FIGURE OF THE See also:

• EARTH
• EARTH (a word common to Teutonic languages, cf. Ger. Erde, Dutch aarde, Swed. and Dan. jord; outside Teutonic it appears only in the Gr. 'pq'e, on the ground; it has been connected by some etymologists with the Aryan root ar-, to plough, which is seen in
• EARTH, FIGURE OF
• FIGURE OF THE EARTH

EARTH  . The determination of the figure of the See also:

• EARTH
• EARTH (a word common to Teutonic languages, cf. Ger. Erde, Dutch aarde, Swed. and Dan. jord; outside Teutonic it appears only in the Gr. 'pq'e, on the ground; it has been connected by some etymologists with the Aryan root ar-, to plough, which is seen in
• EARTH, FIGURE OF
• EARTH, FIGURE OF THE

earth is a problem of the highest importance in See also:

• ASTRONOMY (from Gr. &arpov, a star, and v ,usiv, to classify or arrange)

astronomy, inasmuch as the See also:

• DIAMETER (from the Cr. &a, through, µErpov, measure)

diameter of the earth is the unit to which all See also:

• CELESTIAL, OR STELLAR

celestial distances must be referred . See also:

• HISTORICAL

Historical . Reasoning from the See also:

• UNIFORM

uniform level See also:

• APPEARANCE (from Lat. apparere, to appear)

appearance of the See also:

• HORIZON (Gr. 6pfTwv, dividing)

horizon, the See also:

• VARIATIONS
• VARIATIONS, CALCULUS
• VARIATIONS, CALCULUS OF

variations in See also:

• ALTITUDE (Lat. altitudo, from altus, high)

altitude of the circumpolar stars as one travels towards the See also:

• NORTH
• NORTH, BARONS
• NORTH, MARIANNE (1830—1890)
• NORTH, ROGER (1653-1734)
• NORTH, SIR THOMAS (1535?-16o1?)

north or See also:

• SOUTH
• SOUTH, ROBERT (1634–1716)

south, the disappearance of a See also:

• SHIP

ship See also:

• STANDING

standing out to See also:

• SEA (in O. Eng. sae, a common Teutonic word; cf. Ger. See, Dutch Zee, &c.; the ultimate source is uncertain)
• SEA, COMMAND OF THE

sea, and perhaps other phenomena, the earliest astronomers regarded the earth as a See also:

• SPHERE (Gr. vclsa?pa, a ball or globe)

sphere, and they endeavoured to ascertain its dimensions . See also:

• ARISTOTLE
• ARISTOTLE (384-322 B.C.)

Aristotle relates that the mathematicians had found the circumference to be 400,000 stadia (about 46,000 See also:

• MILES, NELSON APPLETON (1839— )

miles) . But Eratosthenes (c . 250 B.C.) appears to have been the first who entertained an accurate See also:

• IDEA (Gr. Ibia, connected with i&eiv, to see; cf. Lat. species from specere, to look at)

idea of the principles on which the determination of the figure of the earth really depends, and attempted to reduce them to practice . His results were very inaccurate, but his method is the same as that which is followed at the See also:

• PRESENT

present See also:

• DAY (O. Eng. dreg, Ger. Tag; according to the New English Dictionary, " in no way related to the Lat. dies")
• DAY, JOHN (1574-1640?)
• DAY, THOMAS (1748-1789)

day—depending, in fact,on the comparison of a See also:

• LINE

line measured on the earth's See also:

• SURFACE

surface with the corresponding arc of the heavens . He observed that at Syene in Upper See also:

• EGYPT

Egypt, on the day of the summer See also:

• SOLSTICE (Lat. solstitium, from sal, sun, and sistere, to stand still)

solstice, the See also:

• SUN
• SUN (0. Eng. sonne, Ger. sonne. Fr. soleil, Lat. sol, Gr. ijXior, from which comes helio- in various English compounds)

sun was exactly See also:

• VERTICAL (from Lat. vertex, highest point)

vertical, whilst at See also:

• ALEXANDRIA
• ALEXANDRIA (Arab. Iskenderia)

Alexandria at the same See also:

• SEASON (0. Fr. seson, seison, mod. saison, Lat. satio, sowing time, the spring, from serere, to sow; in Late Lat. the word is found with its present meaning, the spring being considered as particularly the season of the year)

season of the See also:

• YEAR

year its See also:

• ZENITH (from the Arabic)

zenith distance was 7° 12', or one-fiftieth of the circumference of a circle . He assumed that these places were on the same See also:

• MERIDIAN
• MERIDIAN (from the Lat. meridianus, pertaining to the south or mid-day)

meridian; and, reckoning their distance apart as 5000 stadia, he inferred that the circumference of the earth was 250,000 stadia (about 29,000 miles) . A similar See also:

• ATTEMPT (Lat. adtemptare, attentare, to try)

attempt was made by See also:

• POSIDONIUS (c. 130-50 B.C.)

Posidonius, who adopted a method which differed from that of Eratosthenes only in using a See also:

• STAR

star instead of the sun . He obtained 240,000 stadia (about 27,600 miles) for the circumference .

See also:

• PTOLEMY (CLAUDIUS PTOLEMAEUS)

Ptolemy in his See also:

• GEOGRAPHY (Gr. yil, earth, and ypiickty, to write)

Geography assigns the length of the degree as 500 stadia . The See also:

• ARABS

Arabs also investigated the question of the earth's magnitude . The See also:

• CALIPH, CALIF

caliph Abdallah al See also:

• MAMUN (c. 786-833)

Mamun (A.D . 814), having fixed on a spot in the plains of See also:

• MESOPOTAMIA (Mevoaora)

Mesopotamia, despatched one See also:

• COMPANY

company of astronomers northwards and another southwards, measuring the See also:

• JOURNEY (through O. Fr. jornee or journee, mod. Fr. journee, from med. Lat. diurnata, Lat. diurnus, of or belonging to dies, day)

journey by rods, until each found the altitude of the See also:

• POLE (FAMILY)
• POLE, REGINALD (1500-1558)
• POLE, RICHARD DE LA (d. 1525)
• POLE, WILLIAM (1814—1900)

pole to have changed one degree . But the result of this measurement does not appear to have been very satisfactory . From this See also:

• TIME (0. Eng. Lima, cf. Icel. timi, Swed. timme, hour, Dan. time; from the root also seen in " tide," properly the time of between the flow and ebb of the sea, cf. O. Eng. getidan, to happen, " even-tide," &c.; it is not directly related to Lat. tempus)
• TIME, MEASUREMENT OF
• TIME, STANDARD

time the subject seems to have attracted no See also:

• ATTENTION (from Lat. ad-tendo, await, expect; the condition of being " stretched " or " tense ")

attention until about 1500, when See also:

• JEAN

Jean See also:

• FERNEL, JEAN FRANCOIS (1497-1558)

Fernel (1497-1558), a Frenchman, measured a distance in the direction of the meridian near See also:

• PARIS
• PARIS (also called ALEXANDROS)
• PARIS, ALEXIS PAULIN (1800-x881)
• PARIS, BRUNO PAULIN GASTON (1839-1903)
• PARIS, FRANCOIS DE (169o-1727)
• PARIS, LOUIS PHILIPPE ALBERT
• PARIS, TREATIES OF (1814-1815)

Paris by counting the number of revolutions of the See also:

• WHEEL (0. Eng. hweol, hweohl, &c., cognate with Icel. hjol, Dan. hiul, the Indo-European root is seen in Sanskrit chakra, Gr. r in Aos, circle, whence " cycle ")
• WHEEL, BREAKING ON THE

wheel of a See also:

• CARRIAGE

carriage . His astronomical observations were made with a triangle used as a quadrant, and his resulting length of a degree was very near the truth . Willebrord See also:

• SNELL, HANNAH (1723–1992)
• SNELL, JOHN (1629-1679)
• SNELL, WILLEBRORD (1591-1626)

Snell' substituted a See also:

• CHAIN (through the O. Fr. citable, chcene, &c., from Lat. catena)

chain of triangles for actual linear measurement . He measured his See also:

• BASE

base line on the frozen surface of the meadows near See also:

• LEIDEN

Leiden, and measured the angles of his triangles, which See also:

• LAY

lay between See also:

• ALKMAAR

Alkmaar and See also:

• BERGEN

Bergen-op-Zoom, with a quadrant and semicircles . He took the precaution of ' Eratosthenes Batavus, seu de terrae ambitus See also:

• VERA, AUGUSTO (1813-1885)

vera quantitate suscitatus, a Willebrordo Snellio, Lugduni-Batavorum 0617) . II comparing his See also:

• STANDARD
• STANDARD, BATTLE OF THE

standard with that of the See also:

• FRENCH
• FRENCH, DANIEL CHESTER (1850– )
• FRENCH, NICHOLAS (1604-1678)

French, so that his result was expressed in toises (the length of the toise is about 6.39 See also:

• ENGLISH

English ft.) . The See also:

• WORK

work was recomputed and reobserved by P. von See also:

• MUSSCHENBROEK, PIETER VAN (1692–1761)

Musschenbroek in 1729 .

In 1637 an Englishman, See also:

• RICHARD
• RICHARD (d. 1184)
• RICHARD, FRANCOIS MARIE BENJAMIN (18,9-1908)
• RICHARD, HENRY (1812-1888)
• RICHARD, ST

Richard See also:

• NORWOOD

Norwood, published a determination of the figure of the earth in a See also:

• VOLUME

volume entitled The See also:

• SEAMAN, OWEN (1861– )

Seaman's Practice, contayning a Fundamentall Probleme in See also:

• NAVIGATION (from Lat. navis, ship, and agere, to move)

Navigation experimentally verified, namely, touching the Compasse of the Earth and Sea and the quantity of a Degree in our English See also:

• MEASURES

Measures . He observed on the 1th of See also:

• JUNE

June 1633 the sun's meridian altitude in See also:

• LONDON

London as 62° 1', and on the 6th of June 1635, his meridian altitude in See also:

• YORK
• YORK (HOUSE OF)
• YORK, EDMUND OF LANGLEY, DUKE OF (1341-1402)
• YORK, EDWARD
• YORK, FREDERICK AUGUSTUS, DUKE OF (1763-1827)
• YORK, RICHARD, DUKE

York as 590 33' . He measured the distance between these places partly with a chain and partly by pacing . By this means, through See also:

• COMPENSATION (from Lat. compensare, to weigh one thing against another)

compensation of errors, he arrived at 367,176 ft. for the degree—a very See also:

• FAIR

fair result . The application of the See also:

• TELESCOPE

telescope to angular See also:

• INSTRUMENTS

instruments was the next important step . Jean See also:

• PICARD, LOUIS JOSEPH ERNEST (1821-1877)

Picard was the first who in 1669, with the telescope, using such precautions as the nature of the operation requires, measured an arc of meridian . He measured with wooden rods a base line of 5663 toises, and a second or base of verification of 3902 toises; his triangulation extended from Malvoisine, near Paris, to Sourdon, near See also:

• AMIENS

Amiens . The angles of the triangles were measured with a quadrant furnished with a telescope having See also:

• CROSS

cross-wires . The difference of See also:

• LATITUDE (Lat. latitudo, latus, broad)

latitude of the terminal stations was determined by observations made with a sector on a star in See also:

• CASSIOPEIA

Cassiopeia, giving 1° 22' 55" for the See also:

• AMPLITUDE (from Lat. amplus, large)

amplitude . The terrestrial measurement gave 78,850 toises,whenceheinferred for the length of the degree 57,060 toises . Hitherto See also:

• GEODETIC

geodetic observations had been confined to the determination of the magnitude of the earth considered as a sphere, but. a See also:

• DISCOVERY

discovery made by Jean Richer (d . 1696) turned the attention of mathematicians to its deviation from a spherical See also:

• FORM (Lat. forma)

form .

This astronomer, having been sent by the See also:

• ACADEMY, GREEK
• ACADEMY, ROYAL

Academy of Sciences of Paris to the See also:

• ISLAND (O.E. ieg =isle, +land')

island of See also:

• CAYENNE

Cayenne, in South See also:

• AMERICA

America, for the purpose of investigating the amount of astronomical See also:

• REFRACTION
• REFRACTION (Lat. refringere, to break open or apart)

refraction and other astronomical See also:

• OBJECTS

objects, observed that his See also:

• CLOCK

clock, which had been regulated at Paris to See also:

• BEAT (a word common in various forms to the Teutonic languages; it is connected with the similar Romanic words derived from the Late Lat. battere)

beat seconds, lost about two minutes and a See also:

• HALF

half daily at Cayenne, and that in See also:

• ORDER
• ORDER (through Fr. ordre, for earlier ordene, from Lat. ordo, ordinis, rank, service, arrangement; the ultimate source is generally taken to be the root seen in Lat. oriri, rise, arise, begin; cf. " origin ")
• ORDER, HOLY

order to bring it to measure mean See also:

• SOLAR, SOLLER (Lat. solarium, Fr.• galetas, Ital. solaio)

solar time it was necessary to shorten the pendulum by more than a line (about , th of an in.) . This fact, which was scarcely credited till it had been confirmed by the subsequent observations of Varin and See also:

• DESHAYES, GERARD PAUL (1795–1875)

Deshayes on the coasts of See also:

• AFRICA
• AFRICA, ROMAN

Africa and America, was first explained in the third See also:

• BOOK

book of See also:

• NEWTON
• NEWTON, ALFRED (1829–1907)
• NEWTON, JOHN (1725-1807)
• NEWTON, JOHN (1823-1895)
• NEWTON, SIR CHARLES THOMAS (1816-1894)
• NEWTON, SIR ISAAC (1642-1727)

Newton's Principia, who showed that it could only be referred to a diminution of gravity arising either from a protuberance of the See also:

• EQUATORIAL

equatorial parts of the earth and consequent increase of the distance from the centre, or from the counteracting effect of the centrifugal force . About the same time (1673) appeared See also:

• CHRISTIAN
• CHRISTIAN, WILLIAM (1608-1663)

Christian See also:

• HUYGENS, CHRISTIAAN (1629-1695)
• HUYGENS, SIR CONSTANTIJN (1596-1687)

Huygens' De Horologio Oscillatorio, in which for the first time were found correct notions on the subject of centrifugal force . It does not, however, appear that they were applied to the theoretical investigation of the figure of the earth before the publication of Newton's Principia . In 1690 Huygens published his De Causa Gravitatis, which contains an investigation of the figure of the earth on the supposition that the attraction of every particle is towards the centre . Between 1684 and 1718 J. and D . See also:

• CASSINI

Cassini, starting from Picard's base, carried a triangulation northwards from Paris to See also:

• DUNKIRK
• DUNKIRK (Fr. Dunkerque)

Dunkirk and southwards from Paris to Collioure . They measured a base of 7246 toises near See also:

• PERPIGNAN

Perpignan, and a somewhat shorter base near Dunkirk; and from the See also:

• NORTHERN

northern portion of the arc, which had an amplitude of 20 12' 9", obtained for the length of a degree 56,96o toises; while from the See also:

• SOUTHERN

southern portion, of which the amplitude was 6° 18' 57", they obtained 57,097 toises . The immediate inference from this was that, the degree diminishing with increasing latitude, theearth must be a prolate See also:

• SPHEROID (Gr. c4aipa-etbis, like a sphere)

spheroid . This conclusion was totally opposed to the theoretical investigations of Newton and Huygens, and accordingly the Academy of Sciences of Paris determined to apply a decisive test by the measurement of arcs at a See also:

• GREAT

great distance from each other—one in the neighbourhood of the See also:

• EQUATOR (Late Lat. aequator, from aequare, to make equal)

equator, the other in a high latitude . Thus arose the celebrated expeditions of the Frenchacademicians . In May 1735 See also:

• LOUIS
• LOUIS (804–876)
• LOUIS (893–911)
• LOUIS, JOSEPH DOMINIQUE, BARON (1755-1837)
• LOUIS, or LEWIS (from the Frankish Chlodowich, Chlodwig, Latinized as Chlodowius, Lodhuwicus, Lodhuvicus, whence-in the Strassburg oath of 842-0. Fr. Lodhuwigs, then Chlovis, Loys and later Louis, whence Span. Luiz and—through the Angevin kings—Hungarian

Louis See also:

• GODIN, JEAN BAPTISTE ANDRE (1817–1888)

Godin, See also:

• PIERRE

Pierre See also:

• BOUGUER, PIERRE (1698-1758)

Bouguer and See also:

• CHARLES
• CHARLES (1270-1325)
• CHARLES (1421-1461)
• CHARLES (1525-1574)
• CHARLES (c. 1319-1364)
• CHARLES (Fr. Charles; Span. Carlos; Ital. Carlo; Ger. Karl; derived from O.H.G. Charal,latinized as Carolus, meaning originally " man ": cf. Mod.Ger., Kerl," fellow," A.S. ceorl, Mod. Eng. " churl ")
• CHARLES (KARL EITEL ZEPHYRIN LUDWIG; in Rum. CAROL)
• CHARLES [KARL ALEXANDER] (1712-1780)
• CHARLES, ELIZABETH (1828-1896)
• CHARLES, JACQUES ALEXANDRE CESAR (1746-1823)
• CHARLES, THOMAS (1755-1814)

Charles See also:

• MARIE

Marie de la Condamine, under the auspices of Louis XV., proceeded to See also:

• PERU
• PERU (apparently from Biru, a small river on the west coast of Colombia, where Pizarro landed)

Peru, where, assisted by two See also:

• SPANISH

Spanish See also:

• OFFICERS

officers, after ten years of laborious exertion, they measured an arc of 3° 7', the northern end near the equator .

The second party consisted of Pierre Louis See also:

• MOREAU, GUSTAVE (1826-1898)
• MOREAU, HEGESIPPE (18ro-1838)
• MOREAU, JEAN VICTOR MARIE (1763-1813)

Moreau de See also:

• MAUPERTUIS, PIERRE LOUIS MOREAU DE (1698–1759)

Maupertuis, See also:

• ALEXIS
• ALEXIS, WILLIBALD

Alexis See also:

• CLAUDE, JEAN (1619-1687)

Claude See also:

• CLAIRAULT (or CLAIRAUT), ALEXIS CLAUDE (1713-1765)

Clairault, Charles See also:

• ETIENNE, CHARLES GUILLAUME (1778-1845)

Etienne Louis See also:

• CAMUS, ARMAND GASTON (1740-1804)
• CAMUS, CHARLES ETIENNE LOUIS (1699-1768)
• CAMUS, FRANCOIS JOSEPH DES (1672-1732)

Camus, Pierre Charles See also:

• LEMONNIER, ANTOINE LOUIS CAMILLE (1844– )
• LEMONNIER, PIERRE CHARLES (1715—1799)

Lemonnier, and Reginaud Outhier, who reached the Gulf of See also:

• BOTHNIA, GULF OF

Bothnia in See also:

• JULY

July 1736; they were in some respects more fortunate than the first party, inasmuch as they completed the measurement of an arc near the polar circle of 57' amplitude and returned within sixteen months from the date of their departure . The measurement of Bouguer and De la Condamine was executed with great care, and on See also:

• ACCOUNT
• ACCOUNT (through O. Fr. acont, Late Lat. comptum, cornputare, to calculate)

account of the locality, as well as the manner in which all the details were conducted, it has always been regarded as a most valuable determination . The southern limit was at Tarqui, the northern at Cotchesqui . A base of 6272 toises was measured in the vicinity of See also:

• QUITO

Quito, near the northern extremity of the arc, and a second base of 5260 toises near the southern extremity . The mountainous nature of the See also:

• COUNTRY (from the Mid. Eng. contre or contrie, and O. Fr. cuntree; Late Lat. contrata, showing the derivation from contra, opposite, over against, thus the tract of land which fronts the sight, cf. Ger. Gegend, neighbourhood)

country made the work very laborious, in some cases the difference of heights of two neighbouring stations exceeding 1 mile; and they had much trouble with their instruments, those with which they were to "determine the latitudes proving untrustworthy . But they succeeded by simultaneous observations of the same star at the two extremities of the arc in obtaining very fair results . The whole length of the arc amounted to 176,945 toises, while the diff erenceof iatitudeswas3° 7' 3" . In consequence of a misunderstanding that arose between De la Condamine and Bouguer, their operations were conducted separately, and each wrote a full account of the expedition . Bouguer's book was published in 1749; that of De la Condamine in 1751 . The toise used in this measure was afterwards regarded as the standard toise, and is always referred to as the Toise of Peru . The party of Maupertuis, though their work was quickly despatched, had also to contend with great difficulties . Not being able to make use of the small islands in the Gulf of Bothnia for the trigonometrical stations, they were forced to penetrate into the forests of See also:

• LAPLAND, or LAPPLAND

Lapland, commencing operations at Torneh, a See also:

• CITY (through Fr. cite, from Lat. civitas)

city situated on the mainland near the extremity of the gulf .

From this, the southern extremity of their arc, they carried a chain of triangles northward to the See also:

• MOUNTAIN (0. Fr. montaigne; popular Lat. montanea, an adjectival form from the classical mons, montis, whence Eng. " mount," a form usually used along with the name of an individual mountain, e.g. Mt Everest)
• MOUNTAIN, THE (La Montagne)

mountain Kittis, which they selected as the northern See also:

• TERMINUS

terminus . The latitudes were determined by observations with a sector (made by See also:

• GEORGE
• GEORGE, 8TH MARQUESS OF TWEEDDALE (1787-1876)
• GEORGE, HENRY (1839—1897)
• GEORGE, LAKE
• GEORGE, SAINT (d. 303)

George See also:

• GRAHAM, SIR GERALD (1831–1899)
• GRAHAM, SIR JAMES ROBERT GEORGE
• GRAHAM, SYLVESTER (1794-1851)
• GRAHAM, THOMAS (1805-1869)

Graham) of the zenith distance of a and b Draconis . The base line was measured on the frozen surface of the See also:

• RIVER

river Tornea about the See also:

• MIDDLE

middle of the arc; two parties measured it separately, and they differed by about 4 in . The result of the whole was that the difference of latitudes of the terminal stations was 57' 29" •6, and the length of the arc 55,023 toises . In this expedition, as well as in that to Peru, observations were made with a pendulum to determine the force of gravity; and these observations coincided with the geodetic results in proving that the earth was an oblate and not prolate spheroid . In 1740 was published in the Paris Memoires an account, by Cassini de Thury, of a remeasurement by himself and See also:

• NICOLAS
• NICOLAS, SIR NICHOLAS HARRIS (1799–1848)

Nicolas Louis de See also:

• LACAILLE, NICOLAS LOUIS DE (1713-1762)

Lacaille of the meridian of Paris . With a view to determine more accurately the variation of the degree along the meridian, they divided the distance from Dunkirk to Collioure into four partial arcs of about two degrees each, by observing the latitude at five stations . The results previously obtained by J. and D . Cassini were not confirmed, but, on the contrary, the length of the degree derived from these partial arcs showed on the whole an increase with an increasing latitude . Cassini and Iacaille also measured an arc of parallel across the mouth of the See also:

• RHONE
• RHONE (Fr. Rhone, Lat. Rhodanus)

Rhone . The difference of time of the extremities was determined by the observers at either end noting the instant of a See also:

• SIGNAL

signal given by flashing See also:

• GUNPOWDER

gunpowder at a point near the middle of the arc . While at the Cape of See also:

• GOOD, JOHN MASON (1764-1827)

Good See also:

• HOPE, ANTHONY
• HOPE, THOMAS (c. 1770-1831)

Hope in 1752, engaged in various astronomical observations, Lacaille measured an arc of meridian of 1° 13' 17", which gave him for the length of the degree 57,037 toises—an unexpected result, which has led to the remeasurement of the arc by See also:

• SIR

Sir See also:

• THOMAS
• THOMAS (c. 1654-1720)
• THOMAS (d. 110o)
• THOMAS, ARTHUR GORING (1850-1892)
• THOMAS, CHARLES LOUIS AMBROISE (1811-1896)
• THOMAS, GEORGE (c. 1756-1802)
• THOMAS, GEORGE HENRY (1816-187o)
• THOMAS, ISAIAH (1749-1831)
• THOMAS, PIERRE (1634-1698)
• THOMAS, SIDNEY GILCHRIST (1850-1885)
• THOMAS, ST
• THOMAS, THEODORE (1835-1905)
• THOMAS, WILLIAM (d. 1554)

Thomas Maclear (see See also:

• GEODESY (from the Gr. y, the earth, and SatEty, to divide)

GEODESY) .

Passing over the measurements made between See also:

• ROME
• ROME (Roma)

Rome and See also:

• RIMINI

Rimini and on the plains of See also:

• PIEDMONT (Ital. Piemonte; Low Lat. Pedemons and Pedemontium)

Piedmont by the See also:

• JESUITS

Jesuits Ruggiero Giuseppe See also:

• BOSCOVICH, ROGER JOSEPH (1711?-1787)

Boscovich and Giovanni Battista See also:

• BECCARIA, GIOVANNI BATTISTA (1716-1781)

Beccaria, and also the arc measured with See also:

• DEAL

deal rods in North America by Charles See also:

• MASON, FRANCIS (1799—1874)
• MASON, GEORGE (1725—1792)
• MASON, GEORGE HEMMING (1818–1872)
• MASON, JAMES MURRAY (1798-1871)
• MASON, JOHN (1586-1635)
• MASON, JOHN YOUNG (1799-1859)
• MASON, LOWELL (1792—1872)
• MASON, SIR JOHN (1503–1566)
• MASON, SIR JOSIAH (1795-1881)
• MASON, WILLIAM (1725—1797)

Mason and See also:

• JEREMIAH

Jeremiah See also:

• DIXON
• DIXON, GEORGE (1755 ?–1800)
• DIXON, HENRY HALL (1822-1870)
• DIXON, RICHARD WATSON (1833-1900)
• DIXON, WILLIAM HEPWORTH (1821-1879)

Dixon, we come to the commencement of the English triangulation . In 1783, in consequence of a See also:

• REPRESENTATION

representation from Cassini de Thury on the advantages that would accrue from the geodetic connexion of Paris and See also:

• GREENWICH

Greenwich, See also:

• GENERAL
• GENERAL (Lat. generalis, of or relating to a genus, kind or class)

General See also:

• WILLIAM
• WILLIAM (1143-1214)
• WILLIAM (1227-1256)
• WILLIAM (1J33-1584)
• WILLIAM (A.S. Wilhelm, O. Norse Vilhidlmr; O. H. Ger. Willahelm, Willahalm, M. H. Ger. Willehelm, Willehalm, Mod.Ger. Wilhelm; Du. Willem; O. Fr. Villalme, Mod. Fr. Guillaume; from " will," Goth. vilja, and " helm," Goth. hilms, Old Norse hidlmr, meaning
• WILLIAM (c. 1130-C. 1190)
• WILLIAM, 13TH

William See also:

• ROY, WILLIAM (1726-1790)

Roy was, with the See also:

• KING
• KING (O. Eng. cyning, abbreviated into cyng, cing; cf. O. H. G. chun- kuning, chun- kunig, M.H.G. kiinic, kiinec, kiinc, Mod. Ger. Konig, O. Norse konungr, kongr, Swed. konung, kung)
• KING [OF OCKHAM], PETER KING, 1ST BARON (1669-1734)
• KING, CHARLES WILLIAM (1818-1888)
• KING, CLARENCE (1842–1901)
• KING, EDWARD (1612–1637)
• KING, EDWARD (1829–1910)
• KING, HENRY (1591-1669)
• KING, RUFUS (1755–1827)
• KING, THOMAS (1730–1805)
• KING, WILLIAM (1650-1729)
• KING, WILLIAM (1663–1712)

king's approval, appointed by the Royal Society to conduct the operations on the See also:

• PART

part of See also:

• ENGLAND
• ENGLAND, THE CHURCH OF

England, See also:

• COUNT (Lat. comes, gen. comitis, Fr. comte, Ital. conte, Span. conde)

Count Cassini, Mechain and See also:

• DELAMBRE, JEAN BAPTISTE JOSEPH (1749-1822)

Delambre being appointed on the French See also:

• SIDE
• SIDE (mod. Eski Adalia)

side . A precision previously unknown was attained by the use of See also:

• RAMSDEN, JESSE (1735-1800)

Ramsden's See also:

• THEODOLITE

theodolite, which was the first to make the spherical excess of triangles measurable . The wooden rods with which the first base was measured were replaced by See also:

• GLASS
• GLASS (O.E. glees, cf. Ger. Glas, perhaps derived from an old Teutonic root gla-, a variant of glo-, having the general sense of shining, cf. " glare," " glow ")
• GLASS, STAINED

glass rods, which were afterwards rejected for the See also:

• STEEL, FLORA ANNIE (1847– )

steel chain of Ramsden . (For further details see Account of the Trigonometrical Survey of England and See also:

• WALES (Cymru, Gwalia, Cambria)

Wales.) Shortly after this, the See also:

• NATIONAL

National See also:

• CONVENTION (Lat. conventio, an assembly or agreement, from convenire, to come together)
• CONVENTION, THE NATIONAL

Convention of See also:

• FRANCE
• FRANCE, ANATOLE (1844– )

France, having agreed to remodel their See also:

• SYSTEM

system of weights and measures, See also:

• CHOSE (Fr. for " thing ")

chose for their unit of length the ten-millionth part of the meridian quadrant . In order to obtain this length precisely, the re-measurement of the French meridian was resolved on, and deputed to J . B . J . Delambre and Pierre See also:

• FRANCOIS (1558-1614)

Francois See also:

• ANDRE, JOHN (1751-178o)

Andre Mechain . The details of this operation will be found in the Base du systeme metrique decimale . The arc was subsequently extended by Jean See also:

• BAPTISTE

Baptiste See also:

• BIOT, JEAN BAPTISTE (1774-1862)

Biot and Dominique Francois Jean See also:

• ARAGO, DOMINIQUE FRANCOIS JEAN (1786-1853)

Arago to the island of See also:

• IVIZA, IBIZA

Iviza . Operations for the connexion of England with the See also:

• CONTINENT (from Lat. continere, "to hold together "; hence " connected," " continuous ")

continent of See also:

• EUROPE

Europe were resumed in 1821 to 1823 by See also:

• HENRY
• HENRY (1129-1195)
• HENRY (c. 1108-1139)
• HENRY (c. 1174–1216)
• HENRY (Fr. Henri; Span. Enrique; Ger. Heinrich; Mid. H. Ger. Heinrich and Heimrich; O.H.G. Haimi- or Heimirih, i.e. " prince, or chief of the house," from O.H.G. heim, the Eng. home, and rih, Goth. reiks; compare Lat. rex " king "—" rich," therefore " mig
• HENRY, EDWARD LAMSON (1841– )
• HENRY, JAMES (1798-1876)
• HENRY, JOSEPH (1797-1878)
• HENRY, MATTHEW (1662-1714)
• HENRY, PATRICK (1736–1799)
• HENRY, PRINCE OF BATTENBERG (1858-1896)
• HENRY, ROBERT (1718-1790)
• HENRY, VICTOR (1850– )
• HENRY, WILLIAM (1795-1836)

Henry See also:

• KATER, HENRY (1777-1835)

Kater and Thomas See also:

• FREDERICK
• FREDERICK (Mod. Ger. Friedrich; Ital. Federigo; Fr. Frederic and Federic; M.H.G. Friderich; O.H.G. Fridurih, " king or lord of peace," from O.H.G. fridu, A.S. frith, " peace," and rfh " rich," " a ruler," for derivation of which see HENRY)

Frederick See also:

• COLBY, THOMAS FREDERICK (1784-1852)

Colby on the English side, and F .

J . D . Arago and Claude Louis Mathieu on the French . The publication in 1838 of See also:

• FRIEDRICH, JOHANN (1836— )

Friedrich Wilhelm See also:

• BESSEL, FRIEDRICH WILHELM (1784-1846)

Bessel's Gradmessung in Ostpreussen marks an era in the See also:

• SCIENCE (Lat. scientia, from scire; to learn, know)

science of geodesy . Here we find the method of least squares applied to the calculation of a network of triangles and the reduction of the observations generally . The systematic manner in which all the observations were taken with the view of securing final results of extreme accuracy is admirable . The triangulation, which was a small one, extended about a degree and a half along the shores of the Baltic in a N.N.E. direction . The angles were observed with theodolites of 12 and 15 in. diameter, and the latitudes determined by means of the transit See also:

• INSTRUMENT (Lat. instrumentum, from insincere, to build up, furnish, arrange, prepare)

instrument in the See also:

• PRIME, PRIMER AND PRIMING

prime vertical—a method much used in See also:

• GERMANY
• GERMANY (Ger. Deutschland)

Germany . (The base apparatus is described in the See also:

• ARTICLE (from Lat. articulus, a joint)

article GEODESY.) The See also:

• PRINCIPAL

principal triangulation of Great See also:

• BRITAIN (Gr. Hperavuml vi7vot, Bperravia; Lat. Britannia, rarely Britannia)

Britain and See also:

• IRELAND
• IRELAND, CHURCH
• IRELAND, CHURCH OF
• IRELAND, JOHN (1761-1842)
• IRELAND, JOHN (1838- )
• IRELAND, WILLIAM HENRY (1777-1835)

Ireland, which was commenced in 1783 under General Roy, for the more immediate purpose of connecting the observatories of Greenwich and Paris, had been gradually extended, under the successive direction of See also:

• COLONEL (derived either from Lat. column, Fr. colonne, column, or Lat. corona, a crown)

Colonel E . See also:

• WILLIAMS
• WILLIAMS, JOHN (1582-1650)
• WILLIAMS, JOHN (1796-1839)
• WILLIAMS, ROGER (c. 1604–1684)
• WILLIAMS, ROWLAND (1817-1870)
• WILLIAMS, SIR WILLIAM FENWICK

Williams, General W . Mudge, General T . F .

Colby, Colonel L . A . See also:

• HALL
• HALL (generally known as SCHWABISCH-HALL, tc distinguish it from the small town of Hall in Tirol and Bad-Hall, a health resort in Upper Austria)
• HALL (O.E. heall, a common Teutonic word, cf. Ger. Halle)
• HALL, BASIL (1788-1844)
• HALL, CARL CHRISTIAN (1812–1888)
• HALL, CHARLES FRANCIS (1821-1871)
• HALL, CHRISTOPHER NEWMAN (1816—19oz)
• HALL, EDWARD (c. 1498-1547)
• HALL, FITZEDWARD (1825-1901)
• HALL, ISAAC HOLLISTER (1837-1896)
• HALL, JAMES (1793–1868)
• HALL, JAMES (1811–1898)
• HALL, JOSEPH (1574-1656)
• HALL, MARSHALL (1790-1857)
• HALL, ROBERT (1764-1831)
• HALL, SAMUEL CARTER (5800-5889)
• HALL, SIR JAMES (1761-1832)
• HALL, WILLIAM EDWARD (1835-1894)

Hall, and Colonel Sir Henry See also:

• JAMES
• JAMES (Gr. 'IlrKw,l3or, the Heb. Ya`akob or Jacob)
• JAMES (JAMES FRANCIS EDWARD STUART) (1688-1766)
• JAMES, 2ND EARL OF DOUGLAS AND MAR(c. 1358–1388)
• JAMES, DAVID (1839-1893)
• JAMES, EPISTLE OF
• JAMES, GEORGE PAYNE RAINSFOP
• JAMES, HENRY (1843— )
• JAMES, JOHN ANGELL (1785-1859)
• JAMES, THOMAS (c. 1573–1629)
• JAMES, WILLIAM (1842–1910)
• JAMES, WILLIAM (d. 1827)

James; it was finished in 1851 . The number of stations is about 250 . At 32 of these the latitudes were determined with Ramsden's and See also:

• AIRY, SIR GEORGE BIDDELL (1801-1892)

Airy's zenith sectors . The theodolites used for this work were, in addition to the two great theodolites of Ramsden which were used by General Roy and See also:

• CAPTAIN (derived from Lat. ca put, head, through the Low Lat. capitanus)

Captain Kater, a smaller theodolite of 18 in. diameter by the same mechanician, and another of 24 in. diameter by Messrs See also:

• TROUGHTON, EDWARD (1753-1835)

Troughton and See also:

• SIMMS, WILLIAM GILMORE (1806–1870)

Simms . Observations for determination of See also:

• ABSOLUTE (Lat. absolvere, to loose, set free)

absolute See also:

• AZIMUTH (from the Arabic)

azimuth were made with those instruments at a large number of stations; the stars a, 3, and X Ursae Minoris and 51 Cephei being those observed always at the greatest azimuths . At six of these stations the probable See also:

• ERROR (Lat. error, from errare, to wander, to err)

error of the result is under 0.4", at twelve under 0.5", at See also:

• THIRTY

thirty-four under o•7": so that the absolute azimuth of the whole network is determined with extreme accuracy . Of the seven base lines which have been measured, five were by means of steel chains and two with Colby's compensation bars (see GEODESY) . The triangulation was computed by least squares . The See also:

• TOTAL

total number of equations of See also:

• CONDITION (Lat. condicio, from condicere, to agree upon, arrange; not connected with conditio, from condere, conditum, to put together)

condition for the triangulation is 920; if therefore the whole had been reduced in one See also:

• MASS (O.E. maesse; Fr. messe; Ger. Messe; Ital. messa; from eccl. Lat. missa)
• MASS, IN

mass, as it should have been, the See also:

• SOLUTION (from Lat. solvere, to loosen, dissolve)

solution of an See also:

• EQUATION (from Lat. aequatio, aequare, to equalize)

equation of 920 unknownquantities would have occurred as a part of the work . To avoid this an approximation was resorted to; the triangulation was divided into twenty-one parts or figures; four of these, not adjacent, were first adjusted by the method explained, and the corrections thus determined in these figures carried into the equations of condition of the adjacent figures .

The See also:

• AVERAGE

average number of equations in a figure is 44; the largest equation is one of 77 unknown quantities . The vertical See also:

• LIMB

limb of Airy's zenith sector is read by four microscopes, and in the See also:

• COMPLETE

complete observation of a star there are 10 See also:

• MICROMETER (from Gr. wcp6c, small, jArpov, a measure)

micrometer readings and 12 level readings . The instrument is portable; and a complete determination of latitude, affected with the mean of the See also:

• DECLINATION (from Lat. declinare, to decline)

declination errors of two stars, is effected by two micrometer readings and four level readings . The observation consists in measuring with the telescope micrometer the difference of zenith distances of two stars which cross the meridian, one to the north and the other to the south of the observer at zenith distances which differ by not much more than ro' or 15', the See also:

• INTERVAL

interval of the times of transit being not less than one nor more than twenty minutes . The advantages are that, with simplicity in the construction of the instrument and facility in the manipulation, refraction is eliminated (or nearly so, as the stars are generally selected within 25° of the zenith), and there is no large divided circle . The telescope, which is counterpoised on one side of the vertical See also:

• AXIS (Lat. for " axle ")

axis, has a small circle for finding, and there is also a small See also:

• HORIZONTAL

horizontal circle . This instrument is universally used in See also:

• AMERICAN

American geodesy . The principal work containing the methods and results of these operations was published in 1858 with the See also:

• TITLE (0. Fr. title, mod. titre, from Lat. titulus)

title " See also:

• ORDNANCE (a syncopated form of " ordinance " or " ordonnance," so spelt in this sense since the 17th century)

Ordnance Trigonometrical Survey of Great Britain and Ireland . Account of the observations and calculations of the principal triangulation and of the figure, dimensions and mean specific gravity of the earth as derived therefrom . See also:

• DRAWN

Drawn up by Captain See also:

• ALEXANDER
• ALEXANDER (1461-1506)
• ALEXANDER (ALEXANDER OF BATTENBERG) (1857-1893)
• ALEXANDER (ALEXANDER OsxENOVici) (1876-1903)
• ALEXANDER, ARCHIBALD (1772—1851)
• ALEXANDER, FRANCIS (1800—1881)
• ALEXANDER, GEORGE (1858— )
• ALEXANDER, JOHN WHITE (1856- )
• ALEXANDER, JOSEPH ADDISON (18o9—186o)
• ALEXANDER, SIR JAMES EDWARD (1803—1885)
• ALEXANDER, WILLIAM (1824— )
• ALEXANDER, WILLIAM LINDSAY (1808—1884)

Alexander See also:

• ROSS
• ROSS, ALEXANDER (1699-1784)
• ROSS, GEORGE WILLIAM (1841- )
• ROSS, JOHN, or KOOESKOOWE (179o-1866)
• ROSS, ROBERT (1766-1814)
• ROSS, SIR HEW DALRYMPLE (1779-1868)
• ROSS, SIR JAMES CLARK (1800-1862)
• ROSS, SIR JOHN (1777-1856)

Ross See also:

• CLARKE, ADAM (1762?—1832)
• CLARKE, CHARLES COWDEN (1787-1877)
• CLARKE, EDWARD DANIEL (1769–1822)
• CLARKE, JAMES FREEMAN (1810–1888)
• CLARKE, JOHN SLEEPER (1833–1899)
• CLARKE, MARCUS ANDREW HISLOP (1846–1881)
• CLARKE, MARY ANNE (c.1776–1852)
• CLARKE, SAMUEL (1675–1729)
• CLARKE, SIR ANDREW (1824-1902)
• CLARKE, SIR EDWARD GEORGE (1841– )
• CLARKE, THOMAS SHIELDS (1866- )
• CLARKE, WILLIAM BRANWHITE (1798-1878)

Clarke, R.E., F.R.A.S., under the direction of Lieut.-Colonel H . James, R.E., F.R.S., M.R.I.A., &c." A supplement appeared in 1862: " See also:

• EXTENSION (Lat. ex, out ; tendere, to stretch)

Extension of the Triangulation of the Ordnance Survey into France and See also:

• BELGIUM
• BELGIUM (Fr. Belgique; Flem. Belgic)

Belgium, with the measurement of an arc of parallel in 52° N. from See also:

• VALENTIA, SIR FRANCIS ANNESLEY, VISCOUNT (1585-166o)

Valentia in Ireland to See also:

• MOUNT, WILLIAM SIDNEY (1807-1868)

Mount Kemmel in Belgium . Published by .

. . See also:

• COL
• COL (Fr. for " neck," Lat. collum)

Col . Sir Henry James." Extensive operations for See also:

• SURVEYING

surveying See also:

• INDIA
• INDIA, FRENCH

India and determining the figure of the earth were commenced in 1800 . Colonel W . Lambton started the great meridian arc at Punnae in latitude 8° 9', and, following generally the methods of the English survey, he carried his triangulation as far north as 20° 30' . The work was continued by Sir George (then Captain) See also:

• EVEREST, MOUNT
• EVEREST, SIR GEORGE (1790—1866)

Everest, who carried it to the latitude of 29° 30' . Two ,admirable volumes by Sir George Everest, published in 1830 and in 1847, give the details of this undertaking . The survey was afterwards prosecuted by Colonel T . T . See also:

• WALKER, FRANCIS AMASA (1840-1897)
• WALKER, FREDERICK (184o--1875)
• WALKER, GEORGE (c. 1618-169o)
• WALKER, HENRY OLIVER (1843— )
• WALKER, HORATIO (1858– )
• WALKER, JOHN (1732—1807)
• WALKER, OBADIAH (1616-1699)
• WALKER, ROBERT (d. c. 1658)
• WALKER, ROBERT JAMES (1801-1869)
• WALKER, SEARS COOK (1805—1853)
• WALKER, THOMAS (1784—1836)
• WALKER, WILLIAM (1824-1860)

Walker, R.E., who made valuable contributions to geodesy . The working out of the See also:

• INDIAN

Indian chains of triangle by the method of least squares presents See also:

• PECULIAR

peculiar difficulties, but, enormous in extent as the work was, it has been thoroughly carried out . The ten base lines on which the survey depends were measured with Colby's compensation bars .

The survey is detailed in eighteen volumes, published at See also:

• DEHRA

Dehra Dun, and entitled Account of the Operations of the Great Trigonometrical Survey of India . Of these the first nine were published under the direction of Colonel Walker; and the See also:

• REMAINDER, REVERSION

remainder by Colonels Strahan and St G . C . See also:

• GORE
• GORE, CATHERINE GRACE FRANCES (1799-1861)
• GORE, CHARLES (1853– )

Gore, See also:

• MAJOR
• MAJOR (Lat. for " greater ")
• MAJOR (or MAIR), JOHN (1470-1550)

Major S . G . Burrard and others . Vol. i., 187o, treats of the base lines; vol. ii., 1879, See also:

• HISTORY

history and general descriptions of the principal triangulation and of its reduction; vol. v., 1879, pendulum operations (Captains T . P . Basevi and W . T . Heaviside); vols. xi., 189o, and xviii., 1906, latitudes; vols. ix., 1883, x., 1887, xv., 1893, longitudes; vol. xvii., 1901, the Indo-See also:

• EUROPEAN

European See also:

• LONGITUDE (from Lat. longitudo, " length ")

longitude-arcs from See also:

• KARACHI, or KURRACHEE

Karachi to Greenwich . The other volumes contain the triangulations .

In 186o Friedrich Georg Wilhelm See also:

• STRUVE, FRIEDRICH GEORG WILHELM (1793-1864)

Struve published his Arc du meridien de 25° 20' entre le See also:

• DANUBE (Ger. Donau, Hungarian Duna, Rumanian Dunarea, Lat. Danubius or Danuvius, and in the lower part of its course Ister)

Danube et la Mer Glaciale mesure depuis 1816 jusqu'en 1855 . The latitudes of the thirteen astronomical stations of this arc were determined partly with vertical circles and partly by means of the transit instrument in the prime vertical . The triangulation, a great part of which, however, is a See also:

• SIMPLE

simple chain of triangles, is reduced by the method of least squares, and the probable errors of the resulting distances of See also:

• PARALLELS

parallels is given; the probable error of the whole arc in length is 6.2 toises . Ten base lines were measured . The sum of the lengths of the ten measured bases is 29,863 toises, so that the average length of a base line is 19,100 ft . The azimuths were observed at fourteen stations . In high latitudes the determination of the meridian is a See also:

• MATTER

matter of great difficulty; nevertheless the azimuths at all the northern stations were successfully determined,—the probable error of the result at Fuglenaes being t o" '53• Before proceeding with the See also:

• MODERN

modern developments of geodetic measurements and their application to the figure of the earth, we must discuss the " See also:

• MECHANICAL

mechanical theory," which is indispensable for a full understanding of the subject . Mechanical Theory . Newton, by applying his theory of See also:

• GRAVITATION (from Lat. gravis, heavy)

gravitation, combined with the so-called centrifugal force, to the earth, and assuming that an oblate See also:

• ELLIPSOID

ellipsoid of rotation is a form of See also:

• EQUILIBRIUM (from the Lat. aequus, equal, and libra, a balance)

equilibrium for a homogeneous fluid rotating with uniform angular velocity, obtained the ratio of the axes 229 : 230, and the See also:

• LAW
• LAW (0. Eng. lagu, M. Eng. lawe; from an old Teutonic root lag, " lie," what lies fixed or evenly; cf. Lat. lex, Fr. loi)
• LAW, JOHN (1671—1729)
• LAW, WILLIAM (1686-1761)

law of variation of gravity on the surface . A few years later Huygens published an investigation of the figure of the earth, supposing the attraction of every particle to be towards the centre of the earth, obtaining as a result that the proportion of the axes should be 578 : 579 . In 1740 See also:

• COLIN, ALEXANDRE (1526-1612)

Colin See also:

• MACLAURIN, COLIN (1698–1746)

Maclaurin, in his De causa physica fluxus et refluxus maxis, demonstrated that the oblate ellipsoid of revolution is a figure which satisfies the conditions of equilibrium in the See also:

• CASE
• CASE, JOHN (d. 1600)

case of a revolving homogeneous fluid mass, whose particles attract one another according to the law of the inverse square of the distance; he gave the equation connecting the See also:

• ELLIPTICITY

ellipticity with the proportion of the centrifugal force at the equator to gravity, and determined the attraction on a particle situated anywhere on the surface of such a See also:

• BODY

body . In 1743 Clairault published his Theorie de la figure de la terre, which contains a remarkable theorem (" ClairauIt's Theorem "), establishing a relation between the ellipticity of the earth and the variation of gravity from the equator to the poles .

Assuming that the earth is composed of concentric ellipsoidal strata having a See also:

• COMMON

common axis of rotation, each stratum homogeneous in itself, but the ellipticities and densities of the successive strata varying according to any law, and that the superficial stratum has the same form as if it were fluid, he proved that See also:

• GIG

gig g+e=2 , where g, g' are the amounts of gravity at the equator and at the pole respectively, e the ellipticity of the meridian (or "flattening "), and m the ratio of the centrifugal force at the equator to g . He also proved that the increase of gravity in proceeding from the equator to the poles is as the square of the sine of the latitude . This, taken with the former theorem, gives the means of deter-See also:

• MINING

mining the earth's ellipticity from observation of the relative force of gravity at any two places . P . S . See also:

• LAPLACE, PIERRE SIMON, MARQUIS DE (1749—1827)

Laplace, who devoted much attention to the subject, remarks on Clairault's work that " the importance of all his results and the elegance with which they are presented See also:

• PLACE (through Fr. from Lat. platea, street; Gr. IrAar6s, wide)

place this work amongst the most beautiful of mathematical productions " (See also:

• ISAAC (Hebrew for " he laughs," on explanatory references to the name, see ABRAHAM)

Isaac See also:

• TODHUNTER, ISAAC (1820-1884)

Todhunter's History of the Mathematical Theories of Attraction and the Figure of the Earth, vol. i. p . 229) . The problem of the figure of the earth treated as a question of See also:

• MECHANICS

mechanics or See also:

• HYDROSTATICS (Gr. iibwp, water, and the root Ora-, to cause to stand)

hydrostatics is one of great difficulty, and it would be quite impracticable but for the circumstance that the surface differs but little from a sphere . In order to See also:

• EXPRESS (through the French from the past participle of the Lat. exprimere, to press out, by transference used of representing objects in painting or sculpture, or of thoughts, &c. in words)

express the forces at any point of the body arising from the attraction of its particles, the form of the surface is required, but this form is the very one which it is the See also:

• OBJECT

object of the investigation to discover; hence the complexity of the subject, and even with all the present resources of mathematicians only a partial and imperfect solution can be obtained . We may here briefly indicate the line of reasoning by which some of the most important results may be obtained . If X, Y, Z be the components parallel to three rectangular axes of the forces acting on a particle of a fluid mass at the point x, y, z, then, p being the pressure there, and p the See also:

• DENSITY (Lat. densus, thick)

density, dp = p(Xdx+Ydy+Zdz) ; and for equilibrium the necessary conditions are, that p(Xdx+ Ydy+Zdz) be a complete See also:

• DIFFERENTIAL

differential, and at the See also:

• FREE

free surface Xdx+ Ydy+Zdz=o . This equation implies that the resultant of the forces is normal to the surface at every point, and in a homogeneous fluid it is obviously the differential equation of all surfaces of equal pressure .

If the fluid be heterogeneous then it is to be remarked that for forces of attraction according to the See also:

• ORDINARY (med. Lat. ordinarius, Fr. ordinaire)

ordinary law of gravitation, if X, Y, Z be the components of the attraction of a mass whose potential is V, then Xdx+Ydy+Zdz = dz dx+dy dy+ dz dz, which is a complete differential . And in the case of a fluid rotating with uniform velocity, in which the so-called centrifugal force enters as a force acting on each particle proportional to its distance from the axis of rotation, the corresponding part of Xdx+Ydy+Zdz is obviously a complete differential . Therefore for the forces with which we are now concerned Xdx+Ydy+Zdz =dU, where U is some See also:

• FUNCTION

function of x, y, z, and it is necessary for equilibrium that dp=pdU be a complete differential; that is, p must be a function of U or a function of p, and so also p a function of U . So that dU=o is the differential equation of surfaces of equal pressure and density . We may now show that a homogeneous fluid mass in the form of an oblate ellipsoid of revolution having a uniform velocity of rotation can be in equilibrium . It may be proved that the attraction of the ellipsoid x2+y2+z2(1+E2) =c2(i+E2) upon a particle P of its mass at x, y, z has for components X=—Ax, Y=—Ay, Z=—Cz, 2 ( ,3 ) A = 2irk p tan-le —E2 C =4wk2p ( 1 E2 E2— 1 Eg tan s) , and k2 the See also:

• CONSTANT, BENJAMIN (1845-1902)

constant of attraction . Besides the attraction of the mass of the ellipsoid, the centrifugal force at P has for components +xw2, +yw2, o; then the condition of fluid equilibrium is (A —See also:

• W2(P — R1)

w2)xdx+(A — w2)ydy+Czdz =o, which by integration gives (A — w2) (x2+y2) +Cz2 = constant . This is the equation of an ellipsoid of rotation, and therefore the equilibrium is possible . The equation coincides with that of the surface of the fluid mass if we make Aw2C/(1+E2), 2irk2p 3E2 E2tan 1E—E, In the case of the earth, which is nearly spherical, we obtain by expanding the expression for w2 in See also:

• POWERS, HIRAM (1805-1873)

powers of E2, rejecting the higher powers, and remarking that the ellipticity e =E2, w2/2 irk2p = 4E2/ 15 =8e/15 . Now if m be the ratio of the centrifugal force to the intensity of gravity at the equator, and a=c(1+e), then m= See also:

• AWE, LOCH