed by the reflection of a periodic electric spark in two mirrors1 one on the support and the other on the pendulum itself4 ,illustration3 ,figure #bd4--,,thomas ,c4 ,,mendenhall 7#ahda-aibd74 ,although largely self-educated1 he became the first professor of physics and mechanics at the ,ohio ,agricultural and ,mechanical ,college 7later ,ohio ,state ,university71 and was subsequently connected with several other universities4 ,in #ahgh1 while teaching at the ,tokyo ,imperial ,university in ,japan1 he made gravity measurements between ,tokyo and ,fujiyama from which he calculated the mean density of the earth4 ,while superintendent of the ,u4,s4 ,coast and ,geodetic ,survey1 #ahhi-id1 he developed the pendulum apparatus which bears his name4 ,von ,sterneck and ,mendenhall ,pendulums ,while scientists who had used the ,repsold-,bessel pendulum apparatus discussed its defects and limitations for gravity surveys1 ,maj4 ,robert von ,sterneck of ,austria-,hungary began to develop an excellent apparatus for the rapid determination of relative values of gravity4#hb ,maj4 von ,sterneck's apparatus contained a nonreversible pendulum #a_/d-meter in length1 and #a_/b-second time of swing4 ,the pendulum was hung by a single knife edge1 which rested on a plate that was supported by a tripod4 ,the pendulum was swung in a chamber from which air was exhausted and which could be maintained at any desired temperature4 ,times of swing were determined by the observation of coincidences of the pendulum with chronometer signals4 ,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 rest4 ,the chronometer signals worked a relay that gave a horizontal spark which was reflected into the telescope from the mirrors4 ,when the pendulum was at rest1 the image of the spark in both mirrors appeared on the horizontal cross wire in the telescope1 and during oscillation of the pendulum the two images appeared in that position upon coincidence4 ,in view of the reduced size of the pendulum1 the chamber in which it was swung was readily portable1 and with an improved method of observing coincidences1 relative determinations of gravity could be made with rapidity and accuracy4 ,by #ahhg ,maj4 von ,sterneck had perfected his apparatus1 and it was widely adopted in ,europe for relative determinations of gravity4 ,he used his apparatus in extensive gravity surveys and also applied it in the silver mines in ,saxony and ,bohemia1 by the previously described methods of ,airy1 for investigations into the internal constitution of the earth4 ,on ,july #a1 #ahhi1 ,thomas ,corwin ,mendenhall became superintendent of the ,u4,s4 ,coast and ,geodetic ,survey4 ,earlier1 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 ,tokyo4 ,supt4 ,mendenhall1 with the cooperation of members of his staff in ,washington1 designed a new pendulum apparatus of the ,von ,sterneck type1 and in ,october #ahij he ordered construction of the first model4#hc ,like the ,von ,sterneck apparatus1 the ,mendenhall pendulum apparatus employed a nonreversible1 invariable pendulum #a_/d-meter in length and of slightly more than #a_/b-second in time of swing4 ,initially1 the knife edge was placed in the head of the pendulum and hung on a fixed plane support1 but after some experimentation ,mendenhall attached the plane surface to the pendulum and hung it on a fixed knife edge4 ,an apparatus was provided with a set of three pendulums1 so that if discrepancies appeared in the results1 the pendulum at fault could be detected4 ,there was also a dummy pendulum which carried a thermometer4 ,a pendulum was swung in a receiver in which the pressure and temperature of the air were controlled4 ,the time of swing was measured by coincidences with the beat of a chronometer4 ,the coincidences were determined by an optical method with the aid of a flash apparatus4 ,illustration3 ,figure #be4--,,mendenhall',s #a_/d-,,meter 7#a_/b-,,second7 ,,apparatus4 ,shown on the left is the flash apparatus and1 on the right1 the vacuum chamber within which the pendulum is swung4 ,the flash apparatus consists of a kerosene lantern and a telescope1 mounted on a box containing an electromagnetically operated shutter4 ,the operation of the shutter is controlled by a chronograph 7not shown71 so that it emits a slit of light at regular intervals4 ,the telescope is focused on two mirrors within the apparatus1 one fixed1 the other attached to the top of the pendulum4 ,it is used to observe the reflection of the flashes from these mirrors4 ,when the two reflections are aligned1 a coincidence is marked on the chronograph tape4 ,the second telescope attached to the bottom of the vacuum chamber is for observing the amplitude of the pendulum swing4 ,the flash apparatus was contained in a light metal box which supported an observing telescope and which was mounted on a stand4 ,within the box was an electromagnet whose coils were connected with a chronometer circuit and whose armature carried a long arm that moved two shutters1 in both of which were horizontal slits of the same size4 ,the shutters were behind the front face of the box1 which also had a horizontal slit4 ,a flash of light from an oil lamp or an electric spark was emitted from the box when the circuit was broken1 but not when it was closed4 ,when the circuit was broken a spring caused the arm to rise1 and the shutters were actuated so that the three slits came into line and a flash of light was emitted4 ,a small circular mirror was set in each side of the pendulum head1 so that from either face of the pendulum the image of the illuminated slit could be reflected into the field of the observing telescope4 ,a similar mirror was placed parallel to these two mirrors and rigidly attached to the support4 ,the chronometer signals broke the circuit1 causing the three slits momentarily to be in line1 and when the images of the slit in the two mirrors coincided1 a coincidence was observed4 ,a coincidence occurred whenever the pendulum gained or lost one oscillation on the beat of the chronometer4 ,the relative intensity of gravity was determined by observations with the first ,mendenhall apparatus at ,washington1 ,d4,c41 at stations on the ,pacific ,coast and in ,alaska1 and at the ,stevens ,institute1 ,hoboken1 ,new ,jersey1 between ,march and ,october #ahia4 ,illustration3 ,figure #bf4--,,vacuum ,,receiver within which the ,mendenhall pendulum is swung4 ,the pressure is reduced to about #ej mm4 to reduce the disturbing effect of air resistance4 ,when the apparatus is sealed1 the pendulum is lifted on the knife edge by the lever q and is started to swing by the lever r4 ,the arc of swing is only about #a deg4 ,the stationary mirror is shown at g4 ,the pendulum shown in outline in the center1 is only about #i4g inches long4 ,under ,supt4 ,mendenhall's direction a smaller1 #a_/d-second1 pendulum apparatus was also constructed and tested1 but did not offer advantages over the #a_/b-second apparatus1 which therefore continued in use4 ,in accordance with ,peirce's theory of the flexure of the stand under oscillations of the pendulum1 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 ,survey4 ,initially1 a statical method was used1 but during #aijh-aiji members of the ,survey adapted the ,michelson interferometer for the determinations of flexure during oscillations from the shift of fringes4#hd ,the first ,mendenhall pendulums were made of bronze1 but about #aibj invar was chosen because of its small coefficient of expansion4 ,about #aicj1 ,lt4 ,e4 ,j4 ,brown of the ,coast and ,geodetic ,survey made significant improvements in the ,mendenhall apparatus1 and the new form came to be known as the ,brown ,pendulum ,apparatus4#he ,illustration3 ,figure #bg4--,,the ,,michelson ,,interferometer4 ,the horizontal component of the force acting on the knife edge through the swinging pendulum causes the support to move in unison with the pendulum1 and thereby affects the period of the oscillation4 ,this movement is the so-called flexure of the pendulum support1 and must be taken into account in the most accurate observations4 ,in #aijg1 the ,michelson interferometer was adapted to this purpose by the ,u4,s4 ,coast and ,geodetic ,survey4 ,as shown here1 the interferometer1 resting on a wooden beam1 is introduced into the path of a light beam reflected from a mirror on the vacuum chamber4 ,movement of that mirror causes a corresponding movement in the interference fringes in the interferometer1 which can be measured4 ,the original ,von ,sterneck apparatus and that of ,mendenhall provided for the oscillation of one pendulum at a time4 ,after the adoption of the ,von ,sterneck pendulum in ,europe1 there were developed stands on which two or four pendulums hung at the same time4 ,this procedure provided a convenient way to observe more than one invariable pendulum at a station for the purpose of detecting changes in length4 ,prof4 ,m4 ,haid of ,karlsruhe in #ahif described a four-pendulum apparatus1#hf and ,dr4 ,schumann of ,potsdam subsequently described a two-pendulum apparatus4#hg ,illustration3 ,figure #bh4--;,,apparatus ,,which ,,was ,,developed ,,in #aibi by the ,gulf ,research and ,development ,company1 ,harmarville1 ,pennsylvania4 ,it was designed to achieve an accuracy within one ten-millionth of the true value of gravity1 and represents the extreme development of pendulum apparatus for relative gravity measurement4 ,the pendulum was designed so that the period would be a minimum4 ,the case 7the top is missing in this photograph7 is dehumidified and its temperature and electrostatic condition are controlled4 ,specially designed pendulum-lifting and -starting mechanisms are used4 ,the problem of flexure of the case is overcome by the ,faye-,peirce method 7see text7 in which two dynamically matched pendulums are swung simultaneously1 #ahj deg4 apart in phase4 ,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 stand4 ,this method of determining the correction for flexure was a development from a ,wippverfahren invented at the ,geodetic ,institute in ,potsdam4 ,a dynamometer was used to impart periodic impulses to the stand1 and the effect was observed upon a pendulum initially at rest4 ,refinements of this method led to the development of a method used by ,lorenzoni in #ahhe-ahhf to determine the flexure of the stand by action of an auxiliary pendulum upon the principal pendulum4 ,dr4 ,schumann1 in #ahii1 gave a mathematical theory of such determinations1#hh and in his paper cited the mathematical methods of ,peirce and ,cellerier for the theory of ,faye's proposal at ,stuttgart in #ahgg to swing two similar pendulums on the same support with equal amplitudes and in opposite phases4 ,illustration3 ,figure #bi4--,,the ,,gulf ,,pendulum is about #aj4g inches long1 and has a period of 4#hi second4 ,it is made of fused quartz which is resistant to the influence of temperature change and to the earth's magnetism4 ,quartz pendulums are subject to the influence of electrostatic charge1 and provision is made to counteract this through the presence of a radium salt in the case4 ,the bearings are made of ,pyrex glass4 ,in #aijb1 ,dr4 ,p4 ,furtwaengler#hi presented the mathematical theory of coupled pendulums in a paper in which he referred to ,faye's proposal of #ahgg and reported that the difficulties predicted upon its application had been found not to occur4 ,finally1 during the gravity survey of ,holland in the years #aiac-aiba1 in view of instability of supports caused by the mobility of the soil1 ,f4 ,a4 ,vening ,meinesz adopted ,faye's proposed method of swinging two pendulums on the same support4#ij ,the observations were made with the ordinary ,stueckrath apparatus1 in which four ,von ,sterneck pendulums swung two by two in planes perpendicular to each other4 ,this successful application of the method--which had been proposed by ,faye and had been demonstrated theoretically to be sound by ,peirce1 who also published a design for its application--was rapidly followed for pendulum apparatus for relative determinations by ,potsdam1#ia ,cambridge 7,england71#ib ,gulf ,oil and ,development ,company1#ic and the ,dominion ,observatory at ,ottawa4#id ,heiskanen and ,vening ,meinesz state3 ,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 phases2 it is clear then the flexure is zero4#ie ,in view of the fact that the symmetrical reversible pendulum is named for ,bessel1 who created the theory and a design for its application by ,repsold1 it appears appropriate to call the method of eliminating flexure by swinging two pendulums on the same support the ,faye-,peirce method4 ,its successful application was made possible by ,maj4 von ,sterneck's invention of the short1 #a_/d-meter pendulum4 ,illustration3 ,figure #cj4--,,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 gravity1 but are somehow compensated for4 ,the most satisfactory solution to this still unanswered question has been the theory of isostasy1 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 parts1 as they float on the earth's fluid core4 ,the metals of different density floating in mercury in this diagram illustrate isostasy according to the theory of ,pratt and ,hayford4 ,absolute ,value of ,gravity at ,potsdam ,the development of the reversible pendulum in the #aith century culminated in the absolute determination of the intensity of gravity at ,potsdam by ,kuehnen and ,furtwaengler of the ,royal ,prussian ,geodetic ,institute1 which then became the world base for gravity surveys4#if ,we have previously seen that in #ahfi the ,geodetic ,institute--founded by ,lt4 ,gen4 ,baeyer--had acquired a ,repsold-,bessel reversible pendulum which was swung by ,dr4 ,albrecht under the direction of ,dr4 ,bruhns4 ,dissatisfaction with this instrument was expressed by ,baeyer in #ahge to ,charles ,s4 ,peirce1 who then1 by experiment and mathematical analysis of the flexure of the stand under oscillations of the pendulum1 determined that previously reported results with the ,repsold apparatus required correction4 ,dr4 ,f4 ,r4 ,helmert1 who in #ahhg succeeded ,baeyer as director of the ,institute1 secured construction of a building for the ,institute in ,potsdam1 and under his direction the scientific study of the intensity of gravity was pursued with vigor4 ,in #ahid1 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 stiffness4 ,dr4 ,kuehnen of the ,institute discovered that the departure from expectations was the result of the flexure of the pendulum staff itself during oscillations4#ig ,peirce1 in #ahhc1 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 staff4#ih ,by experiment1 he also found an even greater flexure for the ,repsold pendulum4 ,in order to eliminate this source of error1 ,peirce designed a pendulum with knives that extended from each side of the cylindrical staff1 and he received authorization from the superintendent of the ,coast and ,geodetic ,survey to arrange for the construction of such pendulums by ,gautier in ,paris4 ,peirce1 who had made his plans in consultation with ,gautier1 was called home before the pendulums were completed1 and these new instruments remained undelivered4 ,in a memoir titled ,effect of the flexure of a pendulum upon its period of oscillation1#ii ,peirce determined analytically the effect on the period of a pendulum with a single elastic connection between two rigid parts of the staff4 ,thus1 ,peirce discovered experimentally the flexure of the staff and derived for a simplified case the effect on the period4 ,it is not known if he ever found the integrated effect of the continuum of elastic connections in the pendulum4 ,lorenzoni1 in #ahif1 offered a solution to the problem1 and ,almansi1 in #ahii1 gave an extended analysis4 ,after the independent discovery of the problem at the ,geodetic ,institute1 ,dr4 ,helmert took up the problem and criticized the theories of ,peirce and ,lorenzoni4 ,he then presented his own theory of flexure in a comprehensive memoir4#ajj ,in view of the previous neglect of the flexure of the pendulum staff in the reduction of observations1 ,helmert directed that the ,geodetic ,institute make a new absolute determination of the intensity of gravity at ,potsdam4 ,for this purpose1 ,kuehnen and ,furtwaengler used the following reversible pendulums which had been constructed by the firm of ,a4 ,repsold and ,sons in ,hamburg3 #a4 ,the seconds pendulum of the ,geodetic ,institute procured in #ahfi4 #b4 ,a seconds pendulum from the ,astronomical ,observatory1 ,padua4 #c4 ,a heavy1 seconds pendulum from the ,imperial and ,royal ,military-,geographical ,institute1 ,vienna4 #d4 ,a light1 seconds pendulum from the ,imperial and ,royal ,military-,geographical ,institute4 #e4 ,a #a_/b-second1 reversible pendulum of the ,geodetic ,institute procured in #ahib4 ,work was begun in #ahih1 and in #aijf ,kuehnen and ,furtwaengler published their monumental memoir1 ,bestimmung der ,absoluten ,groesze der ,schwerkraft zu ,potsdam mit ,reversionspendeln4 ,the acceleration of gravity in the pendulum room of the ,geodetic ,institute was determined to be #iha4bgd - #j4jjc cm_/sec#b4 ,in view of the exceptionally careful and thorough determination at the ,institute1 ,potsdam was accepted as the world base for the absolute value of the intensity of gravity4 ,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 7,t#a7b_/7,t#b7b g#b_/;g#a4 ,thus1 in #aijj1 ,assistant ,g4 ,r4 ,putnam of the ,coast and ,geodetic ,survey swung ,mendenhall pendulums at the ,washington base and at ,potsdam1 and by transfer from ,potsdam determined the intensity of gravity at the ,washington base to be #ihj4aab cm_/sec#b4aja ,in #aicc1 ,lt4 ,e4 ,j4 ,brown made comparative measurements with improved apparatus and raised the value at the ,washington base to #ihj4aah cm_/sec#b4ajb ,in view of discrepancies between the results of various relative determinations1 the ,coast and ,geodetic ,survey in #aibh requested the ,national ,bureau of ,standards to make an absolute determination for ,washington4 ,heyl and ,cook used reversible pendulums made of fused silica having a period of approximately #a second4 ,their result1 published in #aicf1 was interpreted to indicate that the value at ,potsdam was too high by #bj parts in #a million4#ajc ,this estimate was lowered slightly by ,sir ,harold ,jeffreys of ,cambridge1 ,england1 who recomputed the results of ,heyl and ,cook by different methods4#ajd ,illustration3 ,figure #ca4--,,map ,,showing ,,the ,,distribution of gravity stations throughout the ,united ,states as of ,december #aijh4 ,illustration3 ,figure #cb4--,,map ,,showing ,,the ,,distribution of gravity stations throughout the ,united ,states in #aibc4 ,in #aici1 ,j4 ,s4 ,clark published the results of a determination of gravity with pendulums of a non-ferrous ,y-alloy#aje at the ,national ,physical ,laboratory at ,teddington1 ,england1 and1 after recomputation of results by ,jeffreys1 the value was found to be #ab4h parts in #a million less than the value obtained by transfer from ,potsdam4 ,dr4 ,hugh ,l4 ,dryden of the ,national ,bureau of ,standards1 and ,dr4 ,a4 ,berroth of the ,geodetic ,institute at ,potsdam1 have recomputed the ,potsdam data by different methods of adjustment and concluded that the ,potsdam value was too high by about #ab parts in a million4#ajf ,determination of gravity at ,leningrad by ,russian scientists likewise has indicated that the #aijf ,potsdam value is too high4 ,in the light of present information1 it appears justifiable to reduce the ,potsdam value of #iha4bgd by 4#jac cm_/sec#b for purposes of comparison4 ,if the ,brown transfer from ,potsdam in #aicc was taken as accurate1 the value for the ,washington base would be #ihj4aje cm_/sec#b4 ,in this connection1 it is of interest to note that the value given by ,charles ,s4 ,peirce for the comparable ,smithsonian base in ,washington1 as determined by him from comparative methods in the #ahhj's and reported in the ,annual ,report of the ,superintendent of the ,coast and ,geodetic ,survey for the year #ahij-ahia1 was #ihj4ajag cm_/sec#b4ajg ,this value would appear to indicate that ,peirce's pendulums1 observations1 and methods of reduction of data were not inferior to those of the scientists of the ,royal ,prussian ,geodetic ,institute at ,potsdam4 ,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 ,,ii4 ,in a paper published in ,june #aieg1 ,a4 ,h4 ,cook1 ,metrology ,division1 ,national ,physical ,laboratory1 ,teddington1 ,england1 stated3 ,at present about a dozen new absolute determinations are in progress or are being planned4 ,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 ,metrology1 ,leningrad 7,v ,n ,i ,i ,m7 and are planned at ,potsdam4 ,a method using a very long pendulum was tried out in ,russia about #aiaj and again more recently and there are plans for similar work in ,finland4 ,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 pressure4 ,similar experiments have been completed in ,leningrad and are in progress at the ,physikalisch-,technische ,bundesanstalt 7,brunswick7 and at the ,national ,research ,council 7,ottawa71 and analogous experiments are being prepared at the ,national ,physical ,laboratory and at the ,national ,bureau of ,standards4 ,finally1 ,professor ,medi1 ,director of the ,istituto ,nazionale di ,geofisica 7,rome71 is attempting to measure the focal length of the paraboloidal surface of a liquid in a rotating dish4#ajh ,application of ,gravity ,surveys ,we have noted previously that in the ancient and early modern periods1 the earth was presupposed to be spherical in form4 ,determination of the figure of the earth consisted in the measurement of the radius by the astronomical-geodetic method invented by ,eratosthenes4 ,since the earth was assumed to be spherical1 gravity was inferred to be constant over the surface of the earth4 ,this conclusion appeared to be confirmed by the determination of the length of the seconds pendulum at various stations in ,europe by ,picard and others4 ,the observations of ,richer in ,south ,america1 the theoretical discussions of ,newton and ,huygens1 and the measurements of degrees of latitude in ,peru and ,sweden demonstrated that the earth is an oblate spheroid4 ,illustration3 ,figure #cc4--;,,gravity ,,characteristics ,,of ,,the ,,globe4 ,deductions as to the distribution of matter in the earth can be made from gravity measurements4 ,this globe shows worldwide variations in gravity as they now appear from observations at sea 7in submarines7 as well as on land4 ,it is based on data from the ,institute of ,geodesy at ,ohio ,state ,university4 ,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 earth4 ,accordingly1 determinations of the intensity of gravity became of value to the geodesist as a means of determining the figure of the earth4 ,newton1 on the basis of the meager data available to him1 calculated the ellipticity of the earth to be #a_/bcj 7the ellipticity is defined by 7a-b7_/a1 where a is the equatorial radius and b the polar radius74 ,observations of the intensity of gravity were made on the historic missions to ,peru and ,sweden4 ,bouguer and ,la ,condamine found that at the equator at sea level the seconds pendulum was #a4bf ,paris-lines shorter than at ,paris4 ,maupertuis found that in northern ,sweden a certain pendulum clock gained #ei4a seconds per day on its rate in ,paris4 ,then ,clairaut1 from the assumption that the earth is a spheroid of equilibrium1 derived a theorem from which the ellipticity of the earth can be derived from values of the intensity of gravity4 ,illustration3 ,figure #cd4--;,,an ,,exhibit ,,of ,,gravity ,,apparatus at the ,smithsonian ,institution4 ,suspended on the wall1 from left to right1 are the invariable pendulums of ,mendenhall 7#a_/b-second71 ,peirce 7#ahgc-ahgd71 and ,peirce 7#ahha-ahhb72 the double pendulum of ,edward ,kuebel 7see fig4 #ae1 p4 #cai71 and the reversible pendulum of ,peirce4 ,on the display counter1 from left to right1 are the vacuum chamber1 telescope and flash apparatus for the ,mendenhall #a_/d-second apparatus4 ,shown below these are the four pendulums used with the ,mendenhall apparatus1 the one on the right having a thermometer attached4 ,at bottom1 right1 is the ,gulf apparatus 7cover removed7 mentioned in the text1 shown with one quartz pendulum4 ,early in the #aith century a systematic series of observations began to be conducted in order to determine the intensity of gravity at stations all over the world4 ,kater invariable pendulums1 of which #ac examples have been mentioned in the literature1 were used in surveys of gravity by ,kater1 ,sabine1 ,goldingham1 and other ,british pendulum swingers4 ,as has been noted previously1 a ,kater invariable pendulum was used by ,adm4 ,luetke of ,russia on a trip around the world4 ,the ,french also sent out expeditions to determine values of gravity4 ,after several decades of relative inactivity1 ,capts4 ,basevi and ,heaviside of the ,indian ,survey carried out an important series of observations from #ahfe to #ahgc with ,kater invariable pendulums and the ,russian ,repsold-,bessel pendulums4 ,in #ahha-ahhb ,maj4 ,j4 ,herschel swung ,kater invariable pendulums nos4 #d1 #f 7#ahba71 and #aa at stations in ,england and then brought them to the ,united ,states in order to make observations which would connect ,american and ,english base stations4#aji ,the extensive sets of observations of gravity provided the basis of calculations of the ellipticity of the earth4 ,col4 ,a4 ,r4 ,clarke in his ,geodesy 7,london1 #ahhj7 calculated the ellipticity from the results of gravity surveys to be #a_/7bib4b - #a4e74 ,of interest is the calculation by ,charles ,s4 ,peirce1 who used only determinations made with ,kater invariable pendulums and corrected for elevation1 atmospheric effect1 and expansion of the pendulum through temperature4#aaj ,he calculated the ellipticity of the earth to be #a_/7bia4e - #j4i74 ,the #aith century witnessed the culmination of the ellipsoidal era of geodesy1 but the rapid accumulation of data made possible a better approximation to the figure of the earth by the geoid4 ,the geoid is defined as the average level of the sea1 which is thought of as extended through the continents4 ,the basis of geodetic calculations1 however1 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 equator1 and of latitude4 ,the general assembly of the ,international ,union of ,geodesy and ,geophysics1 which was founded after ,world ,war ,i to continue the work of ,die ,internationale ,erdmessung1 adopted in #aibd an international reference ellipsoid1#aaa of which the ellipticity1 or flattening1 is ,hayford's value #a_/big4 ,in #aicj1 the general assembly adopted a correlated ,international ,gravity ,formula of the form gamma gamma,e7#a betasin#b phi epsilonsin#b #bphi7 where gamma is normal gravity at latitude phi1 gamma,e is the value of gravity at sea level at the equator1 beta is a parameter which is computed on the basis of ,clairaut's theorem from the flattening value of the meridian1 and epsilon is a constant which is derived theoretically4 ,the plumb line is perpendicular to the geoid1 and the components of angle between the perpendiculars to geoid and reference ellipsoid are deflections of the vertical4 ,the geoid is above the ellipsoid of reference under mountains and it is below the ellipsoid on the oceans1 where the geoid coincides with mean sea level4 ,in physical geodesy1 gravimetric data are used for the determination of the geoid and components of deflections of the vertical4 ,for this purpose1 one must reduce observed values of gravity to sea level by various reductions1 such as free-air1 ,bouguer1 isostatic reductions4 ,if g#j is observed gravity reduced to sea level and gamma is normal gravity obtained from the ,international ,gravity ,formula1 then ,deltag g#j - gamma is the gravity anomaly4#aab ,in #ahdi1 ,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 earth4 ,vening ,meinesz further derived formulae for the calculation of components of the deflection of the vertical4 ,geometrical geodesy1 which was based on astronomical-geodetic methods1 could give information only concerning the external form of the figure of the earth4 ,the gravimetric methods of physical geodesy1 in conjunction with methods such as those of seismology1 enable scientists to test hypotheses concerning the internal structure of the earth4 ,heiskanen and ,vening ,meinesz summarize the present-day achievements of the gravimetric method of physical geodesy by stating#aac that it alone can give3 #a4 ,the flattening of the reference ellipsoid4 #b4 ,the undulations ,n of the geoid4 #c4 ,the components of the deflection of the vertical xi and eta at any point1 oceans and islands included4 #d4 ,the conversion of existing geodetic systems to the same world geodetic system4 #e4 ,the reduction of triangulation base lines from the geoid to the reference ellipsoid4 #f4 ,the correction of errors in triangulation in mountainous regions due to the effect of the deflections of the vertical4 #g4 ,geophysical applications of gravity measurements1 e4g41 the isostatic study of the earth's interior and the exploration of oil fields and ore deposits4 ,with astronomical observations or with existing triangulations1 the gravimetric method can accomplish further results4 ,heiskanen and ,vening ,meinesz state3 ,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 geodesy1 i4e41 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 system4 ,it can also give invaluable help in the computation of the reference ellipsoid4#aad ,summary ,since the creation of classical mechanics in the #agth century1 the pendulum has been a basic instrument for the determination of the intensity of gravity1 which is expressed as the acceleration of a freely falling body4 ,basis of theory is the simple pendulum1 whose time of swing under gravity is proportional to the square root of the length divided by the acceleration due to gravity4 ,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 seconds1 the intensity of gravity also has been expressed in terms of the length of the seconds pendulum4 ,the reversible compound pendulum has served for the absolute determination of gravity by means of a theory developed by ,huygens4 ,invariable compound pendulums with single axes also have been used to determine relative values of gravity by comparative times of swing4 ,the history of gravity pendulums begins with the ball or simple pendulum of ,galileo as an approximation to the ideal simple pendulum4 ,determinations of the length of the seconds pendulum by ,french scientists culminated in a historic determination at ,paris by ,borda and ,cassini1 from the corrected observations with a long ball pendulum4 ,in the #aith century1 ,bessel found the length of the seconds pendulum at ,koenigsberg and ,berlin by observations with a ball pendulum and by original theoretical considerations4 ,during the century1 however1 the compound pendulum came to be preferred for absolute and relative determinations4 ,capt4 ,henry ,kater1 at ,london1 constructed the first convertible compound for an absolute determination of gravity1 and then he designed an invariable compound pendulum1 examples of which were used for relative determinations at various stations in ,europe and elsewhere4 ,bessel demonstrated theoretically the advantages of a reversible compound pendulum which is symmetrical in form and is hung by interchangeable knives4 ,the firm of ,a4 ,repsold and ,sons in ,hamburg constructed pendulums from the specifications of ,bessel for ,european gravity surveys4 ,charles ,s4 ,peirce in #ahge received delivery in ,hamburg of a ,repsold-,bessel pendulum for the ,u4,s4 ,coast ,survey and observed with it in ,geneva1 ,paris1 ,berlin1 and ,london4 ,upon an initial stimulation from ,baeyer1 founder of ,die ,europaeische ,gradmessung1 ,peirce demonstrated by experiment and theory that results previously obtained with the ,repsold apparatus required correction1 because of the flexure of the stand under oscillations of the pendulum4 ,at the ,stuttgart conference of the geodetic association in #ahgg1 ,herve ,faye proposed to solve the problem of flexure by swinging two similar pendulums from the same support with equal amplitudes and in opposite phases4 ,peirce1 in #ahgi1 demonstrated theoretically the soundness of the method and presented a design for its application1 but the double pendulum was rejected at that time4 ,peirce also designed and had constructed four examples of a new type of invariable1 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 fluid4 ,commandant ,defforges1 of ,france1 also designed and used cylindrical reversible pendulums1 but of different length so that the effect of flexure was eliminated in the reduction of observations4 ,maj4 ,robert von ,sterneck1 of ,austria-,hungary1 initiated a new era in gravity research by the invention of an apparatus with a short pendulum for relative determinations of gravity4 ,stands were then constructed in ,europe on which two or four pendulums were hung at the same time4 ,finally1 early in the present century1 ,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 ,holland4 ,the #bjth century has witnessed increasing activity in the determination of absolute and relative values of gravity4 ,gravimeters have been perfected and have been widely used for rapid relative determinations1 but the compound pendulums remain as indispensable instruments4 ,mendenhall's replacement of knives by planes attached to nonreversible pendulums has been used also for reversible ones4 ,the ,geodetic ,institute at ,potsdam is presently applying the ,faye-,peirce method to the reversible pendulum4#aae ,pendulums have been constructed of new materials1 such as invar1 fused silica1 and fused quartz4 ,minimum pendulums for precise relative determinations have been constructed and used4 ,reversible pendulums have been made with ,i cross sections for better stiffness4 ,with all these modifications1 however1 the foundations of the present designs of compound pendulum apparatus were created in the #aith century4 ,,footnotes3 #a ,the basic historical documents have been collected1 with a bibliography of works and memoirs published from #afbi to the end of #ahhe1 in ,collection de memoires relatifs a la physique1 publies par la ,societe francaise de ,physique hereinafter referred to as ,collection de memoires3 vol4 #d1 ,memoires sur le pendule1 precedes d'une bibliographie 7,paris3 ,gauthier-,villars1 #ahhi72 and vol4 #e1 ,memoires sur le pendule1 part #b 7,paris3 ,gauthier-,villars1 #ahia74 ,important secondary sources are3 ,c4 ,,wolf1 ,introduction historique1 pp4 #a-db in vol4 #d1 above2 and ,,george ,,biddell ,,airy1 ,figure of the ,earth1 pp4 #afe-bdj in vol4 #e of ,encyclopaedia metropolitana 7,london1 #ahde74 #b ,galileo ,galilei's principal statements concerning the pendulum occur in his ,discourses ,concerning ,two ,new ,sciences1 transl4 from ,italian and ,latin into ,english by ,henry ,crew and ,alfonso de ,salvio 7,evanston3 ,northwestern ,university ,press1 #aici71 pp4 #ie-ig1 #agj-agb4 #c ,p4 ,,marin ,,mersenne1 ,cogitata physico-mathematica 7,paris1 #afdd71 p4 #dd4 #d ,,christiaan ,,huygens1 ,horologium oscillatorium1 sive de motu pendulorum ad horologia adaptato demonstrationes geometricae 7,paris1 #afgc71 proposition #bj4 #e ,the historical events reported in the present section are from ,,airy1 ,figure of the ,earth4 #f ,,abbe ,,jean ,,picard1 ,la ,mesure de la terre 7,paris1 #afga74 ,,john ,w4 ,,olmsted1 ,the ',application' of ,telescopes to ,astronomical ,instruments1 #affg-affi1 ,isis 7#aidi71 vol4 #dj1 p4 #bac4 #g ,the toise as a unit of length was #f ,paris feet or about #a1idi millimeters4 #h ,,jean ,,richer1 ,observations astronomiques et physiques faites en l'isle de ,caienne 7,paris1 #afgi74 ,,john ,w4 ,,olmsted1 ,the ,expedition of ,jean ,richer to ,cayenne #afgb-afgc1 ,isis 7#aidb71 vol4 #cd1 pp4 #aag-abh4 #i ,the ,paris foot was #a4jff ,english feet1 and there were #ab lines to the inch4 #aj ,,christiaan ,,huygens1 ,de la cause de la pesanteur1 ,divers ouvrages de mathematiquesmathematiques et de physique par ,,mm4 de l',academie ,royale,royal des ,sciences 7,paris1 #afic71 p4 #cje4 #aa ,,isaac ,,newton1 ,philosophiae naturalis principia mathematica 7,london1 #afhg71 vol4 #c1 propositions #ah-bj4 #ab ,,pierre ,,bouguer1 ,la figure de la terre1 determinee par les observations de ,messieurs ,bouguer et de ,la ,condamine1 envoyes par ordre du ,roy au ,perou1 pour observer aux environs de l'equateur 7,paris1 #agdi74 #ac ,p4 ,l4 ,,moreau ,,de ,,maupertuis1 ,la figure de la terre determinee par les observations de ,messieurs de ,maupertuis1 ,clairaut1 ,camus1 ,le ,monnier1 l',abbe ,outhier et ,celsius1 faites par ordre du ,roy au cercle polaire 7,paris1 #agch74 #ad ,paris1 #agdc4 #ae ,,george ,,gabriel ,,stokes1 ,on ,attraction and on ,clairaut's ,theorem1 ,cambridge and ,dublin ,mathematical ,journal 7#ahdi71 vol4 #d1 p4 #aid4 #af ,see ,collection de memoires1 vol4 #d1 p4 ,b-#cd1 and ,j4 ,h4 ,,poynting and ,,sir ,j4 ,j4 ,,thomson1 ,properties of ,matter 7,london1 #aibg71 p4 #bd4 #ag ,,poynting and ,,thomson1 ibid41 p4 #bb4 #ah ,,charles ,m4 ,,de ,,la ,,condamine1 ,de la mesure du pendule a ,saint ,domingue1 ,collection de memoires1 vol4 #d1 pp4 #c-af4 #ai ,,pere ,r4 ,j4 ,,boscovich1 ,opera pertinentia ad ,opticam et ,astronomiam 7,bassani1 #aghe71 vol4 #e1 no4 #c4 #bj ,j4 ,c4 ,,borda and ,j4 ,d4 ,,cassini ,,de ,,thury1 ,experiences pour connaitre la longueur du pendule qui bat les secondes a ,paris1 ,collection de memoires1 vol4 #d1 pp4 #ag-fd4 #ba ,f4 ,w4 ,,bessel1 ,untersuchungen ueber die ,laenge des einfachen ,secundenpendels1 ,abhandlungen der ,koeniglichen ,akademie der ,wissenschaften zu ,berlin1 #ahbf 7,berlin1 #ahbh74 #bb ,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 ,arago4 #bc ,l4 ,g4 ,,du ,,buat1 ,principes d'hydraulique 7,paris1 #aghf74 ,see excerpts in ,collection de memoires1 pp4 ,b-#fd to ,b-#fg4 #bd ,,capt4 ,,henry ,,kater1 ,an ,account of ,experiments for ,determining the ,length of the ,pendulum ,vibrating ,seconds in the ,latitude of ,london1 ,philosophical ,transactions of the ,royal ,society of ,london 7#ahah71 vol4 #ajh1 p4 #cc4 ,hereinafter abbreviated ,phil4 ,trans4 #be ,m4 ,g4 ,,de ,,prony1 ,methode pour determiner la longueur du pendule simple qui bat les secondes1 ,collection de memoires1 vol4 #d1 pp4 #fe-gf4 #bf ,collection de memoires1 vol4 #d1 p4 ,b-#gd4 #bg ,phil4 ,trans4 7#ahai71 vol4 #aji1 p4 #ccg4 #bh ,,john ,,herschel1 ,notes for a ,history of the ,use of ,invariable ,pendulums1 ,the ,great ,trigonometrical ,survey of ,india 7,calcutta1 #ahgi71 vol4 #e4 #bi ,,capt4 ,,edward ,,sabine1 ,an ,account of ,experiments to ,determine the ,figure of the ,earth1 ,phil4 ,trans4 7#ahbh71 vol4 #aah1 p4 #gf4 #cj ,,john ,,goldingham1 ,observations for ,ascertaining the ,length of the ,pendulum at ,madras in the ,east ,indies1 ,phil4 ,trans4 7#ahbb71 vol4 #aab1 p4 #abg4 #ca ,,basil ,,hall1 ,letter to ,captain ,kater ,communicating the ,details of ,experiments made by him and ,mr4 ,henry ,foster with an ,invariable ,pendulum1 ,phil4 ,trans4 7#ahbc71 vol4 #aac1 p4 #baa4 #cb ,see ,collection de memoires1 vol4 #d1 p4 ,b-#ajc4 #cc ,ibid41 p4 ,b-#hh4 #cd ,ibid41 p4 ,b-#id4 #ce ,,francis ,,baily1 ,on the ,correction of a ,pendulum for the ,reduction to a ,vacuum1 ,together with ,remarks on ,some ,anomalies ,observed in ,pendulum ,experiments1 ,phil4 ,trans4 7#ahcb71 vol4 #abb1 pp4 #cii-dib4 ,see also ,collection de memoires1 vol4 #d1 pp4 ,b-#aje1 ,b-#aab1 ,b-#aae1 ,b-#aaf1 and ,b-#aag4 #cf ,one was of case brass and the other of rolled iron1 #fh in4 long1 #b in4 wide1 and #a_/b in4 thick4 ,triangular knife edges #b in4 long were inserted through triangular apertures #ai4g in4 from the center towards each end4 ,these pendulums seem not to have survived4 ,there is1 however1 in the collection of the ,u4,s4 ,national ,museum1 a similar brass pendulum1 #cg-e_/h in4 long 7fig4 #ae7 stamped with the name of ,edward ,kuebel 7#ahbj-if71 who maintained an instrument business in ,washington1 ,d4,c41 from about #ahdi4 ,the history of this instrument is unknown4 #cg ,see ,baily's remarks in the ,monthly ,notices of the ,royal ,astronomical ,society 7#ahci71 vol4 #d1 pp4 #ada-adc4 ,see also letters mentioned in footnote #ch4 #ch ,this document1 together with certain manuscript notes on the pendulum experiments and six letters between ,wilkes and ,baily1 is in the ,u4,s4 ,national ,archives1 ,navy ,records ,gp4 #cg4 ,these were the source materials for the information presented here on the ,expedition4 ,we are indebted to ,miss ,doris ,ann ,esch and ,mr4 ,joseph ,rudmann of the staff of the ,u4,s4 ,national ,museum for calling our attention to this early ,american pendulum work4 #ci ,g4 ,b4 ,,airy1 ,account of ,experiments ,undertaken in the ,harton ,colliery1 for the ,purpose of ,determining the ,mean ,density of the ,earth1 ,phil4 ,trans4 7#ahef71 vol4 #adf1 p4 #big4 #dj ,t4 ,c4 ,,mendenhall1 ,measurements of the ,force of ,gravity at ,tokyo1 and on the ,summit of ,fujiyama1 ,memoirs of the ,science ,department1 ,university of ,tokyo 7#ahha71 no4 #e4 #da ,j4 ,t4 ,,walker1 ,account of ,operations of ,the ,great ,trigonometrical ,survey of ,india 7,calcutta1 #ahgi71 vol4 #e1 app4 no4 #b4 #db ,,bessel1 op4 cit4 7footnote #ba71 article #ca4 #dc ,c4 ,a4 ,f4 ,,peters1 ,briefwechsel zwischen ,c4 ,f4 ,gauss und ,h4 ,c4 ,schumacher 7,altona1 ,germany1 #ahfj71 ,band #b1 p4 #c4 ,the correction required if the times of swing are not exactly the same is said to have been given also by ,bohnenberger4 #dd ,f4 ,w4 ,,bessel1 ,construction eines symmetrisch geformten ,pendels mit reciproken ,axen1 von ,bessel1 ,astronomische ,nachrichten 7#ahdi71 vol4 #cj1 p4 #a4 #de ,e4 ,,plantamour1 ,experiences faites a ,geneve avec le pendule a reversion1 ,memoires de la ,societe de ,physique et d'histoire naturelle de ,geneve1 #ahfe 7,geneva1 #ahff71 vol4 #ah1 p4 #cji4 #df ,ibid41 pp4 #cji-daf4 #dg ,c4 ,,cellerier1 ,note sur la ,mesure de la ,pesanteur par le ,pendule1 ,memoires de la ,societe de ,physique et d'histoire naturelle de ,geneve1 #ahfe 7,geneva1 #ahff71 vol4 #ah1 pp4 #aig-bah4 #dh ,a4 ,,sawitsch1 ,les variations de la pesanteur dans les provinces occidentales de l',empire russe1 ,memoirs of the ,royal ,astronomical ,society 7#ahgb71 vol4 #ci1 p4 #ai4 #di ,j4 ,j4 ,,baeyer1 ,ueber die ,groesse und ,figur der ,erde 7,berlin1 #ahfa74 #ej ,comptes-rendus de la ,conference ,geodesique ,internationale reunie a ,berlin du #ae-bb ,octobre #ahfd 7,neuchatel1 #ahfe74 #ea ,ibid41 part ,,iii1 subpart ,e4 #eb ,bericht ueber die ,verhandlungen der vom #cj ,september bis #g ,october #ahfg zu ,berlin abgehaltenen allgemeinen ,conferenz der ,europaeischen ,gradmessung 7,berlin1 #ahfh74 ,see report of fourth session1 ,october #c1 #ahfg4 #ec ,c4 ,,bruhns and ,,albrecht1 ,bestimmung der ,laenge des ,secundenpendels in ,bonn1 ,leiden und ,mannheim1 ,astronomisch-,geodaetische ,arbeiten im ,jahre #ahgj 7,leipzig3 ,veroeffentlichungen des ,koeniglichen ,preussischen ,geodaetischen ,instituts1 #ahga74 #ed ,bericht ueber die ,verhandlungen der vom #bc bis #bh ,september #ahgd in ,dresden abgehaltenen vierten allgemeinen ,conferenz der ,europaeischen ,gradmessung 7,berlin1 #ahge74 ,see report of second session1 ,september #bd1 #ahgd4 #ee ,,carolyn ,,eisele1 ,charles ,s4 ,peirce--,nineteenth-,century ,man of ,science1 ,scripta ,mathematica 7#aiei71 vol #bd1 p4 #cje4 ,for the account of the work of ,peirce1 the authors are greatly indebted to this pioneer paper on ,peirce's work on gravity4 ,it is worth noting that the history of pendulum work in ,north ,america goes back to the celebrated ,mason and ,dixon1 who made observations of the going rate of a clock at the forks of the river ,brandiwine in ,pennsylvania1 in #agff-fg4 ,these observations were published in ,phil4 ,trans4 7#agfh71 vol4 #eh1 pp4 #cbi-cce4 #ef ,the pendulums with conical bobs are described and illustrated in ,e4 ,d4 ,,preston1 ,determinations of ,gravity and the ,magnetic ,elements in ,connection with the ,united ,states ,scientific ,expedition to the ,west ,coast of ,africa1 #ahhi-ij1 ,report of the ,superintendent of the ,coast and ,geodetic ,survey for #ahhi-ij 7,washington1 #ahia71 app4 no4 #ab4 #eg ,,eisele1 op4 cit4 7footnote #ee71 p4 #caa4 #eh ,the record of ,peirce's observations in ,europe during #ahge-gf is given in ,c4 ,s4 ,,peirce1 ,measurements of ,gravity at ,initial ,stations in ,america and ,europe1 ,report of the ,superintendent of the ,coast ,survey for #ahge-gf 7,washington1 #ahgi71 pp4 #bjb-ccg and #daj-daf4 ,peirce's report is dated ,december #ac1 #ahgh1 by which time the name of the ,survey had been changed to ,u4,s4 ,coast and ,geodetic ,survey4 #ei ,verhandlungen der vom #bj bis #bi ,september #ahge in ,paris ,vereinigten ,permanenten ,commission der ,europaeischen ,gradmessung 7,berlin1 #ahgf74 #fj ,ibid4 ,see report for fifth session1 ,september #be1 #ahge4 #fa ,the experiments at the ,stevens ,institute1 ,hoboken1 were reported by ,peirce to the ,permanent ,commission which met in ,hamburg1 ,september #d-h1 #ahgh1 and his report was published in the general ,bericht for #ahgh in the ,verhandlungen der vom #d bis #h ,september #ahgh in ,hamburg ,vereinigten ,permanenten ,commission der ,europaeischen ,gradmessung 7,berlin1 #ahgi71 pp4 #aaf-abj4 ,assistant ,j4 ,e4 ,hilgard attended for the ,u4,s4 ,coast and ,geodetic ,survey4 ,the experiments are described in detail in ,c4 ,s4 ,,peirce1 ,on the ,flexure of ,pendulum ,supports1 ,report of the ,superintendent of the ,u4,s4 ,coast and ,geodetic ,survey for #ahhj-ha 7,washington1 #ahhc71 app4 no4 #ad1 pp4 #cei-dda4 #fb ,verhandlungen der vom #e bis #aj ,oktober #ahgf in ,brussels ,vereinigten ,permanenten ,commission der ,europaeischen ,gradmessung 7,berlin1 #ahgg74 ,see report of third session1 ,october #g1 #ahgf4 #fc ,verhandlungen der vom #bg ,september bis #b ,oktober #ahgg zu ,stuttgart abgehaltenen fuenften allgemeinen ,conferenz der ,europaeischen ,gradmessung 7,berlin1 #ahgh74 #fd ,verhandlung der vom #af bis #bj ,september #ahgi in ,genf ,vereinigten ,permanenten ,commission der ,europaeischen ,gradmessung 7,berlin1 #ahhj74 #fe ,assistants' ,reports1 ,u4,s4 ,coast and ,geodetic ,survey1 #ahgi-hj4 ,peirce's paper was published in the ,american ,journal of ,science 7#ahgi71 vol4 #ah1 p4 #aab4 #ff ,comptes-rendus de l',academie des ,sciences 7,paris1 #ahgi71 vol4 #hi1 p4 #dfb4 #fg ,verhandlungen der vom #ac bis #af ,september #ahhj zu ,muenchen abgehaltenen sechsten allgemeinen ,conferenz der ,europaeischen ,gradmessung 7,berlin1 #ahha74 #fh ,ibid41 app4 #b4 #fi ,ibid41 app4 #b;a4 #gj ,verhandlungen der vom #aa bis zum #ae ,september #ahhb im ,haag ,vereinigten ,permanenten ,commission der ,europaeischen ,gradmessung 7,berlin1 #ahhc74 #ga ,verhandlungen der vom #ae bis #bd ,oktober #ahhc zu ,rom abgehaltenen siebenten allgemeinen ,conferenz der ,europaeischen ,gradmessung 7,berlin1 #ahhd74 ,gen4 ,cutts attended for the ,u4,s4 ,coast and ,geodetic ,survey4 #gb ,ibid41 app4 #f4 ,see also1 ,zeitschrift fuer ,instrumentenkunde 7#ahhd71 vol4 #d1 pp4 #cjc and #cgi4 #gc ,op4 cit4 7footnote #fg74 #gd ,report of the ,superintendent of the ,u4,s4 ,coast and ,geodetic ,survey for #ahhj-ha 7,washington1 #ahhc71 p4 #bf4 #ge ,report of the ,superintendent of the ,u4,s4 ,coast and ,geodetic ,survey for #ahhi-ij 7,washington1 #ahia71 app4 no4 #ab4 #gf ,report of the ,superintendent of the ,u4,s4 ,coast and ,geodetic ,survey for #ahha-hb 7,washington1 #ahhc74 #gg ,transactions of the ,cambridge ,philosophical ,society 7#ahef71 vol4 #i1 part #b1 p4 #h4 ,also published in ,mathematical and ,physical ,papers 7,cambridge1 #aija71 vol4 #c1 p4 #a4 #gh ,peirce's comparison of theory and experiment is discussed in a report on the ,peirce memoir by ,,william ,,ferrel1 dated ,october #ai1 #ahij1 ,martinsburg1 ,west ,virginia4 ,u4,s4 ,coast and ,geodetic ,survey1 ,special ,reports1 #ahhg-ahia 7,,ms1 ,national ,archives1 ,washington74 #gi ,the stations at which observations were conducted with the ,peirce pendulums are recorded in the reports of the ,superintendent of the ,u4,s4 ,coast and ,geodetic ,survey from #ahha to #ahij4 #hj ,comptes-rendus de l',academie des ,sciences 7,paris1 #ahhj71 vol4 #ij1 p4 #adja4 ,,herve ,,faye's report1 dated ,june #ba1 #ahhj1 is in the same ,comptes-rendus1 p4 #adfc4 #ha ,,commandant ,c4 ,,defforges1 ,sur l',intensite absolue de la pesanteur1 ,journal de ,physique 7#ahhh71 vol4 #ag1 pp4 #bci1 #cdg1 #dee4 ,see also1 ,,defforges1 ,observations du pendule1 ,memorial du ,depot general de la ,guerre 7,paris1 #ahid71 vol4 #ae4 ,in the latter work1 ,defforges described a pendulum reversible inversable1 which he declared to be truly invariable and therefore appropriate for relative determinations4 ,the knives remained fixed to the pendulums1 and the effect of interchanging knives was obtained by interchanging weights within the pendulum tube4 #hb ,papers by ,,maj4 ,,von ,,sterneck in ,mitteilungen des ,k4 u4 ,k4 ,militaer-geographischen ,instituts1 ,wien1 #ahhb-hg2 see1 in particular1 vol4 #g 7#ahhg74 #hc ,t4 ,c4 ,,mendenhall1 ,determinations of ,gravity with the ,new ,half-,second ,pendulum4441 ,report of the ,superintendent of the ,u4,s4 ,coast and ,geodetic ,survey for #ahij-ia 7,washington1 #ahib71 part #b1 pp4 #ejc-efd4 #hd ,w4 ,h4 ,,burger1 ,the ,measurement of the ,flexure of ,pendulum ,supports with the ,interferometer1 ,report of the ,superintendent of the ,u4,s4 ,coast and ,geodetic ,survey for #aiji-aj 7,washington1 #aiaa71 app4 no4 #f4 #he ,e4 ,j4 ,,brown1 ,a ,determination of the ,relative ,values of ,gravity at ,potsdam and ,washington 7,special ,publication ,no4 #bjd1 ,u4,s4 ,coast and ,geodetic ,survey2 ,washington1 #aicf74 #hf ,m4 ,,haid1 ,neues ,pendelstativ1 ,zeitschrift fuer ,instrumentenkunde 7,july #ahif71 vol4 #af1 p4 #aic4 #hg ,,dr4 ,r4 ,,schumann1 ,ueber eine ,methode1 das ,mitschwingen bei relativen ,schweremessungen zu bestimmen1 ,zeitschrift fuer ,instrumentenkunde 7,january #ahig71 vol4 #ag1 p4 #g4 ,the design for the stand is similar to that of ,peirce's of #ahgi4 #hh ,,dr4 ,r4 ,,schumann1 ,ueber die ,verwendung zweier ,pendel auf gemeinsamer ,unterlage zur ,bestimmung der ,mitschwingung1 ,zeitschrift fuer ,mathematik und ,physik 7#ahii71 vol4 #dd1 p4 #dd4 #hi ,p4 ,,furtwaengler1 ,ueber die ,schwingungen zweier ,pendel mit annaehernd gleicher ,schwingungsdauer auf gemeinsamer ,unterlage1 ,sitzungsberichte der ,koeniglicher ,preussischen ,akademie der ,wissenschaften zu ,berlin 7,berlin1 #aijb7 pp4 #bde-bec4 ,peirce investigated the plan of swinging two pendulums on the same stand 7,report of the ,superintendent of the ,u4,s4 ,coast and ,geodetic ,survey for #ahhj-ha1 ,washington1 #ahhc1 p4 #bf2 also in ,,charles ,,sanders ,,peirce1 ,collected ,papers1 #f4bgc74 ,at a conference on gravity held in ,washington during ,may #ahhb1 ,peirce again advanced the method of eliminating flexure by hanging two pendulums on one support and oscillating them in antiphase 7,report of a conference on gravity determinations held in ,washington1 ,d4,c41 in ,may1 #ahhb1 ,report of the ,superintendent of the ,u4,s4 ,coast and ,geodetic ,survey for #ahha-hb1 ,washington1 #ahhc1 app4 no4 #bb1 pp4 #ejc-eaf74 #ij ,f4 ,a4 ,,vening ,,meinesz1 ,observations de pendule dans les ,pays-,bas 7,delft1 #aibc74 #ia ,a4 ,,berroth1 ,schweremessungen mit zwei und vier gleichzeitig auf demselben ,stativ schwingenden ,pendeln1 ,zeitschrift fuer ,geophysik1 vol4 #a 7#aibd-be71 no4 #c1 p4 #ic4 #ib ,pendulum ,apparatus for ,gravity ,determinations1 ,engineering 7#aibf71 vol4 #abb1 pp4 #bga-bgb4 #ic ,,malcolm ,w4 ,,gay1 ,relative ,gravity ,measurements ,using ,precision ,pendulum ,equipment1 ,geophysics 7#aidj71 vol4 #e1 pp4 #agf-aia4 #id ,l4 ,g4 ,d4 ,,thompson1 ,an ,improved ,bronze ,pendulum ,apparatus for ,relative ,gravity ,determinations1 published by ,dominion ,observatory 7,ottawa1 #aiei71 vol4 #ba1 no4 #c1 pp4 #ade-agf4 #ie ,w4 ,a4 ,,heiskanen and ,f4 ,a4 ,,vening ,,meinesz1 ,the ,earth and its ,gravity ,field 7,mc,graw3 ,new ,york1 #aieh74 #if ,f4 ,,kuehnen and ,p4 ,,furtwaengler1 ,bestimmung der ,absoluten ,groesze der ,schwerkraft zu ,potsdam mit ,reversionspendeln 7,berlin3 ,veroeffentlichungen des ,koeniglichen ,preussischen ,geodaetischen ,instituts1 #aijf71 new ser41 no4 #bg4 #ig ,reported by ,dr4 ,f4 ,kuehnen to the fifth session1 ,october #i1 #ahie1 of the ,eleventh ,general ,conference1 ,die ,internationale ,erdmessung1 held in ,berlin from ,september #be to ,october #ab1 #ahie4 ,a footnote states that ,assistant ,o4 ,h4 ,tittmann1 who represented the ,united ,states1 subsequently reported ,peirce's prior discovery of the influence of the flexure of the pendulum itself upon the period 7,report of the ,superintendent of the ,u4,s4 ,coast and ,geodetic ,survey for #ahhc-hd1 ,washington1 #ahhe1 app4 #af1 pp4 #dhc-dhe74 #ih ,assistants' ,reports1 ,u4,s4 ,coast and ,geodetic ,survey1 #ahhc-hd 7,,ms1 ,national ,archives1 ,washington74 #ii ,c4 ,s4 ,,peirce1 ,effect of the ,flexure of a ,pendulum ,upon its ,period of ,oscillation1 ,report of the ,superintendent of the ,u4,s4 ,coast and ,geodetic ,survey for #ahhc-hd 7,washington1 #ahhe71 app4 no4 #af4 #ajj ,f4 ,r4 ,,helmert1 ,beitraege zur ,theorie des ,reversionspendels 7,potsdam3 ,veroeffentlichungen des ,koeniglichen ,preussischen ,geodaetischen ,instituts1 #ahih74 #aja ,j4 ,a4 ,,duerksen1 ,pendulum ,gravity ,data in the ,united ,states 7,special ,publication ,no4 #bdd1 ,u4,s4 ,coast and ,geodetic ,survey2 ,washington1 #aidi74 #ajb ,ibid41 p4 #b4 ,see also1 ,e4 ,j4 ,,brown1 loc4 cit4 7footnote #he74 #ajc ,,paul ,r4 ,,heyl and ,,guy ,s4 ,,cook1 ,the ,value of ,gravity at ,washington1 ,journal of ,research1 ,national ,bureau of ,standards 7#aicf71 vol4 #ag1 p4 #hje4 #ajd ,,sir ,,harold ,,jeffreys1 ,the ,absolute ,value of ,gravity1 ,monthly ,notices of the ,royal ,astronomical ,society1 ,geophysical ,supplement 7,london1 #aidi71 vol4 #e1 p4 #cih4 #aje ,j4 ,s4 ,,clark1 ,the ,acceleration ,due to ,gravity1 ,phil4 ,trans4 7#aici71 vol4 #bch1 p4 #fe4 #ajf ,,hugh ,l4 ,,dryden1 ,a ,reexamination of the ,potsdam ,absolute ,determination of ,gravity1 ,journal of ,research1 ,national ,bureau of ,standards 7#aidb71 vol4 #bi1 p4 #cjc2 and ,a4 ,,berroth1 ,das ,fundamentalsystem der ,schwere im ,lichte neuer ,reversionspendelmessungen1 ,bulletin ,geodesique 7#aidi71 no4 #ab1 pp4 #ahc-bjd4 #ajg ,t4 ,c4 ,,mendenhall1 op4 cit4 7footnote #hc71 p4 #ebb4 #ajh ,a4 ,h4 ,,cook1 ,recent ,developments in the ,absolute ,measurement of ,gravity1 ,bulletin ,geodesique 7,june #a1 #aieg71 no4 #dd1 pp4 #cd-ei4 #aji ,see footnote #hi4 #aaj ,c4 ,s4 ,,peirce1 ,on the ,deduction of the ,ellipticity of the ,earth1 from ,pendulum ,experiments1 ,report of the ,superintendent of the ,u4,s4 ,coast and ,geodetic ,survey for #ahhj-ha 7,washington1 #ahhc71 app4 no4 #ae1 pp4 #ddb-def4 #aaa ,,heiskanen and ,,vening ,,meinesz1 op4 cit4 7footnote #ie71 p4 #gd4 #aab ,ibid41 p4 #gf4 #aac ,ibid41 p4 #cji4 #aad ,ibid41 p4 #caj4 #aae ,k4 ,,reicheneder1 ,method of the ,new ,measurements at ,potsdam by ,means of the ,reversible ,pendulum1 ,bulletin ,geodesique 7,march #a1 #aiei71 no4 #ea1 p4#gb4 ,u4,s4 ,,government ,,printing ,,office3 #aife ,for sale by the ,superintendent of ,documents1 ,u4,s4 ,government ,printing ,office ,washington1 ,d4,c41 #bjdjb--,price #gj cents4 ,,index ,airy1 ,g4 ,b41 #cai1 #cbd1 #ccb ,albrecht1 ,karl ,theodore1 #cbb1 #cch ,al-,mamun1 seventh calif of ,bagdad1 #cjf ,almansi1 ,emilio1 #cci ,aristotle1 #cjf ,baeyer1 ,j4 ,j41 #cba1 #cbb1 #cbd1 #cch1 #cdf ,baily1 ,francis1 #cag ,basevi1 ,james ,palladio1 #cde ,berroth1 ,a41 #cdb ,bessel1 ,friedrich ,wilhelm1 #cac1 #cad1 #cai1 #cbj1 #cbd1 #cbe1 #cch1 #cdf ,biot1 ,jean ,baptiste1 #cbe1 #cbi ,bohnenberger1 ,johann ,gottlieb ,friedrich1 #cae ,borda1 ,j4 ,c41 #caa1 #cab1 #cae1 #cbe1 #cbi1 #cdf ,boscovitch1 ,pere ,r4 ,j41 #caj1 #caa ,bouguer1 ,pierre1 #cjg1 #cji1 #cbg1 #cdc1 #cde ,brahe1 ,tycho1 #cjf ,brown1 ,e4 ,j41 #ccd1 #cci ,browne1 ,henry1 #cjd1 #cad ,bruhns1 ,c41 #cbb1 #cbd1 #cch ,brunner ,brothers 7,paris71 #cbi ,cassini1 ,giovanni-,domenico1 #cjf1 #cjg ,cassini1 ,jacques1 #cjf ,cassini de ,thury1 ,j4 ,d41 #caa1 #cab1 #cae1 #cbe1 #cbi1 #cdf ,cellerier1 ,charles1 #cbj1 #cba1 #cbe1 #cbf1 #cbi1 #ccf ,clairaut1 ,alexis ,claude1 #cjh1 #cji1 #cdc1 #cde ,clark1 ,j4 ,s41 #cdb ,clarke1 ,a4 ,r41 #cde ,colbert1 ,jean ,baptiste1 #cjf ,cook1 ,a4 ,h41 #cdb ,cook1 ,guy ,s41 #cci1 #cdb ,defforges1 ,c41 #cad1 #cbi1 #cdf ,de ,freycinet1 ,louis ,claude de ,saulses1 #cag ,de la ,hire1 ,gabriel ,philippe1 #cjf ,de ,prony1 ,m4 ,g41 #cad ,dryden1 ,hugh ,l41 #cdb ,du ,buat1 ,l4 ,g41 #cad ,duperry1 ,capt4 ,louis ,isidore1 #cag ,eratosthenes1 #cjf1 #cjh1 #cdb ,eudoxus of ,cnidus1 #cjf ,faye1 ,herve1 #cbe1 #ccf1 #cdf1 #cdg ,fernel1 ,jean1 #cjf ,furtwaengler1 ,p41 #ccg ,galilei1 ,galileo1 #cjd1 #cje1 #cdf ,gauss1 ,c4 ,f41 #cbj ,gautier1 ,p41 #cci ,godin1 ,louis1 #cjg ,goldingham1 ,john1 #caf1 #cde ,greely1 ,a4 ,w41 #cbi ,gulf ,oil and ,development ,company1 #cch ,haid1 ,m41 #cce ,hall1 ,basil1 #caf ,heaviside1 ,w4 ,j41 #cba1 #cde ,heiskanen1 ,w4 ,a41 #cch1 #cde1 #cdf ,helmert1 ,f4 ,r41 #cch1 #cci ,helmholtz1 ,hermann von1 #cbf ,herschel1 ,john1 #cai1 #cbh1 #cde ,heyl1 ,paul ,r41 #cci1 #cdb ,hirsch1 ,adolph1 #cbb1 #cbd ,huygens1 ,christiaan1 #cjd1 #cje1 #cjg1 #cad1 #cdb1 #cdf ,ibanez1 ,carlos1 #cbe ,jeffreys1 ,sir ,harold1 #cdb ,jones1 ,thomas1 #cah ,kater1 ,henry1 #cjd1 #cad1 #cbe1 #cbg1 #cbi1 #cde1 #cdf ,kuehnen1 ,f41 #cch1 #cci ,la ,condamine1 ,charles ,marie de1 #cjg1 #caj1 #caa1 #cdc ,laplace1 ,marquis ,pierre ,simon de1 #cji1 #cac1 #cbj ,lorenzoni1 ,giuseppe1 #ccf1 #cci ,luetke1 ,count ,feodor ,petrovich1 #caf1 #cde ,maupertius1 ,p4 ,l4 ,moreau de1 #cjh1 #cdc ,maxwell1 ,james ,clerk1 #cbd ,medi1 ,enrico1 #cdb ,mendenhall1 ,thomas ,corwin1 #cai1 #cca1 #ccb1 #ccd1 #cdg ,mersenne1 ,p4 ,marin1 #cje ,newton1 ,sir ,isaac1 #cjc1 #cjg1 #cjh1 #cdb1 #cdc ,norwood1 ,richard1 #cjf ,oppolzer1 ,theodor von1 #cbb1 #cbd ,patterson1 ,carlile ,pollock1 #cbe1 #cbf ,peirce1 ,charles ,sanders1 #cad1 #cbb1 #ccb1 #ccf1 #cdb1 #cde ,peters1 ,c4 ,a4 ,f41 #cbb1 #cbd ,picard1 ,abbe ,jean1 #cjf1 #cjh1 #cdb ,plantamour1 ,e41 #cai1 #cbd ,posidonius1 #cjf ,preston1 ,e4 ,d41 #cbh1 #cbi ,putnam1 ,g4 ,r41 #cci ,pythagoras1 #cjf ,repsold1 ,a41 and ,sons 7,hamburg71 #cbj1 #cbb1 #cch1 #cci1 #cdf ,richer1 ,jean1 #cjg1 #cdb ,sabine1 ,capt4 ,edward1 #cae1 #cbe1 #cbi1 #cde ,sawitsch1 ,a41 #cba1 #cbb ,schumacher1 ,h4 ,c41 #cbj ,schumann1 ,r41 #cce1 #ccf ,snell1 ,willebrord1 #cjf ,sterneck1 ,robert von1 #cca1 #ccb1 #cce1 #cch1 #cdf ,stokes1 ,george ,gabriel1 #cbd1 #cbh1 #cbi1 #cde1 #cdf ,ulloa1 ,antonio de1 #cjh ,vening ,meinesz1 ,f4 ,a41 #ccg1 #cch1 #cde ,volet1 ,charles1 #cdb ,wilkes1 ,charles1 #cag1 #cah 99 99 99 99 99 ,transcriber's note3 ,footnotes have been moved to the end of the paper4 ,illustrations and the ,,glossary ,,of ,,gravity ,,terminology section have been moved to avoid breaks in paragraphs4 ,minor punctuation errors have been corrected without note4 ,typographical errors and inconsistencies have been corrected as follows3 ,p4 #cbj 'difference ,t#a - ,t#b is sufficiently'--had 'sufficlently4' ,p4 #cba 'faites a ,geneve avec le pendule a reversion'--had 'reversion4' ,p4 #cbf ',schwere mit ,hilfe verschiedener ,apparate'--had 'verschiedene4' ,p4 #cbh 'between the yard and the meter4'--closing quote mark deleted4 ,p4 #ccd ',mendenhall apparatus were part of'--'was' changed to 'were4' ,p4 #cdb 'of the ,geodetic ,institute at ,potsdam'--had ',postdam4' ,p4 #cde ',the gravimetric methods of physical'--had 'mtehods4' ,footnote #a ',societe francaise de ,physique'--had ',francaise4' ,footnote #c ',cogitata physico-mathematica'--had 'physica4' ,footnote #aj 'mathematiques et de physique par ,,mm4 de l',academie ,royale'--had 'mathematiques1' ',royal4' ,footnote #ab 'par ordre du ,roy au ,perou1 pour observer'--had ',perou1 pour observir4' ,footnote #ai ',opticam et ,astronomiam'--had ',astronomian4' ,footnote #bj 'connaitre la longueur du pendule qui'--had 'connaitre la longuer4' ,footnote #ba ',abhandlungen der ,koeniglichen ,akademie'--had ',koenigliche4' ,footnote #be 'pour determiner la longueur du pendule'--had 'longeur4' ,footnote #da ',survey of ,india 7,calcutta1 #ahgi7'-- had ',surey4' ,footnotes #de and #dg ',societe de ,physique et d'histoire'--had 'd'historire4' ,footnote #di ',ueber die ,groesse und ,figur der ,erde'--had ',grosse4' ,footnote #ec ',bestimmung der ,laenge'--had ',lange'2 ',astronomisch-,geodaetische ,arbeiten'--had ',astronomische'2 ',veroeffentlichungen des ,koeniglichen'--had ',koenigliche4' ,footnote #ee '7#agfh71 vol4 #eh1 pp4 #cbi-cce4'--;had '#cbi-bce4' ,footnote #ff ',comptes-rendus de l',academie'--had ',l',academie4' ,footnote #ha ',sur l',intensite absolue'--had 'l',intensite4' ,footnote #hi ',sitzungsberichte der ,koeniglicher'--had ',koenigliche4' ,footnote #ajj ',veroeffentlichungen des ,koeniglichen' had ',veroeffentlichungen ,koenigliche4' ,capitalisation of ',von'_/'von' has been regulaized to 'von' for all personal names1 except at the beginning of a sentence1 and when referring to the ,von ,sterneck pendulum4