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diff --git a/35024-h/35024-h.htm b/35024-h/35024-h.htm new file mode 100644 index 0000000..8e8bae5 --- /dev/null +++ b/35024-h/35024-h.htm @@ -0,0 +1,4115 @@ +<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN" + "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"> +<html xmlns="http://www.w3.org/1999/xhtml"> +<head> +<meta http-equiv="Content-Type" content="text/html; charset=ISO-8859-1" /> +<title>The Project Gutenberg eBook of Development of Gravity Pendulums in the 19th Century, by Victor Fritz Lenzen and Robert P. Multhauf</title> + <style type="text/css"> + +/*note this is a standard style for this group of papers, therefore there may be unused classes in this document*/ + +body {margin-left: 10%; margin-right: 10%;} +h1 {text-align:right;} +h1.pg {text-align: center;} +h2,h3,h5,h6 {text-align: center; clear: both;} +h4 {text-align: left; clear: both;} +h4.appx {text-align: center;} +p {margin-top: .75em; text-align: justify; margin-bottom: .75em;} +hr {width: 33%; margin-top: 2em; margin-bottom: 2em; margin-left: auto; margin-right: auto; clear: both;} +table {margin-left: auto; margin-right: auto; border:0px; border-collapse:collapse;} +td {vertical-align:top; padding:5px; text-align: left;} + +.blockquot {margin-left: 5%; margin-right: 10%; } +.blockquotn {margin-left: 20%; margin-right: 20%; } +.rnum {position: absolute; right:10%; text-align:left;} +.right15 {position: absolute; right:15%; text-align:left;} +.right40 {position: absolute; right:40%; text-align:left;} +.right25 {position: absolute; right:25%; text-align:left;} +.center {text-align: center;} +.mono {font-family: "courier new",courier, monospace;} +.right {text-align: right;} +.smcap {font-variant: small-caps;} +.caption {font-weight: bold; text-align: center;} +.caption2 {font-weight: bold; text-align: justify;} +p.caption, p.caption2 {margin-top:0.25em;} +.hang {margin-right:4%; padding-left: 2.5em; text-indent: -2.5em;} +.hang2 {padding-left: 1em; text-indent: -2.5em;} +.hang3 {padding-left: 1.5em; text-indent: -1.5em;} +.tnote {padding-bottom: .5em; padding-top: .5em; padding-left: .5em; padding-right: .5em; + margin-left: 1em; margin-top: 1em; font-size: smaller; background: #eeeeee; border: solid 1px;} +.sidebar_rt {float: right; width:46%; clear: right; margin-left: 1em; margin-bottom: 1em; margin-top: 1em; + margin-right: 0; padding-left: 1.5em; padding-right:1.5em; border: solid 1px;} +.sidebar_lf {float: left; width:46%; clear: left; margin-left: 0; margin-bottom: 1em; margin-top: 1em; + margin-right: 1em; padding-left: 1.5em; padding-right:1.5em; border: solid 1px;} +.sidebar {margin: 1em; padding-left: 1.5em; padding-right:1.5em; border: solid 1px;} +.nowrap {white-space: nowrap;} + +/*correction highlighting */ +ins.mycorr {text-decoration: none; border-bottom: 1px dotted #FF0000;} + +/* page numbers */ +.pagenum {position: absolute; left: 92%; font-size: smaller; text-align: right;} + +/* Images */ +.figcenter {margin: auto; text-align: center;} +.figleft {float: left; clear: left; margin-left: 0; margin-bottom: 0.5em; margin-top: 0.5em; + margin-right: 1em; padding: 0; text-align: center;} +.figright { float: right; clear: right; margin-left: 1em; margin-bottom: 0.5em; margin-top: 0.5em; + margin-right: 0; padding: 0; text-align: center;} + +/* Footnotes */ +.footnotes {border: dashed 1px;} +.footnote {margin-left: 10%; margin-right: 10%; font-size: 0.9em;} +.footnote .label {position: absolute; right: 84%; text-align: right;text-decoration: none;} +.fnanchor {vertical-align: super; font-size: .8em; text-decoration: none;} +.fntable {text-decoration: none; font-size: .8em; } + +/* Poetry - do not think ths will be needed?*/ +.poem {margin-left:10%; margin-right:10%; text-align: left;} +.poem br {display: none;} +.poem .stanza {margin: 1em 0em 1em 0em;} +.poem span.i0 {display: block; margin-left: 0em; padding-left: 3em; text-indent: -3em;} +.poem span.i2 {display: block; margin-left: 2em; padding-left: 3em; text-indent: -3em;} +.poem span.i4 {display: block; margin-left: 4em; padding-left: 3em; text-indent: -3em;} + + hr.full { width: 100%; + margin-top: 3em; + margin-bottom: 0em; + margin-left: auto; + margin-right: auto; + height: 4px; + border-width: 4px 0 0 0; /* remove all borders except the top one */ + border-style: solid; + border-color: #000000; + clear: both; } + pre {font-size: 85%;} + </style> +</head> +<body> +<h1 class="pg">The Project Gutenberg eBook, Development of Gravity Pendulums in the 19th +Century, by Victor Fritz Lenzen and Robert P. Multhauf</h1> +<pre> +This eBook is for the use of anyone anywhere at no cost and with +almost no restrictions whatsoever. You may copy it, give it away or +re-use it under the terms of the Project Gutenberg License included +with this eBook or online at <a href = "http://www.gutenberg.org">www.gutenberg.org</a></pre> +<p>Title: Development of Gravity Pendulums in the 19th Century</p> +<p> Contributions from the Museum of History and Technology, Papers 34-44 On Science and Technology, Smithsonian Institution, 1966</p> +<p>Author: Victor Fritz Lenzen and Robert P. Multhauf</p> +<p>Release Date: January 21, 2011 [eBook #35024]</p> +<p>Language: English</p> +<p>Character set encoding: ISO-8859-1</p> +<p>***START OF THE PROJECT GUTENBERG EBOOK DEVELOPMENT OF GRAVITY PENDULUMS IN THE 19TH CENTURY***</p> +<p> </p> +<h3>E-text prepared by Chris Curnow, Joseph Cooper, Louise Pattison,<br /> + and the Online Distributed Proofreading Team<br /> + (http://www.pgdp.net)</h3> +<p> </p> + +<div class="tnote"> +<h3>Transcriber’s Note:</h3> + +<p>This is Paper 44 from the <i>Smithsonian Institution United States +National Museum Bulletin 240</i>, comprising Papers 34-44, which will +also be available as a complete e-book.</p> + +<p>The front material, introduction and relevant index entries from +the <i>Bulletin</i> are included in each single-paper e-book.</p> + +<p><a href="#corrections_44">Corrections</a> to typographical errors are underlined +<ins class="mycorr" title="Original: like thsi">like this</ins>. +Hover the cursor over the marked text to view the original text.</p> +</div> +<p> </p> +<hr class="full" /> +<p> </p> +<p> </p> +<p> </p> + +<h1>SMITHSONIAN INSTITUTION<br /> +UNITED STATES NATIONAL MUSEUM<br /> +BULLETIN 240</h1> + +<div class="figright"> + <img src="images/cover.png" alt="Smithsonian Press Logo" title="" /> +</div> + +<p class="right" style="clear:both;">SMITHSONIAN PRESS<br /></p> + +<p>MUSEUM OF HISTORY AND TECHNOLOGY</p> + +<p style="font-size: 2em; font-weight: bold;" class="smcap">Contributions<br /> +From the<br /> +Museum<br /> +of History and<br /> +Technology</p> + +<p style="font-size: 1.25em;"><em>Papers 34-44<br /> +On Science and Technology</em></p> + +<p>SMITHSONIAN INSTITUTION · WASHINGTON, D.C. 1966</p> + +<hr style="width: 65%;" /> + +<p class="center" style="font-size: 1.25em;"><em>Publications of the United States National Museum</em></p> + +<p>The scholarly and scientific publications of the United States National Museum +include two series, <cite>Proceedings of the United States National Museum</cite> and <cite>United States +National Museum Bulletin</cite>.</p> + +<p>In these series, the Museum publishes original articles and monographs dealing +with the collections and work of its constituent museums—The Museum of Natural +History and the Museum of History and Technology—setting forth newly acquired +facts in the fields of anthropology, biology, history, geology, and technology. Copies +of each publication are distributed to libraries, to cultural and scientific organizations, +and to specialists and others interested in the different subjects.</p> + +<p>The <cite>Proceedings</cite>, begun in 1878, are intended for the publication, in separate +form, of shorter papers from the Museum of Natural History. These are gathered +in volumes, octavo in size, with the publication date of each paper recorded in the +table of contents of the volume.</p> + +<p>In the <cite>Bulletin</cite> series, the first of which was issued in 1875, appear longer, separate +publications consisting of monographs (occasionally in several parts) and volumes +in which are collected works on related subjects. <cite>Bulletins</cite> are either octavo or +quarto in size, depending on the needs of the presentation. Since 1902 papers relating +to the botanical collections of the Museum of Natural History have been +published in the <cite>Bulletin</cite> series under the heading <cite>Contributions from the United States +National Herbarium</cite>, and since 1959, in <cite>Bulletins</cite> titled “Contributions from the Museum +of History and Technology,” have been gathered shorter papers relating to the collections +and research of that Museum.</p> + +<p>The present collection of Contributions, Papers 34-44, comprises Bulletin 240. +Each of these papers has been previously published in separate form. The year of +publication is shown on the last page of each paper.</p> + +<p class="right"><span class="smcap">Frank A. Taylor</span><br /> +<em>Director, United States National Museum</em></p> + +<hr style="width: 65%;" /> +<p><span class="pagenum"><a name="Page_301" id="Page_301">[Pg 301]</a></span></p> + +<h1><a name="Paper_44" id="Paper_44"></a><span class="smcap">Contributions from<br /> +The Museum of History and Technology</span>:<br /> +<span class="smcap">Paper</span> 44<br /> +<br /><br /> +<span class="smcap">Development of Gravity Pendulums in the 19th Century<br /> +<br /></span> +</h1> +<p><span class="rnum" style="font-size: larger;"><i>Victor F. Lenzen</i> and <i>Robert P. Multhauf</i></span><br /><br/> +</p> +<p><span class="rnum">GALILEO, HUYGENS, AND NEWTON <a href="#Page_304">304</a></span><br /> +</p> +<p><span class="rnum">FIGURE OF THE EARTH <a href="#Page_306">306</a></span><br /> +</p> +<p><span class="rnum">EARLY TYPES OF PENDULUMS <a href="#Page_309">309</a></span><br /> +</p> +<p><span class="rnum">KATER’S CONVERTIBLE AND INVARIABLE PENDULUMS <a href="#Page_314">314</a></span><br /> +</p> +<p><span class="rnum">REPSOLD-BESSEL REVERSIBLE PENDULUM <a href="#Page_320">320</a></span><br /> +</p> +<p><span class="rnum">PEIRCE AND DEFFORGES INVARIABLE, REVERSIBLE PENDULUMS <a href="#Page_327">327</a></span><br /> +</p> +<p><span class="rnum">VON STERNECK AND MENDENHALL PENDULUMS <a href="#Page_331">331</a></span><br /> +</p> +<p><span class="rnum">ABSOLUTE VALUE OF GRAVITY AT POTSDAM <a href="#Page_338">338</a></span><br /> +</p> +<p><span class="rnum">APPLICATION OF GRAVITY SURVEYS <a href="#Page_342">342</a></span><br /> +</p> +<p><span class="rnum">SUMMARY <a href="#Page_346">346</a></span><br /> +</p> +<p><span class="pagenum"><a name="Page_302" id="Page_302">[Pg 302]</a></span></p> + + + + +<p style="clear:both;"><br /><br /> +<i>Victor F. Lenzen</i> and<br /> +<i>Robert P. Multhauf</i><br /> +</p> + +<h2> +DEVELOPMENT OF GRAVITY PENDULUMS<br /> +IN THE 19th CENTURY</h2> + +<div class="figcenter" style="width: 600px;"><a name="fig_1" id="fig_1"></a> +<img src="images/i002.png" width="600" height="456" alt="Figure 1." title="Figure 1." /> + +<p class="caption2">Figure 1.—<span class="smcap">A study of the figure of the earth was</span> one of the earliest projects of the +French Academy of Sciences. In order to test the effect of the earth’s rotation on its +gravitational force, the Academy in 1672 sent Jean Richer to the equatorial island of +Cayenne to compare the rate of a clock which was known to have kept accurate time in +Paris. Richer found that the clock lost 2 minutes and 28 seconds at Cayenne, indicating +a substantial decrease in the force of gravity on the pendulum. Subsequent pendulum +experiments revealed that the period of a pendulum varied not only with the latitude but +also regionally, under the influence of topographical features such as mountains. It +became clear that the measurement of gravity should be made a part of the work of the +geodetic surveyor.</p> +</div> + +<p><span class="pagenum"><a name="Page_303" id="Page_303">[Pg 303]</a></span></p> + +<div class="blockquotn"><p><i>The history of gravity pendulums dates back to the +time of Galileo. After the discovery of the variation of +the force of gravity over the surface of the earth, gravity +measurement became a major concern of physics and +geodesy. This article traces the history of the development +of instruments for this purpose.</i></p> + +<p><span class="smcap">THE AUTHORS</span>: <i>Victor F. Lenzen is Professor of +Physics, Emeritus, at the University of California at Berkeley +and Robert P. Multhauf is Chairman of the Department +of Science and Technology in the Smithsonian +Institution’s Museum of History and Technology.</i></p></div> + +<p>The intensity of gravity, or the acceleration of a +freely falling body, is an important physical quantity +for the several physical sciences. The intensity of +gravity determines the weight of a standard pound +or kilogram as a standard or unit of force. In physical +experiments, the force on a body may be measured +by determining the weight of a known mass which +serves to establish equilibrium against it. Thus, in +the absolute determination of the ampere with a current +balance, the force between two coils carrying +current is balanced by the earth’s gravitational force +upon a body of determinable mass. The intensity of +gravity enters into determinations of the size of the +earth from the angular velocity of the moon, its +distance from the earth, and Newton’s inverse square +law of gravitation and the laws of motion. Prediction +of the motion of an artificial satellite requires an +accurate knowledge of gravity for this astronomical +problem.</p> + +<p>The gravity field of the earth also provides data for +a determination of the figure of the earth, or geoid, +but for this problem of geodesy relative values of +gravity are sufficient. If <i>g</i> is the intensity of gravity +at some reference station, and Δ<i>g</i> is the difference +between intensities at two stations, the values of +gravity in geodetic calculations enter as ratios (Δ<i>g</i>)/<i>g</i> +over the surface of the earth. Gravimetric investigations +in conjunction with other forms of geophysical +investigation, such as seismology, furnish data to test<span class="pagenum"><a name="Page_304" id="Page_304">[Pg 304]</a></span> +hypotheses concerning the internal structure of the +earth.</p> + +<p>Whether the intensity of gravity is sought in absolute +or relative measure, the most widely used instrument +for its determination since the creation of classical +mechanics has been the pendulum. In recent decades, +there have been invented gravity meters based +upon the principle of the spring, and these instruments +have made possible the rapid determination of +relative values of gravity to a high degree of accuracy. +The gravity meter, however, must be calibrated at +stations where the absolute value of gravity has been +determined by other means if absolute values are +sought. For absolute determinations of gravity, the +pendulum historically has been the principal instrument +employed. Although alternative methods of +determining absolute values of gravity are now in use, +the pendulum retains its value for absolute determinations, +and even retains it for relative determinations, +as is exemplified by the Cambridge Pendulum Apparatus +and that of the Dominion Observatory at +Ottawa, Ontario.</p> + +<p>The pendulums employed for absolute or relative +determinations of gravity have been of two basic +types. The first form of pendulum used as a physical +instrument consisted of a weight suspended by a fiber, +cord, or fine wire, the upper end of which was attached +to a fixed support. Such a pendulum may be called +a “simple” pendulum; the enclosure of the word +simple by quotation marks is to indicate that such a +pendulum is an approximation to a simple, or mathematical +pendulum, a conceptual object which consists +of a mass-point suspended by a weightless +inextensible cord. If <i>l</i> is the length of the simple +pendulum, the time of swing (half-period in the sense +of physics) for vibrations of infinitely small amplitude, +as derived from Newton’s laws of motion and the +hypothesis that weight is proportional to mass, is +<i>T</i> = <span style="white-space: nowrap;">π√(<i>l</i>/<i>g</i>).</span></p> + +<p>The second form of pendulum is the compound, or +physical, pendulum. It consists of an extended +solid body which vibrates about a fixed axis under +the action of the weight of the body. A compound +pendulum may be constituted to oscillate about one +axis only, in which case it is nonreversible and +applicable only for relative measurements. Or a +compound pendulum may be constituted to oscillate +about two axes, in which case it is reversible (or +“convertible”) and may be used to determine absolute +values of gravity. Capt. Henry Kater, F.R.S., +during the years 1817-1818 was the first to design, +construct, and use a compound pendulum for the +absolute determination of gravity. He constructed +a convertible pendulum with two knife edges and +with it determined the absolute value of gravity at +the house of Henry Browne, F.R.S., in Portland +Place, London. He then constructed a similar +compound pendulum with only one knife edge, and +swung it to determine relative values of gravity at a +number of stations in the British Isles. The 19th +century witnessed the development of the theory and +practice of observations with pendulums for the +determination of absolute and relative values of +gravity.</p> + + + +<hr style="width: 65%;" /> +<h3>Galileo, Huygens, and Newton</h3> + + +<p>The pendulum has been both an objective and an +instrument of physical investigation since the foundations +of classical mechanics were fashioned in the +17th century.<a name="FNanchor_1_1" id="FNanchor_1_1"></a><a href="#Footnote_1_1" class="fnanchor">[1]</a> It is tradition that the youthful +Galileo discovered that the period of oscillation of a +pendulum is constant by observations of the swings +of the great lamp suspended from the ceiling in the +cathedral of Pisa.<a name="FNanchor_2_2" id="FNanchor_2_2"></a><a href="#Footnote_2_2" class="fnanchor">[2]</a> The lamp was only a rough +approximation to a simple pendulum, but Galileo +later performed more accurate experiments with a +“simple” pendulum which consisted of a heavy ball +suspended by a cord. In an experiment designed to +confirm his laws of falling bodies, Galileo lifted the +ball to the level of a given altitude and released it. +The ball ascended to the same level on the other side +of the vertical equilibrium position and thereby +confirmed a prediction from the laws. Galileo also +discovered that the period of vibration of a “simple” +pendulum varies as the square root of its length, a<span class="pagenum"><a name="Page_305" id="Page_305">[Pg 305]</a></span> +result which is expressed by the formula for the time +of swing of the ideal simple pendulum. He also +used a pendulum to measure lapse of time, and he +designed a pendulum clock. Galileo’s experimental +results are important historically, but have required +correction in the light of subsequent measurements +of greater precision.</p> + +<p>Mersenne in 1644 made the first determination of +the length of the seconds pendulum,<a name="FNanchor_3_3" id="FNanchor_3_3"></a><a href="#Footnote_3_3" class="fnanchor">[3]</a> that is, the length +of a simple pendulum that beats seconds (half-period +in the sense of physics). Subsequently, he +proposed the problem to determine the length of +the simple pendulum equivalent in period to a given +compound pendulum. This problem was solved +by Huygens, who in his famous work <i>Horologium +oscillatorium</i> ... (1673) set forth the theory of the +compound pendulum.<a name="FNanchor_4_4" id="FNanchor_4_4"></a><a href="#Footnote_4_4" class="fnanchor">[4]</a></p> + +<p>Huygens derived a theorem which has provided +the basis for the employment of the reversible compound +pendulum for the absolute determination of +the intensity of gravity. The theorem is that a given +compound pendulum possesses conjugate points on +opposite sides of the center of gravity; about these +points, the periods of oscillation are the same. For +each of these points as center of suspension the other +point is the center of oscillation, and the distance +between them is the length of the equivalent simple +pendulum. Earlier, in 1657, Huygens independently +had invented and patented the pendulum clock, +which rapidly came into use for the measurement +of time. Huygens also created the theory of centripetal +force which made it possible to calculate the +effect of the rotation of the earth upon the observed +value of gravity.</p> + +<p>The theory of the gravity field of the earth was +founded upon the laws of motion and the law of +gravitation by Isaac Newton in his famous <i>Principia</i> +(1687). It follows from the Newtonian theory +of gravitation that the acceleration of gravity as +determined on the surface of the earth is the +resultant of two factors: the principal factor is the +gravitational attraction of the earth upon bodies, +and the subsidiary factor is the effect of the rotation +of the earth. A body at rest on the surface of the +earth requires some of the gravitational attraction +for the centripetal acceleration of the body as it is +carried in a circle with constant speed by the rotation +of the earth about its axis. If the rotating earth is +used as a frame of reference, the effect of the rotation +is expressed as a centrifugal force which acts to +diminish the observed intensity of gravity.</p> + +<div class="sidebar"> +<p class="center smcap">Glossary of Gravity Terminology</p> + +<p class="hang">ABSOLUTE GRAVITY: the value of the acceleration of gravity, +also expressed by the length of the seconds pendulum.</p> + +<p class="hang">RELATIVE GRAVITY: the value of the acceleration of gravity +relative to the value at some standard point.</p> + +<p class="hang">SIMPLE PENDULUM: see theoretical pendulum.</p> + +<p class="hang">THEORETICAL PENDULUM: a heavy bob (point-mass) at the +end of a weightless rod.</p> + +<p class="hang">SECONDS PENDULUM: a theoretical or simple pendulum of such +length that its time of swing (half-period) is one second. +(This length is about one meter.)</p> + +<p class="hang">GRAVITY PENDULUM: a precisely made pendulum used for +the measurement of gravity.</p> + +<p class="hang">COMPOUND PENDULUM: a pendulum in which the supporting +rod is not weightless; in other words, any actual pendulum.</p> + +<p class="hang">CONVERTIBLE PENDULUM: a compound pendulum having +knife edges at different distances from the center of gravity. +Huygens demonstrated (1673) that if such a pendulum +were to swing with equal periods from either knife edge, +the distance between those knife edges would be equal to +the length of a theoretical or simple pendulum of the same +period.</p> + +<p class="hang">REVERSIBLE PENDULUM: a convertible pendulum which is +also symmetrical in form.</p> + +<p class="hang">INVARIABLE PENDULUM: a compound pendulum with only +one knife edge, used for relative measurement of gravity.</p> + +</div> + +<p>From Newton’s laws of motion and the hypothesis +that weight is proportional to mass, the formula for +the half-period of a simple pendulum is given by +<i>T</i> = <span class="nowrap">π√(<i>l</i>/<i>g</i>)</span>. If a simple pendulum beats seconds, +1 = <span class="nowrap">π√(λ/<i>g</i>)</span>, where λ is the length of the seconds +pendulum. From <i>T</i> = <span class="nowrap">π√(<i>l</i>/<i>g</i>)</span> and 1 = <span class="nowrap">π√(λ/<i>g</i>)</span>, it follows +that λ = <span class="nowrap"><i>l</i>/<i>T</i><sup>2</sup></span>. Then <i>g</i> = <span class="nowrap">π<sup>2</sup>λ</span>. Thus, the intensity of +gravity can be expressed in terms of the length of +the seconds pendulum, as well as by the acceleration +of a freely falling body. During the 19th century, +gravity usually was expressed in terms of the length +of the seconds pendulum, but present practice is to +express gravity in terms of <i>g</i>, for which the unit is +the gal, or one centimeter per second per second.<span class="pagenum"><a name="Page_306" id="Page_306">[Pg 306]</a></span></p> + +<div class="figcenter" style="width: 800px;"><a name="fig_2" id="fig_2"></a> +<img src="images/i010.png" width="800" height="275" alt="Figure 2." title="Figure 2." /> + +<p class="caption2">Figure 2.—<span class="smcap">This drawing, from Richer’s</span> <i>Observations astronomiques et physiques faites en l’isle de +Caďenne</i> (Paris, 1679), shows most of the astronomical instruments used by Richer, namely, +one of the two pendulum clocks made by Thuret, the 20-foot and the 5-foot telescopes and +the large quadrant. The figure may be intended as a portrait of Richer. This drawing +was done by Sebastian Le Clerc, a young illustrator who made many illustrations of the +early work of the Paris Academy.</p> +</div> + + + +<hr style="width: 65%;" /> +<h3>Figure of the Earth</h3> + + +<p>A principal contribution of the pendulum as a +physical instrument has been the determination of the +figure of the earth.<a name="FNanchor_5_5" id="FNanchor_5_5"></a><a href="#Footnote_5_5" class="fnanchor">[5]</a> That the earth is spherical in +form was accepted doctrine among the ancient +Greeks. Pythagoras is said to have been the first to +describe the earth as a sphere, and this view was +adopted by Eudoxus and Aristotle.</p> + +<p>The Alexandrian scientist Eratosthenes made the +first estimate of the diameter and circumference of +a supposedly spherical earth by an astronomical-geodetic +method. He measured the angle between +the directions of the rays of the sun at Alexandria +and Syene (Aswan), Egypt, and estimated the distance +between these places from the length of time +required by a caravan of camels to travel between +them. From the central angle corresponding to the +arc on the surface, he calculated the radius and hence +the circumference of the earth. A second measurement +was undertaken by Posidonius, who measured +the altitudes of stars at Alexandria and Rhodes and +estimated the distance between them from the time +required to sail from one place to the other.</p> + +<p>With the decline of classical antiquity, the doctrine +of the spherical shape of the earth was lost, and only +one investigation, that by the Arabs under Calif +Al-Mamun in A.D. 827, is recorded until the 16th +century. In 1525, the French mathematician Fernel +measured the length of a degree of latitude between +Paris and Amiens by the revolutions of the wheels of +his carriage, the circumference of which he had determined. +In England, Norwood in 1635 measured +the length of an arc between London and York with a +chain. An important forward step in geodesy was +the measurement of distance by triangulation, first +by Tycho Brahe, in Denmark, and later, in 1615, by +Willebrord Snell, in Holland.</p> + +<p>Of historic importance, was the use of telescopes in +the triangulation for the measurement of a degree of +arc by the Abbé Jean Picard in 1669.<a name="FNanchor_6_6" id="FNanchor_6_6"></a><a href="#Footnote_6_6" class="fnanchor">[6]</a> He had been +commissioned by the newly established Academy of +Sciences to measure an arc corresponding to an angle +of 1°, 22′, 55″ of the meridian between Amiens +and Malvoisine, near Paris. Picard proposed to the +Academy the measurement of the meridian of Paris +through all of France, and this project was supported +by Colbert, who obtained the approval of the King. +In 1684, Giovanni-Domenico Cassini and De la Hire +commenced a trigonometrical measure of an arc +south of Paris; subsequently, Jacques Cassini, the son<span class="pagenum"><a name="Page_307" id="Page_307">[Pg 307]</a></span> +of Giovanni-Domenico, added the arc to the north +of Paris. The project was completed in 1718. The +length of a degree of arc south of Paris was found to +be greater than the length north of Paris. From the +difference, 57,097 toises<a name="FNanchor_7_7" id="FNanchor_7_7"></a><a href="#Footnote_7_7" class="fnanchor">[7]</a> minus 56,960 toises, it was +concluded that the polar diameter of the earth is larger +than the equatorial diameter, i.e., that the earth is a +prolate spheroid (fig. <a href="#fig_3">3</a>).</p> + +<div class="figright" style="width: 300px;"><a name="fig_3" id="fig_3"></a> +<img src="images/i013.png" width="300" height="341" alt="Figure 3." title="Figure 3." /> + +<p class="caption2">Figure 3.—<span class="smcap">Measurements of the length</span> of a degree +of latitude which were completed in different parts +of France in 1669 and 1718 gave differing results +which suggested that the shape of the earth is not +a sphere but a prolate spheroid (1). But Richer’s +pendulum observation of 1672, as explained by +Huygens and Newton, indicated that its shape is +that of an oblate spheroid (2). The disagreement +is reflected in this drawing. In the 1730’s it +was resolved in favor of the latter view by two +French geodetic expeditions for the measurement +of degrees of latitude in the equatorial and polar +regions (Ecuador—then part of Peru—and Lapland).</p> +</div> +<p>Meanwhile, Richer in 1672 had been sent to +Cayenne, French Guiana, to make astronomical +observations and to measure the length of the seconds +pendulum.<a name="FNanchor_8_8" id="FNanchor_8_8"></a><a href="#Footnote_8_8" class="fnanchor">[8]</a> He took with him a pendulum clock +which had been adjusted to keep accurate time in +Paris. At Cayenne, however, Richer found that the +clock was retarded by 2 minutes and 28 seconds per +day (fig. <a href="#fig_1">1</a>). He also fitted up a “simple” pendulum to +vibrate in seconds and measured the length of this +seconds pendulum several times every week for 10 +months. Upon his return to Paris, he found that +the length of the “simple” pendulum which beat +seconds at Cayenne was 1-1/4 Paris lines<a name="FNanchor_9_9" id="FNanchor_9_9"></a><a href="#Footnote_9_9" class="fnanchor">[9]</a> shorter than +the length of the seconds pendulum at Paris. Huygens +explained the reduction in the length of the seconds +pendulum—and, therefore, the lesser intensity of +gravity at the equator with respect to the value at +Paris—in terms of his theory of centripetal force as +applied to the rotation of the earth and pendulum.<a name="FNanchor_10_10" id="FNanchor_10_10"></a><a href="#Footnote_10_10" class="fnanchor">[10]</a></p> + +<p>A more complete theory was given by Newton in +the <i>Principia</i>.<a name="FNanchor_11_11" id="FNanchor_11_11"></a><a href="#Footnote_11_11" class="fnanchor">[11]</a> Newton showed that if the earth is +assumed to be a homogeneous, mutually gravitating +fluid globe, its rotation will result in a bulging at the +equator. The earth will then have the form of an +oblate spheroid, and the intensity of gravity as a +form of universal gravitation will vary with position +on the surface of the earth. Newton took into +account gravitational attraction and centrifugal action, +and he calculated the ratio of the axes of the +spheroid to be 230:229. He calculated and prepared +a table of the lengths of a degree of latitude and of +the seconds pendulum for every 5° of latitude from +the equator to the pole. A discrepancy between his +predicted length of the seconds pendulum at the +equator and Richer’s measured length was explained +by Newton in terms of the expansion of the scale +with higher temperatures near the equator.</p> + +<p>Newton’s theory that the earth is an oblate spheroid +was confirmed by the measurements of Richer, but +was rejected by the Paris Academy of Sciences, for +it contradicted the results of the Cassinis, father and +son, whose measurements of arcs to the south and +north of Paris had led to the conclusion that the +earth is a prolate spheroid. Thus, a controversy +arose between the English scientists and the Paris +Academy. The conflict was finally resolved by the +results of expeditions sent by the Academy to Peru +and Sweden. The first expedition, under Bouguer, +La Condamine, and Godin in 1735, went to a region<span class="pagenum"><a name="Page_308" id="Page_308">[Pg 308]</a></span> +in Peru, and, with the help of the Spaniard Ullo, +measured a meridian arc of about 3°7′ near Quito, +now in Ecuador.<a name="FNanchor_12_12" id="FNanchor_12_12"></a><a href="#Footnote_12_12" class="fnanchor">[12]</a> The second expedition, with +Maupertuis and Clairaut in 1736, went to Lapland +within the Arctic Circle and measured an arc of +about 1° in length.<a name="FNanchor_13_13" id="FNanchor_13_13"></a><a href="#Footnote_13_13" class="fnanchor">[13]</a> The northern arc of 1° was +found to be longer than the Peruvian arc of 1°, and +thus it was confirmed that the earth is an oblate +spheroid, that is, flattened at the poles, as predicted +by the theory of Newton.</p> + +<div class="figcenter" style="width: 600px;"><a name="fig_4" id="fig_4"></a> +<img src="images/i015.png" width="600" height="390" alt="Figure 4." title="Figure 4." /> + +<p class="caption2">Figure 4.—<span class="smcap">The direct use of a clock</span> to measure the force of gravity was found to be limited in accuracy +by the necessary mechanical connection of the pendulum to the clock, and by the unavoidable difference +between the characteristics of a clock pendulum and those of a theoretical (usually called “simple”) pendulum, +in which the mass is concentrated in the bob, and the supporting rod is weightless.</p> + +<p class="caption2">After 1735, the clock was used only to time the swing of a detached pendulum, by the method of “coincidences.” +In this method, invented by J. J. Mairan, the length of the detached pendulum is first accurately +measured, and the clock is corrected by astronomical observation. The detached pendulum is then +swung before the clock pendulum as shown here. The two pendulums swing more or less out of phase, +coming into coincidence each time one has gained a vibration. By counting the number of coincidences +over several hours, the period of the detached pendulum can be very accurately determined. The length +and period of the detached pendulum are the data required for the calculation of the force of gravity.</p> +</div> + +<p>The period from Eratosthenes to Picard has been +called the spherical era of geodesy; the period from +Picard to the end of the 19th century has been called +the ellipsoidal period. During the latter period the +earth was conceived to be an ellipsoid, and the determination +of its ellipticity, that is, the difference of +equatorial radius and polar radius divided by the +equatorial radius, became an important geodetic +problem. A significant contribution to the solution +of this problem was made by determinations of +gravity by the pendulum.</p> + +<p>An epoch-making work during the ellipsoidal era<span class="pagenum"><a name="Page_309" id="Page_309">[Pg 309]</a></span> +of geodesy was Clairaut’s treatise, <i>Théorie de la figure +de la terre</i>.<a name="FNanchor_14_14" id="FNanchor_14_14"></a><a href="#Footnote_14_14" class="fnanchor">[14]</a> On the hypothesis that the earth is a +spheroid of equilibrium, that is, such that a layer of +water would spread all over it, and that the internal +density varies so that layers of equal density are +coaxial spheroids, Clairaut derived a historic +theorem: If γ<sub>E</sub>, γ<sub>P</sub> are the values of gravity at the +equator and pole, respectively, and <i>c</i> the centrifugal +force at the equator divided by γ<sub>E</sub>, then the ellipticity +α = <span class="nowrap">(5/2)<i>c</i> - (γ<sub>P</sub> - γ<sub>E</sub>)/γ<sub>E</sub>.</span></p> + +<p>Laplace showed that the surfaces of equal density +might have any nearly spherical form, and Stokes +showed that it is unnecessary to assume any law of +density as long as the external surface is a spheroid of +equilibrium.<a name="FNanchor_15_15" id="FNanchor_15_15"></a><a href="#Footnote_15_15" class="fnanchor">[15]</a> It follows from Clairaut’s theorem +that if the earth is an oblate spheroid, its ellipticity +can be determined from relative values of gravity +and the absolute value at the equator involved in <i>c</i>. +Observations with nonreversible, invariable compound +pendulums have contributed to the application +of Clairaut’s theorem in its original and contemporary +extended form for the determination of the figure and +gravity field of the earth.</p> + + + +<hr style="width: 65%;" /> +<h3>Early Types of Pendulums</h3> + + +<p>The pendulum employed in observations of gravity +prior to the 19th century usually consisted of a small +weight suspended by a filament (figs. 4-6). The +pioneer experimenters with “simple” pendulums +changed the length of the suspension until the pendulum +beat seconds. Picard in 1669 determined the +length of the seconds pendulum at Paris with a +“simple” pendulum which consisted of a copper ball +an inch in diameter suspended by a fiber of pite +from jaws (pite was a preparation of the leaf of a +species of aloe and was not affected appreciably by +moisture).</p> + +<p>A celebrated set of experiments with a “simple” +pendulum was conducted by Bouguer<a name="FNanchor_16_16" id="FNanchor_16_16"></a><a href="#Footnote_16_16" class="fnanchor">[16]</a> in 1737 in +the Andes, as part of the expedition to measure the +Peruvian arc. The bob of the pendulum was a double +truncated cone, and the length was measured from +the jaw suspension to the center of oscillation of the +thread and bob. Bouguer allowed for change of +length of his measuring rod with temperature and +also for the buoyancy of the air. He determined the +time of swing by an elementary form of the method +of coincidences. The thread of the pendulum was +swung in front of a scale and Bouguer observed how +long it took the pendulum to lose a number of vibrations +on the seconds clock. For this purpose, he +noted the time when the beat of the clock was heard +and, simultaneously, the thread moved past the +center of the scale. A historic aspect of Bouguer’s +method was that he employed an “invariable” +pendulum, that is, the length was maintained the +same at the various stations of observation, a procedure +that has been described as having been +invented by Bouguer.</p> + +<p>Since <i>T</i> = <span class="nowrap">π√(<i>l</i>/<i>g</i>)</span>, it follows that <span class="nowrap"><i>T</i><sub>1</sub><sup>2</sup>/<i>T</i><sub>2</sub><sup>2</sup></span> = <span class="nowrap"><i>g</i><sub>2</sub>/<i>g</i><sub>1</sub></span>. +Thus, if the absolute value of gravity is known at one +station, the value at any other station can be determined +from the ratio of the squares of times of swing +of an invariable pendulum at the two stations. From +the above equation, if <i>T</i><sub>1</sub> is the time of swing at a +station where the intensity of gravity is <i>g</i>, and <i>T</i><sub>2</sub> +is the time at a station where the intensity is <i>g</i> + Δ<i>g</i>, +then <span class="nowrap">(Δ<i>g</i>)/<i>g</i></span> = <span class="nowrap">(<i>T</i><sub>1</sub><sup>2</sup>/<i>T</i><sub>2</sub><sup>2</sup>) - 1.</span></p> + +<p>Bouguer’s investigations with his invariable pendulum +yielded methods for the determination of the internal +structure of the earth. On the Peruvian +expedition, he determined the length of the seconds +pendulum at three stations, including one at Quito, +at varying distances above sea level. If values of +gravity at stations of different elevation are to be +compared, they must be reduced to the same level, +usually to sea level. Since gravity decreases with +height above sea level in accordance with the law of +gravitation, a free-air reduction must be applied to +values of gravity determined above the level of the +sea. Bouguer originated the additional reduction for +the increase in gravity on a mountain or plateau +caused by the attraction of the matter in a plate. +From the relative values of gravity at elevated stations +in Peru and at sea level, Bouguer calculated +that the mean density of the earth was 4.7 times +greater than that of the <i>cordilleras</i>.<a name="FNanchor_17_17" id="FNanchor_17_17"></a><a href="#Footnote_17_17" class="fnanchor">[17]</a> For greater +accuracy in the study of the internal structure of the +earth, in the 19th century the Bouguer plate reduction<span class="pagenum"><a name="Page_310" id="Page_310">[Pg 310]</a></span> +came to be supplemented by corrections for irregularities +of terrain and by different types of isostatic +reduction.</p> + +<p>La Condamine, who like Bouguer was a member of +the Peruvian expedition, conducted his own pendulum +experiments (fig. <a href="#fig_4">4</a>). He experimented in 1735 at +Santo Domingo en route to South America,<a name="FNanchor_18_18" id="FNanchor_18_18"></a><a href="#Footnote_18_18" class="fnanchor">[18]</a> then +at various stations in South America, and again at +Paris upon his return to France. His pendulum consisted +of a copper ball suspended by a thread of pite. +For experimentation the length initially was about 12 +feet, and the time of swing 2 seconds, but then the +length was reduced to about 3 feet with time of swing +1 second. Earlier, when it was believed that gravity +was constant over the earth, Picard and others had +proposed that the length of the seconds pendulum be +chosen as the standard. La Condamine in 1747 +revived the proposal in the form that the length of the +seconds pendulum at the equator be adopted as the +standard of length. Subsequently, he investigated the +expansion of a toise of iron from the variation in the +period of his pendulum. In 1755, he observed the +pendulum at Rome with Boscovich. La Condamine’s +pendulum was used by other observers and finally was +lost at sea on an expedition around the world. The +knowledge of the pendulum acquired by the end of the +18th century was summarized in 1785 in a memoir by +Boscovich.<a name="FNanchor_19_19" id="FNanchor_19_19"></a><a href="#Footnote_19_19" class="fnanchor">[19]</a></p> + +<div class="figcenter" style="width: 740px;"> +<div class="figleft" style="width: 300px;"><a name="fig_5" id="fig_5"></a> +<img src="images/i021.png" width="300" height="531" alt="Figure 5." title="Figure 5." /> +</div> +<div class="figright" style="width: 400px;"> +<p class="caption2">Figure 5.—<span class="smcap">An apparatus for the practice measurement</span> +of the length of the pendulum devised on the basis of a +series of preliminary experiments by C. M. de la Condamine +who, in the course of the French geodetic expedition +to Peru in 1735, devoted a 3-month sojourn on the island +of Santo Domingo to pendulum observations by Mairan’s +Method. In this arrangement, shown here, a vertical rod +of ironwood is used both as the scale and as the support +for the apparatus, having at its top the brass pendulum +support (A) and, below, a horizontal mirror (O) which +serves to align the apparatus vertically through visual +observation of the reflection of the pointer projecting from +A. The pendulum, about 37 inches long, consists of a +thread of pite (a humidity-resistant, natural fiber) and +a copper ball of about 6 ounces. Its exact length is +determined by adjusting the micrometer (S) so that the +ball nearly touches the mirror. It will be noted that the +clock pendulum would be obscured by the scale. La +Condamine seems to have determined the times of +coincidence by visual observation of the occasions on +which “the pendulums swing parallel.” (Portion of plate +1, <i>Mémoires publiés par la Société française de Physique</i>, vol. 4.)</p> +</div> +</div> +<p><span class="pagenum"><a name="Page_311" id="Page_311">[Pg 311]</a></span></p> + +<div class="figcenter" style="width: 600px;"><a name="fig_6" id="fig_6"></a> +<img src="images/i023.png" width="600" height="675" alt="Figure 6." title="Figure 6." /> + +<p class="caption2">Figure 6.—<span class="smcap">The result</span> of early pendulum +experiments was often expressed in terms of the +length of a pendulum which would have a +period of one second and was called “the seconds +pendulum.” In 1792, J. C. Borda and J. D. +Cassini determined the length of the seconds +pendulum at Paris with this apparatus. The +pendulum consists of a platinum ball about 1-1/2 +inches in diameter, suspended by a fine iron +wire. The length, about 12 feet, was such that +its period would be nearly twice as long as that +of the pendulum of the clock (A). The interval +between coincidences was determined by observing, +through the telescope at the left, the +times when the two pendulums emerge together +from behind the screen (M). The exact length +of the pendulum was measured by a platinum +scale (not shown) equipped with a vernier and +an auxiliary copper scale for temperature +correction.</p> + +<p class="caption2">When, at the end of the 18th century, the +French revolutionary government established +the metric system of weights and measures, the +length of the seconds pendulum at Paris was +considered, but not adopted, as the unit of +length. (Plate 2, <i>Mémoires publiés par la Société +française de Physique</i>, vol. 4.)</p> +</div> + +<p>The practice with the “simple” pendulum on the +part of Picard, Bouguer, La Condamine and others +in France culminated in the work of Borda and Cassini +in 1792 at the observatory in Paris<a name="FNanchor_20_20" id="FNanchor_20_20"></a><a href="#Footnote_20_20" class="fnanchor">[20]</a> (fig. <a href="#fig_6">6</a>). The<span class="pagenum"><a name="Page_312" id="Page_312">[Pg 312]</a></span> +experiments were undertaken to determine whether +or not the length of the seconds pendulum should be +adopted as the standard of length by the new government +of France. The bob consisted of a platinum +ball 16-1/6 Paris lines in diameter, and 9,911 grains +(slightly more than 17 ounces) in weight. The bob +was held to a brass cup covering about one-fifth of +its surface by the interposition of a small quantity of +grease. The cup with ball was hung by a fine iron +wire about 12 Paris feet long. The upper end of the +wire was attached to a cylinder which was part of a +wedge-shaped knife edge, on the upper surface of +which was a stem on which a small adjustable weight +was held by a screw thread. The knife edge rested on +a steel plate. The weight on the knife-edge apparatus +was adjusted so that the apparatus would vibrate +with the same period as the pendulum. Thus, the +mass of the suspending apparatus could be neglected +in the theory of motion of the pendulum about the +knife edge.</p> + +<div class="figright" style="width: 400px;"><a name="fig_7" id="fig_7"></a> +<img src="images/i024.png" width="300" height="751" alt="Figure 7." title="Figure 7." /> + +<p class="caption2">Figure 7.—<span class="smcap">Results of experiments</span> in the determination +of the length of the seconds pendulum at Königsberg by a +new method were reported by F. W. Bessel in 1826 and +published in 1828. With this apparatus, he obtained two +sets of data from the same pendulum, by using two different +points of suspension. The pendulum was about 10 +feet long. The distance between the two points of suspension +(<i>a</i> and <i>b</i>) was 1 toise (about six feet). A micrometric +balance (<i>c</i>) below the bob was used to determine the increase +in length due to the weight of the bob. He projected +the image of the clock pendulum (not shown) onto +the gravity pendulum by means of a lens, thus placing the +clock some distance away and eliminating the disturbing +effect of its motion. (Portion of plate 6, <i>Mémoires publiés +par la Société française de Physique</i>, vol. 4.)</p> +</div> + +<p>In the earlier suspension from jaws there was uncertainty +as to the point about which the pendulum +oscillated. Borda and Cassini hung their pendulum +in front of a seconds clock and determined the time of +swing by the method of coincidences. The times on +the clock were observed when the clock gained or +lost one complete vibration (two swings) on the pendulum. +Suppose that the wire pendulum makes <i>n</i> +swings while the clock makes <span class="nowrap">2<i>n</i> + 2</span>. If the clock +beats seconds exactly, the time of one complete +vibration is 2 seconds, and the time of swing of the +wire pendulum is <i>T</i> = <span class="nowrap">(2<i>n</i> + 2)/<i>n</i></span> = <span class="nowrap">2(1 + 1/<i>n</i>)</span>. An error +in the time caused by uncertainty in determining the +coincidence of clock and wire pendulum is reduced<span class="pagenum"><a name="Page_313" id="Page_313">[Pg 313]</a></span> +by employing a long interval of observation 2<i>n</i>. The +whole apparatus was enclosed in a box, in order to +exclude disturbances from currents of air. Corrections +were made for buoyancy, for amplitude of swing and +for variations in length of the wire with temperature. +The final result was that the length of the seconds +pendulum at the observatory in Paris was determined +to be 440.5593 Paris lines, or 993.53 mm., reduced +to sea level 993.85 mm. Some years later the methods +of Borda were used by other French investigators, +among whom was Biot who used the platinum ball of +Borda suspended by a copper wire 60 cm. long.</p> + +<p>Another historic “simple” pendulum was the one +swung by Bessel (fig. <a href="#fig_7">7</a>) for the determination of +gravity at Königsberg 1825-1827.<a name="FNanchor_21_21" id="FNanchor_21_21"></a><a href="#Footnote_21_21" class="fnanchor">[21]</a> The pendulum +consisted of a ball of brass, copper, or ivory that was +suspended by a fine wire, the upper end of which +was wrapped and unwrapped on a horizontal +cylinder as support. The pendulum was swung +first from one point and then from another, exactly +a “toise de Peru”<a name="FNanchor_22_22" id="FNanchor_22_22"></a><a href="#Footnote_22_22" class="fnanchor">[22]</a> higher up, the bob being at +the same level in each case (fig. <a href="#fig_7">7</a>). Bessel found the +period of vibration of the pendulum by the method +of coincidences; and in order to avoid disturbances +from the comparison clock, it was placed at some +distance from the pendulum under observation.</p> + +<p>Bessel’s experiments were significant in view of the +care with which he determined the corrections. He +corrected for the stiffness of the wire and for the lack +of rigidity of connection between the bob and wire. +The necessity for the latter correction had been +pointed out by Laplace, who showed that through +the circumstance that the pull of the wire is now on +one side and now on the other side of the center of +gravity, the bob acquires angular momentum about +its center of gravity, which cannot be accounted for +if the line of the wire, and therefore the force that it +exerts, always passed through the center. In addition +to a correction for buoyancy of the air considered +by his predecessors, Bessel also took account of the +inertia of the air set in motion by the pendulum.</p> + +<div class="figcenter" style="width: 600px;"><a name="fig_8" id="fig_8"></a> +<img src="images/i025.png" width="600" height="597" alt="Figure 8." title="Figure 8." /> + +<p class="caption2">Figure 8.—<span class="smcap">Mode of suspension</span> of Bessel’s pendulum is +shown here. The iron wire is supported by the thumbscrew +and clamp at the left, but passes over a pin at the +center, which is actually the upper terminal of the pendulum. +Bessel found this “cylinder of unrolling” superior +to the clamps and knife edges of earlier pendulums. The +counterweight at the right is part of a system for supporting +the scale in such a way that it is not elongated by its own +weight.</p> + +<p class="caption2">With this apparatus, Bessel determined the ratio of +the lengths of the two pendulums and their times of vibration. +From this the length of the seconds pendulum was +calculated. His method eliminated the need to take into +account such sources of inaccuracy as flexure of the pendulum +wire and imperfections in the shape of the bob. +(Portion of plate 7, <i>Mémoires publiés par la Société française +de Physique</i>, vol. 4.)</p> +</div> + +<p><span class="pagenum"><a name="Page_314" id="Page_314">[Pg 314]</a></span></p> + +<div class="figcenter" style="width: 600px;"><a name="fig_9" id="fig_9"></a> +<img src="images/i027.png" width="300" height="379" alt="Figure 9." title="Figure 9." /> + +<p class="caption2">Figure 9.—<span class="smcap">Friedrich Wilhelm Bessel</span> (1784-1846), +German mathematician and astronomer. He became +the first superintendent of the Prussian observatory +established at Königsberg in 1810, and +remained there during the remainder of his life. +So important were his many contributions to precise +measurement and calculation in astronomy that he +is often considered the founder of the “modern” age +in that science. This characteristic also shows in +his venture into geodesy, 1826-1830, one product +of which was the pendulum experiment reported in +this article.</p> +</div> + +<p>The latter effect had been discovered by Du Buat in +1786,<a name="FNanchor_23_23" id="FNanchor_23_23"></a><a href="#Footnote_23_23" class="fnanchor">[23]</a> but his work was unknown to Bessel. The +length of the seconds pendulum at Königsberg, +reduced to sea level, was found by Bessel to be +440.8179 lines. In 1835, Bessel determined the +intensity of gravity at a site in Berlin where observations +later were conducted in the Imperial Office of +Weights and Measures by Charles S. Peirce of the +U.S. Coast Survey.</p> + + + +<hr style="width: 65%;" /> +<h3>Kater’s Convertible and Invariable Pendulums</h3> + + +<div class="figright" style="width: 360px;"><a name="fig_10" id="fig_10"></a> +<img src="images/i029.png" width="300" height="372" alt="Figure 10." title="Figure 10." /> + +<p class="caption2">Figure 10.—<span class="smcap">Henry Kater</span> (1777-1835), +English army officer and +physicist. His scientific career began +during his military service in +India, where he assisted in the +“great trigonometrical survey.” +Returned to England because of +bad health, and retired in 1814, he +pioneered (1818) in the development +of the convertible pendulum +as an alternative to the approximation +of the “simple” pendulum for +the measurement of the “seconds +pendulum.” Kater’s convertible +pendulum and the invariable pendulum +introduced by him in 1819 +were the basis of English pendulum +work. (<i>Photo courtesy National Portrait +Gallery, London.</i>)</p> +</div> + +<div class="figright" style="width: 360px;"> +<div class="figleft" style="width: 75px;"><a name="fig_11" id="fig_11"></a> +<img src="images/i030.png" width="75" height="710" alt="Figure 11." title="Figure 11." /> +</div> + +<div class="figright" style="width: 250px;"> +<p class="caption2">Figure 11.—<span class="smcap">The attempt to approximate</span> +the simple (theoretical) pendulum in +gravity experiments ended in 1817-18 +when Henry Kater invented the compound +convertible pendulum, from which +the equivalent simple pendulum could be +obtained according to the method of +Huygens (see text, p. <a href="#Page_314">314</a>). Developed in +connection with a project to fix the +standard of English measure, Kater’s pendulum +was called "compound" because it +was a solid bar rather than the fine wire +or string with which earlier experimenters +had tried to approximate a "weightless" +rod. It was called convertible because it +is alternately swung from the two knife +edges (<i>a</i> and <i>b</i>) at opposite ends. The +weights (<i>f</i> and <i>g</i>) are adjusted so that the +period of the pendulum is the same from +either knife edge. The distance between +the two knife edges is then equal to the +length of the equivalent simple pendulum.</p> +</div> +</div> + +<p>The systematic survey of the gravity field of the +earth was given a great impetus by the contributions +of Capt. Henry Kater, F.R.S. In 1817, he designed, +constructed, and applied a convertible compound +pendulum for the absolute determination of gravity +at the house of Henry Browne, F.R.S., in Portland +Place, London.<a name="FNanchor_24_24" id="FNanchor_24_24"></a><a href="#Footnote_24_24" class="fnanchor">[24]</a> Kater’s convertible pendulum (fig. +11) consisted of a brass rod to which were attached a +flat circular bob of brass and two adjustable weights, +the smaller of which was adjusted by a screw. The +convertibility of the pendulum was constituted by the +provision of two knife edges turned inwards on +opposite sides of the center of gravity. The pendulum +was swung on each knife edge, and the adjustable +weights were moved until the times of swing were the +same about each knife edge. When the times were +judged to be the same, the distance between the +knife edges was inferred to be the length of the +equivalent simple pendulum, in accordance with +Huygens’ theorem on conjugate points of a compound +pendulum. Kater determined the time of swing by +the method of coincidences (fig. <a href="#fig_12">12</a>). He corrected +for the buoyancy of the air. The final value of the +length of the seconds pendulum at Browne’s house in +London, reduced to sea level, was determined to be +39.13929 inches.</p> + +<p>The convertible compound pendulum had been +conceived prior to its realization by Kater. In 1792, +on the occasion of the proposal in Paris to establish +the standard of length as the length of the seconds +pendulum, Baron de Prony had proposed the employment +of a compound pendulum with three axes of +oscillation.<a name="FNanchor_25_25" id="FNanchor_25_25"></a><a href="#Footnote_25_25" class="fnanchor">[25]</a> In 1800, he proposed the convertible +compound pendulum with knife edges about which +the pendulum could complete swings in equal times. +De Prony’s proposals were not accepted and his +papers remained unpublished until 1889, at which +time they were discovered by Defforges. The French +decision was to experiment with the ball pendulum, +and the determination of the length of the seconds<span class="pagenum"><a name="Page_315" id="Page_315">[Pg 315]</a></span> +pendulum was carried out by Borda and Cassini by +methods previously described. Bohnenberger in his +<i>Astronomie</i> (1811),<a name="FNanchor_26_26" id="FNanchor_26_26"></a><a href="#Footnote_26_26" class="fnanchor">[26]</a> made the proposal to employ a +convertible pendulum for the absolute determination +of gravity; thus, he has received credit for priority in +publication. Capt. Kater independently conceived +of the convertible pendulum and was the first to +design, construct, and swing one.</p> + +<p>After his observations with the convertible pendulum, +Capt. Kater designed an invariable compound +pendulum with a single knife edge but otherwise similar +in external form to the convertible pendulum<a name="FNanchor_27_27" id="FNanchor_27_27"></a><a href="#Footnote_27_27" class="fnanchor">[27]</a> +(fig. <a href="#fig_13">13</a>). Thirteen of these Kater invariable pendulums +have been reported as constructed and swung +at stations throughout the world.<a name="FNanchor_28_28" id="FNanchor_28_28"></a><a href="#Footnote_28_28" class="fnanchor">[28]</a> Kater himself +swung an invariable pendulum at a station in London +and at various other stations in the British Isles. +Capt. Edward Sabine, between 1820 and 1825, made +voyages and swung Kater invariable pendulums at +stations from the West Indies to Greenland and +Spitzbergen.<a name="FNanchor_29_29" id="FNanchor_29_29"></a><a href="#Footnote_29_29" class="fnanchor">[29]</a> In 1820, Kater swung a Kater invariable +pendulum at London and then sent it to Goldingham, +who swung it in 1821 at Madras, India.<a name="FNanchor_30_30" id="FNanchor_30_30"></a><a href="#Footnote_30_30" class="fnanchor">[30]</a> Also +in 1820, Kater supplied an invariable pendulum to +Hall, who swung it at London and then made observations +near the equator and in the Southern Hemisphere, +and at London again in 1823.<a name="FNanchor_31_31" id="FNanchor_31_31"></a><a href="#Footnote_31_31" class="fnanchor">[31]</a> The same +pendulum, after its knives were reground, was delivered +to Adm. Lütke of Russia, who observed +gravity with it on a trip around the world between +1826 and 1829.<a name="FNanchor_32_32" id="FNanchor_32_32"></a><a href="#Footnote_32_32" class="fnanchor">[32]</a></p> + +<p><span class="pagenum"><a name="Page_316" id="Page_316">[Pg 316]</a></span></p> + +<div class="figcenter" style="width: 600px;"><a name="fig_12" id="fig_12"></a> +<img src="images/i031.png" width="600" height="562" alt="Figure 12." title="Figure 12." /> + +<p class="caption2">Figure 12.—<span class="smcap">The Kater convertible pendulum</span> in use is placed before a clock, whose +pendulum bob is directly behind the extended “tail” of the Kater pendulum. A white +spot is painted on the center of the bob of the clock pendulum. The observing telescope, +left, has a diaphragm with a vertical slit of such width that its view is just filled by the tail +of the Kater pendulum when it is at rest. When the two pendulums are swinging, the +white spot on the clock pendulum can be seen on each swing except that in which the two +pendulums are in coincidence; thus, the coincidences are determined. (Portion of plate +5, <i>Mémoires publiés par la Société française de Physique</i>, vol. 4.)<span class="pagenum"><a name="Page_317" id="Page_317">[Pg 317]</a></span></p> +</div> + +<div class="figcenter" style="width: 600px;"><a name="fig_13" id="fig_13"></a> +<img src="images/i034.png" width="600" height="420" alt="Figure 13." title="Figure 13." /> + +<p class="caption2">Figure 13.—<span class="smcap">This drawing accompanied</span> John Goldingham’s report on the work done in India with Kater’s +invariable pendulum. The value of gravity obtained, directly or indirectly, in terms of the simple pendulum, +is called “absolute.” Once absolute values of gravity were established at a number of stations, it became +possible to use the much simpler “relative” method for the measurement of gravity at new stations. +Because it has only one knife edge, and does not involve the adjustments of the convertible pendulum, this +one is called “invariable.” In use, it is first swung at a station where the absolute value of gravity has been +established, and this period is then compared with its period at one or more new stations. Kater developed +an invariable pendulum in 1819, which was used in England and in Madras, India, in 1821.</p> +</div> + +<div class="figright" style="width: 350px;"><a name="fig_14" id="fig_14"></a> +<img src="images/i037.png" width="300" height="400" alt="Figure 14." title="Figure 14." /> + +<p class="caption2">Figure 14.—<span class="smcap">Vacuum chamber for use</span> with +the Kater pendulum. Of a number of +extraneous effects which tend to disturb the +accuracy of pendulum observations the most +important is air resistance. Experiments reported +by the Greenwich (England) observatory +in 1829 led to the development of a +vacuum chamber within which the pendulum +was swung.</p> +</div> + +<p>While the British were engaged in swinging the +Kater invariable pendulums to determine relative +values of the length of the seconds pendulum, or +of gravity, the French also sent out expeditions. +Capt. de Freycinet made initial observations at Paris +with three invariable brass pendulums and one +wooden one, and then carried out observations at +Rio de Janeiro, Cape of Good Hope, Île de France, +Rawak (near New Guinea), Guam, Maui, and various +other places.<a name="FNanchor_33_33" id="FNanchor_33_33"></a><a href="#Footnote_33_33" class="fnanchor">[33]</a> A similar expedition was conducted +in 1822-1825 by Captain Duperry.<a name="FNanchor_34_34" id="FNanchor_34_34"></a><a href="#Footnote_34_34" class="fnanchor">[34]</a></p> + +<p>During the years from 1827 to 1840, various types +of pendulum were constructed and swung by Francis +Baily, a member of the Royal Astronomical Society, +who reported in 1832 on experiments in which no less +than 41 different pendulums were swung in vacuo, +and their characteristics determined.<a name="FNanchor_35_35" id="FNanchor_35_35"></a><a href="#Footnote_35_35" class="fnanchor">[35]</a> In 1836, +Baily undertook to advise the American Lt. Charles +Wilkes, who was to head the United States Exploring<span class="pagenum"><a name="Page_318" id="Page_318">[Pg 318]</a></span> +Expedition of 1838-1842, on the procurement of +pendulums for this voyage. Wilkes ordered from the +London instrument maker, Thomas Jones, two unusual +pendulums, which Wilkes described as “those +considered the best form by Mr. Baily for traveling +pendulums,” and which Baily, himself, described as +“precisely the same as the two invariable pendulums +belonging to this [Royal Astronomical] Society,” +except for the location of the knife edges.</p> + +<div class="figright" style="width: 350px;"><a name="fig_15" id="fig_15"></a> +<img src="images/i039.png" width="200" height="680" alt="Figure 15." title="Figure 15." /> + +<p class="caption2">Figure 15.—<span class="smcap">One of Francis Baily’s pendulums</span> +(62-1/2 inches long), shown on the left, is now in the +possession of the Science Museum, London, and, +right, two views of a similar pendulum (37-5/8 inches +long) made in the late 19th century by Edward +Kübel, Washington, D.C., which is no. 316,876 in +the collection of the U.S. National Museum. +Among a large number of pendulums tried by +Baily in London (1827-1840), was one which +resembles the reversible pendulum superficially, but +which is actually an invariable pendulum having +knife edges at both ends. The purpose was apparently +economy, since it is equivalent to two separate +invariable pendulums. This is the type of pendulum +used on the U.S. Exploring Expedition of +1838-1842. It is not known what use was made of +the Kübel pendulum.</p> +</div> + +<p>The unusual feature of these pendulums was in +their symmetry of mass as well as of form. They were +made of bars, of iron in one case, and of brass in the +other, and each had two knife edges at opposite ends +equidistant from the center. Thus, although they +resembled reversible pendulums, their symmetry of +mass prevented their use as such, and they were rather +equivalent to four separate invariable pendulums.<a name="FNanchor_36_36" id="FNanchor_36_36"></a><a href="#Footnote_36_36" class="fnanchor">[36]</a></p> + +<p>Wilkes was taught the use of the pendulum by +Baily, and conducted experiments at Baily’s house, +where the latter had carried out the work reported on +in 1832. The subsequent experiments made on the +U.S. Exploring Expedition were under the charge of +Wilkes, himself, who made observations on 11 separate +occasions, beginning with that in London (1836) and +followed by others in New York, Washington, D.C., +Rio de Janeiro, Sydney, Honolulu, “Pendulum Peak” +(Mauna Loa), Mount Kanoha, Nesqually (Oregon +Territory), and, finally, two more times in Washington, +D.C. (1841 and 1845).</p> + +<p>Wilkes’ results were communicated to Baily, who +appears to have found the work defective because of +insufficient attention to the maintenance of temperature +constancy and to certain alterations made to the +pendulums.<a name="FNanchor_37_37" id="FNanchor_37_37"></a><a href="#Footnote_37_37" class="fnanchor">[37]</a> The results were also to have been +included in the publications of the Expedition, but +were part of the unpublished 24th volume. Fortunately +they still exist, in what appears to be a +printer’s proof.<a name="FNanchor_38_38" id="FNanchor_38_38"></a><a href="#Footnote_38_38" class="fnanchor">[38]</a></p> + +<p>The Kater invariable pendulums were used to +investigate the internal constitution of the earth. +Airy sought to determine the density of the earth by +observing the times of swing of pendulums at the top +and bottom of a mine. The first experiments were +made in 1826 at the Dolcoath copper mine in Cornwall, +and failed when the pendulum fell to the bottom.<span class="pagenum"><a name="Page_319" id="Page_319">[Pg 319]</a></span> +In 1854, the experiments were again undertaken in the +Harton coalpit, near Sunderland.<a name="FNanchor_39_39" id="FNanchor_39_39"></a><a href="#Footnote_39_39" class="fnanchor">[39]</a> Gravity at the +surface was greater than below, because of the attraction +of a shell equal to the depth of the pit. From +the density of the shell as determined from specimens +of rock, Airy found the density of the earth to be 6-1/2 +times greater than that of water. T. C. Mendenhall, +in 1880, used a Kater convertible pendulum in an +invariable manner to compare values of gravity on +Fujiyama and at Tokyo, Japan.<a name="FNanchor_40_40" id="FNanchor_40_40"></a><a href="#Footnote_40_40" class="fnanchor">[40]</a> He used a “simple” +pendulum of the Borda type to determine the absolute +value of gravity at Tokyo. From the values of gravity +on the mountain and at Tokyo, and an estimate of +the volume of the mountain, he estimated the mean +density of the earth as 5.77 times greater than that +of water.</p> + +<p>In 1879, Maj. J. Herschel, R.E., stated:</p> + +<blockquote><p>The years from 1840 to 1865 are a complete blank, if +we except Airy’s relative density experiments in 1854. +This pause was broken simultaneously in three different +ways. Two pendulums of the Kater pattern were sent +to India; two after Bessel’s design were set to work in +Russia; and at Geneva, Plantamour’s zealous experiments +with a pendulum of the same kind mark the +commencement of an era of renewed activity on the +European continent.<a name="FNanchor_41_41" id="FNanchor_41_41"></a><a href="#Footnote_41_41" class="fnanchor">[41]</a></p></blockquote> + +<p>With the statement that Kater invariable pendulums +nos. 4 and 6 (1821) were used in India between 1865 +and 1873, we now consider the other events mentioned +by Herschel.</p> + + +<hr style="width: 65%;" /><p><span class="pagenum"><a name="Page_320" id="Page_320">[Pg 320]</a></span></p> +<h3>Repsold-Bessel Reversible Pendulum</h3> + + +<p>As we have noted, Bessel made determinations of +gravity with a ball (“simple”) pendulum in the +period 1825-1827 and in 1835 at Königsberg and +Berlin, respectively. In the memoir on his observations +at Königsberg, he set forth the theory of the +symmetrical compound pendulum with interchangeable +knife edges.<a name="FNanchor_42_42" id="FNanchor_42_42"></a><a href="#Footnote_42_42" class="fnanchor">[42]</a> Bessel demonstrated theoretically +that if the pendulum were symmetrical with respect +to its geometrical center, if the times of swing about +each axis were the same, the effects of buoyancy and +of air set in motion would be eliminated. Laplace +had already shown that the knife edge must be +regarded as a cylinder and not as a mere line of +support. Bessel then showed that if the knife edges +were equal cylinders, their effects were eliminated +by inverting the pendulum; and if the knife edges +were not equal cylinders, the difference in their effects +was canceled by interchanging the knives and again +determining the times of swing in the so-called erect +and inverted positions. Bessel further showed that +it is unnecessary to make the times of swing exactly +equal for the two knife edges.</p> + +<p>The simplified discussion for infinitely small oscillations +in a vacuum is as follows: If <i>T</i><sub>1</sub> and <i>T</i><sub>2</sub> +are the times of swing about the knife edges, and if +<i>h</i><sub>1</sub> and <i>h</i><sub>2</sub> are distances of the knife edges from the +center of gravity, and if <i>k</i> is the radius of gyration +about an axis through the center of gravity, then +from the equation of motion of a rigid body oscillating +about a fixed axis under gravity <span class="nowrap"><i>T</i><sub>1</sub><sup>2</sup></span> = <span class="nowrap">π<sup>2</sup>(<i>k</i><sup>2</sup> + <i>h</i><sub>1</sub><sup>2</sup>)/<i>g</i><i>h</i><sub>1</sub></span>, +<span class="nowrap"><i>T</i><sub>2</sub><sup>2</sup></span> = <span class="nowrap">π<sup>2</sup>(<i>k</i><sup>2</sup> + <i>h</i><sub>2</sub><sup>2</sup>)/<i>g</i><i>h</i><sub>2</sub></span>. Then <span class="nowrap">(<i>h</i><sub>1</sub><i>T</i><sub>1</sub><sup>2</sup> +- <i>h</i><sub>2</sub><i>T</i><sub>2</sub><sup>2</sup>)/(<i>h</i><sub>1</sub> - <i>h</i><sub>2</sub>)</span> = <span class="nowrap"><span class="nowrap">(π<sup>2</sup>/<i>g</i>)(<i>h</i><sub>1</sub> + <i>h</i><sub>2</sub>) = τ<sup>2</sup>.</span></span></p> + +<p>τ is then the time of swing of a simple pendulum +of length <span class="nowrap"><i>h</i><sub>1</sub> + <i>h</i><sub>2</sub></span>. If the difference <span class="nowrap"><i>T</i><sub>1</sub> - <i>T</i><sub>2</sub></span> is <a name="corr_44_01" id="corr_44_01"></a><ins class="mycorr" title="Original: sufficlently">sufficiently</ins> +small, τ = <span class="nowrap">(<i>h</i><sub>1</sub><i>T</i><sub>1</sub> - <i>h</i><sub>2</sub><i>T</i><sub>2</sub>)/(<i>h</i><sub>1</sub> - <i>h</i><sub>2</sub>)</span>. Prior to its publication +by Bessel in 1828, the formula for the time of +swing of a simple pendulum of length <span class="nowrap"><i>h</i><sub>1</sub> + <i>h</i><sub>2</sub></span> in terms +of <i>T</i><sub>1</sub>, <i>T</i><sub>2</sub> had been given by C. F. Gauss in a letter +to H. C. Schumacher dated November 28, 1824.<a name="FNanchor_43_43" id="FNanchor_43_43"></a><a href="#Footnote_43_43" class="fnanchor">[43]</a></p> + +<p>The symmetrical compound pendulum with interchangeable +knives, for which Bessel gave a posthumously +published design and specifications,<a name="FNanchor_44_44" id="FNanchor_44_44"></a><a href="#Footnote_44_44" class="fnanchor">[44]</a> has +been called a reversible pendulum; it may thereby +be distinguished from Kater’s unsymmetrical convertible +pendulum. In 1861, the Swiss Geodetic +Commission was formed, and in one of its first sessions +in 1862 it was decided to add determinations of +gravity to the operations connected with the measurement—at +different points in Switzerland—of the arc +of the meridian traversing central Europe.<a name="FNanchor_45_45" id="FNanchor_45_45"></a><a href="#Footnote_45_45" class="fnanchor">[45]</a> It was +decided further to employ a reversible pendulum of +Bessel’s design and to have it constructed by the firm +of A. Repsold and Sons, Hamburg. It was also +decided to make the first observations with the pendulum +in Geneva; accordingly, the Repsold-Bessel pendulum +(fig. <a href="#fig_16">16</a>) was sent to Prof. E. Plantamour, +director of the observatory at Geneva, in the autumn +of 1864.<a name="FNanchor_46_46" id="FNanchor_46_46"></a><a href="#Footnote_46_46" class="fnanchor">[46]</a></p> + +<p>The Swiss reversible pendulum was about 560 mm. +in length (distance between the knife edges) and the +time of swing was approximately 3/4-second. At the +extremities of the stem of the pendulum were movable +cylindrical disks, one of which was solid and heavy, +the other hollow and light. It was intended by the +mechanicians that equality of times of oscillation +about the knife edges would be achieved by adjusting +the position of a movable disk. The pendulum was +hung by a knife edge on a plate supported by a +tripod and having an attachment from which a +measuring rod could be suspended so that the distance +between the knife edges could be measured by a +comparator. Plantamour found it impracticable to +adjust a disk until the times of swing about each +knife edge were equal. His colleague, Charles Cellérier,<a name="FNanchor_47_47" id="FNanchor_47_47"></a><a href="#Footnote_47_47" class="fnanchor">[47]</a> +<span class="pagenum"><a name="Page_321" id="Page_321">[Pg 321]</a></span>then showed that if <span class="nowrap">(<i>T</i><sub>1</sub> - <i>T</i><sub>2</sub>)/<i>T</i><sub>1</sub></span> is sufficiently +small so that one can neglect its square, one can +determine the length of the seconds pendulum from +the times of swing about the knife edges by a theory +which uses the distances of the center of gravity from +the respective knife edges. Thus, a role for the position +of the center of gravity in the theory of the +reversible pendulum, which had been set forth earlier +by Bessel, was discovered independently by Cellérier +for the Swiss observers of pendulums.</p> + +<p>In 1866, Plantamour published an extensive memoir +“Expériences faites ŕ Genčve avec le pendule ŕ <a name="corr_44_02" id="corr_44_02"></a><ins class="mycorr" title="Original: reversion">réversion</ins>.” +Another memoir, published in 1872, presented +further results of determinations of gravity +in Switzerland. Plantamour was the first scientist +in western Europe to use a Repsold-Bessel reversible +pendulum and to work out methods for its employment.</p> + +<p>The Russian Imperial Academy of Sciences acquired +two Repsold-Bessel pendulums, and observations +with them were begun in 1864 by Prof. Sawitsch, +University of St. Petersburg, and others.<a href="#Footnote_48_48" class="fnanchor">[48]</a> In 1869, +the Russian pendulums were loaned to the India +Survey in order to enable members of the Survey to +supplement observations with the Kater invariable +pendulums nos. 4 and 6 (1821). During the transport +of the Russian apparatus to India, the knives became +rusted and the apparatus had to be reconditioned. +Capt. Heaviside of the India Survey observed with +both pendulums at Kew Observatory, near London, +in the spring of 1874, after which the Russian pendulums +were sent to Pulkowa (Russia) and were +used for observations there and in the Caucasus.</p> + +<p>The introduction of the Repsold-Bessel reversible +pendulum for the determination of gravity was +accompanied by the creation of the first international +scientific association, one for geodesy. In 1861, Lt. +Gen. J. J. Baeyer, director of the Prussian Geodetic +Survey, sent a memorandum to the Prussian minister +of war in which he proposed that the independent +geodetic surveys of the states of central Europe be +coordinated by the creation of an international +organization.<a name="FNanchor_49_49" id="FNanchor_49_49"></a><a href="#Footnote_49_49" class="fnanchor">[49]</a> In 1862, invitations were sent to the +various German states and to other states of central +Europe. The first General Conference of the association, +initially called <i>Die Mittel-Europäische Gradmessung</i>, +also <i>L’Association Géodésique Internationale</i>, was<span class="pagenum"><a name="Page_322" id="Page_322">[Pg 322]</a></span> +held from the 15th to the 22d of October 1864 in +Berlin.<a name="FNanchor_50_50" id="FNanchor_50_50"></a><a href="#Footnote_50_50" class="fnanchor">[50]</a> The Conference decided upon questions +of organization: a general conference was to be held +ordinarily every three years; a permanent commission +initially consisting of seven members was to be the +scientific organ of the association and to meet annually; +a central bureau was to be established for the +reception, publication, and distribution of reports +from the member states.</p> + +<div class="figright" style="width: 350px;"><a name="fig_16" id="fig_16"></a> +<img src="images/i043.png" width="300" height="415" alt="Figure 16." title="Figure 16." /> + +<p class="caption2">Figure 16.—<span class="smcap">From a Design Left by Bessel</span>, +this portable apparatus was developed in +1862 by the firm of Repsold in Hamburg, +whose founder had assisted Bessel in the +construction of his pendulum apparatus +of 1826. The pendulum is convertible, +but differs from Kater’s in being geometrically +symmetrical and, for this +reason, Repsold’s is usually called “reversible.” +Just to the right of the pendulum +is a standard scale. To the left is a +“vertical comparator” designed by Repsold +to measure the distance between the +knife edges of the pendulum. To make +this measurement, two micrometer microscopes +which project horizontally through +the comparator are alternately focused on +the knife edges and on the standard scale.</p> +</div> + +<p>Under the topic “Astronomical Questions,” the +General Conference of 1864 resolved that there +should be determinations of the intensity of gravity +at the greatest possible number of points of the +geodetic network, and recommended the reversible +pendulum as the instrument of observation.<a name="FNanchor_51_51" id="FNanchor_51_51"></a><a href="#Footnote_51_51" class="fnanchor">[51]</a> At the +second General Conference, in Berlin in 1867, on the +basis of favorable reports by Dr. Hirsch, director of +the observatory at Neuchâtel, of Swiss practice with +the Repsold-Bessel reversible pendulum, this instrument +was specifically recommended for determinations +of gravity.<a name="FNanchor_52_52" id="FNanchor_52_52"></a><a href="#Footnote_52_52" class="fnanchor">[52]</a> The title of the association was +changed to <i>Die Europäische Gradmessung</i>; in 1886, it +became <i>Die Internationale Erdmessung</i>, under which +title it continued until World War I.</p> + +<p>On April 1, 1866, the Central Bureau of <i>Die Europäische +Gradmessung</i> was opened in Berlin under the +presidency of Baeyer, and in 1868 there was founded +at Berlin, also under his presidency, the Royal +Prussian Geodetic Institute, which obtained regular +budgetary status on January 1, 1870. A reversible +pendulum for the Institute was ordered from A. +Repsold and Sons, and it was delivered in the spring +of 1869. The Prussian instrument was symmetrical +geometrically, as specified by Bessel, but different in +form from the Swiss and Russian pendulums. The +distance between the knife edges was 1 meter, and +the time of swing approximately 1 second. The +Prussian Repsold-Bessel pendulum was swung at +Leipzig and other stations in central Europe during +the years 1869-1870 by Dr. Albrecht under the +direction of Dr. Bruhns, director of the observatory +at Leipzig and chief of the astronomical section of the +Geodetic Institute. The results of these first observations +appeared in a publication of the Royal Prussian +Geodetic Institute in 1871.<a name="FNanchor_53_53" id="FNanchor_53_53"></a><a href="#Footnote_53_53" class="fnanchor">[53]</a></p> + +<p>Results of observations with the Russian Repsold-Bessel +pendulums were published by the Imperial +Academy of Sciences. In 1872, Prof. Sawitsch +reported the work for western Europeans in “Les +variations de la pesanteur dans les provinces occidentales +de l’Empire russe.”<a name="FNanchor_48_48" id="FNanchor_48_48"></a><a href="#Footnote_48_48" class="fnanchor">[48]</a> In November 1873, the +Austrian Geodetic Commission received a Repsold-Bessel +reversible pendulum and on September 24, +1874, Prof. Theodor von Oppolzer reported on observations +at Vienna and other stations to the Fourth +General Conference of <i>Die Europäische Gradmessung</i> in +Dresden.<a name="FNanchor_54_54" id="FNanchor_54_54"></a><a href="#Footnote_54_54" class="fnanchor">[54]</a> At the fourth session of the Conference, +on September 28, 1874, a Special Commission, +consisting of Baeyer, as chairman, and Bruhns, Hirsch, +Von Oppolzer, Peters, and Albrecht, was appointed +to consider (under Topic 3 of the program): “Observations +for the determination of the intensity of +gravity,” the question, “Which Pendulum-apparatuses +are preferable for the determination of many points?”</p> + +<p>After the adoption of the Repsold-Bessel reversible +pendulum for gravity determinations in Europe, work +in the field was begun by the U.S. Coast Survey under +the superintendency of Prof. Benjamin Peirce. There +is mention in reports of observations with pendulums +prior to Peirce’s direction to his son Charles on November +30, 1872, “to take charge of the Pendulum Experiments +of the Coast Survey and to direct and inspect all +parties engaged in such experiments and as often +as circumstances will permit, to take the field with +a party....”<a name="FNanchor_55_55" id="FNanchor_55_55"></a><a href="#Footnote_55_55" class="fnanchor">[55]</a> Systematic and important gravity +work by the Survey was begun by Charles Sanders<span class="pagenum"><a name="Page_323" id="Page_323">[Pg 323]</a></span> +Peirce. Upon receiving notice of his appointment, +the latter promptly ordered from the Repsolds a +pendulum similar to the Prussian instrument. Since +the firm of mechanicians was engaged in making +instruments for observations of the transit of Venus in +1874, the pendulum for the Coast Survey could not be +constructed immediately. Meanwhile, during the +years 1873-1874, Charles Peirce conducted a party +which made observations of gravity in the Hoosac +Tunnel near North Adams, and at Northampton and +Cambridge, Massachusetts. The pendulums used +were nonreversible, invariable pendulums with conical +bobs. Among them was a silver pendulum, but +similar pendulums of brass were used also.<a name="FNanchor_56_56" id="FNanchor_56_56"></a><a href="#Footnote_56_56" class="fnanchor">[56]</a></p> + +<div class="figcenter" style="width: 640px;"> +<div class="figleft" style="width: 300px;"><a name="fig_17" id="fig_17"></a> +<img src="images/i046.png" width="300" height="524" alt="Figure 17." title="Figure 17." /> + +<p class="caption2">Figure 17.—<span class="smcap">Repsold-Bessel reversible pendulum</span> apparatus +as made in 1875, and used in the gravity work of +the U.S. Coast and Geodetic Survey. Continental +geodesists continued to favor the general use of convertible +pendulums and absolute determinations of gravity, while +their English colleagues had turned to invariable pendulums +and relative determinations, except for base stations. +Perhaps the first important American contribution to +gravity work was C. S. Peirce’s demonstration of the +error inherent in the Repsold apparatus through flexure +of the stand.</p> +</div> + +<div class="figright" style="width: 300px;"><a name="fig_18" id="fig_18"></a> +<img src="images/i047.png" width="300" height="390" alt="Figure 18." title="Figure 18." /> + +<p class="caption2">Figure 18.—<span class="smcap">Charles Sanders Peirce</span> (1839-1914), +son of Benjamin Peirce, Perkins Professor of +Astronomy and Mathematics at Harvard College. +C. S. Peirce graduated from Harvard in 1859. +From 1873 to 1891, as an assistant at the U.S. +Coast and Geodetic Survey, he accomplished the +important gravimetric work described in this article. +Peirce was also interested in many other fields, +but above all in the logic, philosophy, and history +of science, in which he wrote extensively. His +greatest fame is in philosophy, where he is regarded +as the founder of pragmatism.</p> +</div> +</div> + +<p style="clear:both;"><span class="pagenum"><a name="Page_324" id="Page_324">[Pg 324]</a></span>In 1874, Charles Peirce expressed the desire to be +sent to Europe for at least a year, beginning about +March 1, 1875, “to learn the use of the new convertible +pendulum and to compare it with those of the European +measure of a Degree and the Swiss and to compare” +his “invariable pendulums in the manner which +has been used by swinging them in London and +Paris.”<a name="FNanchor_57_57" id="FNanchor_57_57"></a><a href="#Footnote_57_57" class="fnanchor">[57]</a></p> + +<p>Charles S. Peirce, assistant, U.S. Coast Survey, +sailed for Europe on April 3, 1875, on his mission to +obtain the Repsold-Bessel reversible pendulum ordered +for the Survey and to learn the methods of +using it for the determination of gravity. In England, +he conferred with Maxwell, Stokes, and Airy concerning +the theory and practice of research with +pendulums. In May, he continued on to Hamburg +and obtained delivery from the Repsolds of the pendulum +for the Coast Survey (fig. <a href="#fig_17">17</a>). Peirce then +went to Berlin and conferred with Gen. Baeyer, who +expressed doubts of the stability of the Repsold stand +for the pendulum. Peirce next went to Geneva, +where, under arrangements with Prof. Plantamour, +he swung the newly acquired pendulum at the +observatory.<a name="FNanchor_58_58" id="FNanchor_58_58"></a><a href="#Footnote_58_58" class="fnanchor">[58]</a></p> + +<p>In view of Baeyer’s expressed doubts of the rigidity +of the Repsold stand, Peirce performed experiments +to measure the flexure of the stand caused by the +oscillations of the pendulum. His method was to set +up a micrometer in front of the pendulum stand and, +with a microscope, to measure the displacement +caused by a weight passing over a pulley, the friction +of which had been determined. Peirce calculated +the correction to be applied to the length of the seconds +pendulum—on account of the swaying of the stand +during the swings of the pendulum—to amount to +over 0.2 mm. Although Peirce’s measurements of +flexure in Geneva were not as precise as his later +measurements, he believed that failure to correct for +flexure of the stand in determinations previously made +with Repsold pendulums was responsible for appreciable +errors in reported values of the length of the +seconds pendulum.</p> + +<p>The Permanent Commission of <i>Die Europäische +Gradmessung</i> met in Paris, September 20-29, 1875. +In conjunction with this meeting, there was held on +September 21 a meeting of the Special Commission +on the Pendulum. The basis of the discussion by the +Special Commission was provided by reports which +had been submitted in response to a circular sent +out by the Central Bureau to the members on +February 26, 1874.<a name="FNanchor_59_59" id="FNanchor_59_59"></a><a href="#Footnote_59_59" class="fnanchor">[59]</a></p> + +<p>Gen. Baeyer stated that the distance of 1 meter between +the knife edges of the Prussian Repsold-Bessel +pendulum made it unwieldy and unsuited for transport. +He declared that the instability of the stand +also was a source of error. Accordingly, Gen. Baeyer +expressed the opinion that absolute determinations +of gravity should be made at a control station by a +reversible pendulum hung on a permanent, and therefore +stable stand, and he said that relative values of +gravity with respect to the control station should be +obtained in the field by means of a Bouguer invariable +pendulum. Dr. Bruhns and Dr. Peters agreed with +Gen. Baeyer; however, the Swiss investigators, +Prof. Plantamour and Dr. Hirsch reported in defense +of the reversible pendulum as a field instrument, as +did Prof. von Oppolzer of Vienna. The circumstance +that an invariable pendulum is subject to changes in +length was offered as an argument in favor of the +reversible pendulum as a field instrument.</p> + +<p>Peirce was present during these discussions by the +members of the Special Commission, and he reported +that his experiments at Geneva demonstrated that +the oscillations of the pendulum called forth a flexure +of the support which hitherto had been neglected. +The observers who used the Swiss and Austrian +Repsold pendulums contended, in opposition to +Peirce, that the Repsold stand was stable.</p> + +<p>The outcome of these discussions was that the +Special Commission reported to the Permanent Commission +that the Repsold-Bessel reversible pendulum, +except for some small changes, satisfied all requirements +for the determination of gravity. The Special +Commission proposed that the Repsold pendulums +of the several states be swung at the Prussian Eichungsamt +in Berlin where, as Peirce pointed out, Bessel +had made his determination of the intensity of gravity +with a ball pendulum in 1835. Peirce was encouraged +to swing the Coast Survey reversible pendulum at the +stations in France, England, and Germany where<span class="pagenum"><a name="Page_325" id="Page_325">[Pg 325]</a></span> +Borda and Cassini, Kater, and Bessel, respectively, +had made historic determinations. The Permanent +Commission, in whose sessions Peirce also participated, +by resolutions adopted the report of the Special +Commission on the Pendulum.<a name="FNanchor_60_60" id="FNanchor_60_60"></a><a href="#Footnote_60_60" class="fnanchor">[60]</a></p> + +<p>During the months of January and February 1876, +Peirce conducted observations in the Grande Salle du +Meridien at the observatory in Paris where Borda, +Biot, and Capt. Edward Sabine had swung pendulums +early in the 19th century. He conducted observations +in Berlin from April to June 1876 and, by +experiment, determined the correction for flexure to +be applied to the value of gravity previously obtained +with the Prussian instrument. Subsequent observations +were made at Kew. After his return to the +United States on August 26, 1876, Peirce conducted +experiments at the Stevens Institute in Hoboken, +New Jersey, where he made careful measurements of +the flexure of the stand by statical and dynamical +methods. In Geneva, he had secured the construction +of a vacuum chamber in which the pendulum +could be swung on a support which he called the +Geneva support. At the Stevens Institute, Peirce +swung the Repsold-Bessel pendulum on the Geneva +support and determined the effect of different pressures +and temperatures on the period of oscillation of +the pendulum. These experiments continued into +1878.<a name="FNanchor_61_61" id="FNanchor_61_61"></a><a href="#Footnote_61_61" class="fnanchor">[61]</a></p> + +<p>Meanwhile, the Permanent Commission met October +5-10, 1876, in Brussels and continued the discussion +of the pendulum.<a name="FNanchor_62_62" id="FNanchor_62_62"></a><a href="#Footnote_62_62" class="fnanchor">[62]</a> Gen. Baeyer reported on +Peirce’s experiments in Berlin to determine the flexure +of the stand. The difference of 0.18 mm. in the +lengths of the seconds pendulum as determined by +Bessel and as determined by the Repsold instrument +agreed with Peirce’s estimate of error caused by +neglect of flexure of the Repsold stand. Dr. Hirsch, +speaking for the Swiss survey, and Prof. von Oppolzer, +speaking for the Austrian survey, contended, however, +that their stands possessed sufficient stability and that +the results found by Peirce applied only to the stands +and bases investigated by him. The Permanent Commission +proposed further study of the pendulum.</p> + +<p>The Fifth General Conference of <i>Die Europäische +Gradmessung</i> was held from September 27 to October 2, +1877, in Stuttgart.<a name="FNanchor_63_63" id="FNanchor_63_63"></a><a href="#Footnote_63_63" class="fnanchor">[63]</a> Peirce had instructions from +Supt. Patterson of the U.S. Coast Survey to attend +this conference, and on arrival presented a letter of +introduction from Patterson requesting that he, +Peirce, be permitted to participate in the sessions. +Upon invitation from Prof. Plantamour, as approved +by Gen. Ibańez, president of the Permanent Commission, +Peirce had sent on July 13, 1877, from New +York, the manuscript of a memoir titled “De +l’Influence de la flexibilité du trépied sur l’oscillation +du pendule ŕ réversion.” This memoir and others +by Cellérier and Plantamour confirming Peirce’s +work were published as appendices to the proceedings +of the conference. As appendices to Peirce’s contribution +were published also two notes by Prof. von +Oppolzer. At the second session on September 29, +1877, when Plantamour reported that the work of +Hirsch and himself had confirmed experimentally +the independent theoretical work of Cellérier and the +theoretical and experimental work of Peirce on +flexure, Peirce described his Hoboken experiments.</p> + +<p>During the discussions at Stuttgart on the flexure +of the Repsold stand, Hervé Faye, president of the +Bureau of Longitudes, Paris, suggested that the swaying +of the stand during oscillations of the pendulum +could be overcome by the suspension from one support +of two similar pendulums which oscillated with equal +amplitudes and in opposite phases. This proposal +was criticized by Dr. Hirsch, who declared that exact +observation of passages of a “double pendulum” +would be difficult and that two pendulums swinging +so close together would interfere with each other. +The proposal of the double pendulum came up again at +the meeting of the Permanent Commission at Geneva +in 1879.<a name="FNanchor_64_64" id="FNanchor_64_64"></a><a href="#Footnote_64_64" class="fnanchor">[64]</a> On February 17, 1879, Peirce had completed +a paper “On a Method of Swinging Pendulums +for the Determination of Gravity, Proposed by M.<span class="pagenum"><a name="Page_326" id="Page_326">[Pg 326]</a></span> +Faye.” In this paper, Peirce presented the results +of an analytical mechanical investigation of Faye’s +proposal. Peirce set up the differential equations, +found the solutions, interpreted them physically, +and arrived at the conclusion “that the suggestion of +M. Faye ... is as sound as it is brilliant and offers +some peculiar advantages over the existing method of +swinging pendulums.”</p> + +<p>In a report to Supt. Patterson, dated July 1879, +Peirce stated: “I think it is important before making +a new pendulum apparatus to experiment with Faye’s +proposed method.”<a name="FNanchor_65_65" id="FNanchor_65_65"></a><a href="#Footnote_65_65" class="fnanchor">[65]</a> He wrote further: “The +method proves to be perfectly sound in theory, and as +it would greatly facilitate the work it is probably +destined eventually to prevail. We must unfortunately +leave to other surveys the merit of practically +testing and introducing the new method, as our appropriations +are insufficient for us to maintain the leading +position in this matter, which we otherwise might +take.” Copies of the published version of Peirce’s +remarks were sent to Europe. At a meeting of the +Academy of Sciences in Paris on September 1, 1879, +Faye presented a report on Peirce’s findings.<a name="FNanchor_66_66" id="FNanchor_66_66"></a><a href="#Footnote_66_66" class="fnanchor">[66]</a> The +Permanent Commission met September 16-20, 1879, +in Geneva. At the third session on September 19, +by action of Gen. Baeyer, copies of Peirce’s paper on +Faye’s proposed method of swinging pendulums were +distributed. Dr. Hirsch again commented adversely +on the proposal, but moved that the question be +investigated and reported on at the coming General +Conference. The Permanent Commission accepted +the proposal of Dr. Hirsch, and Prof. Plantamour +was named to report on the matter at the General +Conference. At Plantamour’s request, Charles Cellérier +was appointed to join him, since the problem +essentially was a theoretical one.</p> + +<p>The Sixth General Conference of <i>Die Europäische +Gradmessung</i> met September 13-16, 1880, in Munich.<a name="FNanchor_67_67" id="FNanchor_67_67"></a><a href="#Footnote_67_67" class="fnanchor">[67]</a> +Topic III, part 7 of the program was entitled “On Determinations +of Gravity through pendulum observations. +Which construction of a pendulum apparatus +corresponds completely to all requirements of science? +Special report on the pendulum.”</p> + +<div class="figright" style="width: 350px;"><a name="fig_19" id="fig_19"></a> +<img src="images/i054.png" width="200" height="524" alt="Figure 19." title="Figure 19." /> + +<p class="caption2">Figure 19.—<span class="smcap">Three pendulums used in early work</span> +at the U.S. Coast and Geodetic Survey. Shown on +the left is the Peirce invariable; center, the Peirce +reversible; and, right, the Repsold reversible. +Peirce designed the cylindrical pendulum in 1881-1882 +to study the effect of air resistance according +to the theory of G. G. Stokes on the motion of a +pendulum in a viscous field. Three examples of +the Peirce pendulums are in the U.S. National +Museum.</p> +</div> + +<p>The conference received a memoir by Cellérier<a name="FNanchor_68_68" id="FNanchor_68_68"></a><a href="#Footnote_68_68" class="fnanchor">[68]</a> on +the theory of the double pendulum and a report by +Plantamour and Cellérier.<a name="FNanchor_69_69" id="FNanchor_69_69"></a><a href="#Footnote_69_69" class="fnanchor">[69]</a> Cellérier’s mathematical +analysis began with the equations of Peirce and used +the latter’s notation as far as possible. His general +discussion included the results of Peirce, but he +stated that the difficulties to be overcome did not +justify the employment of the “double pendulum.” +He presented an alternative method of correcting for +flexure based upon a theory by which the flexure +caused by the oscillation of a given reversible pendulum +could be determined from the behavior of an +auxiliary pendulum of the same length but of different +weight. This method of correcting for flexure was +recommended to the General Conference by Plantamour +and Cellérier in their joint report. At the +fourth session of the conference on September 16, +1880, the problem of the pendulum was discussed +and, in consequence, a commission consisting of +Faye, Helmholtz, Plantamour (replaced in 1882 by +Hirsch), and Von Oppolzer was appointed to study +apparatus suitable for relative determinations of +gravity.</p> + +<p>The Permanent Commission met September 11-15, +1882, at The Hague,<a name="FNanchor_70_70" id="FNanchor_70_70"></a><a href="#Footnote_70_70" class="fnanchor">[70]</a> and at its last session appointed +Prof. von Oppolzer to report to the Seventh General +Conference on different forms of apparatus for the +determination of gravity. The Seventh Conference +met October 15-24, 1883, in Rome,<a name="FNanchor_71_71" id="FNanchor_71_71"></a><a href="#Footnote_71_71" class="fnanchor">[71]</a> and, at its +eighth session, on October 22, received a comprehensive, +critical review from Prof. von Oppolzer entitled +“Über die Bestimmung der Schwere mit Hilfe <a name="corr_44_16" id="corr_44_16"></a><ins class="mycorr" title="Original: verschiedene">verschiedener</ins> +Apparate.”<a name="FNanchor_72_72" id="FNanchor_72_72"></a><a href="#Footnote_72_72" class="fnanchor">[72]</a> Von Oppolzer especially expounded +the advantages of the Bessel reversible +pendulum, which compensated for air effects by +symmetry of form if the times of swing for both +positions were maintained between the same amplitudes, +and compensated for irregular knife edges by +making them interchangeable. Prof. von Oppolzer +reviewed the problem of flexure of the Repsold stand +and stated that a solution in the right direction was<span class="pagenum"><a name="Page_327" id="Page_327">[Pg 327]</a></span> +the proposal—made by Faye and theoretically +pursued by Peirce—to swing two pendulums from the +same stand with equal amplitudes and in opposite +phases, but that the proposal was not practicable. +He concluded that for absolute determinations of +gravity, the Bessel reversible pendulum was highly +appropriate if one swung two exemplars of different +weight from the same stand for the elimination of +flexure. Prof. von Oppolzer’s important report recognized +that absolute determinations were less accurate +than relative ones, and should be conducted only at +special places.</p> + +<p>The discussions initiated by Peirce’s demonstration +of the flexure of the Repsold stand resulted, finally, +in the abandonment of the plan to make absolute +determinations of gravity at all stations with the +reversible pendulum.</p> + + + + +<hr style="width: 65%;" /> +<h3>Peirce and Defforges Invariable, Reversible Pendulums</h3> + +<p>The Repsold-Bessel reversible pendulum was designed +and initially used to make absolute determinations +of gravity not only at initial stations such as +Kew, the observatory in Paris, and the Smithsonian +Institution in Washington, D.C., but also at stations +in the field. An invariable pendulum with a single +knife edge, however, is adequate for relative determinations. +As we have seen, such invariable pendulums +had been used by Bouguer and Kater, and after +the experiences with the Repsold apparatus had been +recommended again by Baeyer for relative determinations. +But an invariable pendulum is subject to +uncontrollable changes of length. Peirce proposed +to detect such changes in an invariable pendulum in +the field by combining the invariable and reversible +principles. He explained his proposal to Faye in a +letter dated July 23, 1880, and he presented it on +September 16, 1880, at the fourth session of the sixth +General Conference of <i>Die Europäische Gradmessung</i>, +in Munich.<a name="FNanchor_73_73" id="FNanchor_73_73"></a><a href="#Footnote_73_73" class="fnanchor">[73]</a></p> + +<p>As recorded in the Proceedings of the Conference, +Peirce wrote:</p> + +<blockquote><p>But I obviate it in making my pendulum both invariable +and reversible. Every alteration of the pendulum will +be revealed immediately by the change in the difference +of the two periods of oscillation in the two positions. +Once discovered, it will be taken account of by means +of new measures of the distance between the two supports.</p></blockquote><p><span class="pagenum"><a name="Page_328" id="Page_328">[Pg 328]</a></span></p> + +<p>Peirce added that it seemed to him that if the reversible +pendulum perhaps is not the best instrument to +determine absolute gravity, it is, on condition that +it be truly invariable, the best to determine relative +gravity. Peirce further stated that he would wish +that the pendulum be formed of a tube of drawn brass +with heavy plugs of brass equally drawn. The cylinder +would be terminated by two hemispheres; the +knives would be attached to tongues fixed near the +ends of the cylinder.</p> + +<p>During the years 1881 and 1882, four invariable, +reversible pendulums were made after the design of +Peirce at the office of the U.S. Coast and Geodetic +Survey in Washington, D.C. The report of the superintendent +for the year 1880-1881 states:</p> + +<blockquote><p>A new pattern of the reversible pendulum has been invented, +having its surface as nearly as convenient in the +form of an elongated ellipsoid. Three of these instruments +have been constructed, two having a distance of +one meter between the knife edges and the third a distance +of one yard. It is proposed to swing one of the +meter pendulums at a temperature near 32° F. at the +same time that the yard is swung at 60° F., in order to +determine anew the relation between the yard and the +<a name="corr_44_03" id="corr_44_03"></a><ins class="mycorr" title="Original: meter.”">meter.</ins><a name="FNanchor_74_74" id="FNanchor_74_74"></a><a href="#Footnote_74_74" class="fnanchor">[74]</a></p></blockquote> + +<p>The report for 1881-1882 mentions four of these +Peirce pendulums.</p> + +<p>A description of the Peirce invariable, reversible +pendulums was given by Assistant E. D. Preston in +“Determinations of Gravity and the Magnetic Elements +in Connection with the United States Scientific +Expedition to the West Coast of Africa, 1889-90.”<a name="FNanchor_75_75" id="FNanchor_75_75"></a><a href="#Footnote_75_75" class="fnanchor">[75]</a> +The invariable, reversible pendulum, Peirce +no. 4, now preserved in the Smithsonian Institution’s +Museum of History and Technology (fig. <a href="#fig_34">34</a>), may be +taken as typical of the meter pendulums: In the same +memoir, Preston gives the diameter of the tube as +63.7 mm., thickness of tube 1.5 mm., weight 10.680 +kilograms, and distance between the knives 1.000 +meter.</p> + +<p>The combination of invariability and reversibility +in the Peirce pendulums was an innovation for relative +determinations. Indeed, the combination was +criticized by Maj. J. Herschel, R.E., of the Indian +Survey, at a conference on gravity held in Washington +in May 1882 on the occasion of his visit to the United +States for the purpose of connecting English and +American stations by relative determinations with +three Kater invariable pendulums. These three +pendulums have been designated as nos. 4, 6 (1821), +and 11.<a name="FNanchor_76_76" id="FNanchor_76_76"></a><a href="#Footnote_76_76" class="fnanchor">[76]</a></p> + +<div class="figright" style="width: 350px;"><a name="fig_20" id="fig_20"></a> +<img src="images/i057.png" width="300" height="442" alt="Figure 20." title="Figure 20." /> + +<p class="caption2">Figure 20.—<span class="smcap">Support for the Peirce pendulum</span>, +1889. Much of the work of C. S. Peirce was concerned +with the determination of the error introduced +into observations made with the portable apparatus +by the vibration of the stand with the pendulum. +He showed that the popular Bessel-Repsold apparatus +was subject to such an error. His own pendulums +were swung from a simple but rugged wooden +frame to which a hardened steel bearing was fixed.</p> +</div> + +<p>Another novel characteristic of the Peirce pendulums +was the mainly cylindrical form. Prof. George +Gabriel Stokes, in a paper “On the Effect of the<span class="pagenum"><a name="Page_329" id="Page_329">[Pg 329]</a></span> +Internal Friction of Fluids on the Motion of Pendulums”<a name="FNanchor_77_77" id="FNanchor_77_77"></a><a href="#Footnote_77_77" class="fnanchor">[77]</a> +that was read to the Cambridge Philosophical +Society on December 9, 1850, had solved the hydrodynamical +equations to obtain the resistance to the +motions of a sphere and a cylinder in a viscous +fluid. Peirce had studied the effect of viscous +resistance on the motion of his Repsold-Bessel +pendulum, which was symmetrical in form but not +cylindrical. The mainly cylindrical form of his +pendulums (fig. <a href="#fig_19">19</a>) permitted Peirce to predict +from Stokes’ theory the effect of viscosity and to +compare the results with experiment. His report +of November 20, 1889, in which he presented the +comparison of experimental results with the theory +of Stokes, was not published.<a name="FNanchor_78_78" id="FNanchor_78_78"></a><a href="#Footnote_78_78" class="fnanchor">[78]</a></p> + +<p>Peirce used his pendulums in 1883 to establish +a station at the Smithsonian Institution that was to +serve as the base station for the Coast and Geodetic +Survey for some years. Pendulum Peirce no. 1 +was swung at Washington in 1881 and was then taken +by the party of Lieutenant Greely, U.S.A., on an +expedition to Lady Franklin Bay where it was swung +in 1882 at Fort Conger, Grinnell Land, Canada. +Peirce nos. 2 and 3 were swung by Peirce in 1882 +at Washington, D.C.; Hoboken, New Jersey; +Montreal, Canada; and Albany, New York. Assistant +Preston took Peirce no. 3 on a U.S. eclipse +expedition to the Caroline Islands in 1883. Peirce +in 1885 swung pendulums nos. 2 and 3 at Ann +Arbor, Michigan; Madison, Wisconsin; and Ithaca, +New York. Assistant Preston in 1887 swung Peirce +nos. 3 and 4 at stations in the Hawaiian Islands, +and in 1890 he swung Peirce nos. 3 and 4 at stations +on the west coast of Africa.<a name="FNanchor_79_79" id="FNanchor_79_79"></a><a href="#Footnote_79_79" class="fnanchor">[79]</a></p> + +<p>The new pattern of pendulum designed by Peirce +was also adopted in France, after some years of +experience with a Repsold-Bessel pendulum. Peirce +in 1875 had swung his Repsold-Bessel pendulum at +the observatory in Paris, where Borda and Cassini, +and Biot, had made historic observations and where +Sabine also had determined gravity by comparison +with Kater’s value at London. During the spring of +1880, Peirce made studies of the supports for the +pendulums of these earlier determinations and calculated +corrections to those results for hydrodynamic +effects, viscosity, and flexure. On June 14, 1880, +Peirce addressed the Academy of Sciences, Paris, +on the value of gravity at Paris, and compared his results +with the corrected results of Borda and Biot and +with the transferred value of Kater.<a name="FNanchor_80_80" id="FNanchor_80_80"></a><a href="#Footnote_80_80" class="fnanchor">[80]</a></p> + +<p>In the same year the French Geographic Service +of the Army acquired a Repsold-Bessel reversible +pendulum of the smaller type, and Defforges conducted +experiments with it.<a name="FNanchor_81_81" id="FNanchor_81_81"></a><a href="#Footnote_81_81" class="fnanchor">[81]</a> He introduced the +method of measuring flexure from the movement +of interference fringes during motion of the pendulum. +He found an appreciable difference between dynamical +and statical coefficients of flexure and +concluded that the “correction formula of Peirce +and Cellérier is suited perfectly to practice and +represents exactly the variation of period caused +by swaying of the support, on the condition that one +uses the statical coefficient.” Defforges developed a +theory for the employment of two similar pendulums +of the same weight, but of different length, and hung +by the same knives. This theory eliminated the +flexure of the support and the curvature of the knives +from the reduction of observations.</p> + +<p>Pendulums of 1-meter and of 1/2-meter distance +between the knife edges were constructed from +Defforges’ design by Brunner Brothers in Paris +(fig. <a href="#fig_21">21</a>). These Defforges pendulums were cylindrical +in form with hemispherical ends like the Peirce +pendulums, and were hung on knives that projected +from the sides of the pendulum, as in some unfinished +Gautier pendulums designed by Peirce in 1883 in +Paris.<span class="pagenum"><a name="Page_330" id="Page_330">[Pg 330]</a></span></p> + +<div class="figcenter" style="width: 600px;"><a name="fig_21" id="fig_21"></a> +<img src="images/i060.png" width="600" height="688" alt="Figure 21." title="Figure 21." /> + +<p class="caption2">Figure 21.—<span class="smcap">Reversible pendulum apparatus</span> of Defforges, as constructed by Brunner, Paris, +about 1887. The clock and telescope used to observe coincidences are not shown. The +telescope shown is part of an interferometer used to measure flexure of the support. One +mirror of the interferometer is attached to the pendulum support; the other to the separate +masonry pillar at the left.</p> +</div> +<p><span class="pagenum"><a name="Page_331" id="Page_331">[Pg 331]</a></span></p> + +<div class="figcenter" style="width: 615px;"> +<div class="figleft" style="width: 250px;"><a name="fig_22" id="fig_22"></a> +<img src="images/i061-a.png" width="250" height="579" alt="Figure 22." title="Figure 22." /> + +<p class="caption2">Figure 22.—<span class="smcap">Because of the greater simplicity</span> of +its use, the invariable pendulum superseded the +convertible pendulum towards the end of the 19th +century, except at various national base stations +(Kew, Paris, Potsdam, Washington, D.C., etc.). +Shown here are, right to left, a pendulum of the +type used by Peirce at the Hoosac Tunnel in +1873-74, the Mendenhall 1/2-second pendulum of +1890, and the pendulum designed by Peirce in +1881-1882.</p> +</div> + +<div class="figright" style="width: 300px;"><a name="fig_23" id="fig_23"></a> +<img src="images/i061-b.png" width="300" height="395" alt="Figure 23." title="Figure 23." /> + +<p class="caption2">Figure 23.—<span class="smcap">The overall size</span> of portable pendulum +apparatus was greatly reduced with the introduction of +this 1/2-second apparatus in 1887, by the Austrian military +officer, Robert von Sterneck. Used with a vacuum +chamber not shown here, the apparatus is only about 2 +feet high. Coincidences are observed by the reflection of +a periodic electric spark in two mirrors, one on the support +and the other on the pendulum itself.</p> +</div> +</div> + +<div class="figcenter" style="width: 615px;"><a name="fig_24" id="fig_24"></a> +<img style="margin-left:45px;" src="images/i063.png" width="300" height="316" alt="Figure 24." title="Figure 24." /> + +<p class="caption2">Figure 24.—<span class="smcap">Thomas C. Mendenhall</span> (1841-1924). +Although largely self-educated, he became the first +professor of physics and mechanics at the Ohio +Agricultural and Mechanical College (later Ohio +State University), and was subsequently connected +with several other universities. In 1878, while +teaching at the Tokyo Imperial University in +Japan, he made gravity measurements between +Tokyo and Fujiyama from which he calculated +the mean density of the earth. While superintendent +of the U.S. Coast and Geodetic Survey, +1889-94, he developed the pendulum apparatus +which bears his name.</p> + +</div> + + +<hr style="width: 65%;" /> +<h3>Von Sterneck and Mendenhall Pendulums</h3> + + +<p>While scientists who had used the Repsold-Bessel +pendulum apparatus discussed its defects and limitations +for gravity surveys, Maj. Robert von Sterneck +of Austria-Hungary began to develop an excellent +apparatus for the rapid determination of relative +values of gravity.<a name="FNanchor_82_82" id="FNanchor_82_82"></a><a href="#Footnote_82_82" class="fnanchor">[82]</a> Maj. von Sterneck’s apparatus +contained a nonreversible pendulum 1/4-meter in<span class="pagenum"><a name="Page_332" id="Page_332">[Pg 332]</a></span> +length, and 1/2-second time of swing. The pendulum +was hung by a single knife edge, which rested on a +plate that was supported by a tripod. The pendulum +was swung in a chamber from which air was exhausted +and which could be maintained at any desired +temperature. Times of swing were determined by the +observation of coincidences of the pendulum with +chronometer signals. In the final form a small +mirror was attached to the knife edge perpendicular +to the plane of vibration of the pendulum and a second +fixed mirror was placed close to it so that the two +mirrors were parallel when the pendulum was at rest. +The chronometer signals worked a relay that gave a +horizontal spark which was reflected into the telescope +from the mirrors. When the pendulum was at rest, +the image of the spark in both mirrors appeared on +the horizontal cross wire in the telescope, and during +oscillation of the pendulum the two images appeared +in that position upon coincidence. In view of the +reduced size of the pendulum, the chamber in which +it was swung was readily portable, and with an +improved method of observing coincidences, relative +determinations of gravity could be made with rapidity +and accuracy.</p> + +<p>By 1887 Maj. von Sterneck had perfected his +apparatus, and it was widely adopted in Europe for +relative determinations of gravity. He used his +apparatus in extensive gravity surveys and also applied +it in the silver mines in Saxony and Bohemia, by the +previously described methods of Airy, for investigations +into the internal constitution of the earth.</p> + +<p>On July 1, 1889, Thomas Corwin Mendenhall +became superintendent of the U.S. Coast and Geodetic +Survey. Earlier, he had been professor of +physics at the University of Tokyo and had directed +observations of pendulums for the determination of +gravity on Fujiyama and at Tokyo. Supt. Mendenhall, +with the cooperation of members of his staff in +Washington, designed a new pendulum apparatus +of the von Sterneck type, and in October 1890 he +ordered construction of the first model.<a name="FNanchor_83_83" id="FNanchor_83_83"></a><a href="#Footnote_83_83" class="fnanchor">[83]</a></p> + +<p>Like the Von Sterneck apparatus, the Mendenhall +pendulum apparatus employed a nonreversible, invariable +pendulum 1/4-meter in length and of slightly +more than 1/2-second in time of swing. Initially, the +knife edge was placed in the head of the pendulum +and hung on a fixed plane support, but after some +experimentation Mendenhall attached the plane +surface to the pendulum and hung it on a fixed knife +edge. An apparatus was provided with a set of +three pendulums, so that if discrepancies appeared +in the results, the pendulum at fault could be detected. +There was also a dummy pendulum which +carried a thermometer. A pendulum was swung +in a receiver in which the pressure and temperature +of the air were controlled. The time of swing was +measured by coincidences with the beat of a chronometer. +The coincidences were determined by an +optical method with the aid of a flash apparatus.<span class="pagenum"><a name="Page_333" id="Page_333">[Pg 333]</a></span></p> + +<div class="figcenter" style="width: 600px;"><a name="fig_25" id="fig_25"></a> +<img src="images/i065.png" width="600" height="424" alt="Figure 25." title="Figure 25." /> + +<p class="caption2">Figure 25.—<span class="smcap">Mendenhall’s 1/4-meter (1/2-second) apparatus.</span> Shown on the left is the flash +apparatus and, on the right, the vacuum chamber within which the pendulum is swung. +The flash apparatus consists of a kerosene lantern and a telescope, mounted on a box +containing an electromagnetically operated shutter. The operation of the shutter is controlled +by a chronograph (not shown), so that it emits a slit of light at regular intervals. +The telescope is focused on two mirrors within the apparatus, one fixed, the other attached +to the top of the pendulum. It is used to observe the reflection of the flashes from these +mirrors. When the two reflections are aligned, a “coincidence” is marked on the chronograph +tape. The second telescope attached to the bottom of the vacuum chamber is for +observing the amplitude of the pendulum swing.</p> +</div> + +<p>The flash apparatus was contained in a light metal +box which supported an observing telescope and +which was mounted on a stand. Within the box was +an electromagnet whose coils were connected with a +chronometer circuit and whose armature carried a +long arm that moved two shutters, in both of which +were horizontal slits of the same size. The shutters +were behind the front face of the box, which also had +a horizontal slit. A flash of light from an oil lamp or +an electric spark was emitted from the box when the +circuit was broken, but not when it was closed. When +the circuit was broken a spring caused the arm to +rise, and the shutters were actuated so that the three +slits came into line and a flash of light was emitted. +A small circular mirror was set in each side of the +pendulum head, so that from either face of the pendulum +the image of the illuminated slit could be reflected +into the field of the observing telescope. A +similar mirror was placed parallel to these two mirrors +and rigidly attached to the support. The chronometer +signals broke the circuit, causing the three slits +momentarily to be in line, and when the images of +the slit in the two mirrors coincided, a coincidence +was observed. A coincidence occurred whenever the +pendulum gained or lost one oscillation on the beat<span class="pagenum"><a name="Page_334" id="Page_334">[Pg 334]</a></span> +of the chronometer. The relative intensity of gravity +was determined by observations with the first +Mendenhall apparatus at Washington, D.C., at stations +on the Pacific Coast and in Alaska, and at the +Stevens Institute, Hoboken, New Jersey, between +March and October 1891.</p> + +<div class="figcenter" style="width: 500px;"><a name="fig_26" id="fig_26"></a> +<img src="images/i068.png" width="500" height="431" alt="Figure 26." title="Figure 26." /> + +<p class="caption2">Figure 26.—<span class="smcap">Vacuum receiver</span> within which the Mendenhall pendulum is swung. The pressure +is reduced to about 50 mm. to reduce the disturbing effect of air resistance. When +the apparatus is sealed, the pendulum is lifted on the knife edge by the lever <i>q</i> and is +started to swing by the lever <i>r</i>. The arc of swing is only about 1°. The stationary mirror +is shown at <i>g</i>. The pendulum shown in outline in the center, is only about 9.7 inches long.</p> +</div> + +<p>Under Supt. Mendenhall’s direction a smaller, 1/4-second, +pendulum apparatus was also constructed +and tested, but did not offer advantages over the +1/2-second apparatus, which therefore continued in +use.</p> + +<p>In accordance with Peirce’s theory of the flexure +of the stand under oscillations of the pendulum, +determinations of the displacement of the receiver +of the Mendenhall apparatus <a name="corr_44_04" id="corr_44_04"></a><ins class="mycorr" title="Original: was">were</ins> part of a relative +determination of gravity by members of the Coast and +Geodetic Survey. Initially, a statical method was +used, but during 1908-1909 members of the Survey +adapted the Michelson interferometer for the determinations +of flexure during oscillations from the shift of +fringes.<a name="FNanchor_84_84" id="FNanchor_84_84"></a><a href="#Footnote_84_84" class="fnanchor">[84]</a> The first Mendenhall pendulums were +made of bronze, but about 1920 invar was chosen +because of its small coefficient of expansion. About +1930, Lt. E. J. Brown of the Coast and Geodetic +Survey made significant improvements in the Mendenhall +apparatus, and the new form came to be known +as the Brown Pendulum Apparatus.<a name="FNanchor_85_85" id="FNanchor_85_85"></a><a href="#Footnote_85_85" class="fnanchor">[85]</a><span class="pagenum"><a name="Page_335" id="Page_335">[Pg 335]</a></span></p> + +<div class="figcenter" style="width: 600px;"> +<img src="images/i071.png" width="600" height="395" alt="Figure 27." title="Figure 27." /> + +<p class="caption2"><a name="fig_27" id="fig_27"></a>Figure 27.—<span class="smcap">The Michelson interferometer.</span> The horizontal component of the force +acting on the knife edge through the swinging pendulum causes the support to move in +unison with the pendulum, and thereby affects the period of the oscillation. This movement +is the so-called flexure of the pendulum support, and must be taken into account in +the most accurate observations.</p> + +<p class="caption2">In 1907, the Michelson interferometer was adapted to this purpose by the U.S. Coast +and Geodetic Survey. As shown here, the interferometer, resting on a wooden beam, is +introduced into the path of a light beam reflected from a mirror on the vacuum chamber. +Movement of that mirror causes a corresponding movement in the interference fringes +in the interferometer, which can be measured.</p> +</div> + +<p>The original Von Sterneck apparatus and that of +Mendenhall provided for the oscillation of one +pendulum at a time. After the adoption of the Von +Sterneck pendulum in Europe, there were developed +stands on which two or four pendulums hung at the +same time. This procedure provided a convenient +way to observe more than one invariable pendulum +at a station for the purpose of detecting changes in +length. Prof. M. Haid of Karlsruhe in 1896 described +a four-pendulum apparatus,<a name="FNanchor_86_86" id="FNanchor_86_86"></a><a href="#Footnote_86_86" class="fnanchor">[86]</a> and Dr. Schumann of +Potsdam subsequently described a two-pendulum +apparatus.<a name="FNanchor_87_87" id="FNanchor_87_87"></a><a href="#Footnote_87_87" class="fnanchor">[87]</a><span class="pagenum"><a name="Page_336" id="Page_336">[Pg 336]</a></span></p> + +<div class="figcenter" style="width: 600px;"><a name="fig_28" id="fig_28"></a> +<img src="images/i074.png" width="600" height="473" alt="Figure 28." title="Figure 28." /> + +<p class="caption2">Figure 28.—<span class="smcap">Apparatus which was developed in 1929</span> by the Gulf Research and Development +Company, Harmarville, Pennsylvania. It was designed to achieve an accuracy +within one ten-millionth of the true value of gravity, and represents the extreme development +of pendulum apparatus for relative gravity measurement. The pendulum was +designed so that the period would be a minimum. The case (the top is missing in this +photograph) is dehumidified and its temperature and electrostatic condition are controlled. +Specially designed pendulum-lifting and <span class="nowrap">-starting</span> mechanisms are used. The problem of +flexure of the case is overcome by the Faye-Peirce method (see text) in which two +dynamically matched pendulums are swung simultaneously, 180° apart in phase.</p> +</div> + +<p>The multiple-pendulum apparatus then provided a +method of determining the flexure of the stand from +the action of one pendulum upon a second pendulum +hung on the same stand. This method of determining +the correction for flexure was a development from a +“Wippverfahren” invented at the Geodetic Institute +in Potsdam. A dynamometer was used to impart +periodic impulses to the stand, and the effect was +observed upon a pendulum initially at rest. Refinements +of this method led to the development of a +method used by Lorenzoni in 1885-1886 to determine +the flexure of the stand by action of an auxiliary +pendulum upon the principal pendulum. Dr. Schumann, +in 1899, gave a mathematical theory of such +determinations,<a name="FNanchor_88_88" id="FNanchor_88_88"></a><a href="#Footnote_88_88" class="fnanchor">[88]</a> and in his paper cited the mathematical +methods of Peirce and Cellérier for the theory of +Faye’s proposal at Stuttgart in 1877 to swing two +similar pendulums on the same support with equal +amplitudes and in opposite phases.<span class="pagenum"><a name="Page_337" id="Page_337">[Pg 337]</a></span></p> + +<div class="figcenter" style="width: 600px;"><a name="fig_29" id="fig_29"></a> +<img src="images/i077.png" width="600" height="490" alt="Figure 29." title="Figure 29." /> + +<p class="caption2">Figure 29.—<span class="smcap">The Gulf pendulum</span> is about 10.7 inches long, and has a period of .89 second. +It is made of fused quartz which is resistant to the influence of temperature change and to +the earth’s magnetism. Quartz pendulums are subject to the influence of electrostatic +charge, and provision is made to counteract this through the presence of a radium salt in +the case. The bearings are made of Pyrex glass.</p> +</div> + +<p>In 1902, Dr. P. Furtwängler<a name="FNanchor_89_89" id="FNanchor_89_89"></a><a href="#Footnote_89_89" class="fnanchor">[89]</a> presented the mathematical +theory of coupled pendulums in a paper in +which he referred to Faye’s proposal of 1877 and reported +that the difficulties predicted upon its application +had been found not to occur. Finally, during the +gravity survey of Holland in the years 1913-1921, in +view of instability of supports caused by the mobility +of the soil, F. A. Vening Meinesz adopted Faye’s +proposed method of swinging two pendulums on the +same support.<a name="FNanchor_90_90" id="FNanchor_90_90"></a><a href="#Footnote_90_90" class="fnanchor">[90]</a> The observations were made with +the ordinary Stückrath apparatus, in which four Von +Sterneck pendulums swung two by two in planes +perpendicular to each other. This successful application +of the method—which had been proposed by +Faye and had been demonstrated theoretically to be +sound by Peirce, who also published a design for its<span class="pagenum"><a name="Page_338" id="Page_338">[Pg 338]</a></span> +application—was rapidly followed for pendulum apparatus +for relative determinations by Potsdam,<a name="FNanchor_91_91" id="FNanchor_91_91"></a><a href="#Footnote_91_91" class="fnanchor">[91]</a> +Cambridge (England),<a name="FNanchor_92_92" id="FNanchor_92_92"></a><a href="#Footnote_92_92" class="fnanchor">[92]</a> Gulf Oil and Development +Company,<a name="FNanchor_93_93" id="FNanchor_93_93"></a><a href="#Footnote_93_93" class="fnanchor">[93]</a> and the Dominion Observatory at +Ottawa.<a name="FNanchor_94_94" id="FNanchor_94_94"></a><a href="#Footnote_94_94" class="fnanchor">[94]</a> Heiskanen and Vening Meinesz state:</p> + +<blockquote><p>The best way to eliminate the effect of flexure is to use +two synchronized pendulums of the same length swinging +on the same apparatus in the same plane and with +the same amplitudes but in opposite phases; it is clear +then the flexure is zero.<a name="FNanchor_95_95" id="FNanchor_95_95"></a><a href="#Footnote_95_95" class="fnanchor">[95]</a></p></blockquote> + +<p>In view of the fact that the symmetrical reversible +pendulum is named for Bessel, who created the theory +and a design for its application by Repsold, it appears +appropriate to call the method of eliminating flexure +by swinging two pendulums on the same support the +Faye-Peirce method. Its successful application was +made possible by Maj. von Sterneck’s invention of the +short, 1/4-meter pendulum.</p> + +<div class="figcenter" style="width: 600px;"> +<img src="images/i080.png" width="600" height="202" alt="Figure 30." title="Figure 30." /> + +<p class="caption2"><a name="fig_30" id="fig_30"></a>Figure 30.—<span class="smcap">The accumulated data of gravity</span> observations over the earth’s +surface have indicated that irregularities such as mountains do not have +the effect which would be expected in modifying gravity, but are somehow +compensated for. The most satisfactory solution to this still unanswered +question has been the theory of isostasy, according to which variations in +the density of the material in the earth’s crust produce a kind of hydrostatic +equilibrium between its higher and lower parts, as they “float” on the earth’s +fluid core. The metals of different density floating in mercury in this +diagram illustrate isostasy according to the theory of Pratt and Hayford.</p> +</div> + +<hr style="width: 65%;" /> +<h3>Absolute Value of Gravity at Potsdam</h3> + + +<p>The development of the reversible pendulum in the +19th century culminated in the absolute determination +of the intensity of gravity at Potsdam by Kühnen +and Furtwängler of the Royal Prussian Geodetic Institute, +which then became the world base for gravity +surveys.<a name="FNanchor_96_96" id="FNanchor_96_96"></a><a href="#Footnote_96_96" class="fnanchor">[96]</a></p> + +<p>We have previously seen that in 1869 the Geodetic +Institute—founded by Lt. Gen. Baeyer—had acquired +a Repsold-Bessel reversible pendulum which +was swung by Dr. Albrecht under the direction of +Dr. Bruhns. Dissatisfaction with this instrument was +expressed by Baeyer in 1875 to Charles S. Peirce, +who then, by experiment and mathematical analysis +of the flexure of the stand under oscillations of the +pendulum, determined that previously reported results +with the Repsold apparatus required correction. +Dr. F. R. Helmert, who in 1887 succeeded Baeyer as +director of the Institute, secured construction of a +building for the Institute in Potsdam, and under his +direction the scientific study of the intensity of gravity +was pursued with vigor. In 1894, it was discovered +in Potsdam that a pendulum constructed of very +flexible material yielded results which differed +markedly from those obtained with pendulums of<span class="pagenum"><a name="Page_339" id="Page_339">[Pg 339]</a></span> +greater stiffness. Dr. Kühnen of the Institute discovered +that the departure from expectations was the +result of the flexure of the pendulum staff itself during +oscillations.<a name="FNanchor_97_97" id="FNanchor_97_97"></a><a href="#Footnote_97_97" class="fnanchor">[97]</a></p> + +<p>Peirce, in 1883, had discovered that the recesses cut +in his pendulums for the insertion of tongues that +carried the knives had resulted in the flexure of the +pendulum staff.<a name="FNanchor_98_98" id="FNanchor_98_98"></a><a href="#Footnote_98_98" class="fnanchor">[98]</a> By experiment, he also found an +even greater flexure for the Repsold pendulum. In +order to eliminate this source of error, Peirce designed +a pendulum with knives that extended from each +side of the cylindrical staff, and he received authorization +from the superintendent of the Coast and Geodetic +Survey to arrange for the construction of such +pendulums by Gautier in Paris. Peirce, who had +made his plans in consultation with Gautier, was +called home before the pendulums were completed, +and these new instruments remained undelivered.</p> + +<p>In a memoir titled “Effect of the flexure of a pendulum +upon its period of oscillation,”<a name="FNanchor_99_99" id="FNanchor_99_99"></a><a href="#Footnote_99_99" class="fnanchor">[99]</a> Peirce determined +analytically the effect on the period of a pendulum +with a single elastic connection between two +rigid parts of the staff. Thus, Peirce discovered +experimentally the flexure of the staff and derived for +a simplified case the effect on the period. It is not +known if he ever found the integrated effect of the +continuum of elastic connections in the pendulum. +Lorenzoni, in 1896, offered a solution to the problem, +and Almansi, in 1899, gave an extended analysis. +After the independent discovery of the problem at +the Geodetic Institute, Dr. Helmert took up the problem +and criticized the theories of Peirce and Lorenzoni. +He then presented his own theory of flexure in +a comprehensive memoir.<a name="FNanchor_100_100" id="FNanchor_100_100"></a><a href="#Footnote_100_100" class="fnanchor">[100]</a> In view of the previous +neglect of the flexure of the pendulum staff in the +reduction of observations, Helmert directed that the +Geodetic Institute make a new absolute determination +of the intensity of gravity at Potsdam. For this +purpose, Kühnen and Furtwängler used the following +reversible pendulums which had been constructed +by the firm of A. Repsold and Sons in Hamburg:</p> + +<blockquote><p>1. The seconds pendulum of the Geodetic Institute +procured in 1869.</p> + +<p>2. A seconds pendulum from the Astronomical Observatory, +Padua.</p> + +<p>3. A heavy, seconds pendulum from the Imperial and +Royal Military-Geographical Institute, Vienna.</p> + +<p>4. A light, seconds pendulum from the Imperial and +Royal Military-Geographical Institute.</p> + +<p>5. A 1/2-second, reversible pendulum of the Geodetic +Institute procured in 1892.</p></blockquote> + +<p>Work was begun in 1898, and in 1906 Kühnen and +Furtwängler published their monumental memoir, +“Bestimmung der Absoluten Grösze der Schwerkraft +zu Potsdam mit Reversionspendeln.”</p> + +<p>The acceleration of gravity in the pendulum room +of the Geodetic Institute was determined to be 981.274 +ą 0.003 cm/sec<sup>2</sup>. In view of the exceptionally careful +and thorough determination at the Institute, Potsdam +was accepted as the world base for the absolute value +of the intensity of gravity. The absolute value of +gravity at some other station on the Potsdam system +was determined from the times of swing of an invariable +pendulum at the station and at Potsdam by the +relation <i>T</i><sub>1</sub><sup>2</sup>/<i>T</i><sub>2</sub><sup>2</sup> = <i>g</i><sub>2</sub>/<i>g</i><sub>1</sub>. +Thus, in 1900, Assistant G. R. Putnam of the Coast +and Geodetic Survey swung Mendenhall pendulums +at the Washington base and at Potsdam, and by transfer +from Potsdam determined the intensity of gravity +at the Washington base to be 980.112 cm/sec<sup>2</sup>.<a name="FNanchor_101_101" id="FNanchor_101_101"></a><a href="#Footnote_101_101" class="fnanchor">[101]</a> +In 1933, Lt. E. J. Brown made comparative measurements +with improved apparatus and raised the +value at the Washington base to 980.118 cm/sec<sup>2</sup>.<a name="FNanchor_102_102" id="FNanchor_102_102"></a><a href="#Footnote_102_102" class="fnanchor">[102]</a></p> + +<p>In view of discrepancies between the results of +various relative determinations, the Coast and Geodetic +Survey in 1928 requested the National Bureau +of Standards to make an absolute determination for +Washington. Heyl and Cook used reversible pendulums +made of fused silica having a period of +approximately 1 second. Their result, published in +1936, was interpreted to indicate that the value at +Potsdam was too high by 20 parts in 1 million.<a name="FNanchor_103_103" id="FNanchor_103_103"></a><a href="#Footnote_103_103" class="fnanchor">[103]</a> +This estimate was lowered slightly by Sir Harold +Jeffreys of Cambridge, England, who recomputed +the results of Heyl and Cook by different methods.<a name="FNanchor_104_104" id="FNanchor_104_104"></a><a href="#Footnote_104_104" class="fnanchor">[104]</a><span class="pagenum"><a name="Page_340" id="Page_340">[Pg 340]</a></span></p> + +<div class="figcenter" style="width: 700px;"><a name="fig_31" id="fig_31"></a> +<img src="images/i085.png" width="700" height="437" alt="Figure 31." title="Figure 31." /> +<p class="caption">Figure 31.—<span class="smcap">Map showing the distribution</span> of gravity stations throughout the United States as of December 1908.</p> +</div> +<p><span class="pagenum"><a name="Page_341" id="Page_341">[Pg 341]</a></span></p> +<div class="figcenter" style="width: 700px;"><a name="fig_32" id="fig_32"></a> +<img src="images/i086.png" width="700" height="447" alt="Figure 32." title="Figure 32." /> +<p class="caption">Figure 32.—<span class="smcap">Map showing the distribution</span> of gravity stations throughout the United States in 1923.</p> +</div> + +<p><span class="pagenum"><a name="Page_342" id="Page_342">[Pg 342]</a></span> +In 1939, J. S. Clark published the results of a +determination of gravity with pendulums of a non-ferrous +Y-alloy<a name="FNanchor_105_105" id="FNanchor_105_105"></a><a href="#Footnote_105_105" class="fnanchor">[105]</a> at the National Physical Laboratory +at Teddington, England, and, after recomputation +of results by Jeffreys, the value was found to be +12.8 parts in 1 million less than the value obtained by +transfer from Potsdam. Dr. Hugh L. Dryden of the +National Bureau of Standards, and Dr. A. Berroth +of the Geodetic Institute at <a name="corr_44_05" id="corr_44_05"></a><ins class="mycorr" title="Original: Postdam">Potsdam</ins>, have recomputed +the Potsdam data by different methods of adjustment +and concluded that the Potsdam value was too high +by about 12 parts in a million.<a name="FNanchor_106_106" id="FNanchor_106_106"></a><a href="#Footnote_106_106" class="fnanchor">[106]</a> Determination of +gravity at Leningrad by Russian scientists likewise +has indicated that the 1906 Potsdam value is too +high. In the light of present information, it +appears justifiable to reduce the Potsdam value of +981.274 by .013 cm/sec<sup>2</sup> for purposes of comparison. +If the Brown transfer from Potsdam in 1933 was +taken as accurate, the value for the Washington base +would be 980.105 cm/sec<sup>2</sup>. In this connection, it is +of interest to note that the value given by Charles +S. Peirce for the comparable Smithsonian base in +Washington, as determined by him from comparative +methods in the 1880’s and reported in the <i>Annual +Report of the Superintendent of the Coast and Geodetic Survey +for the year 1890-1891</i>, was 980.1017 cm/sec<sup>2</sup>.<a name="FNanchor_107_107" id="FNanchor_107_107"></a><a href="#Footnote_107_107" class="fnanchor">[107]</a> This +value would appear to indicate that Peirce’s pendulums, +observations, and methods of reduction of data +were not inferior to those of the scientists of the +Royal Prussian Geodetic Institute at Potsdam.</p> + +<p>Doubts concerning the accuracy of the Potsdam +value of gravity have stimulated many new determinations +of the intensity of gravity since the end of World +War II. In a paper published in June 1957, A. H. +Cook, Metrology Division, National Physical Laboratory, +Teddington, England, stated:</p> + +<blockquote><p>At present about a dozen new absolute determinations +are in progress or are being planned. Heyl and Cook’s +reversible pendulum apparatus is in use in Buenos Aires +and further reversible pendulum experiments have been +made in the All Union Scientific Research Institute of +Metrology, Leningrad <span class="nowrap">(V N I I M)</span> and are planned at +Potsdam. A method using a very long pendulum was +tried out in Russia about 1910 and again more recently +and there are plans for similar work in Finland. The +first experiment with a freely falling body was that +carried out by Volet who photographed a graduated +scale falling in an enclosure at low air pressure. Similar +experiments have been completed in Leningrad and +are in progress at the Physikalisch-Technische Bundesanstalt +(Brunswick) and at the National Research +Council (Ottawa), and analogous experiments are being +prepared at the National Physical Laboratory and at +the National Bureau of Standards. Finally, Professor +Medi, Director of the Istituto Nazionale di Geofisica +(Rome), is attempting to measure the focal length of +the paraboloidal surface of a liquid in a rotating dish.<a name="FNanchor_108_108" id="FNanchor_108_108"></a><a href="#Footnote_108_108" class="fnanchor">[108]</a></p></blockquote> + + + +<hr style="width: 65%;" /> +<h3>Application of Gravity Surveys</h3> + + +<p>We have noted previously that in the ancient and +early modern periods, the earth was presupposed to +be spherical in form. Determination of the figure of +the earth consisted in the measurement of the radius +by the astronomical-geodetic method invented by +Eratosthenes. Since the earth was assumed to be +spherical, gravity was inferred to be constant over +the surface of the earth. This conclusion appeared +to be confirmed by the determination of the length +of the seconds pendulum at various stations in Europe +by Picard and others. The observations of Richer in +South America, the theoretical discussions of Newton +and Huygens, and the measurements of degrees of +latitude in Peru and Sweden demonstrated that the +earth is an oblate spheroid.</p> + +<div class="figcenter" style="width: 600px;"><a name="fig_33" id="fig_33"></a> +<img src="images/i089.png" width="600" height="597" alt="Figure 33." title="Figure 33." /> + +<p class="caption2">Figure 33.—<span class="smcap">Gravity characteristics of the globe.</span> +Deductions as to the distribution of matter in the earth +can be made from gravity measurements. This globe +shows worldwide variations in gravity as they now appear +from observations at sea (in submarines) as well as on +land. It is based on data from the Institute of Geodesy +at Ohio State University.</p> +</div> + +<p>The theory of gravitation and the theory of central +forces led to the result that the intensity of gravity +is variable over the surface of the earth. Accordingly,<span class="pagenum"><a name="Page_343" id="Page_343">[Pg 343]</a></span> +determinations of the intensity of gravity became of +value to the geodesist as a means of determining +the figure of the earth. Newton, on the basis of the +meager data available to him, calculated the ellipticity +of the earth to be 1/230 (the ellipticity is defined by +(<i>a</i>-<i>b</i>)/<i>a</i>, where <i>a</i> is the equatorial radius and <i>b</i> the polar +radius). Observations of the intensity of gravity +were made on the historic missions to Peru and +Sweden. Bouguer and La Condamine found that at +the equator at sea level the seconds pendulum was +1.26 Paris-lines shorter than at Paris. Maupertuis +found that in northern Sweden a certain pendulum +clock gained 59.1 seconds per day on its rate in +Paris. Then Clairaut, from the assumption that the +earth is a spheroid of equilibrium, derived a theorem +from which the ellipticity of the earth can be derived +from values of the intensity of gravity.<span class="pagenum"><a name="Page_344" id="Page_344">[Pg 344]</a></span></p> + +<div class="figcenter" style="width: 600px;"><a name="fig_34" id="fig_34"></a> +<img src="images/i092.png" width="500" height="720" alt="Figure 34." title="Figure 34." /> + +<p class="caption2">Figure 34.—<span class="smcap">An exhibit of gravity apparatus</span> at the Smithsonian Institution. Suspended +on the wall, from left to right, are the invariable pendulums of Mendenhall (1/2-second), +Peirce (1873-1874), and Peirce (1881-1882); the double pendulum of Edward Kübel +(see fig. <a href="#fig_15">15</a>, p. <a href="#Page_319">319</a>), and the reversible pendulum of Peirce. On the display counter, from +left to right, are the vacuum chamber, telescope and flash apparatus for the Mendenhall +1/4-second apparatus. Shown below these are the four pendulums used with the Mendenhall +apparatus, the one on the right having a thermometer attached. At bottom, right, is +the Gulf apparatus (cover removed) mentioned in the text, shown with one quartz +pendulum.</p> +</div> + +<p><span class="pagenum"><a name="Page_345" id="Page_345">[Pg 345]</a></span> +Early in the 19th century a systematic series of +observations began to be conducted in order to determine +the intensity of gravity at stations all over +the world. Kater invariable pendulums, of which +13 examples have been mentioned in the literature, +were used in surveys of gravity by Kater, Sabine, +Goldingham, and other British pendulum swingers. +As has been noted previously, a Kater invariable +pendulum was used by Adm. Lütke of Russia on +a trip around the world. The French also sent +out expeditions to determine values of gravity. +After several decades of relative inactivity, Capts. +Basevi and Heaviside of the Indian Survey carried +out an important series of observations from 1865 to +1873 with Kater invariable pendulums and the +Russian Repsold-Bessel pendulums. In 1881-1882 +Maj. J. Herschel swung Kater invariable pendulums +nos. 4, 6 (1821), and 11 at stations in England and +then brought them to the United States in order to +make observations which would connect American +and English base stations.<a name="FNanchor_109_109" id="FNanchor_109_109"></a><a href="#Footnote_109_109" class="fnanchor">[109]</a></p> + +<p>The extensive sets of observations of gravity provided +the basis of calculations of the ellipticity of the +earth. Col. A. R. Clarke in his <i>Geodesy</i> (London, +1880) calculated the ellipticity from the results of +gravity surveys to be <span class="nowrap">1/(292.2 ą 1.5)</span>. Of interest is the +calculation by Charles S. Peirce, who used only +determinations made with Kater invariable pendulums +and corrected for elevation, atmospheric effect, +and expansion of the pendulum through temperature.<a name="FNanchor_110_110" id="FNanchor_110_110"></a><a href="#Footnote_110_110" class="fnanchor">[110]</a> +He calculated the ellipticity of the earth +to be <span class="nowrap">1/(291.5 ą 0.9)</span>.</p> + +<p>The 19th century witnessed the culmination of the +ellipsoidal era of geodesy, but the rapid accumulation +of data made possible a better approximation to the +figure of the earth by the geoid. The geoid is +defined as the average level of the sea, which is +thought of as extended through the continents. +The basis of geodetic calculations, however, is an +ellipsoid of reference for which a gravity formula +expresses the value of normal gravity at a point on +the ellipsoid as a function of gravity at sea level at +the equator, and of latitude. The general assembly +of the International Union of Geodesy and Geophysics, +which was founded after World War I to continue +the work of <i>Die Internationale Erdmessung</i>, adopted +in 1924 an international reference ellipsoid,<a name="FNanchor_111_111" id="FNanchor_111_111"></a><a href="#Footnote_111_111" class="fnanchor">[111]</a> of +which the ellipticity, or flattening, is Hayford’s +value 1/297. In 1930, the general assembly adopted +a correlated International Gravity Formula of the form +γ = γ<sub><i>E</i></sub><span class="nowrap">(1 + β(sin<sup>2</sup> φ) + ε(sin<sup>2</sup> 2φ))</span> +where γ is normal gravity at latitude φ, γ<sub><i>E</i></sub> is the +value of gravity at sea level at the equator, β is a +parameter which is computed on the basis of Clairaut’s +theorem from the flattening value of the meridian, +and ε is a constant which is derived theoretically. +The plumb line is perpendicular to the geoid, and +the components of angle between the perpendiculars +to geoid and reference ellipsoid are deflections of +the vertical. The geoid is above the ellipsoid of +reference under mountains and it is below the +ellipsoid on the oceans, where the geoid coincides +with mean sea level. In physical geodesy, gravimetric +data are used for the determination of the +geoid and components of deflections of the vertical. +For this purpose, one must reduce observed values +of gravity to sea level by various reductions, such as +free-air, Bouguer, isostatic reductions. If <i>g</i><sub>0</sub> is observed +gravity reduced to sea level and γ is normal gravity +obtained from the International Gravity Formula, +then Δ<i>g</i> = <span class="nowrap"><i>g</i><sub>0</sub> - γ</span> is the gravity anomaly.<a name="FNanchor_112_112" id="FNanchor_112_112"></a><a href="#Footnote_112_112" class="fnanchor">[112]</a></p> + +<p>In 1849, Stokes derived a theorem whereby the +distance <i>N</i> of the geoid from the ellipsoid of reference +can be obtained from an integration of gravity +anomalies over the surface of the earth. Vening +Meinesz further derived formulae for the calculation +of components of the deflection of the vertical.</p> + +<p>Geometrical geodesy, which was based on astronomical-geodetic +methods, could give information +only concerning the external form of the figure of +the earth. The gravimetric <a name="corr_44_06" id="corr_44_06"></a><ins class="mycorr" title="Original: mtehods">methods</ins> of physical +geodesy, in conjunction with methods such as those +of seismology, enable scientists to test hypotheses +concerning the internal structure of the earth. +Heiskanen and Vening Meinesz summarize the +present-day achievements of the gravimetric method of<span class="pagenum"><a name="Page_346" id="Page_346">[Pg 346]</a></span> +physical geodesy by stating<a name="FNanchor_113_113" id="FNanchor_113_113"></a><a href="#Footnote_113_113" class="fnanchor">[113]</a> that it alone can give:</p> + +<blockquote><p>1. The flattening of the reference ellipsoid.</p> + +<p>2. The undulations <i>N</i> of the geoid.</p> + +<p>3. The components of the deflection of the vertical ζ +and η at any point, oceans and islands included.</p> + +<p>4. The conversion of existing geodetic systems to the +same world geodetic system.</p> + +<p>5. The reduction of triangulation base lines from the +geoid to the reference ellipsoid.</p> + +<p>6. The correction of errors in triangulation in mountainous +regions due to the effect of the deflections of +the vertical.</p> + +<p>7. Geophysical applications of gravity measurements, +e.g., the isostatic study of the earth’s interior and the +exploration of oil fields and ore deposits.</p></blockquote> + +<p>With astronomical observations or with existing +triangulations, the gravimetric method can accomplish +further results. Heiskanen and Vening Meinesz state:</p> + +<blockquote><p>It is the firm conviction of the authors that the gravimetric +method is by far the best of the existing methods +for solving the main problems of geodesy, i.e., to determine +the shape of the geoid on the continents as well as +at sea and to convert the existing geodetic systems to +the world geodetic system. It can also give invaluable +help in the computation of the reference ellipsoid.<a name="FNanchor_114_114" id="FNanchor_114_114"></a><a href="#Footnote_114_114" class="fnanchor">[114]</a></p></blockquote> + + + +<hr style="width: 65%;" /> +<h3>Summary</h3> + +<p>Since the creation of classical mechanics in the 17th +century, the pendulum has been a basic instrument for +the determination of the intensity of gravity, which is +expressed as the acceleration of a freely falling body. +Basis of theory is the simple pendulum, whose time of +swing under gravity is proportional to the square root +of the length divided by the acceleration due to +gravity. Since the length of a simple pendulum +divided by the square of its time of swing is equal to +the length of a pendulum that beats seconds, the intensity +of gravity also has been expressed in terms of +the length of the seconds pendulum. The reversible +compound pendulum has served for the absolute +determination of gravity by means of a theory developed +by Huygens. Invariable compound pendulums +with single axes also have been used to determine +relative values of gravity by comparative times +of swing.</p> + +<p>The history of gravity pendulums begins with the +ball or “simple” pendulum of Galileo as an approximation +to the ideal simple pendulum. Determinations +of the length of the seconds pendulum by French +scientists culminated in a historic determination at +Paris by Borda and Cassini, from the corrected observations +with a long ball pendulum. In the 19th +century, Bessel found the length of the seconds pendulum +at Königsberg and Berlin by observations with +a ball pendulum and by original theoretical considerations. +During the century, however, the compound +pendulum came to be preferred for absolute and relative +determinations.</p> + +<p>Capt. Henry Kater, at London, constructed the first +convertible compound for an absolute determination +of gravity, and then he designed an invariable compound +pendulum, examples of which were used for +relative determinations at various stations in Europe +and elsewhere. Bessel demonstrated theoretically the +advantages of a reversible compound pendulum which +is symmetrical in form and is hung by interchangeable +knives. The firm of A. Repsold and Sons in Hamburg +constructed pendulums from the specifications of +Bessel for European gravity surveys.</p> + +<p>Charles S. Peirce in 1875 received delivery in Hamburg +of a Repsold-Bessel pendulum for the U.S. Coast +Survey and observed with it in Geneva, Paris, Berlin, +and London. Upon an initial stimulation from +Baeyer, founder of <i>Die Europäische Gradmessung</i>, +Peirce demonstrated by experiment and theory that +results previously obtained with the Repsold apparatus +required correction, because of the flexure of +the stand under oscillations of the pendulum. At +the Stuttgart conference of the geodetic association in +1877, Hervé Faye proposed to solve the problem of +flexure by swinging two similar pendulums from the +same support with equal amplitudes and in opposite +phases. Peirce, in 1879, demonstrated theoretically +the soundness of the method and presented a design +for its application, but the “double pendulum” was +rejected at that time. Peirce also designed and had +constructed four examples of a new type of invariable, +reversible pendulum of cylindrical form which made +possible the experimental study of Stokes’ theory of the +resistance to motion of a pendulum in a viscous fluid. +Commandant Defforges, of France, also designed and +used cylindrical reversible pendulums, but of different +length so that the effect of flexure was eliminated in +the reduction of observations. Maj. Robert von +Sterneck, of Austria-Hungary, initiated a new era in +gravity research by the invention of an apparatus with +a short pendulum for relative determinations of +gravity. Stands were then constructed in Europe on<span class="pagenum"><a name="Page_347" id="Page_347">[Pg 347]</a></span> +which two or four pendulums were hung at the same +time. Finally, early in the present century, Vening +Meinesz found that the Faye-Peirce method of swinging +pendulums hung on a Stückrath four-pendulum +stand solved the problem of instability due to the +mobility of the soil in Holland.</p> + +<p>The 20th century has witnessed increasing activity +in the determination of absolute and relative values of +gravity. Gravimeters have been perfected and have +been widely used for rapid relative determinations, +but the compound pendulums remain as indispensable +instruments. Mendenhall’s replacement of knives by +planes attached to nonreversible pendulums has been +used also for reversible ones. The Geodetic Institute at +Potsdam is presently applying the Faye-Peirce method +to the reversible pendulum.<a name="FNanchor_115_115" id="FNanchor_115_115"></a><a href="#Footnote_115_115" class="fnanchor">[115]</a> Pendulums have been +constructed of new materials, such as invar, fused +silica, and fused quartz. Minimum pendulums for +precise relative determinations have been constructed +and used. Reversible pendulums have been made with +“I” cross sections for better stiffness. With all these +modifications, however, the foundations of the present +designs of compound pendulum apparatus were +created in the 19th century.</p> + + + +<div class="footnotes"> +<h3>FOOTNOTES:</h3> + +<div class="footnote"><p><a name="Footnote_1_1" id="Footnote_1_1"></a><a href="#FNanchor_1_1"><span class="label">[1]</span></a> The basic historical documents have been collected, with a +bibliography of works and memoirs published from 1629 to +the end of 1885, in <i>Collection de mémoires relatifs a la physique, +publiés par la Société <a name="corr_44_07" id="corr_44_07"></a><ins class="mycorr" title="Original: Française">française</ins> de Physique</i> [hereinafter referred +to as <i>Collection de mémoires</i>]: vol. 4, <i>Mémoires sur le pendule, précédés +d’une bibliographie</i> (Paris: Gauthier-Villars, 1889); and vol. 5, +<i>Mémoires sur le pendule</i>, part 2 (Paris: Gauthier-Villars, 1891). +Important secondary sources are: <span class="smcap">C. Wolf</span>, “Introduction +historique,” pp. 1-42 in vol. 4, above; and <span class="smcap">George Biddell +Airy</span>, “Figure of the Earth,” pp. 165-240 in vol. 5 of <i>Encyclopaedia +metropolitana</i> (London, 1845).</p></div> + +<div class="footnote"><p><a name="Footnote_2_2" id="Footnote_2_2"></a><a href="#FNanchor_2_2"><span class="label">[2]</span></a> Galileo Galilei’s principal statements concerning the pendulum +occur in his <i>Discourses Concerning Two New Sciences</i>, +transl. from Italian and Latin into English by Henry Crew and +Alfonso de Salvio (Evanston: Northwestern University Press, +1939), pp. 95-97, 170-172.</p></div> + +<div class="footnote"><p><a name="Footnote_3_3" id="Footnote_3_3"></a><a href="#FNanchor_3_3"><span class="label">[3]</span></a> <span class="smcap">P. Marin Mersenne</span>, <i>Cogitata <a name="corr_44_17" id="corr_44_17"></a><ins class="mycorr" title="Original: physica">physico</ins>-mathematica</i> (Paris, +1644), p. 44.</p></div> + +<div class="footnote"><p><a name="Footnote_4_4" id="Footnote_4_4"></a><a href="#FNanchor_4_4"><span class="label">[4]</span></a> <span class="smcap">Christiaan Huygens</span>, <i>Horologium oscillatorium, sive de motu +pendulorum ad horologia adaptato demonstrationes geometricae</i> (Paris, +1673), proposition 20.</p></div> + +<div class="footnote"><p><a name="Footnote_5_5" id="Footnote_5_5"></a><a href="#FNanchor_5_5"><span class="label">[5]</span></a> The historical events reported in the present section are +from <span class="smcap">Airy</span>, “Figure of the Earth.”</p></div> + +<div class="footnote"><p><a name="Footnote_6_6" id="Footnote_6_6"></a><a href="#FNanchor_6_6"><span class="label">[6]</span></a> <span class="smcap">Abbé Jean Picard</span>, <i>La Mesure de la terre</i> (Paris, 1671). +<span class="smcap">John W. Olmsted</span>, “The ‘Application’ of Telescopes to Astronomical +Instruments, 1667-1669,” <i>Isis</i> (1949), vol. 40, p. 213.</p></div> + +<div class="footnote"><p><a name="Footnote_7_7" id="Footnote_7_7"></a><a href="#FNanchor_7_7"><span class="label">[7]</span></a> The toise as a unit of length was 6 Paris feet or about 1,949 +millimeters.</p></div> + +<div class="footnote"><p><a name="Footnote_8_8" id="Footnote_8_8"></a><a href="#FNanchor_8_8"><span class="label">[8]</span></a> <span class="smcap">Jean Richer</span>, <i>Observations astronomiques et physiques faites +en l’isle de Caďenne</i> (Paris, 1679). <span class="smcap">John W. Olmsted</span>, “The +Expedition of Jean Richer to Cayenne 1672-1673,” <i>Isis</i> +(1942), vol. 34, pp. 117-128.</p></div> + +<div class="footnote"><p><a name="Footnote_9_9" id="Footnote_9_9"></a><a href="#FNanchor_9_9"><span class="label">[9]</span></a> The Paris foot was 1.066 English feet, and there were 12 +lines to the inch.</p></div> + +<div class="footnote"><p><a name="Footnote_10_10" id="Footnote_10_10"></a><a href="#FNanchor_10_10"><span class="label">[10]</span></a> <span class="smcap">Christiaan Huygens</span>, “De la cause de la pesanteur,” +<i>Divers ouvrages de mathematiques et de physique par MM. de +l’Académie <a name="corr_44_18" id="corr_44_18"></a><ins class="mycorr" title="Original: Royal">Royale</ins> des Sciences</i> (Paris, 1693), p. 305.</p></div> + +<div class="footnote"><p><a name="Footnote_11_11" id="Footnote_11_11"></a><a href="#FNanchor_11_11"><span class="label">[11]</span></a> <span class="smcap">Isaac Newton</span>, <i>Philosophiae naturalis principia mathematica</i> +(London, 1687), vol. 3, propositions 18-20.</p></div> + +<div class="footnote"><p><a name="Footnote_12_12" id="Footnote_12_12"></a><a href="#FNanchor_12_12"><span class="label">[12]</span></a> <span class="smcap">Pierre Bouguer</span>, <i>La figure de la terre, déterminée par les +observations de Messieurs Bouguer et de La Condamine, envoyés +par ordre du Roy au <a name="corr_44_08" id="corr_44_08"></a><ins class="mycorr" title="Original: Perou, pour observir">Pérou, pour observer</ins> aux environs de l’equateur</i> +(Paris, 1749).</p></div> + +<div class="footnote"><p><a name="Footnote_13_13" id="Footnote_13_13"></a><a href="#FNanchor_13_13"><span class="label">[13]</span></a> <span class="smcap">P. L. Moreau de Maupertuis</span>, <i>La figure de la terre déterminée +par les observations de Messieurs de Maupertuis, Clairaut, Camus, Le +Monnier, l’Abbé Outhier et Celsius, faites par ordre du Roy au cercle +polaire</i> (Paris, 1738).</p></div> + +<div class="footnote"><p><a name="Footnote_14_14" id="Footnote_14_14"></a><a href="#FNanchor_14_14"><span class="label">[14]</span></a> Paris, 1743.</p></div> + +<div class="footnote"><p><a name="Footnote_15_15" id="Footnote_15_15"></a><a href="#FNanchor_15_15"><span class="label">[15]</span></a> <span class="smcap">George Gabriel Stokes</span>, “On Attraction and on Clairaut’s +Theorem,” <i>Cambridge and Dublin Mathematical Journal</i> +(1849), vol. 4, p. 194.</p></div> + +<div class="footnote"><p><a name="Footnote_16_16" id="Footnote_16_16"></a><a href="#FNanchor_16_16"><span class="label">[16]</span></a> See <i>Collection de mémoires</i>, vol. 4, p. B-34, and <span class="smcap">J. H. Poynting</span> +and <span class="smcap">Sir J. J. Thomson</span>, <i>Properties of Matter</i> (London, 1927), +p. 24.</p></div> + +<div class="footnote"><p><a name="Footnote_17_17" id="Footnote_17_17"></a><a href="#FNanchor_17_17"><span class="label">[17]</span></a> <span class="smcap">Poynting</span> and <span class="smcap">Thomson</span>, ibid., p. 22.</p></div> + +<div class="footnote"><p><a name="Footnote_18_18" id="Footnote_18_18"></a><a href="#FNanchor_18_18"><span class="label">[18]</span></a> <span class="smcap">Charles M. de la Condamine</span>, “De la mesure du pendule +ŕ Saint Domingue,” <i>Collection de mémoires</i>, vol. 4, pp. 3-16.</p></div> + +<div class="footnote"><p><a name="Footnote_19_19" id="Footnote_19_19"></a><a href="#FNanchor_19_19"><span class="label">[19]</span></a> <span class="smcap">Pčre R. J. Boscovich</span>, <i>Opera pertinentia ad Opticam et +<a name="corr_44_19" id="corr_44_19"></a><ins class="mycorr" title="Original: Astronomian">Astronomiam</ins></i> (Bassani, 1785), vol. 5, no. 3.</p></div> + +<div class="footnote"><p><a name="Footnote_20_20" id="Footnote_20_20"></a><a href="#FNanchor_20_20"><span class="label">[20]</span></a> <span class="smcap">J. C. Borda</span> and <span class="smcap">J. D. Cassini de Thury</span>, “Expériences +pour <a name="corr_44_09" id="corr_44_09"></a><ins class="mycorr" title="Original: connaitre la longuer">connaître la longueur</ins> du pendule qui bat les secondes ŕ +Paris,” <i>Collection de mémoires</i>, vol. 4, pp. 17-64.</p></div> + +<div class="footnote"><p><a name="Footnote_21_21" id="Footnote_21_21"></a><a href="#FNanchor_21_21"><span class="label">[21]</span></a> <span class="smcap">F. W. Bessel</span>, “Untersuchungen über die Länge des +einfachen Secundenpendels,” <i>Abhandlungen der <a name="corr_44_20" id="corr_44_20"></a><ins class="mycorr" title="Original: Königliche">Königlichen</ins> Akademie +der Wissenschaften zu Berlin, 1826</i> (Berlin, 1828).</p></div> + +<div class="footnote"><p><a name="Footnote_22_22" id="Footnote_22_22"></a><a href="#FNanchor_22_22"><span class="label">[22]</span></a> Bessel used as a standard of length a toise which had been +made by Fortin in Paris and had been compared with the +original of the “toise de Peru” by Arago.</p></div> + +<div class="footnote"><p><a name="Footnote_23_23" id="Footnote_23_23"></a><a href="#FNanchor_23_23"><span class="label">[23]</span></a> <span class="smcap">L. G. du Buat</span>, <i>Principes d’hydraulique</i> (Paris, 1786). See +excerpts in <i>Collection de mémoires</i>, pp. B-64 to B-67.</p></div> + +<div class="footnote"><p><a name="Footnote_24_24" id="Footnote_24_24"></a><a href="#FNanchor_24_24"><span class="label">[24]</span></a> <span class="smcap">Capt. Henry Kater</span>, “An Account of Experiments for +Determining the Length of the Pendulum Vibrating Seconds +in the Latitude of London,” <i>Philosophical Transactions of the +Royal Society of London</i> (1818), vol. 108, p. 33. [Hereinafter +abbreviated <i>Phil. Trans.</i>]</p></div> + +<div class="footnote"><p><a name="Footnote_25_25" id="Footnote_25_25"></a><a href="#FNanchor_25_25"><span class="label">[25]</span></a> <span class="smcap">M. G. de Prony</span>, “Méthode pour déterminer la <a name="corr_44_21" id="corr_44_21"></a><ins class="mycorr" title="Original: longeur">longueur</ins> +du pendule simple qui bat les secondes,” <i>Collection de mémoires</i>, +vol. 4, pp. 65-76.</p></div> + +<div class="footnote"><p><a name="Footnote_26_26" id="Footnote_26_26"></a><a href="#FNanchor_26_26"><span class="label">[26]</span></a> <i>Collection de mémoires</i>, vol. 4, p. B-74.</p></div> + +<div class="footnote"><p><a name="Footnote_27_27" id="Footnote_27_27"></a><a href="#FNanchor_27_27"><span class="label">[27]</span></a> <i>Phil. Trans.</i> (1819), vol. 109, p. 337.</p></div> + +<div class="footnote"><p><a name="Footnote_28_28" id="Footnote_28_28"></a><a href="#FNanchor_28_28"><span class="label">[28]</span></a> <span class="smcap">John Herschel</span>, “Notes for a History of the Use of +Invariable Pendulums,” <i>The Great Trigonometrical Survey of India</i> +(Calcutta, 1879), vol. 5.</p></div> + +<div class="footnote"><p><a name="Footnote_29_29" id="Footnote_29_29"></a><a href="#FNanchor_29_29"><span class="label">[29]</span></a> <span class="smcap">Capt. Edward Sabine</span>, “An Account of Experiments to +Determine the Figure of the Earth,” <i>Phil. Trans.</i> (1828), vol. 118, +p. 76.</p></div> + +<div class="footnote"><p><a name="Footnote_30_30" id="Footnote_30_30"></a><a href="#FNanchor_30_30"><span class="label">[30]</span></a> <span class="smcap">John Goldingham</span>, “Observations for Ascertaining the +Length of the Pendulum at Madras in the East Indies,” <i>Phil. +Trans.</i> (1822), vol. 112, p. 127.</p></div> + +<div class="footnote"><p><a name="Footnote_31_31" id="Footnote_31_31"></a><a href="#FNanchor_31_31"><span class="label">[31]</span></a> <span class="smcap">Basil Hall</span>, “Letter to Captain Kater Communicating +the Details of Experiments made by him and Mr. Henry +Foster with an Invariable Pendulum,” <i>Phil. Trans.</i> (1823), +vol. 113, p. 211.</p></div> + +<div class="footnote"><p><a name="Footnote_32_32" id="Footnote_32_32"></a><a href="#FNanchor_32_32"><span class="label">[32]</span></a> See <i>Collection de mémoires</i>, vol. 4, p. B-103.</p></div> + +<div class="footnote"><p><a name="Footnote_33_33" id="Footnote_33_33"></a><a href="#FNanchor_33_33"><span class="label">[33]</span></a> Ibid., p. B-88.</p></div> + +<div class="footnote"><p><a name="Footnote_34_34" id="Footnote_34_34"></a><a href="#FNanchor_34_34"><span class="label">[34]</span></a> Ibid., p. B-94.</p></div> + +<div class="footnote"><p><a name="Footnote_35_35" id="Footnote_35_35"></a><a href="#FNanchor_35_35"><span class="label">[35]</span></a> <span class="smcap">Francis Baily</span>, “On the Correction of a Pendulum for +the Reduction to a Vacuum, Together with Remarks on Some +Anomalies Observed in Pendulum Experiments,” <i>Phil. Trans.</i> +(1832), vol. 122, pp. 399-492. See also <i>Collection de mémoires</i>, +vol. 4, pp. B-105, B-112, B-115, B-116, and B-117.</p></div> + +<div class="footnote"><p><a name="Footnote_36_36" id="Footnote_36_36"></a><a href="#FNanchor_36_36"><span class="label">[36]</span></a> One was of case brass and the other of rolled iron, 68 in. +long, 2 in. wide, and 1/2 in. thick. Triangular knife edges 2 in. +long were inserted through triangular apertures 19.7 in. from +the center towards each end. These pendulums seem not to +have survived. There is, however, in the collection of the +U.S. National Museum, a similar brass pendulum, 37-5/8 in. +long (fig. <a href="#fig_15">15</a>) stamped with the name of Edward Kübel (1820-96), +who maintained an instrument business in Washington, +D.C., from about 1849. The history of this instrument is +unknown.</p></div> + +<div class="footnote"><p><a name="Footnote_37_37" id="Footnote_37_37"></a><a href="#FNanchor_37_37"><span class="label">[37]</span></a> See Baily’s remarks in the <i>Monthly Notices of the Royal +Astronomical Society</i> (1839), vol. 4, pp. 141-143. See also letters +mentioned in footnote <a href="#Footnote_38_38">38</a>.</p></div> + +<div class="footnote"><p><a name="Footnote_38_38" id="Footnote_38_38"></a><a href="#FNanchor_38_38"><span class="label">[38]</span></a> This document, together with certain manuscript notes on +the pendulum experiments and six letters between Wilkes and +Baily, is in the U.S. National Archives, Navy Records Gp. 37. +These were the source materials for the information presented +here on the Expedition. We are indebted to Miss Doris +Ann Esch and Mr. Joseph Rudmann of the staff of the U.S. +National Museum for calling our attention to this early American +pendulum work.</p></div> + +<div class="footnote"><p><a name="Footnote_39_39" id="Footnote_39_39"></a><a href="#FNanchor_39_39"><span class="label">[39]</span></a> <span class="smcap">G. B. Airy</span>, “Account of Experiments Undertaken in the +Harton Colliery, for the Purpose of Determining the Mean +Density of the Earth,” <i>Phil. Trans.</i> (1856), vol. 146, p. 297.</p></div> + +<div class="footnote"><p><a name="Footnote_40_40" id="Footnote_40_40"></a><a href="#FNanchor_40_40"><span class="label">[40]</span></a> <span class="smcap">T. C. Mendenhall</span>, “Measurements of the Force of Gravity +at Tokyo, and on the Summit of Fujiyama,” <i>Memoirs of the +Science Department, University of Tokyo</i> (1881), no. 5.</p></div> + +<div class="footnote"><p><a name="Footnote_41_41" id="Footnote_41_41"></a><a href="#FNanchor_41_41"><span class="label">[41]</span></a> <span class="smcap">J. T. Walker</span>, <i>Account of Operations of The Great Trigonometrical +<a name="corr_44_10" id="corr_44_10"></a><ins class="mycorr" title="Original: Surey">Survey</ins> of India</i> (Calcutta, 1879), vol. 5, app. no. 2.</p></div> + +<div class="footnote"><p><a name="Footnote_42_42" id="Footnote_42_42"></a><a href="#FNanchor_42_42"><span class="label">[42]</span></a> <span class="smcap">Bessel</span>, op. cit. (footnote <a href="#Footnote_21_21">21</a>), article 31.</p></div> + +<div class="footnote"><p><a name="Footnote_43_43" id="Footnote_43_43"></a><a href="#FNanchor_43_43"><span class="label">[43]</span></a> <span class="smcap">C. A. F. Peters</span>, <i>Briefwechsel zwischen C. F. Gauss und H. C. +Schumacher</i> (Altona, Germany, 1860), <i>Band</i> 2, p. 3. The +correction required if the times of swing are not exactly the +same is said to have been given also by Bohnenberger.</p></div> + +<div class="footnote"><p><a name="Footnote_44_44" id="Footnote_44_44"></a><a href="#FNanchor_44_44"><span class="label">[44]</span></a> <span class="smcap">F. W. Bessel</span>, “Construction eines symmetrisch geformten +Pendels mit reciproken Axen, von Bessel,” <i>Astronomische +Nachrichten</i> (1849), vol. 30, p. 1.</p></div> + +<div class="footnote"><p><a name="Footnote_45_45" id="Footnote_45_45"></a><a href="#FNanchor_45_45"><span class="label">[45]</span></a> <span class="smcap">E. Plantamour</span>, “Expériences faites ŕ Genčve avec le +pendule ŕ réversion,” <i>Mémoires de la Société de Physique et <a name="corr_44_11" id="corr_44_11"></a><ins class="mycorr" title="Original: d’historire">d’histoire</ins> +naturelle de Genčve, 1865</i> (Geneva, 1866), vol. 18, p. 309.</p></div> + +<div class="footnote"><p><a name="Footnote_46_46" id="Footnote_46_46"></a><a href="#FNanchor_46_46"><span class="label">[46]</span></a> Ibid., pp. 309-416.</p></div> + +<div class="footnote"><p><a name="Footnote_47_47" id="Footnote_47_47"></a><a href="#FNanchor_47_47"><span class="label">[47]</span></a> <span class="smcap">C. Cellérier</span>, “Note sur la Mesure de la Pesanteur par +le Pendule,” <i>Mémoires de la Société de Physique et <a name="corr_44_12" id="corr_44_12"></a><ins class="mycorr" title="Original: d’historire">d’histoire</ins> +naturelle de Genčve, 1865</i> (Geneva, 1866), vol. 18, pp. 197-218.</p></div> + +<div class="footnote"><p><a name="Footnote_48_48" id="Footnote_48_48"></a><a href="#FNanchor_48_48"><span class="label">[48]</span></a> <span class="smcap">A. Sawitsch</span>, “Les variations de la pesanteur dans les +provinces occidentales de l’Empire russe,” <i>Memoirs of the +Royal Astronomical Society</i> (1872), vol. 39, p. 19.</p></div> + +<div class="footnote"><p><a name="Footnote_49_49" id="Footnote_49_49"></a><a href="#FNanchor_49_49"><span class="label">[49]</span></a> <span class="smcap">J. J. Baeyer</span>, <i>Über die <a name="corr_44_22" id="corr_44_22"></a><ins class="mycorr" title="Original: Grosse">Grösse</ins> und Figur der Erde</i> (Berlin, +1861).</p></div> + +<div class="footnote"><p><a name="Footnote_50_50" id="Footnote_50_50"></a><a href="#FNanchor_50_50"><span class="label">[50]</span></a> <i>Comptes-rendus de la Conférence Géodésique Internationale +réunie ŕ Berlin du 15-22 Octobre 1864</i> (Neuchâtel, 1865).</p></div> + +<div class="footnote"><p><a name="Footnote_51_51" id="Footnote_51_51"></a><a href="#FNanchor_51_51"><span class="label">[51]</span></a> Ibid., part III, subpart E.</p></div> + +<div class="footnote"><p><a name="Footnote_52_52" id="Footnote_52_52"></a><a href="#FNanchor_52_52"><span class="label">[52]</span></a> <i>Bericht über die Verhandlungen der vom 30 September bis 7 +October 1867 zu Berlin abgehaltenen allgemeinen Conferenz der +Europäischen Gradmessung</i> (Berlin, 1868). See report of fourth +session, October 3, 1867.</p></div> + +<div class="footnote"><p><a name="Footnote_53_53" id="Footnote_53_53"></a><a href="#FNanchor_53_53"><span class="label">[53]</span></a> <span class="smcap">C. Bruhns</span> and <span class="smcap">Albrecht</span>, “Bestimmung der <a name="corr_44_13" id="corr_44_13"></a><ins class="mycorr" title="Original: Lange">Länge</ins> +des Secundenpendels in Bonn, Leiden und Mannheim,” +<i><ins class="mycorr" title="Original: Astronomische">Astronomisch</ins>-Geodätische Arbeiten im Jahre 1870</i> (Leipzig: Veröffentlichungen +des <ins class="mycorr" title="Original: Königliche">Königlichen</ins> Preussischen Geodätischen +Instituts, 1871).</p></div> + +<div class="footnote"><p><a name="Footnote_54_54" id="Footnote_54_54"></a><a href="#FNanchor_54_54"><span class="label">[54]</span></a> <i>Bericht über die Verhandlungen der vom 23 bis 28 September +1874 in Dresden abgehaltenen vierten allgemeinen Conferenz der +Europäischen Gradmessung</i> (Berlin, 1875). See report of second +session, September 24, 1874.</p></div> + +<div class="footnote"><p><a name="Footnote_55_55" id="Footnote_55_55"></a><a href="#FNanchor_55_55"><span class="label">[55]</span></a> <span class="smcap">Carolyn Eisele</span>, “Charles S. Peirce—Nineteenth-Century +Man of Science,” <i>Scripta Mathematica</i> (1959), vol 24, p. 305. +For the account of the work of Peirce, the authors are greatly +indebted to this pioneer paper on Peirce’s work on gravity. It +is worth noting that the history of pendulum work in North +America goes back to the celebrated Mason and Dixon, who +made observations of “the going rate of a clock” at “the forks +of the river Brandiwine in Pennsylvania,” in 1766-67. These +observations were published in <i>Phil. Trans.</i> (1768), vol. 58, +pp. <a name="corr_44_14" id="corr_44_14"></a><ins class="mycorr" title="Original: 329-235.">329-335.</ins></p></div> + +<div class="footnote"><p><a name="Footnote_56_56" id="Footnote_56_56"></a><a href="#FNanchor_56_56"><span class="label">[56]</span></a> The pendulums with conical bobs are described and illustrated +in <span class="smcap">E. D. Preston</span>, “Determinations of Gravity and the +Magnetic Elements in Connection with the United States +Scientific Expedition to the West Coast of Africa, 1889-90,” +<i>Report of the Superintendent of the Coast and Geodetic Survey for 1889-90</i> +(Washington, 1891), app. no. 12.</p></div> + +<div class="footnote"><p><a name="Footnote_57_57" id="Footnote_57_57"></a><a href="#FNanchor_57_57"><span class="label">[57]</span></a> <span class="smcap">Eisele</span>, op. cit. (footnote <a href="#Footnote_55_55">55</a>), p. 311.</p></div> + +<div class="footnote"><p><a name="Footnote_58_58" id="Footnote_58_58"></a><a href="#FNanchor_58_58"><span class="label">[58]</span></a> The record of Peirce’s observations in Europe during 1875-76 +is given in <span class="smcap">C. S. Peirce</span>, “Measurements of Gravity at +Initial Stations in America and Europe,” <i>Report of the Superintendent +of the Coast Survey for 1875-76</i> (Washington, 1879), pp. +202-337 and 410-416. Peirce’s report is dated December 13, +1878, by which time the name of the Survey had been changed +to U.S. Coast and Geodetic Survey.</p></div> + +<div class="footnote"><p><a name="Footnote_59_59" id="Footnote_59_59"></a><a href="#FNanchor_59_59"><span class="label">[59]</span></a> <i>Verhandlungen der vom 20 bis 29 September 1875 in Paris Vereinigten +Permanenten Commission der Europäischen Gradmessung</i> (Berlin, +1876).</p></div> + +<div class="footnote"><p><a name="Footnote_60_60" id="Footnote_60_60"></a><a href="#FNanchor_60_60"><span class="label">[60]</span></a> Ibid. See report for fifth session, September 25, 1875.</p></div> + +<div class="footnote"><p><a name="Footnote_61_61" id="Footnote_61_61"></a><a href="#FNanchor_61_61"><span class="label">[61]</span></a> The experiments at the Stevens Institute, Hoboken, were +reported by Peirce to the Permanent Commission which met +in Hamburg, September 4-8, 1878, and his report was published +in the general <i>Bericht</i> for 1878 in the <i>Verhandlungen +der vom 4 bis 8 September 1878 in Hamburg Vereinigten Permanenten +Commission der Europäischen Gradmessung</i> (Berlin, 1879), pp. 116-120. +Assistant J. E. Hilgard attended for the U.S. Coast and +Geodetic Survey. The experiments are described in detail in +<span class="smcap">C. S. Peirce</span>, “On the Flexure of Pendulum Supports,” <i>Report +of the Superintendent of the U.S. Coast and Geodetic Survey for 1880-81</i> +(Washington, 1883), app. no. 14, pp. 359-441.</p></div> + +<div class="footnote"><p><a name="Footnote_62_62" id="Footnote_62_62"></a><a href="#FNanchor_62_62"><span class="label">[62]</span></a> <i>Verhandlungen der vom 5 bis 10 Oktober 1876 in Brussels Vereinigten +Permanenten Commission der Europäischen Gradmessung</i> +(Berlin, 1877). See report of third session, October 7, 1876.</p></div> + +<div class="footnote"><p><a name="Footnote_63_63" id="Footnote_63_63"></a><a href="#FNanchor_63_63"><span class="label">[63]</span></a> <i>Verhandlungen der vom 27 September bis 2 Oktober 1877 zu +Stuttgart abgehaltenen fünften allgemeinen Conferenz der Europäischen +Gradmessung</i> (Berlin, 1878).</p></div> + +<div class="footnote"><p><a name="Footnote_64_64" id="Footnote_64_64"></a><a href="#FNanchor_64_64"><span class="label">[64]</span></a> <i>Verhandlung der vom 16 bis 20 September 1879 in Genf Vereinigten +Permanenten Commission der Europäischen Gradmessung</i> +(Berlin, 1880).</p></div> + +<div class="footnote"><p><a name="Footnote_65_65" id="Footnote_65_65"></a><a href="#FNanchor_65_65"><span class="label">[65]</span></a> <i>Assistants’ Reports, U.S. Coast and Geodetic Survey, 1879-80.</i> +Peirce’s paper was published in the <i>American Journal of Science</i> +(1879), vol. 18, p. 112.</p></div> + +<div class="footnote"><p><a name="Footnote_66_66" id="Footnote_66_66"></a><a href="#FNanchor_66_66"><span class="label">[66]</span></a> <i>Comptes-rendus de <a name="corr_44_15" id="corr_44_15"></a><ins class="mycorr" title="Original: L’Académie">l’Académie</ins> des Sciences</i> (Paris, 1879), +vol. 89, p. 462.</p></div> + +<div class="footnote"><p><a name="Footnote_67_67" id="Footnote_67_67"></a><a href="#FNanchor_67_67"><span class="label">[67]</span></a> <i>Verhandlungen der vom 13 bis 16 September 1880 zu München +abgehaltenen sechsten allgemeinen Conferenz der Europäischen Gradmessung</i> +(Berlin, 1881).</p></div> + +<div class="footnote"><p><a name="Footnote_68_68" id="Footnote_68_68"></a><a href="#FNanchor_68_68"><span class="label">[68]</span></a> Ibid., app. 2.</p></div> + +<div class="footnote"><p><a name="Footnote_69_69" id="Footnote_69_69"></a><a href="#FNanchor_69_69"><span class="label">[69]</span></a> Ibid., app. 2a.</p></div> + +<div class="footnote"><p><a name="Footnote_70_70" id="Footnote_70_70"></a><a href="#FNanchor_70_70"><span class="label">[70]</span></a> <i>Verhandlungen der vom 11 bis zum 15 September 1882 im Haag +Vereinigten Permanenten Commission der Europäischen Gradmessung</i> +(Berlin, 1883).</p></div> + +<div class="footnote"><p><a name="Footnote_71_71" id="Footnote_71_71"></a><a href="#FNanchor_71_71"><span class="label">[71]</span></a> <i>Verhandlungen der vom 15 bis 24 Oktober 1883 zu Rom abgehaltenen +siebenten allgemeinen Conferenz der Europäischen Gradmessung</i> +(Berlin, 1884). Gen. Cutts attended for the U.S. Coast and +Geodetic Survey.</p></div> + +<div class="footnote"><p><a name="Footnote_72_72" id="Footnote_72_72"></a><a href="#FNanchor_72_72"><span class="label">[72]</span></a> Ibid., app. 6. See also, <i>Zeitschrift für Instrumentenkunde</i> +(1884), vol. 4, pp. 303 and 379.</p></div> + +<div class="footnote"><p><a name="Footnote_73_73" id="Footnote_73_73"></a><a href="#FNanchor_73_73"><span class="label">[73]</span></a> Op. cit. (footnote <a href="#Footnote_67_67">67</a>).</p></div> + +<div class="footnote"><p><a name="Footnote_74_74" id="Footnote_74_74"></a><a href="#FNanchor_74_74"><span class="label">[74]</span></a> <i>Report of the Superintendent of the U.S. Coast and Geodetic +Survey for 1880-81</i> (Washington, 1883), p. 26.</p></div> + +<div class="footnote"><p><a name="Footnote_75_75" id="Footnote_75_75"></a><a href="#FNanchor_75_75"><span class="label">[75]</span></a> <i>Report of the Superintendent of the U.S. Coast and Geodetic +Survey for 1889-90</i> (Washington, 1891), app. no. 12.</p></div> + +<div class="footnote"><p><a name="Footnote_76_76" id="Footnote_76_76"></a><a href="#FNanchor_76_76"><span class="label">[76]</span></a> <i>Report of the Superintendent of the U.S. Coast and Geodetic +Survey for 1881-82</i> (Washington, 1883).</p></div> + +<div class="footnote"><p><a name="Footnote_77_77" id="Footnote_77_77"></a><a href="#FNanchor_77_77"><span class="label">[77]</span></a> <i>Transactions of the Cambridge Philosophical Society</i> (1856), +vol. 9, part 2, p. 8. Also published in <i>Mathematical and Physical +Papers</i> (Cambridge, 1901), vol. 3, p. 1.</p></div> + +<div class="footnote"><p><a name="Footnote_78_78" id="Footnote_78_78"></a><a href="#FNanchor_78_78"><span class="label">[78]</span></a> Peirce’s comparison of theory and experiment is discussed +in a report on the Peirce memoir by <span class="smcap">William Ferrel</span>, dated +October 19, 1890, Martinsburg, West Virginia. <i>U.S. Coast and +Geodetic Survey, Special Reports, 1887-1891</i> (MS, National +Archives, Washington).</p></div> + +<div class="footnote"><p><a name="Footnote_79_79" id="Footnote_79_79"></a><a href="#FNanchor_79_79"><span class="label">[79]</span></a> The stations at which observations were conducted with +the Peirce pendulums are recorded in the reports of the Superintendent +of the U.S. Coast and Geodetic Survey from 1881 to +1890.</p></div> + +<div class="footnote"><p><a name="Footnote_80_80" id="Footnote_80_80"></a><a href="#FNanchor_80_80"><span class="label">[80]</span></a> <i>Comptes-rendus de l’Académie des Sciences</i> (Paris, 1880), +vol. 90, p. 1401. <span class="smcap">Hervé Faye</span>’s report, dated June 21, 1880, is +in the same <i>Comptes-rendus</i>, p. 1463.</p></div> + +<div class="footnote"><p><a name="Footnote_81_81" id="Footnote_81_81"></a><a href="#FNanchor_81_81"><span class="label">[81]</span></a> <span class="smcap">Commandant C. Defforges</span>, “Sur <a name="corr_44_23" id="corr_44_23"></a><ins class="mycorr" title="Original: l’Intensite">l’Intensité</ins> absolue de +la pesanteur,” <i>Journal de Physique</i> (1888), vol. 17, pp. 239, 347, +455. See also, <span class="smcap">Defforges</span>, “Observations du pendule,” +<i>Mémorial du Dépôt général de la Guerre</i> (Paris, 1894), vol. 15. +In the latter work, Defforges described a pendulum “reversible +inversable,” which he declared to be truly invariable and +therefore appropriate for relative determinations. The knives +remained fixed to the pendulums, and the effect of interchanging +knives was obtained by interchanging weights within the +pendulum tube.</p></div> + +<div class="footnote"><p><a name="Footnote_82_82" id="Footnote_82_82"></a><a href="#FNanchor_82_82"><span class="label">[82]</span></a> Papers by <span class="smcap">Maj. von Sterneck</span> in <i>Mitteilungen des K. u. K. +Militär-geographischen Instituts, Wien</i>, 1882-87; see, in particular, +vol. 7 (1887).</p></div> + +<div class="footnote"><p><a name="Footnote_83_83" id="Footnote_83_83"></a><a href="#FNanchor_83_83"><span class="label">[83]</span></a> <span class="smcap">T. C. Mendenhall</span>, “Determinations of Gravity with +the New Half-Second Pendulum ...,” <i>Report of the Superintendent +of the U.S. Coast and Geodetic Survey for 1890-91</i> (Washington, +1892), part 2, pp. 503-564.</p></div> + +<div class="footnote"><p><a name="Footnote_84_84" id="Footnote_84_84"></a><a href="#FNanchor_84_84"><span class="label">[84]</span></a> <span class="smcap">W. H. Burger</span>, “The Measurement of the Flexure of +Pendulum Supports with the Interferometer,” <i>Report of the +Superintendent of the U.S. Coast and Geodetic Survey for 1909-10</i> +(Washington, 1911), app. no. 6.</p></div> + +<div class="footnote"><p><a name="Footnote_85_85" id="Footnote_85_85"></a><a href="#FNanchor_85_85"><span class="label">[85]</span></a> <span class="smcap">E. J. Brown</span>, <i>A Determination of the Relative Values of +Gravity at Potsdam and Washington</i> (Special Publication No. +204, U.S. Coast and Geodetic Survey; Washington, 1936).</p></div> + +<div class="footnote"><p><a name="Footnote_86_86" id="Footnote_86_86"></a><a href="#FNanchor_86_86"><span class="label">[86]</span></a> <span class="smcap">M. Haid</span>, “Neues Pendelstativ,” <i>Zeitschrift für Instrumentenkunde</i> +(July 1896), vol. 16, p. 193.</p></div> + +<div class="footnote"><p><a name="Footnote_87_87" id="Footnote_87_87"></a><a href="#FNanchor_87_87"><span class="label">[87]</span></a> <span class="smcap">Dr. R. Schumann</span>, “Über eine Methode, das Mitschwingen +bei relativen Schweremessungen zu bestimmen,” <i>Zeitschrift +für Instrumentenkunde</i> (January 1897), vol. 17, p. 7. The design +for the stand is similar to that of Peirce’s of 1879.</p></div> + +<div class="footnote"><p><a name="Footnote_88_88" id="Footnote_88_88"></a><a href="#FNanchor_88_88"><span class="label">[88]</span></a> <span class="smcap">Dr. R. Schumann</span>, “Über die Verwendung zweier Pendel +auf gemeinsamer Unterlage zur Bestimmung der Mitschwingung,” +<i>Zeitschrift für Mathematik und Physik</i> (1899), vol. 44, p. 44.</p></div> + +<div class="footnote"><p><a name="Footnote_89_89" id="Footnote_89_89"></a><a href="#FNanchor_89_89"><span class="label">[89]</span></a> <span class="smcap">P. Furtwängler</span>, “Über die Schwingungen zweier +Pendel mit annähernd gleicher Schwingungsdauer auf gemeinsamer +Unterlage,” <i>Sitzungsberichte der <a name="corr_44_24" id="corr_44_24"></a><ins class="mycorr" title="Original: Königliche">Königlicher</ins> Preussischen +Akademie der Wissenschaften zu Berlin</i> (Berlin, 1902) pp. 245-253. +Peirce investigated the plan of swinging two pendulums on the +same stand (<i>Report of the Superintendent of the U.S. Coast and +Geodetic Survey for 1880-81</i>, Washington, 1883, p. 26; also in +<span class="smcap">Charles Sanders Peirce</span>, <i>Collected Papers</i>, 6.273). At a +conference on gravity held in Washington during May 1882, +Peirce again advanced the method of eliminating flexure +by hanging two pendulums on one support and oscillating +them in antiphase (“Report of a conference on gravity determinations +held in Washington, D.C., in May, 1882,” <i>Report +of the Superintendent of the U.S. Coast and Geodetic Survey for 1881-82</i>, +Washington, 1883, app. no. 22, pp. 503-516).</p></div> + +<div class="footnote"><p><a name="Footnote_90_90" id="Footnote_90_90"></a><a href="#FNanchor_90_90"><span class="label">[90]</span></a> <span class="smcap">F. A. Vening Meinesz</span>, <i>Observations de pendule dans les +Pays-Bas</i> (Delft, 1923).</p></div> + +<div class="footnote"><p><a name="Footnote_91_91" id="Footnote_91_91"></a><a href="#FNanchor_91_91"><span class="label">[91]</span></a> <span class="smcap">A. Berroth</span>, “Schweremessungen mit zwei und vier +gleichzeitig auf demselben Stativ schwingenden Pendeln,” +<i>Zeitschrift für Geophysik</i>, vol. 1 (1924-25), no. 3, p. 93.</p></div> + +<div class="footnote"><p><a name="Footnote_92_92" id="Footnote_92_92"></a><a href="#FNanchor_92_92"><span class="label">[92]</span></a> “Pendulum Apparatus for Gravity Determinations,” +<i>Engineering</i> (1926), vol. 122, pp. 271-272.</p></div> + +<div class="footnote"><p><a name="Footnote_93_93" id="Footnote_93_93"></a><a href="#FNanchor_93_93"><span class="label">[93]</span></a> <span class="smcap">Malcolm W. Gay</span>, “Relative Gravity Measurements +Using Precision Pendulum Equipment,” <i>Geophysics</i> (1940), vol. +5, pp. 176-191.</p></div> + +<div class="footnote"><p><a name="Footnote_94_94" id="Footnote_94_94"></a><a href="#FNanchor_94_94"><span class="label">[94]</span></a> <span class="smcap">L. G. D. Thompson</span>, “An Improved Bronze Pendulum +Apparatus for Relative Gravity Determinations,” [published +by] <i>Dominion Observatory</i> (Ottawa, 1959), vol. 21, no. 3, pp. +145-176.</p></div> + +<div class="footnote"><p><a name="Footnote_95_95" id="Footnote_95_95"></a><a href="#FNanchor_95_95"><span class="label">[95]</span></a> <span class="smcap">W. A. Heiskanen</span> and <span class="smcap">F. A. Vening Meinesz</span>, <i>The Earth +and its Gravity Field</i> (McGraw: New York, 1958).</p></div> + +<div class="footnote"><p><a name="Footnote_96_96" id="Footnote_96_96"></a><a href="#FNanchor_96_96"><span class="label">[96]</span></a> <span class="smcap">F. Kühnen</span> and <span class="smcap">P. Furtwängler</span>, <i>Bestimmung der Absoluten +Grösze der Schwerkraft zu Potsdam mit Reversionspendeln</i> (Berlin: +Veröffentlichungen des Königlichen Preussischen Geodätischen +Instituts, 1906), new ser., no. 27.</p></div> + +<div class="footnote"><p><a name="Footnote_97_97" id="Footnote_97_97"></a><a href="#FNanchor_97_97"><span class="label">[97]</span></a> Reported by Dr. F. Kühnen to the fifth session, October 9, +1895, of the Eleventh General Conference, <i>Die Internationale +Erdmessung</i>, held in Berlin from September 25 to October 12, +1895. A footnote states that Assistant O. H. Tittmann, who +represented the United States, subsequently reported Peirce’s +prior discovery of the influence of the flexure of the pendulum +itself upon the period (<i>Report of the Superintendent of the U.S. +Coast and Geodetic Survey for 1883-84</i>, Washington, 1885, app. +16, pp. 483-485).</p></div> + +<div class="footnote"><p><a name="Footnote_98_98" id="Footnote_98_98"></a><a href="#FNanchor_98_98"><span class="label">[98]</span></a> <i>Assistants’ Reports, U.S. Coast and Geodetic Survey, 1883-84</i> +(MS, National Archives, Washington).</p></div> + +<div class="footnote"><p><a name="Footnote_99_99" id="Footnote_99_99"></a><a href="#FNanchor_99_99"><span class="label">[99]</span></a> <span class="smcap">C. S. Peirce</span>, “Effect of the Flexure of a Pendulum Upon +its Period of Oscillation,” <i>Report of the Superintendent of the U.S. +Coast and Geodetic Survey for 1883-84</i> (Washington, 1885), app. +no. 16.</p></div> + +<div class="footnote"><p><a name="Footnote_100_100" id="Footnote_100_100"></a><a href="#FNanchor_100_100"><span class="label">[100]</span></a> <span class="smcap">F. R. Helmert</span>, <i>Beiträge zur Theorie des Reversionspendels</i> +(Potsdam: Veröffentlichungen <a name="corr_44_25" id="corr_44_25"></a><ins class="mycorr" title="Original: Königliche">des Königlichen</ins> Preussischen Geodätischen +Instituts, 1898).</p></div> + +<div class="footnote"><p><a name="Footnote_101_101" id="Footnote_101_101"></a><a href="#FNanchor_101_101"><span class="label">[101]</span></a> <span class="smcap">J. A. Duerksen</span>, <i>Pendulum Gravity Data in the United States</i> +(Special Publication No. 244, U.S. Coast and Geodetic Survey; +Washington, 1949).</p></div> + +<div class="footnote"><p><a name="Footnote_102_102" id="Footnote_102_102"></a><a href="#FNanchor_102_102"><span class="label">[102]</span></a> Ibid., p. 2. See also, <span class="smcap">E. J. Brown</span>, loc. cit. (footnote <a href="#Footnote_85_85">85</a>).</p></div> + +<div class="footnote"><p><a name="Footnote_103_103" id="Footnote_103_103"></a><a href="#FNanchor_103_103"><span class="label">[103]</span></a> <span class="smcap">Paul R. Heyl</span> and <span class="smcap">Guy S. Cook</span>, “The Value of Gravity +at Washington,” <i>Journal of Research, National Bureau of Standards</i> +(1936), vol. 17, p. 805.</p></div> + +<div class="footnote"><p><a name="Footnote_104_104" id="Footnote_104_104"></a><a href="#FNanchor_104_104"><span class="label">[104]</span></a> <span class="smcap">Sir Harold Jeffreys</span>, “The Absolute Value of Gravity,” +<i>Monthly Notices of the Royal Astronomical Society, Geophysical Supplement</i> +(London, 1949), vol. 5, p. 398.</p></div> + +<div class="footnote"><p><a name="Footnote_105_105" id="Footnote_105_105"></a><a href="#FNanchor_105_105"><span class="label">[105]</span></a> <span class="smcap">J. S. Clark</span>, “The Acceleration Due to Gravity,” <i>Phil. +Trans.</i> (1939), vol. 238, p. 65.</p></div> + +<div class="footnote"><p><a name="Footnote_106_106" id="Footnote_106_106"></a><a href="#FNanchor_106_106"><span class="label">[106]</span></a> <span class="smcap">Hugh L. Dryden</span>, “A Reexamination of the Potsdam +Absolute Determination of Gravity,” <i>Journal of Research, +National Bureau of Standards</i> (1942), vol. 29, p. 303; and <span class="smcap">A. +Berroth</span>, “Das Fundamentalsystem der Schwere im Lichte +neuer Reversionspendelmessungen,” <i>Bulletin Géodésique</i> (1949), +no. 12, pp. 183-204.</p></div> + +<div class="footnote"><p><a name="Footnote_107_107" id="Footnote_107_107"></a><a href="#FNanchor_107_107"><span class="label">[107]</span></a> <span class="smcap">T. C. Mendenhall</span>, op. cit. (footnote <a href="#Footnote_83_83">83</a>), p. 522.</p></div> + +<div class="footnote"><p><a name="Footnote_108_108" id="Footnote_108_108"></a><a href="#FNanchor_108_108"><span class="label">[108]</span></a> <span class="smcap">A. H. Cook</span>, “Recent Developments in the Absolute +Measurement of Gravity,” <i>Bulletin Géodésique</i> (June 1, 1957), +no. 44, pp. 34-59.</p></div> + +<div class="footnote"><p><a name="Footnote_109_109" id="Footnote_109_109"></a><a href="#FNanchor_109_109"><span class="label">[109]</span></a> See footnote <a href="#Footnote_89_89">89</a>.</p></div> + +<div class="footnote"><p><a name="Footnote_110_110" id="Footnote_110_110"></a><a href="#FNanchor_110_110"><span class="label">[110]</span></a> <span class="smcap">C. S. Peirce</span>, “On the Deduction of the Ellipticity of the +Earth, from Pendulum Experiments,” <i>Report of the Superintendent +of the U.S. Coast and Geodetic Survey for 1880-81</i> (Washington, +1883), app. no. 15, pp. 442-456.</p></div> + +<div class="footnote"><p><a name="Footnote_111_111" id="Footnote_111_111"></a><a href="#FNanchor_111_111"><span class="label">[111]</span></a> <span class="smcap">Heiskanen</span> and <span class="smcap">Vening Meinesz</span>, op. cit. (footnote <a href="#Footnote_95_95">95</a>), p. +74.</p></div> + +<div class="footnote"><p><a name="Footnote_112_112" id="Footnote_112_112"></a><a href="#FNanchor_112_112"><span class="label">[112]</span></a> Ibid., p. 76.</p></div> + +<div class="footnote"><p><a name="Footnote_113_113" id="Footnote_113_113"></a><a href="#FNanchor_113_113"><span class="label">[113]</span></a> Ibid., p. 309.</p></div> + +<div class="footnote"><p><a name="Footnote_114_114" id="Footnote_114_114"></a><a href="#FNanchor_114_114"><span class="label">[114]</span></a> Ibid., p. 310.</p></div> + +<div class="footnote"><p><a name="Footnote_115_115" id="Footnote_115_115"></a><a href="#FNanchor_115_115"><span class="label">[115]</span></a> <span class="smcap">K. Reicheneder</span>, “Method of the New Measurements at Potsdam by +Means of the Reversible Pendulum,” <i>Bulletin Géodésique</i> (March 1, 1959), +no. 51, p.72.</p></div> + +</div> +<p class="center"><br /><br /> +U.S. GOVERNMENT PRINTING OFFICE: 1965<br /> +<br /> +For sale by the Superintendent of Documents, U.S. Government Printing Office<br /> +Washington, D.C., 20402—Price 70 cents.<br /> +</p> + + + +<hr style="width: 65%;" /> +<h3>INDEX</h3> + + +<p>Airy, G. B., <a href="#Page_319">319</a>, <a href="#Page_324">324</a>, <a href="#Page_332">332</a></p> + +<p>Albrecht, Karl Theodore, <a href="#Page_322">322</a>, <a href="#Page_338">338</a></p> + +<p>Al-Mamun, seventh calif of Bagdad, <a href="#Page_306">306</a></p> + +<p>Almansi, Emilio, <a href="#Page_339">339</a></p> + +<p>Aristotle, <a href="#Page_306">306</a></p> + + +<p>Baeyer, J. J., <a href="#Page_321">321</a>, <a href="#Page_322">322</a>, <a href="#Page_324">324</a>, <a href="#Page_338">338</a>, <a href="#Page_346">346</a></p> + +<p>Baily, Francis, <a href="#Page_317">317</a></p> + +<p>Basevi, James Palladio, <a href="#Page_345">345</a></p> + +<p>Berroth, A., <a href="#Page_342">342</a></p> + +<p>Bessel, Friedrich Wilhelm, <a href="#Page_313">313</a>, <a href="#Page_314">314</a>, <a href="#Page_319">319</a>, <a href="#Page_320">320</a>, <a href="#Page_324">324</a>, <a href="#Page_325">325</a>, <a href="#Page_338">338</a>, <a href="#Page_346">346</a></p> + +<p>Biot, Jean Baptiste, <a href="#Page_325">325</a>, <a href="#Page_329">329</a></p> + +<p>Bohnenberger, Johann Gottlieb Friedrich, <a href="#Page_315">315</a></p> + +<p>Borda, J. C., <a href="#Page_311">311</a>, <a href="#Page_312">312</a>, <a href="#Page_315">315</a>, <a href="#Page_325">325</a>, <a href="#Page_329">329</a>, <a href="#Page_346">346</a></p> + +<p>Boscovitch, Pčre R. J., <a href="#Page_310">310</a>, <a href="#Page_311">311</a></p> + +<p>Bouguer, Pierre, <a href="#Page_307">307</a>, <a href="#Page_309">309</a>, <a href="#Page_327">327</a>, <a href="#Page_343">343</a>, <a href="#Page_345">345</a></p> + +<p>Brahe, Tycho, <a href="#Page_306">306</a></p> + +<p>Brown, E. J., <a href="#Page_334">334</a>, <a href="#Page_339">339</a></p> + +<p>Browne, Henry, <a href="#Page_304">304</a>, <a href="#Page_314">314</a></p> + +<p>Bruhns, C., <a href="#Page_322">322</a>, <a href="#Page_324">324</a>, <a href="#Page_338">338</a></p> + +<p>Brunner Brothers (Paris), <a href="#Page_329">329</a></p> + + +<p>Cassini, Giovanni-Domenico, <a href="#Page_306">306</a>, <a href="#Page_307">307</a></p> + +<p>Cassini, Jacques, <a href="#Page_306">306</a></p> + +<p>Cassini de Thury, J. D., <a href="#Page_311">311</a>, <a href="#Page_312">312</a>, <a href="#Page_315">315</a>, <a href="#Page_325">325</a>, <a href="#Page_329">329</a>, <a href="#Page_346">346</a></p> + +<p>Cellérier, Charles, <a href="#Page_320">320</a>, <a href="#Page_321">321</a>, <a href="#Page_325">325</a>, <a href="#Page_326">326</a>, <a href="#Page_329">329</a>, <a href="#Page_336">336</a></p> + +<p>Clairaut, Alexis Claude, <a href="#Page_308">308</a>, <a href="#Page_309">309</a>, <a href="#Page_343">343</a>, <a href="#Page_345">345</a></p> + +<p>Clark, J. S., <a href="#Page_342">342</a></p> + +<p>Clarke, A. R., <a href="#Page_345">345</a></p> + +<p>Colbert, Jean Baptiste, <a href="#Page_306">306</a></p> + +<p>Cook, A. H., <a href="#Page_342">342</a></p> + +<p>Cook, Guy S., <a href="#Page_339">339</a>, <a href="#Page_342">342</a></p> + + +<p>Defforges, C., <a href="#Page_314">314</a>, <a href="#Page_329">329</a>, <a href="#Page_346">346</a></p> + +<p>De Freycinet, Louis Claude de Saulses, <a href="#Page_317">317</a></p> + +<p>De la Hire, Gabriel Philippe, <a href="#Page_306">306</a></p> + +<p>De Prony, M. G., <a href="#Page_314">314</a></p> + +<p>Dryden, Hugh L., <a href="#Page_342">342</a></p> + +<p>Du Buat, L. G., <a href="#Page_314">314</a></p> + +<p>Duperry, Capt. Louis Isidore, <a href="#Page_317">317</a></p> + + +<p>Eratosthenes, <a href="#Page_306">306</a>, <a href="#Page_308">308</a>, <a href="#Page_342">342</a></p> + +<p>Eudoxus of Cnidus, <a href="#Page_306">306</a></p> + + +<p>Faye, Hervé, <a href="#Page_325">325</a>, <a href="#Page_336">336</a>, <a href="#Page_346">346</a>, <a href="#Page_347">347</a></p> + +<p>Fernel, Jean, <a href="#Page_306">306</a></p> + +<p>Furtwängler, P., <a href="#Page_337">337</a></p> + + +<p>Galilei, Galileo, <a href="#Page_304">304</a>, <a href="#Page_305">305</a>, <a href="#Page_346">346</a></p> + +<p>Gauss, C. F., <a href="#Page_320">320</a></p> + +<p>Gautier, P., <a href="#Page_339">339</a></p> + +<p>Godin, Louis, <a href="#Page_307">307</a></p> + +<p>Goldingham, John, <a href="#Page_316">316</a>, <a href="#Page_345">345</a></p> + +<p>Greely, A. W., <a href="#Page_329">329</a></p> + +<p>Gulf Oil and Development Company, <a href="#Page_338">338</a></p> + + +<p>Haid, M., <a href="#Page_335">335</a></p> + +<p>Hall, Basil, <a href="#Page_316">316</a></p> + +<p>Heaviside, W. J., <a href="#Page_321">321</a>, <a href="#Page_345">345</a></p> + +<p>Heiskanen, W. A., <a href="#Page_338">338</a>, <a href="#Page_345">345</a>, <a href="#Page_346">346</a></p> + +<p>Helmert, F. R., <a href="#Page_338">338</a>, <a href="#Page_339">339</a></p> + +<p>Helmholtz, Hermann von, <a href="#Page_326">326</a></p> + +<p>Herschel, John, <a href="#Page_319">319</a>, <a href="#Page_328">328</a>, <a href="#Page_345">345</a></p> + +<p>Heyl, Paul R., <a href="#Page_339">339</a>, <a href="#Page_342">342</a></p> + +<p>Hirsch, Adolph, <a href="#Page_322">322</a>, <a href="#Page_324">324</a></p> + +<p>Huygens, Christiaan, <a href="#Page_304">304</a>, <a href="#Page_305">305</a>, <a href="#Page_307">307</a>, <a href="#Page_314">314</a>, <a href="#Page_342">342</a>, <a href="#Page_346">346</a></p> + + +<p>Ibańez, Carlos, <a href="#Page_325">325</a></p> + + +<p>Jeffreys, Sir Harold, <a href="#Page_342">342</a></p> + +<p>Jones, Thomas, <a href="#Page_318">318</a></p> + + +<p>Kater, Henry, <a href="#Page_304">304</a>, <a href="#Page_314">314</a>, <a href="#Page_325">325</a>, <a href="#Page_327">327</a>, <a href="#Page_329">329</a>, <a href="#Page_345">345</a>, <a href="#Page_346">346</a></p> + +<p>Kühnen, F., <a href="#Page_338">338</a>, <a href="#Page_339">339</a></p> + + +<p>La Condamine, Charles Marie de, <a href="#Page_307">307</a>, <a href="#Page_310">310</a>, <a href="#Page_311">311</a>, <a href="#Page_343">343</a></p> + +<p>Laplace, Marquis Pierre Simon de, <a href="#Page_309">309</a>, <a href="#Page_313">313</a>, <a href="#Page_320">320</a></p> + +<p>Lorenzoni, Giuseppe, <a href="#Page_336">336</a>, <a href="#Page_339">339</a></p> + +<p>Lütke, Count Feodor Petrovich, <a href="#Page_316">316</a>, <a href="#Page_345">345</a></p> + + +<p>Maupertius, P. L. Moreau de, <a href="#Page_308">308</a>, <a href="#Page_343">343</a></p> + +<p>Maxwell, James Clerk, <a href="#Page_324">324</a></p> + +<p>Medi, Enrico, <a href="#Page_342">342</a></p> + +<p>Mendenhall, Thomas Corwin, <a href="#Page_319">319</a>, <a href="#Page_331">331</a>, <a href="#Page_332">332</a>, <a href="#Page_334">334</a>, <a href="#Page_347">347</a></p> + +<p>Mersenne, P. Marin, <a href="#Page_305">305</a></p> + + +<p>Newton, Sir Isaac, <a href="#Page_303">303</a>, <a href="#Page_307">307</a>, <a href="#Page_308">308</a>, <a href="#Page_342">342</a>, <a href="#Page_343">343</a></p> + +<p>Norwood, Richard, <a href="#Page_306">306</a></p> + + +<p>Oppolzer, Theodor von, <a href="#Page_322">322</a>, <a href="#Page_324">324</a></p> + + +<p>Patterson, Carlile Pollock, <a href="#Page_325">325</a>, <a href="#Page_326">326</a></p> + +<p>Peirce, Charles Sanders, <a href="#Page_314">314</a>, <a href="#Page_322">322</a>, <a href="#Page_332">332</a>, <a href="#Page_336">336</a>, <a href="#Page_342">342</a>, <a href="#Page_345">345</a></p> + +<p>Peters, C. A. F., <a href="#Page_322">322</a>, <a href="#Page_324">324</a></p> + +<p>Picard, Abbé Jean, <a href="#Page_306">306</a>, <a href="#Page_308">308</a>, <a href="#Page_342">342</a></p> + +<p>Plantamour, E., <a href="#Page_319">319</a>, <a href="#Page_324">324</a></p> + +<p>Posidonius, <a href="#Page_306">306</a></p> + +<p>Preston, E. D., <a href="#Page_328">328</a>, <a href="#Page_329">329</a></p> + +<p>Putnam, G. R., <a href="#Page_339">339</a></p> + +<p>Pythagoras, <a href="#Page_306">306</a></p> + + +<p>Repsold, A., and Sons (Hamburg), <a href="#Page_320">320</a>, <a href="#Page_322">322</a>, <a href="#Page_338">338</a>, <a href="#Page_339">339</a>, <a href="#Page_346">346</a></p> + +<p>Richer, Jean, <a href="#Page_307">307</a>, <a href="#Page_342">342</a></p> + + +<p>Sabine, Capt. Edward, <a href="#Page_315">315</a>, <a href="#Page_325">325</a>, <a href="#Page_329">329</a>, <a href="#Page_345">345</a></p> + +<p>Sawitsch, A., <a href="#Page_321">321</a>, <a href="#Page_322">322</a></p> + +<p>Schumacher, H. C., <a href="#Page_320">320</a></p> + +<p>Schumann, R., <a href="#Page_335">335</a>, <a href="#Page_336">336</a></p> + +<p>Snell, Willebrord, <a href="#Page_306">306</a></p> + +<p>Sterneck, Robert von, <a href="#Page_331">331</a>, <a href="#Page_332">332</a>, <a href="#Page_335">335</a>, <a href="#Page_338">338</a>, <a href="#Page_346">346</a></p> + +<p>Stokes, George Gabriel, <a href="#Page_324">324</a>, <a href="#Page_328">328</a>, <a href="#Page_329">329</a>, <a href="#Page_345">345</a>, <a href="#Page_346">346</a></p> + + +<p>Ulloa, Antonio de, <a href="#Page_308">308</a></p> + + +<p>Vening Meinesz, F. A., <a href="#Page_337">337</a>, <a href="#Page_338">338</a>, <a href="#Page_345">345</a></p> + +<p>Volet, Charles, <a href="#Page_342">342</a></p> + + +<p>Wilkes, Charles, <a href="#Page_317">317</a>, <a href="#Page_318">318</a><br /><br /></p> + +<div class="tnote"><a name="corrections_44" id="corrections_44"></a> +<h3>Transcriber’s Corrections</h3> + +<p>Formatting of equations has been altered from the original to display them +‘in line,’ and brackets have been added to clarify expressions where necessary.</p> + +<p>Footnotes have been moved to the end of the paper. Illustrations and the +<span class="smcap">Glossary of Gravity Terminology</span> section have been moved to avoid breaks +in paragraphs. Minor punctuation errors have been corrected without +note. Typographical errors and inconsistencies have been corrected as +follows:</p> +<div style="margin-left: 2em;"> + <p class="hang">P. <a href="#corr_44_01">320</a> ‘difference <i>T</i><sub>1</sub> - <i>T</i><sub>2</sub> is sufficiently’—had ‘sufficlently.’</p> + <p class="hang">P. <a href="#corr_44_02">321</a> ‘faites ŕ Genčve avec le pendule ŕ réversion’—had ‘reversion.’</p> + <p class="hang">P. <a href="#corr_44_16">326</a> ‘Schwere mit Hilfe verschiedener Apparate’—had ‘verschiedene.’</p> + <p class="hang">P. <a href="#corr_44_03">328</a> ‘between the yard and the meter.’—closing quote mark deleted.</p> + <p class="hang">P. <a href="#corr_44_04">334</a> ‘Mendenhall apparatus were part of’—‘was’ changed to ‘were.’</p> + <p class="hang">P. <a href="#corr_44_05">342</a> ‘of the Geodetic Institute at Potsdam’—had ‘Postdam.’</p> + <p class="hang">P. <a href="#corr_44_06">345</a> ‘The gravimetric methods of physical’—had ‘mtehods.’</p> + <p class="hang">Footnote <a href="#corr_44_07">1</a> ‘Société française de Physique’—had ‘Française.’</p> + <p class="hang">Footnote <a href="#corr_44_17">3</a> ‘Cogitata physico-mathematica’—had ‘physica.’</p> + <p class="hang">Footnote <a href="#corr_44_18">10</a> ‘mathématiques et de physique par MM. de l’Académie Royale’—had ‘mathematiques,’ ‘Royal.’</p> + <p class="hang">Footnote <a href="#corr_44_08">12</a> ‘par ordre du Roy au Pérou, pour observer’—had ‘Perou, pour observir.’</p> + <p class="hang">Footnote <a href="#corr_44_19">19</a> ‘Opticam et Astronomiam’—had ‘Astronomian.’</p> + <p class="hang">Footnote <a href="#corr_44_09">20</a> ‘connaître la longueur du pendule qui’—had ‘connaitre la longuer.’</p> + <p class="hang">Footnote <a href="#corr_44_20">21</a> ‘Abhandlungen der Königlichen Akademie’—had ‘Königliche.’</p> + <p class="hang">Footnote <a href="#corr_44_21">25</a> ‘pour déterminer la longueur du pendule’—had ‘longeur.’</p> + <p class="hang">Footnote <a href="#corr_44_10">41</a> ‘Survey of India (Calcutta, 1879)’— had ‘Surey.’</p> + <p class="hang">Footnotes <a href="#corr_44_11">45</a> and <a href="#corr_44_12">47</a> ‘Société de Physique et d’histoire’—had + ‘d’historire.’</p> + <p class="hang">Footnote <a href="#corr_44_22">49</a> ‘Über die Grösse und Figur der Erde’—had ‘Grosse.’</p> + <p class="hang">Footnote <a href="#corr_44_13">53</a> ‘Bestimmung der Länge’—had ‘Lange’; ‘Astronomisch-Geodätische Arbeiten’—had +‘Astronomische’; ‘Veröffentlichungen des Königlichen’—had ‘Königliche.’</p> + <p class="hang">Footnote <a href="#corr_44_14">55</a> ‘(1768), vol. 58, pp. 329-335.’—had ‘329-235.’</p> + <p class="hang">Footnote <a href="#corr_44_15">66</a> ‘Comptes-rendus de l’Académie’—had ‘L’Académie.’</p> + <p class="hang">Footnote <a href="#corr_44_23">81</a> ‘Sur l’Intensité absolue’—had ‘l’Intensite.’</p> + <p class="hang">Footnote <a href="#corr_44_24">89</a> ‘Sitzungsberichte der Königlicher’—had ‘Königliche.’</p> + <p class="hang">Footnote <a href="#corr_44_25">100</a> ‘Veröffentlichungen des Königlichen’ had ‘Veröffentlichungen Königliche.’</p> +</div> +<p>Capitalisation of ‘Von’/‘von’ has been regulaized to ‘von’ for all +personal names, except at the beginning of a sentence, and when +referring to the Von Sterneck pendulum.</p> +</div> + +<p> </p> +<p> </p> +<hr class="full" /> +<p>***END OF THE PROJECT GUTENBERG EBOOK DEVELOPMENT OF GRAVITY PENDULUMS IN THE 19TH CENTURY***</p> +<p>******* This file should be named 35024-h.txt or 35024-h.zip *******</p> +<p>This and all associated files of various formats will be found in:<br /> +<a href="http://www.gutenberg.org/dirs/3/5/0/2/35024">http://www.gutenberg.org/3/5/0/2/35024</a></p> +<p>Updated editions will replace the previous one--the old editions +will be renamed.</p> + +<p>Creating the works from public domain print editions means that no +one owns a United States copyright in these works, so the Foundation +(and you!) can copy and distribute it in the United States without +permission and without paying copyright royalties. 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