651:(1792) approximated a simple pendulum by using a metal sphere suspended by a light wire. If the wire had negligible mass, the center of oscillation was close to the center of gravity of the sphere. But even finding the center of gravity of the sphere accurately was difficult. In addition, this type of pendulum inherently wasn't very accurate. The sphere and wire didn't swing back and forth as a rigid unit, because the sphere acquired a slight
854:. After corrections, he found that the mean length of the solar seconds pendulum at London, at sea level, at 62 °F (17 °C), swinging in vacuum, was 39.1386 inches. This is equivalent to a gravitational acceleration of 9.81158 m/s. The largest variation of his results from the mean was 0.00028 inches (7.1 μm). This represented a precision of gravity measurement of 0.7×10 (7
31:
683:, of a rigid (compound) pendulum. If a pendulum were hung upside down from a second pivot point that could be adjusted up and down on the pendulum's rod, and the second pivot were adjusted until the pendulum had the same period as it did when swinging right side up from the first pivot, the second pivot would be at the center of oscillation, and the distance between the two pivot points would be
671:, Huygens had also proved that the pivot point and the center of oscillation were interchangeable. That is, if any pendulum is suspended upside down from its center of oscillation, it has the same period of swing, and the new center of oscillation is the old pivot point. The distance between these two conjugate points was equal to the length of a simple pendulum with the same period.
917:
702:
460:
874:
1139:
1563:
in the spring of 1875 to proceed to Europe for the purpose of making pendulum experiments to chief initial stations for operations of this sort, in order to bring the determinations of the forces of gravity in
America into communication with those of other parts of the world; and also for the purpose
635:
and depends on the mass distribution along the length of the pendulum. The problem was there was no way to find the location of the center of oscillation in a real pendulum accurately. It could theoretically be calculated from the shape of the pendulum if the metal parts had uniform density, but the
786:
on the pendulum shaft. Since gravity only varies by a maximum of 0.5% over the Earth, and in most locations much less than that, the weight had to be adjusted only slightly. Moving the weight toward one of the pivots decreased the period when hung from that pivot, and increased the period when hung
189:
The Kater's pendulum consists of a rigid metal bar with two pivot points, one near each end of the bar. It can be suspended from either pivot and swung. It also has either an adjustable weight that can be moved up and down the bar, or one adjustable pivot, to adjust the periods of swing. In use, it
1567:
The determination of gravity by the reversible pendulum was subject to two types of error. On the one hand the resistance of the air and on the other hand the movements that the oscillations of the pendulum imparted to its plane of suspension. These movements were particularly important with the
795:
To use, the pendulum was hung from a bracket on a wall, with the knife blade pivots supported on two small horizontal agate plates, in front of a precision pendulum clock to time the period. It was swung first from one pivot, and the oscillations timed, then turned upside down and swung from the
825:
when the two pendulums were swinging in synchronism. He measured the distance between the pivot blades with a microscope comparator, to an accuracy of 10 in. (2.5 ÎĽm). As with other pendulum gravity measurements, he had to apply small corrections to the result for a number of variable factors:
1576:
found a way to demonstrate the movements of the pendulum's suspension plane by an ingenious process of optical amplification. Isaac-Charles Élisée Cellérier, a mathematician from Geneva and
Charles Sanders Peirce would independently develop a correction formula that allowed the use of the
1503:
In addition, Bessel showed that if the pendulum was made with a symmetrical shape, but internally weighted on one end, the error caused by effects of air resistance would cancel out. Also, another error caused by the non-zero radius of the pivot knife edges could be made to cancel out by
769:
on one end. For a low friction pivot he used a pair of short triangular 'knife' blades attached to the rod. In use the pendulum was hung from a bracket on the wall, supported by the edges of the knife blades resting on flat agate plates. The pendulum had two of these knife blade pivots
385:
779:
Kater found that making one of the pivots adjustable caused inaccuracies, making it hard to keep the axis of both pivots precisely parallel. Instead he permanently attached the knife blades to the rod, and adjusted the periods of the pendulum by a small movable weight
1542:
made his famous measurements, the determination of gravity by means of devices of various kinds employed in different countries, in order to compare them and thus to have the equation of their scales, after an in-depth discussion in which an
American scholar,
527:; it was thicker at the equator because of the Earth's rotation. Since the surface was farther from the Earth's center at Cayenne than at Paris, gravity was weaker there. After that discovery was made, freeswinging pendulums started to be used as precision
678:
in 1816 to reform
British measures, Kater had been contracted by the House of Commons to determine accurately the length of the seconds pendulum in London. He realized Huygens' principle could be used to find the center of oscillation, and so the length
612:
in equation (1) above was the length of an ideal mathematical 'simple pendulum' consisting of a point mass swinging on the end of a massless cord. However the 'length' of a real pendulum, a swinging rigid body, known in mechanics as a
1338:, cannot be measured with comparable accuracy. They are found by balancing the pendulum on a knife edge to find its center of gravity, and measuring the distances of each of the pivots from the center of gravity. However, because
952:
78:
do not have to be determined, allowing a greater accuracy. For about a century, until the 1930s, Kater's pendulum and its various refinements remained the standard method for measuring the strength of the Earth's gravity during
531:, taken on voyages to different parts of the world to measure the local gravitational acceleration. The accumulation of geographical gravity data resulted in more and more accurate models of the overall shape of the Earth.
1507:
Bessel didn't construct such a pendulum, but in 1864 Adolf
Repsold, under contract to the Swiss Geodetic Commission, developed a symmetric pendulum 56 cm long with interchangeable pivot blades, with a period of about
169:
211:
1538:
dealt with the best instrument to be used for the determination of gravity. The association decided in favor of the reversion pendulum and it was resolved to redo in Berlin, in the station where
194:
timed, and then turned upside down and swung from the other pivot, and the period timed. The movable weight (or pivot) is adjusted until the two periods are equal. At this point the period
889:. To be useful, it was necessary to find the exact location (latitude and longitude) of the 'station' where a gravity measurement was taken, so pendulum measurements became part of
1568:
apparatus designed by the
Repsold brothers on the indications of Bessel, because the pendulum had a large mass in order to counteract the effect of the viscosity of the air. While
1438:
1387:
2258:
Kater, Henry (June 1818) An
Account of the Experiments for determining the length of the pendulum vibrating seconds in the latitude of London, The Edinburgh Review, Vol. 30, p.407
544:, a pendulum with a period of two seconds, so each swing takes one second. It can be seen from equation (1) that for a seconds pendulum, the length is simply proportional to
1479:
1336:
1239:
586:
1295:
1267:
1198:
1170:
534:
Pendulums were so universally used to measure gravity that, in Kater's time, the local strength of gravity was usually expressed not by the value of the acceleration
600:
of pendulums could be measured very precisely by timing them with precision clocks set by the passage of stars overhead. Prior to Kater's discovery, the accuracy of
802:
was adjusted with the adjusting screw, and the process repeated until the pendulum had the same period when swung from each pivot. By putting the measured period
1134:{\displaystyle T^{2}={\frac {T_{1}^{2}+T_{2}^{2}}{2}}+{\frac {T_{1}^{2}-T_{2}^{2}}{2}}\left({\frac {h_{1}+h_{2}}{h_{1}-h_{2}}}\right)\,\qquad \qquad \qquad (2)}
1581:
625:, showed that a real pendulum had the same period as a simple pendulum with a length equal to the distance between the pivot point and a point called the
1750:
United States
National Museum Bulletin 240: Contributions from the Museum of History and Technology reprinted in Bulletin of the Smithsonian Institution
1997:"Report from Charles S. Peirce on his second European trip for the Anual Report of the Superintendent of the U. S. Coast Survey, New York, 18.05.1877"
198:
is equal to the period of an 'ideal' simple pendulum of length equal to the distance between the pivots. From the period and the measured distance
1556:
776:, facing one another, about a meter (40 in) apart, so that a swing of the pendulum took approximately one second when hung from each pivot.
787:
from the other pivot. This also had the advantage that the precision measurement of the separation between the pivots had to be made only once.
380:{\displaystyle g={\frac {8\pi ^{2}}{{\dfrac {T_{1}^{2}+T_{2}^{2}}{\ell _{1}+\ell _{2}}}+{\dfrac {T_{1}^{2}-T_{2}^{2}}{\ell _{1}-\ell _{2}}}}}}
2118:
2069:
120:
1484:
Therefore, the pendulum doesn't have to be adjustable at all, it can simply be a rod with two pivots. As long as each pivot is close to the
1585:
1535:
2095:"From a Regional Project to an International Organization: The "Baeyer-Helmert-Era" of the International Association of Geodesy 1862–1916"
842:
altitude, which reduced the gravitational force with distance from the center of the Earth. Gravity measurements are always referenced to
861:
In 1824, the
British Parliament made Kater's measurement of the seconds pendulum the official backup standard of length for defining the
2048:
Discursos leidos ante la Real
Academia de Ciencias Exactas Fisicas y Naturales en la recepcion pĂşblica de Don Joaquin Barraquer y Rovira
924:
Repeatedly timing each period of a Kater pendulum, and adjusting the weights until they were equal, was time-consuming and error-prone.
1481:
doesn't have to be determined with high accuracy, and the balancing procedure described above is sufficient to give accurate results.
1883:
The Metric System of Weights and Measures Compared with British Standard Weights and Measures in a Complete Set of Comparative Tables
1643:
1942:
1915:
1784:
839:
atmospheric pressure, which reduced the effective mass of the pendulum by the buoyancy of the displaced air, increasing the period
904:
Reversible pendulums remained the standard method used for absolute gravity measurements until they were superseded by free-fall
1853:
The wording of the Act indicates that the pendulum definition is to be used to restore the yard if the prototype is destroyed.
1592:. Under Ibáñez's presidency, the International Geodetic Association acquired a global dimension with the accession of the
644:
2022:"Comptes rendus hebdomadaires des séances de l'Académie des sciences / publiés... par MM. les secrétaires perpétuels"
897:
of much of the world that were being done during the 19th century. In particular, Kater's pendulums were used in the
501:
minutes per day slower than at Paris, or equivalently the length of a pendulum with a swing of one second there was
202:
between the pivots, the acceleration of gravity can be calculated with great precision from the equation (1) above.
1846:
1519:
898:
92:
67:
1644:"An account of experiments for determining the length of the pendulum vibrating seconds in the latitude of London"
817:
Kater performed 12 trials. He measured the period of his pendulum very accurately using the clock pendulum by the
2191:"A profile of General Carlos Ibáñez e Ibáñez de Ibero: first president of the International Geodetic Association"
2136:"The International Association of Geodesy 1862 to 1922: from a regional project to an international organization"
1617:
636:
metallurgical quality and mathematical abilities of the time didn't allow the calculation to be made accurately.
1613:
1492:
of the equivalent simple pendulum can be calculated with equation (2), and the gravity can be calculated from
1200:
are the distances of the two pivots from the pendulum's center of gravity. The distance between the pivots,
480:
2260:
Has detailed account of experiment, description of pendulum, value determined, interest of French scientists
2247:
1564:
of making a careful study of the methods of pursuing these researches in the different countries of Europe.
1392:
1341:
1560:
1544:
1523:
1518:
second. The Repsold pendulum was used extensively by the Swiss and Russian Geodetic agencies, and in the
1485:
862:
648:
627:
75:
2202:
2147:
519:, or 2.6 mm, shorter than at Paris. It was realized by the scientists of the day, and proven by
1770:
1698:
1589:
1548:
1443:
1300:
1203:
554:
1799:
2273:
2226:
2171:
894:
881:
The large increase in gravity measurement accuracy made possible by Kater's pendulum established
694:
first proposed a reversible pendulum in 1800, but his work was not published until 1889. In 1811
618:
38:
was timed by comparing its swing with the pendulum in the precision clock behind it. The sight
487:. Through the observations he made in the following year, Richer determined that the clock was
467:
The first person to discover that gravity varied over the Earth's surface was French scientist
2218:
2163:
2114:
2075:
2065:
1957:
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1932:
1911:
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1811:
1780:
1774:
834:
698:
again discovered it, but Kater independently invented it and was first to put it in practice.
632:
614:
71:
1272:
1244:
1175:
1147:
655:
during each swing. Also the wire stretched elastically during the pendulum's swing, changing
2257:
2210:
2155:
2106:
1907:
1569:
1552:
1539:
925:
851:
691:
652:
540:
523:
in 1687, that this was due to the fact that the Earth was not a perfect sphere but slightly
2094:
1745:
1730:
928:
showed in 1826 that this was unnecessary. As long as the periods measured from each pivot,
524:
1971:
1903:
Revolution in Measurement: Western European Weights and Measures Since the Age of Science
1776:
Revolution in Measurement: Western European Weights and Measures since the Age of Science
1580:
President of the Permanent Commission of the European Arc Measurement from 1874 to 1886,
2206:
2151:
1440:, the second term on the right in the above equation is small compared to the first, so
1673:
1573:
484:
43:
17:
2267:
2230:
2175:
1593:
675:
476:
1996:
695:
520:
1901:
1724:
30:
2050:(in Spanish). Madrid: Imprenta de la Viuda e Hijo de D.E. Aguado. pp. 70–73.
34:
Kater's original pendulum, showing use, from Kater's 1818 paper. The pendulum's
640:
468:
59:
2021:
916:
2159:
1612:. As a result of the work of the International Geodetic Association, in 1901,
905:
882:
528:
70:. Its advantage is that, unlike previous pendulum gravimeters, the pendulum's
63:
2222:
2167:
2135:
2079:
1605:
890:
843:
701:
459:
191:
99:
80:
35:
1841:
An Act for ascertaining and establishing Uniformity of Weights and Measures
2214:
1848:
The Statutes of the United Kingdom of Great Britain and Ireland, Volume 27
2110:
855:
604:
measurements was limited by the difficulty of measuring the other factor
55:
2190:
1881:
886:
472:
164:{\displaystyle T=2\pi {\sqrt {\frac {L}{g}}}\qquad \qquad \qquad (1)\,}
2064:(in French). Encyclopedia Universalis. 1996. pp. Vol 10, p. 302.
830:
the non-zero width of the pendulum's swing, which increased the period
749:
Kater built a pendulum consisting of a brass rod about 2 meters long,
1597:
639:
To get around this problem, most early gravity researchers, such as
83:
surveys. It is now used only for demonstrating pendulum principles.
873:
1609:
1601:
872:
458:
2020:
texte, Académie des sciences (France) Auteur du (January 1880).
1746:"Paper 44: Development of gravity pendulums in the 19th century"
796:
other pivot, and the oscillations timed again. The small weight
690:
Kater was not the first to have this idea. French mathematician
205:
The acceleration due to gravity by Kater's pendulum is given by
946:
of the equivalent simple pendulum can be calculated from them:
833:
temperature, which caused the length of the rod to vary due to
404:
are the time periods of oscillations when it is suspended from
2093:
Torge, Wolfgang (2016). Rizos, Chris; Willis, Pascal (eds.).
1972:"Bulletin de la Société des Sciences Naturelles de Neuchâtel"
712:
opposing knife edge pivots from which pendulum is suspended
1572:
was carrying out a series of experiments with this device,
1522:. Other widely used pendulums of this design were made by
1752:. Washington: Smithsonian Institution Press. p. 307
1488:
of the other, so the two periods are close, the period
617:, is more difficult to define. In 1673 Dutch scientist
1446:
1395:
1344:
1303:
1275:
1247:
1206:
1178:
1150:
955:
893:. Kater's pendulums were taken on the great historic
806:, and the measured distance between the pivot blades
763:
inches wide and one-eighth inch thick, with a weight
557:
309:
240:
214:
123:
1680:. Dept. of Physics and Astronomy, Georgia State Univ
538:
now used, but by the length at that location of the
1723:Poynting, John Henry; Joseph John Thompson (1907).
726:
coarse adjustment weight clamped to rod by setscrew
1473:
1432:
1381:
1330:
1289:
1261:
1233:
1192:
1164:
1133:
580:
483:, assigned the task of making measurements with a
379:
163:
62:in 1817, published 29th January 1818 for use as a
2105:. Cham: Springer International Publishing: 3–18.
2101:. International Association of Geodesy Symposia.
1843:, British Parliament, 17 June 1824, reprinted in
1779:. New York: Diane Publishing. pp. 107–110.
1577:observations made with this type of gravimeter.
719:fine adjustment weight moved by adjusting screw
1830:. New York: Harper & Brothers. p. 109.
1616:found, mainly by gravimetry, parameters of the
58:invented by British physicist and army captain
1744:Victor F., Lenzen; Robert P. Multauf (1964).
8:
1729:. London: Charles Griffin & Co. p.
1906:. American Philosophical Society. pp.
877:Gravimeter with variant of Repsold pendulum
621:in his mathematical analysis of pendulums,
471:, who in 1671 was sent on an expedition to
1241:, can be measured with great accuracy.
608:, the length of the pendulum, accurately.
2046:Ibáñez e Ibáñez de Ibero, Carlos (1881).
1866:The Civil Engineer's Pocket-book, 18th Ed
1551:could be inferred from variations of the
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1451:
1445:
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1423:
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1378:
1372:
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1349:
1343:
1327:
1321:
1308:
1302:
1286:
1280:
1274:
1258:
1252:
1246:
1230:
1224:
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1155:
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1021:
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999:
994:
981:
976:
969:
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954:
577:
568:
556:
446:from the center of gravity respectively.
364:
351:
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160:
136:
122:
915:
850:He gave his result as the length of the
700:
674:As part of a committee appointed by the
631:, which is located under the pendulum's
29:
2252:using Kater's pendulum, U. of Sheffield
1637:
1635:
1633:
1629:
1851:. London: Andrew Strahan. p. 759.
1827:Elements of Natural Philosophy, 4th Ed
91:A pendulum can be used to measure the
1433:{\displaystyle T_{1}^{2}+T_{2}^{2}\,}
1382:{\displaystyle T_{1}^{2}-T_{2}^{2}\,}
865:if the yard prototype was destroyed.
814:could be calculated very accurately.
7:
2062:Géodésie in Encyclopedia Universalis
1530:International Association of Geodesy
1886:. Effingham Wilson. pp. xvii.
1937:. Walter de Gruyter. p. 177.
1705:. Amrita Vishwa Vidyapeetham. 2011
1586:International Geodetic Association
1584:became the first president of the
821:; timing the interval between the
592:Inaccuracy of gravimeter pendulums
455:Gravity measurement with pendulums
25:
942:, are close in value, the period
432:are the distances of knife edges
190:is swung from one pivot, and the
27:Reversible free swinging pendulum
1863:Trautwine, John Cresson (1907).
810:, into the period equation (1),
66:instrument to measure the local
2028:(in French). pp. 1463–1466
1588:(1887–1891) after the death of
1504:interchanging the knife edges.
1121:
1120:
1119:
150:
149:
148:
1128:
1122:
157:
151:
98:because for narrow swings its
54:is a reversible free swinging
1:
1900:Zupko, Ronald Edward (1990).
1726:A Textbook of Physics, 4th Ed
1620:remarkably close to reality.
1582:Carlos Ibáñez Ibáñez de Ibero
1474:{\displaystyle h_{1}-h_{2}\,}
1331:{\displaystyle h_{1}-h_{2}\,}
1234:{\displaystyle h_{1}+h_{2}\,}
912:Repsold–Bessel pendulum
645:Charles Marie de la Condamine
581:{\displaystyle g=\pi ^{2}L\,}
2248:The Accurate Measurement of
1958:Poynting & Thompson 1907
1812:Poynting & Thompson 1907
1557:United States Coast Survey's
1547:, took part. Indeed, as the
1297:, and thus their difference
705:Drawing of Kater's pendulum
596:In Kater's time, the period
463:A Kater's pendulum and stand
2254:Has derivation of equations
1703:Virtual Amrita Laboratories
1534:The 1875 Conference of the
659:slightly during the cycle.
174:So by measuring the length
2290:
899:Great Trigonometric Survey
2160:10.1007/s00190-004-0423-0
1970:Zuerich, ETH-Bibliothek.
1800:Lenzen & Multauf 1964
2189:Soler, T. (1997-02-01).
2134:Torge, W. (2005-04-01).
1978:(in French). p. 256
1934:Geodesy: An Introduction
1931:Torge, Wolfgang (2001).
1614:Friedrich Robert Helmert
1540:Friedrich Wilhelm Bessel
1536:European Arc Measurement
1389:is so much smaller than
669:Horologium Oscillatorium
662:
623:Horologium Oscillatorium
1290:{\displaystyle h_{2}\,}
1262:{\displaystyle h_{1}\,}
1193:{\displaystyle h_{2}\,}
1165:{\displaystyle h_{1}\,}
93:acceleration of gravity
68:acceleration of gravity
18:Repsold–Bessel pendulum
1880:Rutter, Henry (1866).
1845:Raithby, John (1824).
1561:Charles Sanders Peirce
1545:Charles Sanders Peirce
1475:
1434:
1383:
1332:
1291:
1263:
1235:
1194:
1166:
1135:
921:
878:
819:method of coincidences
791:Experimental procedure
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696:Friedrich Bohnenberger
582:
464:
381:
165:
47:
2215:10.1007/s001900050086
1869:. Wiley. p. 216.
1824:Elias Loomis (1864).
1642:Kater, Henry (1818).
1559:direction instructed
1486:center of oscillation
1476:
1435:
1384:
1333:
1292:
1264:
1236:
1195:
1167:
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919:
885:as a regular part of
876:
704:
649:Jean-Charles de Borda
628:center of oscillation
583:
481:Académie des Sciences
462:
382:
166:
76:center of oscillation
33:
2111:10.1007/1345_2015_42
1771:Zupko, Ronald Edward
1672:Nave, C. R. (2005).
1444:
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1342:
1301:
1273:
1245:
1204:
1176:
1148:
953:
740:pointers for reading
555:
212:
121:
2207:1997JGeod..71..176S
2152:2005JGeod..78..558T
1648:Phil. Trans. R. Soc
1590:Johann Jacob Baeyer
1549:figure of the Earth
1428:
1410:
1377:
1359:
1049:
1031:
1004:
986:
344:
326:
275:
257:
186:can be calculated.
2195:Journal of Geodesy
2140:Journal of Geodesy
1699:"Kater's Pendulum"
1526:and C. Defforges.
1471:
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619:Christiaan Huygens
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261:
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48:
42:was used to avoid
2120:978-3-319-30895-1
2071:978-2-85229-290-1
1674:"Simple Pendulum"
1654:(33). London: 109
1112:
1054:
1009:
920:Repsold pendulum.
835:thermal expansion
633:center of gravity
615:compound pendulum
418:respectively and
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371:
302:
146:
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72:centre of gravity
16:(Redirected from
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1639:
1570:Emile Plantamour
1553:seconds pendulum
1517:
1516:
1512:
1480:
1478:
1477:
1472:
1469:
1468:
1456:
1455:
1439:
1437:
1436:
1431:
1427:
1422:
1409:
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1388:
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1358:
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1199:
1197:
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1188:
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1097:
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1086:
1085:
1084:
1072:
1071:
1061:
1055:
1050:
1048:
1043:
1030:
1025:
1015:
1010:
1005:
1003:
998:
985:
980:
970:
965:
964:
926:Friedrich Bessel
895:geodetic surveys
852:seconds pendulum
800:
784:
774:
767:
762:
761:
757:
754:
738:
731:
724:
717:
710:
692:Gaspard de Prony
663:Kater's solution
653:angular momentum
587:
585:
584:
579:
573:
572:
541:seconds pendulum
514:
513:
509:
506:
500:
499:
495:
492:
479:, by the French
386:
384:
383:
378:
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356:
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241:
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222:
170:
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147:
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106:depends only on
52:Kater's pendulum
21:
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2244:
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2055:
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2031:
2029:
2019:
2018:
2014:
2005:
2003:
1995:
1994:
1990:
1981:
1979:
1969:
1968:
1964:
1956:
1952:
1945:
1930:
1929:
1925:
1918:
1899:
1898:
1894:
1879:
1878:
1874:
1862:
1861:
1857:
1844:
1839:
1835:
1823:
1822:
1818:
1810:
1806:
1798:
1794:
1787:
1769:
1768:
1764:
1755:
1753:
1743:
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1722:
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1708:
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1697:
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1683:
1681:
1671:
1670:
1666:
1657:
1655:
1641:
1640:
1631:
1626:
1532:
1520:Survey of India
1514:
1510:
1509:
1460:
1447:
1442:
1441:
1391:
1390:
1340:
1339:
1317:
1304:
1299:
1298:
1276:
1271:
1270:
1248:
1243:
1242:
1220:
1207:
1202:
1201:
1179:
1174:
1173:
1151:
1146:
1145:
1101:
1088:
1087:
1076:
1063:
1062:
1056:
1016:
971:
956:
951:
950:
941:
934:
914:
871:
798:
793:
782:
772:
765:
759:
755:
752:
750:
747:
736:
734:
729:
727:
722:
720:
715:
713:
708:
706:
665:
594:
564:
553:
552:
511:
507:
504:
502:
497:
493:
490:
488:
457:
452:
445:
438:
431:
424:
417:
410:
403:
396:
360:
347:
346:
311:
291:
278:
277:
242:
238:
227:
223:
210:
209:
182:of a pendulum,
119:
118:
110:and its length
89:
28:
23:
22:
15:
12:
11:
5:
2287:
2285:
2277:
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2255:
2243:
2242:External links
2240:
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2201:(3): 176–188.
2181:
2146:(9): 558–568.
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2099:IAG 150 Years
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2042:
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676:Royal Society
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486:
482:
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477:French Guiana
474:
470:
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37:
32:
19:
2249:
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2194:
2184:
2143:
2139:
2129:
2102:
2098:
2088:
2061:
2056:
2047:
2041:
2030:. Retrieved
2025:
2015:
2004:. Retrieved
2000:
1991:
1980:. Retrieved
1975:
1965:
1953:
1933:
1926:
1902:
1895:
1887:
1882:
1875:
1865:
1858:
1847:
1840:
1836:
1826:
1819:
1807:
1795:
1775:
1765:
1754:. Retrieved
1749:
1739:
1725:
1718:
1707:. Retrieved
1702:
1693:
1682:. Retrieved
1678:Hyperphysics
1677:
1667:
1656:. Retrieved
1651:
1647:
1579:
1566:
1555:length, the
1533:
1506:
1502:
1497:
1493:
1489:
1483:
1143:
943:
936:
929:
923:
903:
880:
860:
849:
823:coincidences
822:
818:
816:
811:
807:
803:
797:
794:
781:
778:
771:
764:
748:
745:The pendulum
735:
728:
721:
714:
707:
689:
684:
680:
673:
668:
667:However, in
666:
656:
647:(1735), and
638:
626:
622:
609:
605:
601:
597:
595:
545:
539:
535:
533:
521:Isaac Newton
516:
466:
440:
433:
426:
419:
412:
405:
398:
391:
389:
204:
199:
195:
188:
183:
179:
175:
173:
111:
107:
103:
95:
90:
51:
49:
39:
2001:www.unav.es
1976:E-Periodica
906:gravimeters
641:Jean Picard
529:gravimeters
469:Jean Richer
178:and period
87:Description
60:Henry Kater
2032:2021-10-25
2006:2021-10-25
1982:2021-10-25
1756:2009-01-28
1709:2019-01-26
1684:2009-02-20
1658:2008-11-25
1624:References
1500:with (1).
901:of India.
883:gravimetry
64:gravimeter
2274:Pendulums
2231:119447198
2223:1432-1394
2176:120943411
2168:1432-1394
1888:pendulum.
1618:ellipsoid
1606:Argentina
1458:−
1361:−
1315:−
1099:−
1033:−
891:surveying
856:milligals
844:sea level
566:π
362:ℓ
358:−
349:ℓ
328:−
293:ℓ
280:ℓ
229:π
134:π
102:of swing
2268:Category
2080:36747385
1802:, p. 315
1773:(1990).
643:(1669),
81:geodetic
56:pendulum
2203:Bibcode
2148:Bibcode
2026:Gallica
1960:, p. 15
1814:, p. 12
1513:⁄
887:geodesy
758:⁄
510:⁄
496:⁄
473:Cayenne
450:History
2229:
2221:
2174:
2166:
2117:
2078:
2068:
1941:
1914:
1783:
1598:Mexico
525:oblate
515:Paris
390:where
192:period
100:period
40:(left)
36:period
2227:S2CID
2172:S2CID
1610:Japan
1602:Chile
1144:Here
783:(b,c)
517:lines
2219:ISSN
2164:ISSN
2115:ISBN
2076:OCLC
2066:ISBN
1939:ISBN
1912:ISBN
1781:ISBN
1608:and
1496:and
1269:and
1172:and
935:and
863:yard
733:bob
439:and
425:and
411:and
397:and
74:and
2211:doi
2156:doi
2107:doi
2103:143
1908:179
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869:Use
858:).
799:(b)
773:(a)
766:(d)
737:(e)
730:(d)
723:(c)
716:(b)
709:(a)
2270::
2225:.
2217:.
2209:.
2199:71
2197:.
2193:.
2170:.
2162:.
2154:.
2144:78
2142:.
2138:.
2113:.
2097:.
2074:.
2024:.
1999:.
1974:.
1910:.
1748:.
1731:20
1701:.
1676:.
1650:.
1646:.
1632:^
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1600:,
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687:.
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50:A
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2178:.
2158::
2150::
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2109::
2082:.
2035:.
2009:.
1985:.
1947:.
1920:.
1789:.
1759:.
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1712:.
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1661:.
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967:=
962:2
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933:1
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846:.
812:g
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681:L
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602:g
598:T
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570:2
562:=
559:g
546:g
536:g
512:4
508:1
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444:2
441:K
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219:=
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155:1
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131:2
128:=
125:T
112:L
108:g
104:T
96:g
46:.
20:)
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