An Account of the Mode Proposed to be Followed in Determining the Relative Situation of the Royal...

An Account of the Mode Proposed to be Followed in Determining the Relative Situation ofthe Royal Observatories of Greenwich and Paris. By Major-General William Roy, F. R. S. andA. S.Author(s): William RoySource: Philosophical Transactions of the Royal Society of London, Vol. 77 (1787), pp. 188-469Published by: The Royal SocietyStable URL: http://www.jstor.org/stable/106720 .

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XX1X Sg Aao8nZ of tEg Moda protefed to bef5Flowed in /drX

Greenusich ard Paris 13y iWSorSGetrural WilKala- RoyS 075: Ft. T and A So

Read Fehruary z X 7 g7

oTNwQ- yetarS have atar]:y elapfed flnce an acco-un-t of the- 9 meafuretneIat of a lDaCe oi1 LIounflow-EIeath was laid

before the Royal Society being the firit pat of an operatioa ordered lDy his MaJeRy to be executed fbr- the ilntmediate ptlrF poSe of aCcertainirlg the relative fituatsons of the Royal OSfer- vatories of Greellwi-ch and Paris r but whofe cvhief a-lld ulti-* Inate oVjed has always been conEldered o-f a Ilill more itnpor tatlt nature, tlatmelys the l-aying the fouIldation of a general irvey of the }3ritiffi I[latads.

Wrhen tlle operatiotl commenced in x784, it was taot d-oubted9 that ill 786 at latell:, we 31hould have beerl alile to have pro ¢eeded with the feries of tria+<.gles frolll EIounflow-Heath to the neighbouthood of Dover9 but the corltrivance and cotsIl:rll&ionf of an illfirumentg new <3f its kind propoSed to be m-ade uSe of:,- and more particularly the rlicety of its dierifion9 whereby it is; hoped the angles- may be- detertnined to a degree of preciiRoa hit;herto utlexampl¢4 laave re9^<uired1 much lu0Xe time than Mr

z }taxss>

[ I88 X .



- Gen Aots Abco?>n!, &C0 Si

RAM3DEN hilufblf at firR izlagllled; With-out lmearlitlg to diEcl)poillt this itzgeniorls artiR was perlzaps-- iM the outWe-t too reLmifs a}d dilatryG, atlJ SCCifvlEtltS hCt<>'tllg happex3ed svhet-lrtlle

>torkrn^-LtnlfEip sas a]l*t>asty f-al advatlccads hichS 1at cotld llot forefce or prvve.<t, t11e execu;tion has thereby beeta greatly reZ tal djtlt I-Xo-<vev t X v X tce the iARrutmetat may at pr-efbnt be coRw iSdcted >s twarly fi;tsill<Cd (fuch parts afs yer letmait- to be per feE1cd Izeilwg, Ol-Sly of tlze finalLer lilld) rve nway fawrly collcludeg tht.t eal ly itz the et<fuitlg Iulallmet ) or as We<An - as tlle wveatller i tl-is C0u11tly \67i] perlazSit, the tligrononaetrlcal operatlon miay be bOgU11. IL] til;s fiate of thingsS 1 have therefore judged that it miglat be proper to lay before the Society a nlcort-Ik@tch of the lllode n71aich is propoled to be followed itl ful- N 1b*P; fillillg;his M.jeRy's corninatlcls, accompatliecl by-a \>X:%; very {light general lmap of the countly, only col- 9t

leAed from the comtnon furveysg but llill fuffi- / : cient to {hew llearly the difpofition of the trianglesX <

that will be made uSe of in forlming the jurEtions \ t3etsveen the lneridarls of the tsvo ObSertatortes ; <

In this buEi$wefs it will be underfiood7 that I rz-ean; J / t-o adhere to ia:ch prirlcit?les as have l\ten un.ivere \ fally received and admitted as- juIt \

In every feries of-triangles ̂7vhere each angle i:s^ / to be adually obServed with the fame itl-rumentX; g

lthey IhouldS as near as the circumllatlces will \ permit, be equilateral ee' for wers it po{Elble-toN >

thoofe the :Llatioils itl fuch a mantlerw as that-each / atagle {henld be exadtly 6s degrees tghe half num 4. ber of triangles- in the feries multiplied by th-ez length of one fldev would9 as in the aanexed 4< fiE;ure givc at oncc the total- diRance; mt -only the fldes of

th.e



T g O O R o v M Hr c a w t < W

tIe oIka1e -o-r ladder woeuld lte perfedly parallel, but tlse diagolal Repss nzarking the pro$reIi from one extremity to the otherS would lDe alternat@ly go throu;llout the vvhole letlg;th. The fitA lide is ippoefed to be fbund by the mPafisrelmellt of a bait HS of alJout half lts lettlgth; and the laI} flde to be verifiedl by ich another bati R at the oppoSite extrernlty

It1 a!ny particlvllar caIZS where otllyttwo agles of a tl^ila£lgd

tCAtl 5t 22uAIty obEerved7 ttheSe ffiould be as near as poSSle eact-l 45 degreets tt ny rat their I;lln thould llOt diSer tnucEa Son poO i f8r the leSs the cotnput:ed ang;le difirs from °°s the lets cElance tllere will be of a:ay coLafitlerable error ill the itlter fiEtion.

tJ/gAZe7-l9$ SOtn its levelne as well as otE1er advatl tagbous circurnilataces attendirag its fatuations sxtllicll tlde bare illipeEtiotl of-the map svill rellder olvious Ibemit3g to me to afford tlle lDeR lbaS of verification for the 1aIt- triat}gle I have g;i6te£< the IErles tlwe fhorteIt direEtion fi*om t30utlf10sv1E3[eath to tha¢t palt of Kerlt. The ht tla11d Ila.tiorls occupy in gelaeral the heiglats svhicll extend acrofh the Wealds rlaoSe fon tlze leEt are placed on the tgreat range of chalk hIlls wI1ich el1d ot1 our fide of the Cha1anel} between FolkRone aawd Wal mer"Caftle alad re commetlce orl the oppofte fide betnvee Cape }31antrCz and Calais.

1t will be perceiveds that I do i30t meatl to make St. Pau1*s a Ratior} ill tlle Ii ite , becauSe in that cafe Harrow and Hamp- Rea*d muR likesviS have been made ui of att three extrernely inconvenient for the reception of tlae $reat inErumentX Beiidesfl, Greellwich Ob{brvatery iDei]g hiddetl frotn the country to the +iuth-weR by zthe Norsvood heights alld Son that to the wfoutllweaR by Shooterss"Hill7 after Vil3g Inade the+detour of Harrow ansl Hax}pfiead alld come a.C>S the finolie of the

Capital:,



propoJH tizitegonometrzecai C:peratXw X g .

^Opical, we ilwould Llill have been obliged to make ui} of the +Wo fl:ations of Norw7ood and Sllooters-hilIS without procuring fo goM an illterSeEtiotl of I3ortlewtwill> called ill the cotnmoa

maps (I believe errotleouIly)- Borleyrhills as is obriained by meatls of tize Ratio1a at the I-:tu>t*slredracreolaouSee But aittaoug1w

toIle of tIIn Itatiorls of the irics adtutaIly fullt wi-tlz1n Lotl0otziS Ilet

vertIlieleIss from thofe in its vcinityiS tat. the Pagoda NorsvoocT5 CIeeIXsvictl OlfervatoryXi atad Shoiotersshilly we Ihall equully have it itlt our posrer- to determine accurately the tituatiorls of Hariosvs t-larnptleadS at<d St. Paulssy as well as maIly other thieS fteeples rvitiliil vlhe limits of tt1G SpitA

=Anot]]etv pritIciple l have elldeavoured to adhere to9 iI] tl]

difpoiltxoll of the triatlgles is this, tlwat after havitlg obtaitaed fJes iw 1CI)gTh ffom : 2 tO lS IBileSn tcontillUe ielM at that 1CILgTh as tnucll rlS the circumRatlces svill perinit n irn if tlley CAtblt tO be reduced cotlflderably below tlwat estelzt7 the obvious ad6tantages of a lollg ba at tlle outit would 1ze loNe! by theN f;bIWquent contraEioll tsvards tlae clofe of the operation

The tvsver of Tet+terden church IJtillg a veC colafpicuous oEje&> tuay lxe feerl every svhere fi*om the fummit of the chalk llillss as br srre as the rlver lMedway It likewiIS n<ay be ietw hrn the caItern extretmity of the Wecond baSe wllereby tlla laR tlXiangleS Sesterde IJig HVingto7z Itol49 isQ propofed > ba verified. This 1QI1o11 is itfelE a very rernarkalule oVje&, vmore accelEble and t11 other rerpOs more proper too.ir the purpofe of a RasonRthan lE<7mlIe church Reepley wh-ich I had at olle tine thoughts of occupyitlg The high ground wlaich lbparates RomIaey-marlh frotn the Wealdsx of Werlt pafles tmmediately behilwd Rvaktngeg that is to fay} to tlie northe weRward of itS and may therefbre probably prevetlt tlat t:q?> o£ the chalk hills Dom being Seen fi:om the well end of tlle baId

X, o.f



X g (z CX Roo>s Ssnt f a

of verfiCatiotl ) IDut lf tt/ff1-Brrt@ QR 3tly od<er polXweaoa +t!ie rat1ge wP.ar itt can !)e -:Eet1 finorn Ruckitage tl<en the Ra tiotl otw tI<e ktaoll as svell as that at Lynltle will become equally unneceflaryS and the triangle af veriScation will bev mXne WSerd2X7 LX TWtterDse

It vill be Eeercetveds that I propoSe to have a Ilatio;El oa

Dirl@bt HeadS a land of confiderable hteightS Som xvlaence tiere is a go0 Vt;t* 05 gile couR of France near hulogne. Frorn thtss pei*at at3d lEntterlees7 svit13 the help of the Indiars Jights9 1 have to do-ubt of -obtaitling a fine lnterSeS^ion of the EgIaal of the ]?avlembe a hill o:f fbme note behind the town tf Boulogne alld one of the fiatiQNS made u@i of w the :French Acadetniciaxls itl the execution of their triatlgles - The AdV2IltaS of obtainiilg a triangle of this magnitude whoS ides aw rerpedtively iti lttlgth about 45} 36 and >5 milesi are

* * Vtoo obvrlous to l-equtre any comlneIat.

trh-e hlgll ctlalk cliS near Folkllone prelrent Dover (:JS t;}oa} being Setl finm Lid} or atly where ln the plain of iRomney-marSh. Hence it will become 1leceffiry to ffirm two frnall triangles to the northward of Tatterleesg inorder to Robtain aLl interSe&;on of ;one of the turrets of the kew of that ;caRle. Of the celweer-of the keep) it will be perceis7ed7 by the firong dotted lilless tllat the Fren-ch Acadetnicians halre prow sured, from tlleir Ratiorls at Calsa Blv8cnez and Ading he7X at1 acute interSeS:ion ( b (;rf ligr de I)outresJ 3zlaking in tbe whole an a£lgle of 3800*I6/ 20//t

tthe pOitltS which are obviouIly the beR br conneAiIlg our trlalagles with thotb of our ncighbours are the Boulemberg; BIaxucnez atld Calais provided we could by ary means obtaia

* Tatterlees Barn7 ln Psexs Map of ga: EeitX X t1le :imxniet of the +hAlk t;lls} 7 sz 7 itt above 1 hP ia

5



prapayeed tr,azoXghozc Opera/ot xr>?3

as ,ood ars interfedion of the la{l as we are certain of getting o£ the tsvo firIl X lJut the breadth cf glle ratage of chalk hills being little different on our f1de fror< wlaat i9t ls orl theirs,, ljy confilling oulfelves to fuch a l)afe as they svill ?;fBrd us, w-e cannot alzy vay obtaill an interfeded alzt:,le at C^alzi;s great>r than about nS>0 or 3o.

Thilaking that poffibly fronu St CENeter's church tl the iile of

Thanet the tower of Notre Datne at Calais tnay be feell, I have extended dotted triangles itlto that part of Ketlt; becauSe, if the unlte(l heigUlts thoulcl I10t be fufficieylt to raife the t;op of

the tower above tlle cllrvature of the fea, wh-ich is the orlly thing, to be doubted, we are always certains that the {ignal of ]31ancnez may, by mealls of the Irsdian lights, le eafily feeniy fce the whole range of chalk hills bellitld (:alais are diI; covered with the naked eye hom the ifle of Thanetj when thc weather is tolerably clear.

tIAYing ill this manner aCcertailled the relative ftuatiolls with regard to the coaR of England9 o-f three pOilltS on the coaft of Fr*.ltlce, formirlg a triangle whofe fid-es a11d allgles twe already -lullow frot*n their trigolaotazetrxcal operatiotlSg we Shall itl l}e S*alllltr lle et.alDled to determltze tIwe I;tuation of

the poirat M tleal t)unkirl> vXanere the meridiall of the Royal Obiervatory cf i}$t'i$ ii.ltCr5C&s a lit<e draxrn from tlle great tower of lL)utlkitwk to tXl-LAt C)5 XO D3- at (5jalais (See Tab. IX.) The diRatlce lKi? o< elle rxlerienian of Paris, will then be l-ad ,, aI:ld that lJeill2; added tO ID34I7 fathomsX tlle diCance of Al tlorthrard fron the 3Reya1 ObSerrratory, we ihall have the totat terreRrial arcX coInpreherlded bSetween the parallels of tile; two ObServatoriesX aili^WCring to a11 arc in the heavells of 2°3&' 2;6", or a dtiFferenGe of la.titude betweell s° z$3tc+OXJ alld 480 50, t4/

Vo LYsSVII9 C c I



X 94 Gew. RoY s -Sccognt of a

In like mataIler the diLtance frorn (Steel<Nvich to P7 on tI<e

parallelofGlxeel-lwich, willthelabereadilycornputed, allfwer--

iIlg to ti-we difEeretlce of lo}agitude bietvreen the tvvo ObSe-rva tories; which as far as can be jludged S>om the map of Kenty correAed for the error iil the direScioll of its meridiatl, amounts to about 2t° zo' zo/', fuppoElug aIways that no uncertaitity renzains xvith regard to the pofitiorl of the point M. But here fonwe remarks become neceffiary, wlich may prolJably fuggeI} t5z rlze AcadevBy of ScieIlces that a further inveRigatioll of this matter tnayf be needful on their part.

By referrillg to tlze s7th page of the fir{} part of M Cts SINI's Booli fLa Merxdianne veriJ2ee)^, it will be feetl, that Dun- lwirk, by one feries o:f triangles, is eaIlward from tlle meridian of Paris I426.53s and by another I4t4t29 toifes, whereof the meatl is I 4zo.4 Xequal to l 5CI 4 fathoms This difftrence of 63 fathoms, or little more tllaIl half a fecotld of longitude betweeo t4he meaD a-nd extreme places o:f M, is- certainWly rery illconfl

slerable. Bu-t ill the 60th page svhereyr ill verifying the meri- 6lian of Paris, by the compatifoll of the aIag;le that- Broulezele makes with the meridian of Dunkilk, and the angle of con- vergence of ol<e meridiars to the other, a diflierence of 21 fie-

conds betweelb I;0° I6 I3/ and 10° t6' 34t', is alledged to be almoRs infenfible we do IlOt think toZ be aU concltlflon fo unex? septioIlable. This, however, is not the only cauSe o£ utlcerw tainty w.ith regard to the juR poEltion of the poirlt M: one of more importance arifes, from the difference that is found by -two fets af triatlgles in the angle of interSeEtion of the meri diaxl. of ParisX with a 1111e drawrl through M from the tower of Dunkirk to thatS of Calaisv

Thus, by pe 53. and 560 of tIat firR part Of ME CASSI-N1'S- bookf Dutlksl^k 1Jc1t1g tht fhtionX Broulezele makes all angle

with



propoJ*ed trigonometrieal Oprtanon. t95 nvith the meridian of I0 I8 25Xi tozvards the fouthXweR: alld theallgle betweenBroulezele alld Hondfcotebeirlg78° I Iz 4z'ts tlleir difference 6,,3 53' I7X' iS the allgle that HondScote is folzth-eaR frotn the tneridinta ; s^71lerefore the colnplenzent of thi$ laft atagle to I 800, viz. I I 3° 6 43" is the atlgle that t-IondScote rnakes svitll the mericlian of Dunkirk produced llorthwXard. - 13y p. 166. of the fecond part, Vul-lkirk beitag the tiOll, ttle

angle ltetssreen HondScol:e and Mont-Cael is {llexvn to l)e SI° 7' I 5// ; tt-<at betsveen MontsCaXel and Dattell 4zo 6' 35"; and by pb I 67. that between Watten alld Calais is 5 I° 40' 20'f

The fum of thefe tllree angles is -X44°-54' IO':'s from whicl deduding 67° 53' I 7" the atlgle that Hondfcote 1s foutll- eaRsvarel from the meridian, there remain 77°o! s3"for tlle angle of Calais fouth-weAward from it, atld the cotnpleme1<t of this angle to I 800, Vi2;o I02° 59' 7'X, 13ecomes the angle that the meridian of Dunkirk produced lworthward makes with a line drawll tllroug;h M to Calais: to which la addiIlg the angle of collvergetlce of olle meridiatl to the other lX 5OX'-, correfponditzg tothediRanceofIsI4fiathoms equalto IX 29//3 of aglXeatcir cle,-Xve {hall have I o3° o' 5 7"2 for the angle svhlch tlle naeridiaLl of Paris produced northward from M nuates with the lille joining Dulll<-irk alad Calaia

Again, by p. {i3e of the tElird part of ASt CASSINI'S book, T3unkirk beitlg tJ.ae IlatiollS the angle that Gravelilza makes with the xneridian Sc)ut}-wSsreRrvard is 7zO I1 48"; atld by p. I2-.

of the faid third paft7 the angle betweell Wattell alld N. D.

Calais is 5 39 50'f * alSo that between (;ravelilles and Calais

xx 56° 4z' o' Now the diffierence between tluefe two l-all 4° 47 50' being add-ed to 7-z° I IX t8", we ffiall have 76° 59' 38"a- and its coSupletnent Io3° 0 23//, for the angles that the m@ridian of Dunkirk makes with the line drawn from thence through t1ac pOxIat M to Calaisv to svhich- laA allgle adding

C: c j the.



X (3 6 SnO FtoYt5 Fa¢; OSS} 0 g

the f;rtner c0tw57ergence 1' $o"IS we llave to3 2' z' for tlle atzgle that t1<e lnerldian of Paris pl*odu-ced noldthward from M makes witll tlle faid liLle; but by the fortner ret Of aIlgles

it was -fourld to be only io3° o' 57z'X9 whereire tlle d;ierence is I I S '.

FROLn M. CASSINI S 1:ook it: appears, that Dunkirk is north from Paris I 2 55 I S*35 tOi{@S, \vX11-cIL IM2kC E 33768 i-thoms o and the point A¢ belng fouth hflom the tower of Dutlkirk 35 X ithorns:, there rernains for thn cliRance of 1M nortilward ftXorn the Royal £)bServatory I334Iy ithoms-* Nots, with tllis difiance as radiusX tlae value of an atgle of It I$tZ iS 48t fathOtusg equal to +" 34"' of. IXo-tagitui Thus the polilt MX inSead of being veItsvard from Dul<-kirk I 5 I 4 ithotnss 0st-ill) by the laR fet of angieSS 0111Y be relmested frorn it I465-Z Xthotzls: wlacrefore the dllEerence betsvee-l tle meat) alld extrerne places of X, ir^^

. . ,

this way of conlidering itx w8ill atno-tlnt tO 244 Sthotus aboutr four tilnos as much as that refultitag fronz tlae com?arifon fiatedL

; tm the sJzth pa. fn the pa3IIel of Greellwicll the extrerne di£Urence will arnorlnt to 58.4 fathotnsS or about 5 feconds o£- longitude :not lntch more than otleFthii part of a fecond o£ ttme

In this irt of urlcertaulty? with rega-rd to the precife pointo of itltsrfeAio-ll of the mer=dian of tlle Royal ObSer-vatory of Paris with the line joinitlgj Dtlilkirk and Calais, the only thing thats canF be d>e otl oiir part, is to contider the amean pofition of WI as JuR, that is to IZy, to fuppoiE it to be I 5 I 4e fathoms weRward fiom the great tower of Dunkirlc; and havitlg conIleded tt with the BritiSh triatlglesX to {hev thenL. what angle sts merldian will make svAh the lirwe drawn Somv Du;wkirk to Calais

C5tpa



prepaJ2d trigorometrical Qperatio* t97

Cotarsi?s of the celeJ?ial- owrc of ]ie meridizatz, co?oup>^ehended between the parallels of i CIreetwsvich anGS Perpignan, wit tbe correJ9ondingportzons, meaMigredandvoantutfdS fJf zbeterre/^;al arc of heJ9id yner;tdxan, bu*7ween M asd Perpigtlall

11l tIae conflderation of this matter it is to be obServed, that M CASSINI has din7ided tlle celeIlial arc bettweerl the parallel

of DUI1kirk or svhich is the fanae thiIlg, betsveell the parallel

of M and Pel-pignata itato four principal feAiotas; t^z that

from 1AvI to Paris, ftaom laris to Bourges from Bourges to

Rodes7 arld frotzz- Redes to Perpxgnan; aiXlglfitlg to each feAionX

the meafured pOftiOll of the correEpondit1g terrefirial arc refult-

ilog froin tahe ttiatgles of tile meridianX

By pages I IO? I I I7 I I2, of the frR part of the book it appears £iom the mean obir:ved zerlith cliRances ofi bur ftars, that the arc of the heavens betmreen the parallels- of Durlkirk and Perpigtl-arl, c, a , {,

contains - - 8 20 2 26->

And fr-om the mean of the obfervations of- a like ullmber of Aars9

the parallel of Rodes is diItant from that of Dl1nkirk, 6 5 ) 5I 14

NVhereSte Rades is north from Perpignan - 1 39 I I I 2

Bourgess by the mean zenith diItances of ttsro RarsX IS fouth om

CE)unkirk, - , ,,, _ 3 56 59 55

Wherefore Bourt,cs is nerth from Perpi-gnan _ 4 23 2 3I

Andbyanothermean, inp Il2*itis - - 4 23 2 35s

1?he mean of which two meatls gives fQr the diRance of their

parallels. - 4 23 2 33;

arh t s :

rerpgnan



Ger. Ro:Y's AG;OW7aZ 0S a

Petrpigtnarl ts buth from the tRoyat ObIErtat0ry at Paris, by tlle

mean zenith d¢Itances vf follr fEats, 6 8 3:l 52 Her2Ce Paris is fowlth from E)unkirk - 2 aI 50 34.

Rodes ss inth from P.al;isX by the mean zenith diRances of two

fwtars _ + 28 59 45 Hence Perpignan is foutla fr(3ni Rodesg I 39 ta 7 And by a brmer refillt it was- found to be, _ I 39 I I I 2

Hence.the mean of the tzre means gis-ess fior the dif:atlee of the

parallels of Rodes and Perpigrxan,, X 39 II 28

The tower of the ,reat cherch at Dunkirk i5 northward from arhat nvas the Sation d the feEtor 844 toifes equal to 89!S Sthoms,

svhich correfpond to anarc in the heavetls of o o P5 t9 Wherefore the touter is noIth from Paris, 2 II 55 53

13llt the point M i5 fouYh from the totrer 35s fathoms? which

¢anfwer to an art in the heavens of _ _ o o 2Q 45

Hence the point M: is northward from theRoyal Obfiervatel7 at Paris - - - _ _ 2 I I 35 8

NowS from theSe d4ZaX together with the latitude of the Royal Ol)fervatory at Greenwich 51° 28f 40/' and that of Paris 48 SQo I 4f' wre ffial-l Ilave tIle latitudes of the feveral fations lCtween Greenwich and Perpiguan, writh their dif- fereIlces, or the celeRial arcs £omprehended l)ctween them, as undertzearllF

Stations Latitudes Dif. or celeRial arc$. o o sa ,8 o 8 ,, 888

Greerlwich Royal ObServatorys 5I 28 4o o 26 50 52 iPolnt M near DunkitkX 5x X 49 8 Paris Royal ObSe5rat°rYay 48 50 14 2 I I 35 8 BourgesX 47 s + +1 I 45 9 Ig Rodes 44 2 I I 3 36 2 43 5 I S

iperpignan (S. yagmes) w 42 42 2 8 X 39 IX 28

This lFatitude of Perpignan 4z° 4-z° ̂ ' 8' is what refult$ fiom the immediato comparifon of the logths of the celeIRial

arcsX,



propo,fgog trigonometrical Operatioss. 1 99

frcs as detertnllled by the zenith diRances - of arsa taken with a feAor of fix feet radiuss all(l where the obServations are fo nearly conflteLltrfLolag tllenlfielves as to leave little dotlbt^

of their accuracy; but iIl the zgoth page of A1. CASSINIsS book, fo ofte1l quoted, as wel1 as itl the t70th page of. his DsJcrzpgio8KGiograp/uiqtxe de la France, pululifiieddirl t783, the latitude ofV Perpig;nan is givell 42¢° 4l' 55",, wlwicll is 7" 8'2' lefs that1 ti1at deduced frotn the obSeXvations, Btitl1out ay rea fo11 tstiat I catl pcrceive bcitlg a{Tigned for tlae reelilAione

Perpignan, the fouthernmoR fiatiot1 of rlle tneridia1l l1ne extendillg, fipln DullItirk*> throllgl8 tlle ullole ki'Iagdotn of;

France is--Eltuated- at no great diLEance from ttie l)ottom of the Pyrelleall; mountaiils, svhere that lofty rallge ellds at the IVIedi- arranean Sea. M;* DE LA CAtILLE \VaS of opillion, tlaat the lplumnlet of the; feAor tnuR kave beeIl affieEl:ed by the attraAio which it-would fufir fiOIB tha£ cauSe; a fuppoEltiotl which, evertheleSs, has been doubeed, fillce tlit obfervatiotls rnade iIl

tShis country on the attraetiotl Qf Schehallio8: fior by thefe its appeared'> that tlle efReEt, aIthough ferlfibleS was but fmallX even when the fedor was placed« as near as pOitblt to the oppo-+ ffte fides of tile mountain. It is itldeed true, that tlae Canazgog,

the IiigDheI} of the Pyrenean rangeu being fituated obliquely l:o tlle rneridian, atld at a co1RElderable diItallce from Perpignan, would not probably occailon much deviation ill the pIummet 9

yet, on the other hands svhen we compare tiZe very t:riffitJg guantity of matter in Schehallion with the itllmenflty of the maSs in the PyrelleesJ in the direEion of the meridian, I can ;l-wot help besng of M. DE LA CAILLE S OpilliOIl, that the plutnZ met of the fedor would be fenElbIy afleEted, that is to fay,, its wollld be drawn to the fouthwardt oute of its perpeIldiculare direAion arld woul thereby give the zenith diAallce of the

ole:,



tOt fOX toY+s A;NUB^f a

pole) or ty other torthertl Rars too littlea atld col+fiequelltty t

]<ltitude too great Until triangles ffiall laave beell cxteIlded l3eyoIld tlle Pyreneesf and the reaOr placed otl t1Xt uth fide of the rangeS the qUatltity of this attraEtiotl xk}y tts clouble or counter-eS&) CAllNOt potElblybeakertainAdX I \it19 h0WY0very

eluly fuppofe it to have l)een lOZi S/ to be dedv-XEted fioom} the latitude of Perpignatl +Xvhich svill thell become 43 4XZ 52//, >Ozly tilree Weconds leIX thatI that a{i:igned to it ir} M CASSINI'S two books lJefbre tnelltioned-. zThus the arc betweetl Rodes XdIld PerpigllatR Wiit be t° 39J jliX 36ofZ, atld the total celeRial art betsveea (:reensvicll atld Perpigilan will be SO 46/ 48fJs SS

trlay be feeti lDy attellclillg to the ftur colurnts tosrards the leitX hatlcl of tlle annexed table of compar1t<.

Witil tegard to the corref2ollditzg terreltrial arc, ullder svhich liead are arratiged the elevetl colulnns towards the right hand of the tables it is tc lfe obServted that various meatwure- ments llave at diffirent times beeaw made itl diSerellt latitudes ot +t-he lelagths of the degrees of tllt tneridianS br the purpofe of oltaining withltz certain litnits ate leaRi the true :figure and dinaent;olls of the earthe The OR eIletltial operatiolls of this

i

fort as havstzg loeen executed svitll luoR care, with the beIt itlfilXumetlts-Sud at the greateIt diIlatlces from eacli otherf have all beell done NVittlitl theSe laR forty or fiRy years + tiatnelyf in Peru ullderEthe equator illA tniddle latitedes itl Fratace and Italy-7 and-in Lapland 1lear the polale circIe The attra>Sion of nloutltaills,, and ura+7oidalble etrors in the tecutiota will ever prextent juil concltlfioIls from belng drawn frotn the compariI8n of tneafuretnelltsXmade too 1lear each other Thelie laR will aiways be fbund to diEr more or leIi among themfelves Solwetitlles earell the refults may become a1DISurd or cvntradiEtory. ^ttl cafes of tl1is fortt a mean of veraL ffiouId nQ doubt be

S taken



propnJ2d rganomerical Oper4tion. 20!

taken fol a mean latitude, flallefs there Ihould be fuflicierlt groullds for rejeAing any :frotu the number, as diffierillg too lullch fron the o-tlael-X lSetlce it iss that philo£ophers are rlot yet agreed itz cDpinion with regard to the figut^e of the eatth >

forxwe colltelld1XlgS t^llat lt has no regular figUlreS that js to fclyX

tzot fucl-ll as srotlld LJe generated l)y the revolution of a cttrve

around its axis. Otllers have fuppofed it to be an ellipSoid s

regularS if both polar lides Illould have tize fam-e (leg;ee of flatl-efi 9 but irregulalos if 011e {llould be flatter than the OTIJeVX And, lalilyX fome fuppoNe it to be a fpheroid differitlg frotn tEe cllipSoidX bllt Iuch as would be formedX llevertllelefts by the resrolutioli cf a curve around its axis; althotlgh in tllis cafe tOO OtlC poldr fide lmay ltot be frnilal toS but mol^e t>r lefs obXate than tize orller

Itw order, $herefore, to put this matter irl its true SiglltS atld to enable every otle to 3tldges by f1mple itlEpeAion otlly rllich of tile tlleories agrees beIt 0vith aSual meafuretawczlltX I have coinputed oll tetl diSere:zt hypotheLfes, and arranged iLl their order, the letlbths of the arc between Greenwich and Perpig- nan, as alSo fome other chief proper/ties of cach figure7 wthich laIt fiil up the fpace towardc the bottotn of tile table. This £node of colleSciilg the refults feened to Lne to 1se thv moIt diRind:t tlaut could be follosved, to avoid that perplexity wlwicll tnuR for ever occur in reScrring back t0? and comparilogs alany umlJers together, computed 011 (liSerbllt fyRerns wllen thc

ullole ale not placed beforw the eye at once. The firIt of the elevetl columnsX or that svhich comes nest

to the celeIlial arc7 co-ltaitls the mea:fured portions of the cor refponding terreRrial arc, as far as they have yet beetl ese- cuted lnhe lulallks at top ca.rinot be fupplied7 until vze thall

VOL. LLXVIIo D d have



20X G6n ROY§S ACC.OUNZ-Qf d

have determine;d the lellgth MPS iu the tuap, being the Epace corxaprehellded between the parallels of M atld CIreenwicht

I1< the fecond colurnil are arranged the computed dlmenfiolls appel^taiaing to thwarth as a IihereS fuppofing its ferni-diameter to lJe a nean betwectl the longell arud IhorteR of M. ROUGUERSS! fecolld fplleroid It is from the tnagtlitude of this fpllere tllat I compute the deglees of a great circle for tlle fldes of fphericalf triallgles ]3y adverting to the errors or diSerences betweell the tneafurement and computations in their refped;ive placesX it will olvYiouny appear that the earth diSrs very conflderably fsorn a lphere: for although the arc M Perpigrlan of 803 ouly ssceeds the tluth by 60p fal:holus n yet an arc- of equal lengtlr Gt tlae equatorS Sviz S*3;> X 374.6 0vould give atl exceSs of 3I2O fithoms; and at the polar circle S°33 x 335.2 would give a defiA of z7gz fatlloms.

After the fpllere follow fevell ellip:Cotds of differetlt degrees of oblatetlefs, fron the fir 7hofe fetni-diameters have to cach otEler the ratio of 7g.o47 to I78.047 to tlle feventh9i where lt Is only that of sAo to 539V On the prlnciples whichs have ferved as the fbundatiotl of the firR anal fecolld it will be necetEary to make fome remarks; but as to the othLlers, a fiv words will fuffice for each

With regard to the firfl: elllpiid> fuppoting the eartll to be holllogeneousX it ls well known, that thei ratio of its femi diameters may be found by compairing mrlth: each other the; leIagths of the pendulurns tllat s7ibrate :Geconds it-l difEereilt latit tudes, which lengths are deduced fiSore the fecoIlds of accelet

fatiOll, that the pendulutn fo adjuRed aad ullalterably fixed as to lengthe at the equatorS would perforrn in 24 llou.rs, oAi

beiIlg f-vacceEvely trarlEported to-diflReretlt latitudes,^ as ir as the pole9 where the force of gravity being the greate the acceleraton wauld likewiiIb 1oc the g;reate. The nalcu

.ladoIl$>



propd zrigoto>tencvZ0perv>o z-O3 atlonS for this purpo£E were firft made foon aRer $- Lord MUL

GRAVE1S return from his AToyage towards the North*Pole 1I} I 773

§ The moR northerrl experItuents hitherto made with the petlduJum} are thofe atSpitzbergeIJs itllatitude 794 50t9 whelcof an accollnt has beengiven Xrt

the Yoyage towards the NorLh Pole in 773# The machine made uk of oil that occaSion beIvnged origlnally to the celebrated watchXmaker, M1+. GRAHANy and was lent for the ptlrpoSe by its preint prt?prietor7 Mr. C UMN[I NG} who has Snee t3bligltlgly permitted it to remaitl br feverd years at two dlfferent tiSts9 in my poeffiont where it -now is Soon aRer the return from Spitzbergen} I atertained its rate of gO1t1g I)Y my olfervatolDr clocka for a great urhile togethe}r7 in very different temperatults; alld thereby bund that the vatiatioll faom heat and cold} nameiy, IRSthS OF a S£tOnd for each degree of FARRENI-IBIr7 was conEw derabW more tharl had been allowed br it} itl determitling the acceleration hDm Londor} to Spitzbergen. Abollt this tirne llkewifet Dr HORSLEY diScovered that an error had been commttted by the aRronornet^ (the late.Mr. ISRAEL LYONS) employed by the Board of Longitude on that Voyage} in-calculating the bherical triangle for the corre&-ion of the time lzetween the I6th and I7tS (?f JuIy,, oa

account of ttie obliqllity of tIle tranfitXinRrtlment. From Dr. I-I-oxstiEssprinted letter orl this Elbjed in I774} it appean, that the diSewnce frole that cani tmotlnted to 37 Xc}nds ia tilat} as giercn by the obServattotl With the infirument and by the watch. And {illte there was reafbn to ibe&> that the telekope had by accident been moyed (one could not te11 how IBUCh)} Xt WS ;ndged faiR to

adhere to the acceleration 7 tto8 as g1Xa by the W>ath& Tho corredlon br ztize greater contl*a&1on of the Reel rod3, 1 bund to be tt*58 to be ftbtra&ed} whereX fore tE1e acceleration fl*olul tondoI1 to Sptzbergen becalne 68!t*) and tlat. om the cquator to Londen being 156/tt ti1e total aecelerationkom the ttlator-ea Spstzbergen ceIlfeqtlerstly was 22tttSEe Nesr ippoSir}gti1elength of the pent dutuin at the CKU4tOI TO t)e} as .bi Bous-Uzr made ILF juR SS*gg+g incluts we flallhave-1tslength at Spitzbetgew39. X 97 S inclzes > andt:hencether;tio of the fim1*diameters (}f the earth) c)nSdered as azz hornogeneous elllpi°lda W1ll bt tE}lt

66 93* t() 92*I il<Rtad 0f 83*7 tt:) S2e7fy whtCh the aceeleratiorl 72tJ7 uncorre&¢d would have given. And here t fiems neceWaly to men+.son ime other miakes of COmpUtttlOIl} itl2vtRtelltlyv falIen into by the afironomer} 1a deducing the ratios of the iinidiameters of the earth} as fiated TR the sy9th page of the Voyap to the North Pole} not htherto notlced tha£ I kilow ofo

D d :., Wt,X.



F - * *

i,- w

Rcceleration

Experimerlts. From the FromlJon o+f e femi diamettlS

e4rfa,t)t to dtjIl to TotlX Spttzberg t& erth

LorldonO Spitzberg perw4tllum

csX C. Ro§s MIGCOW of a

773. I am aware that experiments wltb the perldulum Iwave

not y;et been made with that accuracy that the delicacy of lthelt nature

With the accelerationsX illdicated in that p;zge as ar;fing o1n diErent experi

ments} or dependng an the @ftems of dzfferentpl}lloiphe>rs} tEwe length of the

Spitzbergerl pelldlll.lm, a?ld corliqllentty the ratiow, {llotlld h3>\,tt ROOd as tI1 the

ann-esed table with which it wlll be pei+ceived they diSt tTely cotlf-derably. Sst

Iss^e NEwros's ratlo is placedtaR, as probably diiorlx> moR frQIl} the truth

Saifl tO bt3> t)Ut erro- / n.

Xneollfly} Six I*3VEWTON S 1 ( + 66*9 222.9 S.I96t 194*5 to 1u3*5 giib¢r. AN2BELL s l l + 76*6 23246 39.2o32 iS6. 5 tO I85*5 .

{M. IdarPeRtrvIs s >I56 < +86*5 242*5 39*2I4I I78*8 tO z774g

OTL4MULGRAYE scorred;. | | -F68*5 224 5 39e9 t8 TSX3^I to I9Z, oir ISAAC NERTON'S J t + En4 T88.4 39X X652 230* tQ 229

1n thls mauller the leIl>ths QF the pttadulum havltlgt been XUI1d br alt the

latitlldeS where the be or moR confiRent tspe3ialetts had beeru made on its

ac.celeratiolz and thtfe lengths hRVing bttR icceSvely compaied with cach otlier

from Spxtzbergen to the equator} lI9 refults irl tahe mrhote were produced Btst as

this nnxuber cvmprehended the compariins of thek at the Cape of Good Hizpe and

the Ifle of France} both LI1 bllth latitude thek being thrown Olltg tS well Rs tht

Portv BC11Q pendulum, .and bme fdx others more irregular than the wR} there

3remained at 1aR 75 wilt¢} tlle arithmetical mean of which gave the ratio

s 79.o47 - to X 7 8.0479 as ment:ioned ir; the t-ext. In caX Mr CUMMINGSS machine ffiould at any time hereafter be em#eyed jn

the firne brt of cxperiments} it tnq be proper to obilve} that the bdianzeter of

the braX bal} is 3.906 inches and its weight 637SS Trg grains. The urcight

of its blllk of met*cllry is Ia6g8o Troy grains. Hdtlce t:he welght of merc

JS TQ the we3ght of fllls i11 of brafi,, in air of the heat of 624 of FAHRENHEITS,

as .67875z to x; andthemreightof this brafs to air is as 7673 to . T-tie

eXPer1mentS br thiS purpoiEX, 1 Whlch .DZ. G£0RGE FORDYCE aSlReds were

Rd$ 12 hE OUE OF Mr ALCHQNE his WIeI2s Mint 1n the Tower9 o¢



propQfed trtgonmetrical Operati-onn. moy nature Senls to require, and- whichE at- folneQ future period; Ihould, and probably wills be undertake-n on a more tsterldXed

planX svith the very befl: ala-cllines that cala be conflru&edR for tlle purpofe Ma£zVy ages m0ay elapSe befole meafuremetlts of e any co--tafderable portio< of the furface of the earth ca114 beQ tBslde i£) very high fou thertz latittldesg fo as to determille witE any tolerabIe degree o-f exadne{a wherker the fouthei n feml theroid bt fimAilar to the northert< :ippOElllg looth to be figtlres. of revolution. Bu-t l)y good experimerlts; w}th the pendtllun alorleS eafily reeated in all latitlldesS tlle ratio o£ the ini^ dia-neter of the e(luator to the fetniaxis on both fides rnny be readily (?brailacde Ill tlze mean tirme I thought it miglzt be uSe f;X1 to Ihew the reful-t of .-1 colnpariIbn of tEle rlloI} confifient ex periments <3>f that fort, that havezhithelto been made itl 61ifferent Iatitudes aSter hasling applicd to theSe at Spi:tz-bergetl certairL corred:ionsb which feemed neceiryy as further explaired iIli

tIve note. TIlus it appears that lle aritllmetical meatl of 7--.S co-tBparifbIls between Spitzl)ergell alad the equator givies tle

tlle 8th of April and 6th. of June, I776; and the beam made ufe of; withs lbs. weight in each kale} was ttue to th-ree graiils. The lmachine, in its preferlt StateX although l etter So-t-ted in experiinents- thanS

at was origlllullyf {ince it has a wheel added, whereby it regiRers its own- time5<

alld now gees ir- tz hours without winding upj is no Ionger the falne thatr per° iSrmed at Spitzbergen: for in the interitn of the two times it has been irl myr poiItcIElon} the Ree1 I*Od« hav1vlg by accident been broken, Mr CIJMMING fubRiZ tuted anotherX as like to- the braner as poiElble with r-egard; to filzet but sYhick will, in all probal3llity, be fuSceptlble of a differerlt expanfion and contradtiont from the e5eAs of heat and cold. In tlle application of the machine, l have ufually loaded the top of it with about 2+119S;. weight of lead> inorder to rerlderk the center of fufpenfion as little liable to motion as poElble: yet, notwithRanding;; this precautionX a poiv3ted pltlminet fufpended to the top of tlle fratneS has a. fnuall depee of counver,vib-at;on; to- that of the tall> wh;c-h no doubt muR Br°** dxlce- bmes e*o&.



zoG Gen. RQ§S ACccount of a

ratio of the femiodiameters fornzerly mentioned xvgo47 to t78eO4*7 On this hypothefis the arc MP ffiould conta:a 27350 fatlloms The error on the total arc Mf Perpignaa aluounts to 2078 fathoms. M. BOUGUER'S degree at the equator being adhered to, the 45th of- latitude will-exceed the treth ^16 and that at the equator s48 fatlloms,

The ratio of the femi-diarneters of the Second ellipSoidS which comes nosr to be fpokerl of, has beell obtairled by the comparifo}l of ft}ch meafured lengths of the deglees of the rneridian ill different latitudesX as l<ave been foulld Eto be rrloIt confi{cent svith each otherO Our countryman Mr. NORWOODX was the firll5 of lato tim,s, who made any attempt of this fort But the meafilremerlt, executed by him in the year 1633, I)etsveen Lolldon alld Rrc)rkX has nopretetlcetoexad:t

-lefhj fince hehimfelf tells usX that xe)hen he did not tneayrgrev he paced! Befildesj his degree is as great or even greaterX thar that itl Lapland X and thefe are furely fufficiellt reaSoIls for rejeEtitlg it from the compariCon. The degree meafured by M. LIESSAP{IG in latitude 45° \57' in that part of Poland lately fallell to the lEare of the Emperor arld atlnesed to HutlgaryX beitlg fo snucll {horter thatl degrees to the fouthward of its gives ;roullds to fuSpe2 that fornet error had crept into that operatioll5 or that the plutntnet flad been affeded by the attrac- tiOll of Ileighbouritlg tmouIltaillsX atld therefore is not tuade ufe of on the prefelat occaflon. WIc DE LA CAILLESS degree at the Cape of Good LIope being itl fouth latitudeX and fo much g;reater thatl thoSe of the fame height itl northerll Iatitudes,, is improper likewiCe to be brought into th-e coxnparifon leSt the difference may ha-ve ariSelu from a diffilmilar;ty in the two polar fides of the ellipSoid The degree meafured itl the north of France,, compared with that in AuRriaX comitlg out

abfurd5 '



proysofQd trigonometrical Operation zz,7 abfurd, it has been judged beR to take a mean b@tween thenz for a meall latitude In like manner the latitude-s of the two Italiatl degrees diGering but little from each other9 a mean length has been taken betsveen them for a mean latitude Ac- cordingly tlle latitudes and the meafured lengths of the degrees which i21 tlle Wecotld ellipSoid, have been compared togetherffl, -Xvill appear ash underrweath:

ObServers names Countriese Lastitudese MeaRlred lengffiss O }

BO-UGUER;} Per1J - <3 O - 60484<5

MASONand DIZON, MarY1Rnd, _ - 39 I2. - 60628.S

30SCOWICHS 1tAlY, 43 O 14O 52 r 725 5160773.4;

BECCARIAS PiCdinO2A 44 44 J l 6082 I .3 .J CASSINI, 86-ce Mlddle of FraIlce9 - 4S £) _ 6077746

LIESGANIGV, AUAria, - 48 43124.9 3r 60839 4160833 Q CASSINI &C. IVOrThO5FRal]CeÆ 49 23J - L60826.6) MAU1?ERTUISS &CO Lapland, 66. ao - 61 I94X3

Now theSe fix degteesS being fucceffilvely compared with c.a.ch. other,. fifteen refultJ a.re thereby obtain.ed,. hereof the arithmetical tneatl gives for the ratio of th.e fezli-diameter.s of theellipSoid thatof I9Ze4r83 t.o I9I.483. By advert-iolgto the tableg it will ftTrther. appear. that th.e arc MP Ihould be ia length 27331 fattoms The arc M Perpignan exceeds-thel truth I758> the 4sth of lati.tude 180 arld that abt the polar circle near 88 fatholrXs.

The ratio of t:he femi-diametersY of the third ellipfold z 1 60o6 0 2I5.06 iS obtained by adhering to the meafured lerlgths of the degrees at the equator atld polar circ]e. Accordiagu to this Bypothefis the arc MP Ihould contain 273ox tathorns-! t1<}1ae

arc M Perpigllan exceeds the truth I a88 axld th;at at the 4JS of latitude more thall xaS fithotus*

* Th¢WS



OS Cen. RoY§s Jcat °f a

The ratio of the femivdlameters of the iurth ellipCoid zzz.55 $0 23 I 5S iS the fame, as tnay be feen w reirring to the table with that ailgned l)y A1. Bowour to llis firft fpVlwekrvid where the irlc-rements to the de-grees oiS the nwelhidian above that at the equator are as the Ideond power- QR aufares of the 1:1ES ;-of the latitudess It was intelldeld-clsleSy to utr hosr ftnall the diSerellce is betweell the magllitudes a11X1 natv1re of tlae cul*ves of tlle tsvo ltiguresv rrhe arc ML3 Wsaould cowtain z7z9+ -fithons The arc 1VI Pe-rpigtlatl errs iz tCet- 1 t / 7 fath Llse The ASth degree exceeds the trvXth ] 6 fathoms; a-ld that at the polar circle falls {lac}rt of the ncaftIred 1e-1zgth z- fatholns :M. BOUGIrER9S degree at tlle equator beitlg adhered to as the Ratldard

The ratio of the Ser<zi-diameters of the Efth ellipSoid 230 to 229, is that ai<g;ned to the earth by Sir ISAAC NENVTONO On this hypothefis the arc MP {hou]l cotltaitl 273AI fathoms The arc M Perpiguatl only esteeds the truth zoz fathQtnss, becauSe the Astll degrce of the rneridian is here adhered to as the Ratldard length. But tlaen the degree at the equator falls Ihort of the meaftlrement IOZ fithotns, and that at the polar circle I A6 z ; svllerefore an arc of 83 3t S in the firR cafe, would be defeAive SyoS and in tlle laIt I220 fathoms

The ratio of the femi-diarneters of the fixth ellipSoidS 3IOS3 to 3w.3 is obtained by adherillg to the meafured lexlgths oiS the degree.s at the equator and 4sth of latitude. tThe arc MP Ihould contaill 27zflo fathoms The arc WIz Perpigna-l -ouly exceeds the truth I3I fathoms;-bu£onthisChypotheI;s5 the degree at the polar circle srould be defEdRive 1lear z 1 7 fa- thomsX a£d cotlfequetltly on 8°S the elror snrould l)e ISO7 fathoms.

The 7



iroM>ofed xrigon>neXcat Opet2tir@ 2O-

The fes7enth or laR ellipSoid7 being that of the leaR flatteaPng has for the ratio of its feluidianXeters 54o ato s3yO The atc MP ihould corltairs 27zo6 fathotris The 4sth degree of lat;ttde beillg adhered to as the flatldard the alSc ^M Ai9Crplgtw-at) srouldl only ex;ceed the truth ly 4.6 fattlOtuS 9 butX ota the other l<;llldS the degree at th@ equator triilns itz t^efs t24t- El+ton¢, aXd that at the polar circle beillg defeS;ve rlear 3O39 tileleSorey ilz -tlle firIt cafe, the elror on S°3t would De to3yX and itl t:twe 11aR zsz4 fathorms. lIcllce it is olvious) that the arcs of an ellip foid hoxvever grcat sJl S-nail tE}e degree of its oblatewefs Xu1ay be, will not atly sstay correfipotld svith the tmeafurecl pottions vS the furface of the earth: for if we 1*etaitl the length of M BOUGUERSS degree at the equator as tlae Ilandard, and nLlake the ellipSoid extretnely flat, as in 1!FD . tile figure 7vill become too promitlent ill rniddle latitudes, that i$ to ,2the curv-e will riSe above tlle real ftlrEace of the eartl1 atldS in proportiorl to the excefs Of the radius, 5v:ill always give degrees that exceecl the meafured lengtlle 0n tlae coatrary, lf we glve the elllp foid a f[lall degree of flatlleI; as ill N° 7. at1d adopt the meaw fured length oiS tlle 45th degree as the Randard, the meafured al1d computed arGs svill nearly agree in middle latitudes; but at the equator the curve svill X<e ̂?ery corlfiderably above :the furu face atld svill thereby give clegrees that are toa great; white at

the polar circleSit will fall; below itX at:1d-give -clegrees th-at are too little irl the proportioll of alout 22 :to S X cotlaparedw wt:I the error at tl1e equ!tol*O ' Frorn a11 which we tnay coI1clude that tl1e earth is llQt an ellipCoid.

-The tAro columlls towaxds tlle rigtltlzand of the talules contaitl tlae arcs of twox fplleroids diSerillg from the ellipSoid The filiR is tllat adopred by X, BOUGUER as; hisu fir3R hypo thefisX u?llere the itacrene1lts to the deglecs of the meridiarx

VOL * LXXVIIe E: C a3!l)0v:



z o GenO Ro srs Sccount o,f a alvove that at the equator fiollow the ratio of the Second power or rquares of the flnes of the latitudesS and to which he has fuited his hrIt table of degreesX N° 32. p zgSt. This fpheroid differs-but itlfenfibly as has beetl alteady mentiotzedS from the fUurtll ellipSoidX Tlley have bc)tE;l the fame Semidia7^neters; but the arcs of the fplleroid being fome5Nhat longer than thoS of the ellipSoids the - :former thereby becomes i) a trifl;ng degree? rxlore prominetlt in middle latitudes. ()I1 this hypo- tI-leIls tlje arc MP {hould 1)e in length 27295 fathoms; M Per pigllatwexceedsthe meafuretnent It96 fathoms; alldthe deu g;ree at the equator being adhered to as tlae :Randard, the 4sth errs iru exceSs I87 svhile tIzat at tile polar citcle is defeAis>e only 23 fathoms.

The fecorud fpheroid is that whereotl 1u }30uGugR foundedt his fecond hypotileflsn which fuppofes the ilcrements to the- degrees of the Ineridiat}, above that at the equator, to fbllour the ratio of the fourth pos07er or fquared Xuares of the firles of the latitudes, arzl to svhich he has adapted his Iecotld table of- degrees 19 38. p, 30z0 It will be perceived, that the ratio of the femi-diameters of this fpheroid viz I / 9.4 to I 7So4 diSers little from that a?pertaitling to the fir{t ellipSoid; but here the curve fallillg c-fiderably 0TithinX that is to fay, being Refs pro minent thatl the e:LlipSoid- in mid-dle latitudes the arcs are thereby contrafted in fuch a mantler as to agree s^rithitl 5 fathoms with the meai:ured length of the meridian of France in an extent of abollt 803, comprehetlded between 1M ntar Dur- kirk,, arld Perpignatl fituated at the bottom of the Pyrenean tountains. ]3y infpedion of th: table it will furtller appear at tile errors in the feveral fedions of this arc ale not only fmall) btlt they are Cometimes plus and fometimes minas7 aL

aever failing proof that3 as fax as our prerent da will enable ;



propofed rigotxowergcv-l C@ervio z X X

us to jualge the figure here a{ilg;ned to the earth, notwithRandZ lng what has been alledged to t301e @ cotltrary7 is exceedingly near the truth lNccord11lg to this hJrpothefn7 the diRance M1? on th-e meridiall of Paris, whicil is yet to be determiI@d by our trigotlo- ^netrical operatiollsv thould colataitl z 7 z43 fathoms lDeitlg only 3 5 fathorns lefs tlaatl wllat is giv£n by tEle mean of rlae Ceven cliiXeret-lt ellipCoids3 a fpace not alnoutlting quite to 3' of latitude Tl<e re i5ult of the meafuretnent of tizis fpaceX anfs0rerilzg lto all arc irL the heavens of 26' S°" 5z/'f of latitude will be a further confir mation, or otherwife of the jURnefS Of the tIlec)ly The- degree at the equator being adilered to as the fiatldardS it will be feen from the tableX that the 4sth is defiedive 37.6 whilc that at the polar circle errs in excefs 9.4 fathoms.

* Affr. J. KLOS^rE3tMANNX TuEpe&tor of the Corps of Pages at St. Peteriurk,r, a1} his azanuSclipt Memoir, fotne time fialee tranitnitted to the Royal Societv to ghe SQcademy of Scietlces at Paris9 and alIo to that at Gottingerl, has enaleaw voumd to Illew, that the French trigonoinetrical operations are extreluely Gelro neousa It would feenz} neverthelefs} that he has attempted to prove too mllchw XIe ihould certainly have con}Sned his criticiEm to the trianglec of the meridiaa only, which are diRingllinled frotn the others in M. CASSINI'S lzook by being printed in larger charaders withotlt drawing concluSions frorn very acute anglesX, whichS although iniertecl in the general legider vvere rlot made uSe of in thr

determination irl qtleRiorl. Bllt as the :Royal Academy of Sciences will, rao doubr;

viadicate the cl edit of theil onrn operations? I illall Otlly further remark,, on tho iLiterary Nevirs .fiom Gottingen (AJvclvricht ags dez Gotzingifichea Snzeigen vog gelehrex Sache.g II7 st2*ttt l785), which accompanied the faid A1ellloir, and where it is faid} 4' That M BOUGUER'S hypothefis of the 4th power ill to the 6-X ground} b bon as other degrees wele meafured than thofe on which hc had 4 founded it,' that I confefs myielf to lJe quite of a diflSerent opinion, not

doubtirlg that mthen the comparifon is fairly drawn between this and every other tRem that kas hltherto been fubmitted to the confideratiotl of the public} M }3oUGUERES W;l1 be forlnd to be jllfily entitled to the prefere{lcefi whlch I hare

lhere endeavoured to give it. His works Ihemrs that he was a ma£s of very fu^?erior abilities, eluinent as a xnatheluaticsan, and pelhaps the beR pradcal one that cver esiRedO

19: e z

serld



::2 t Z Se. RoY's SccoanZ af a

BeEldes thefe tsro fpheroids of 1hI. BOUGUER, I hAd COUO puted the arcs of a^Llother betsveen !the ts0709 where the incre: xnerlts to the degrees of the meridian were itl the ratio of t1Xe fiadional povver 3l40^0- of th1e fitles of the latitudes. Thtls the: conlpu-ed a;X-eed accurately w7ith the meafurelul dimenfotas in three plsincipal pa1-tS of the earth9s circumference zlalmlelfr at the equatorX 45° and 66° to' of latltude, at the famG titne that the - arc N[IE3 CO1li!lCC! 272 5 7 fatllotns; bu-t C11} lat30ur of corn- putatiotl lJeit]-g Iaereby greatly augtnelltedn atad ttle error on t}to totql arc betweell 1!d arud Perpigrlan amountillg to 290 fathoms irl eXceSs, this fyRem did not feem to me to deServe to be put ln competition with rlae ilnaplicity of rllat af Ad[* 33otrGuJn, xvhoX indeedn for the falue rea-Cotl9 pre:Selred the fourth tc) the fracZ tiona1 power 3I4Dtt1SX which he tells t7s was tlaat uthich 111 catne nearer the truth. It1 {ho1*t, it mu-R be frotn the refults of future operations executed sIl very high latitudes, atld the meafurement of degrees of longitude orl the equatorX that sve

can hope to have fufficient autl<ority for atly corrediorl or a

melioration of the lzyfl;em of M. 130UGUER

4erxces sf lorga;X>de

Elitherto there has been no particular reference to Coule lines at thc bottom of the table7 containing the cotuput;ed lengths oiS degrecs of longitude on each hypothefisX in three difirerlt- latix $udes9 naz3elyX the equal:ol*7 43° 3zt' and Sto z8' AQ" ND} meafuremeints of degrees of longitude, as fir as I knowX have tver been czecuted with fufficietlt £are and accuracyX except: that ill the fouFh of Fratlce7 as mentiotled in the xt5t1w an o6th pages of MW CssNSs T;-ook which was determilled by

t.1zo



pr90pg £rgQ80MetrXCd/ ?drJXiSS 2 f j

the tepeated explofions of gunpowder in the open air and yund 11:<) co*altain. 4 6X8 toifesX equa1 t° 443S4¢+ fitlzoms. w

attendi3zS t:o the table it >ill be feetls tEwat tEle error irl exceX of M BOLTGUERES TheOrY9 OrL the length ot this degree trigonoZ metrically :neafuredS amoullts only tO T ithorns whicil is ittle rilore thatl ,,til part of a Scond of titile

In fixed ObErvatories where able aRronotners have lJeetl fbr tnaizy years ernployed irl rtpeating tileir (3bfkrvatiotls of the heaverlly bodiesS it ftems furprifrlgS tllat any dolabt lluould 3remaill v^7ith regard to what is called the aiRronorntCa1 dif- £drence vf lotgitudeg orv itl other wvrds the diSrence oftime 113etsveeil thetu; yet i-t has been alledgedS that -an Lll<.ertail£y

of this fort exiAs) evell with regard tQ tt iituatioll 0 Green- WICi1 and Paris, whichS recliolled lSy ts extremes extellds to about IQ or It Secondsy anfwering in the latitede of CJ1MeX1- wich to the enorrnous; diVerence in Iiace of lerNveetl 6oo atld I700 f:athetns t But it will be cotllldered as- Rill more won;* derSIy if letweetl two Bxit; Obfervatories) Greelsvtch atld Oxirdt which have beeD lotlg fupplied svith great a1wd cofily ilwAruments of the very beR kirlds) there Ihould remaitl; a; mt1Certait1tY in this rerpe& of %r or 3 £ecotlds of tixne * ir ilP

the latitude of Gree.uwich 3 iCGOtldS corre@otld to q77) a1ad ;n

that of C)sford to +743 fatllotusO TheIb, holveverS are poit<ts

which mull be lefi- to the refpedive aRrotwor3zers to Settle iw t1<e beIl: s^7ay thg can; arld it is not to be doubredX that the AItrotlo mer Royal will throw a new atld very fatisEadcory 1lght on the ;rnatterX irl the lE)aper which he propoSes a1:u3ut this time tQ Iay before the Royal Society3. alorag wtith M CASSINI)S [emoir

wthich, ir that purpok has now lbeen acar1y two years ixl lais pofleffioz

MTiti



2 E 4 Ger. RoY9s AccoE3 f a WitEa rega(l to the t:rigonometrical operation (Nvhich ma3r

be contidered as infillible becauSe by rneans of the belS of rerificatiolzL it svill prove itSelf; and if fmall errors unavoidably arife in the courEe of a lellg ftlite of triangles the maximunz of thefe may- be always aScertailled) I have no douSt that the diftaslce betweetl- Greerlvicl;l and the poitlt P in tIve map xnay thereby be detertnined to a very fmall nurnber of fithoms perElaps to fifteen or fisteen on a dltEyrerlce of lotlgitude of 85Otit 3° jOf ZoJfs alld therefore to about 2Ith part of a fecond of time on each degree. This, br any uSeful purpo will certai1lly be advnitted to be fufficiently near the truth, and is probably cotfideralely nearer thall it ssrill be brought fbr many years to come by a mean of the beI} obServations of the hea vexaly Lodies if thefS lhould be foutld in &he prefent i:late of tIle mattet to leave it yet doubtful to t:wo or three ficorlds

ltlae afEronomica1 diSretace of titne may likewife be olv tilined by experitnents on the inRautalleous explofioll of light; but thefe I would propoWe to be made libEequently to the trigo- nometri£a1 aperations. The Ilation of lZttle71ees towards the eaRern estremity of our rallgc of chall; hills, or Ibxne point xzear it would fiem to be the moR proper br thoplace of ex plofiotl ltecauSe it can bc feen ftorn B0/le-tig ot1 the lime r-allges atld rlcarly in tIae tneridiarl of Greenwich ObIbrvatory :Xt is not to be doubted that 7NtXerlaes may be reen frorra Ftienne Mndfaill, or even perhaps from that of tEle Br7unfflbeg D rNcO

they are l)oth fltuatiorls on the gontitluatlon of the i[ime range in FranceX the diCanGe 13eing ffiorter too and littIe lalld but chiefly fea9 intervenillg. Let US then fuppofe that the two afironomers with their clocks atlfl t:ranfItintirumcnts are poRed, one at BozS-biElX and the other at the Brwe?bergS while gun powder is repeatedly exploded at lktterle> or while the Indiau

lights



prtoJred trXoromerical Oper"ion*, : X 5 iights are aIterllately exhibitedg and again covered by an extin- guither prepared for rlae purpofeX which operatiotu may be repeated feveral tilBCS the <alme evenintJ; it is certaitlX that a

juR mean beillg taketl lDetss7een the itltiatlts fo marked by the refpedzive clocksS svell regulated hefore-hatld, the diSeretlce of time betsveell tile two extreme ftations will thereby be ob tained to a very conflderable degree of accuracyS and probably more to be relied upon thall that refultitzg from the comparifo Qf the obServations of tile heavellly bodles

Rockets have beetl propofed by forne to be applie£l itl this way t 5ut eveil the largeR f01t carrying up too imall a body of whtte light couldS I thinlis otlly be depended upon for- {hort diflances; and fince the adual error of obServation srouM be the fame on a ffiort as on a long diItatlce remote fiatiSs

oIlly {hould be choitl for concluElve experiments of this nature. It is imagitled7 that by mentas of: a lJalloon- a 1@t of

lightbalIs might be fent up of fuch a volume as to be foenS from their gr@at elevatiotl ill the air at Itatiorls very retuote from the place of explofiot* The balloon would afcetld wth a burraing fu%e attached to the firLl ball, and having a-ttairled the svithed-for heighta it might be made i:R I)elours by means of a cord fixed to it fo.r the puurpofe i The firR ball) o ts explofilUll, would cotumurlicate fire to the fecotldX and fo 1ll

facceIIlot at intervals of time proportionable to the lengths of

* It is to be underRood that the ,>rad:icabillty of Sening the- ballooa

to the eartll by meaus of a rope muR be previoufly aSc-ertaiIled: br if that Ihouicl te blltld ilxipoffilble} without tile vYinsl f.orcing it (lown agai2a to the ground9 then it muR be ferlt wlpRdetached wi.th burnin$ izes fixed to Itv Qf fiill greater dimenEot-sJ iBy meaEls of thefe it :is imagined, that the track of the balloon in the air might be poiIlted out to the rerxlote obServers9 who might perhaps be better eu4abled

therelzy to watch f-4.r a.nd diAinv,uiX the fuccefiive lnItaxl+s of the blxrRing of the

wEte light> . 1



2X tS CSao RoYs AvoUSV af a

ttle filZeSw ttit1S 8 OZ IO iSealltS tnight be tnarked Som the expertnent of a {illgIt lalloor which might l:hen bet hauled dowtz to be teloaded iLo a repetition. But whateater naight be the node adopted as the bR ir con3uRing experitnents of this

-rlatureS the obirvets muR twot only be very attentiv& atld dilit -gent but a1b quick-lighted9 have their clocts nicely regulated indeed and tlle trials tnuR be tnatly tires repeated ltefUre-the tucertaitz eswtl ln this way whtcIl ienzs to be tlle be{l: -modef could be recluced to 1tSi than tTth part of a Second of tiin¢ to htch t may infillibly be ;brouglt by trigonometry

Having in this maurler Ihewtl what prolzable degree of exa&* nefi may lbe cxpeAed itl the trariotlS} but ufua1 ways of- aIde tAitlit.]g the diSerence of IoWitude between the 0-birvatries of Greellwlch aIld atiS aiUd CQmpared the teittS NVith the tncertainty that feems yet tO tSiR ill this tnatter from the Rate of afiror<omical obIbrvatiotls; ler us r<ext fee how Mr0 It-Axs iE)EN S itX0ru-metat is likely to perbrtuS when adually applied to the detertnination-in qeRiOn7 ty; the obferveci-anglebe tsveen tlle pole Ilar in its ea{tern or we{tern azimuth} and a

wry retnote liation whoS-d£Rance Xtn the -1nIltument is PhIloWIl by the Ibrses of triangles>i and diRtnguiffiablet by the Iwdian l;ghts at lwight fbr the purpoi of this particular

i r

o 5a,rV2t10 o

With an ilwRtumenta carryitag teleftopest fb good that &txe pole iLlar may be Iben n1 daylight it is olvious that the 51* *ie&ed angIe bnt-wreen the Ilar irl its eaRern and sreRern azi muths will- give at ot-Rce the -polar diftance of the £iar- -arld the true lneridiatl of the place - as refir;red to ̂ a-ny- knowtw- -ltaZ tions vi{ible at the time of obCervatioR But as cloudy wea ther nzay 0i5I1 preve<t a complete obirvation of this brtb £vonu being oltained} alld fuce much time might be loR ;a

3 attemptirg



propofed Zrggonot9eXrical Op{rJtioX j I 7 attetnptitzg its therefore the declination of the Rar fettled for ally particular period beitzg accurately lstlown@, its apparel}t diLlallce frotn the pole mas, by the eRabliLhed rtllesS beveadily computed for ally propofed dayX as well as the preciCe times of ts greatefl: elongationsS tsvice in 34 hoursX wllen in its eaRerrL atld sveIlertl azilnuths, at svhich times it svill, for feveral mi nutes, appear, as to fenSe, llatiotlaryorwithout motiotl except ln altitude Thefe are, therefore, the beR times for takitlg the angle between the Rar and atly partictllar flation, firlce the obServatiotls may be repeated frequently in tlle fpace of a fes minutesS or ulatil it Shall be perceived that the liar has again approached towards the pole. Now, fuppofe the Ratioll of tha inlErument to be at CaXerleesg tvhofe dill:ance from the perpen dicular to the meridiall of Greenwich alld conGequently from its parallel, is known lty the trigonometrical operation. The latitude of the Ratioll xbecomes kllown likewife; and let the co-latitude lve 38° 54' zo"* Let us likewife fuppofe the diRance c)f Bottls-hill on one flde to be 44too fathoms, equal to 43 z8t.6 of a great circle; and that of the Brunemberg on the other to be 38zSo fithoms, equal to 37' 4%".6 of a great circle, and further, that on thefe two Rations the Indian ligllts are exhibtted for the time propofed. Now let the angle between the meridian and Bottle-hill and that beeween it and the Brunemberg, be obServed by mealls of tlle pole Rar correRed for its diItance for the day; and fuppofe the firIt t:o be 73° xott

* Since this Paper was written, the ARronomer Royal has been fo obliging at

to furnilh me with the mean diRance of the pole Rar from the pole, as fettled at Greenwich by eight obfervations above axld nitle below it, made in the year 1 786; whereby it alopears, that the mean diIlance reduced to the beginning of that year, was I° 50' 8't*3S; and uthe mean annual preceIIion in deelinatie being Ig".55,, confequentlya the mean di$ance for the I: 0 Januaryt X tS7> was ° 49r 8/t 8

vOL, LXXViIfl F f arld



Gen. RoY's SCC0z4?Xt of a 218 and thelal} I25°5t; thusiwe ffiall havetwo^fpbericaltr KaIlgles tocomputes ill each of which twokfides, alld the cotw- tailled allgle, are known atld otwe Edes o)is the coulatitude, ss

commoll to bOthe NowX fium thefe d4tv, making efe of the half fum and half diffierence of the fidesX we thall hasre the angles in thefe tsvo triangles asulldertleath and tEle alzgle of longitude betveeti Bottelerhill and the Brunemberg} equal to that at the pole will be found to be s° S3t 56//*Iw

If from this angle we deduA ab4ut 3o' or 35/, for the fpace that the Bottle-bill fxms to i totthee weRward of the ineridian; of Greenwich, there will tllen-remain I°.5 5' 2+ I" for theeaft lulgitude of the Br28emberg being very nearly that expreXed in the map of the triatlgles which accompanies thI8 paper.

* Bottle-hilI. * o J 8

Half difference - S I 2S I4.05

Half fum - 5z 3z 25n7 __

. .

Angle 7zt Bottle-hlll I03 57 39*I2

AngIeat the pole - I 7 II.Ok

Containedangle _ 75 I0 . 0.

..

Sum of the three angle8 I80 I4 50,I4 " r- - _

Angle of corlYergence 5z ao.88

EXi8 abwe I80° - I4 50.I4

ALngAe of longitude - I 7 I I.02

Brunemberg.

Half difference - 26 44 I2.t

Half fum _ 27 3a 57.3 . _

AngIe at Brunemberg 54 x7 9.5 Angle at the pole _ 48 45IX

Contained angle - 25 5 o

__.

Samofthethreeangles 180 IO 54.6

- .

Angle ofJ conv¢rgentc - 37 5.o.$

Excefs above t80° IO 546

Angle of langstude _ 48 45,.£.

Sum af the two longitudes I° 55' 56". I2.

lt is to be oMewed, that the meridians, which are all parallel to each otller at the equatorj on their deprture;from thence econterge more and mar£ as thqr $pproach towards :the poleX where tohe angle of sonirergence bsoms$ equal to tbe tngXe of longitude, It mayt alfoibe remarked, that the angle of xnvergetlge, augmented by the evefs of the three anglese of the fpherical triangle abwe I.,80°e

is alu ays eqvlal oto the angle of longitudea or that: at the pole. And asHthisSholdt univerfally in all\ latitudes it alffiord a rely ans o£ pro\ving that the sompu*. ttions are juR

3 A$



propd trgonomerxecat er4oforo :X . . . As fir a we are elialbled to judge atw preEnty fiJOt^zl the exa+

minat:iorz- of the divifions of Mr 9.AMSURNSS iNl}rUtMCtX,

there is ev-ery reaIbil to believe} that iiw taking angles ArOUl3

t:ht horizon thenz-ean of INveral repetitiolzs of the ime atlgfe*> as re@rred to diSrent parts of the circumirence of -the cire:let will diiEr very-little from the trut:h7 t little indeed that :ltt- many cais the error will totally V$tRili But in elevat;tg- tiae

telekope- towards the poles let us l:ilppoSe that ail error of g ficonds on eacJl of the coutained arlgles at 1C//erlees lnas beea

committed atad f;Xrthe*> that tven an error of S Secolids of latitude equal to alout 84zg ithoms ot- the merldiall} nav haw been i11ta intoS in eAttnating the colatitude (WE3Xctt tlever can happeri but is ouly here adtnitted to place the example in the tmoRdiAdvantageous circum0ances paffilble) R then whoearer will gtve thelmiElws the trouble tQ recompute tlze tsre

triangles with theSe llew data7 svill find t:llt reIBlt in longitude

not to be varied thereby7 in the firR care abeve tt1 part of a

Sconda or th paxt of a fecond irl time -- and in the 1aR tlOt

quite X fecond or-Ytth part of a fecond in time. H[ence 1E conclude, that the befl: nwede of derermining the dffzzen&es of longitude will lJe by the inArument itSelt applied iw this wayt at} takitzg tTle atigles 1;)e-tween the pole Rar and very remote Rattolls diRi2lguithable at nigllt by the-help of the Itldian- ightSS and sv}zofe diflatace is acturately ktlownO This rnethod

wlilS it is true be liab«lev as well as aRronomical obServationsS to the imperfiEtioris of the irwilrument particularly thofe of the telUficOpeS atid the utiavoidable errOr in its ApptiCatiOn; buty on the other harlda it svill be etltirely firee fiom the lrregula rities of clocks ad the imperfiAious of vifotw in nzarkiing the ln0DailtatleC)US explofioll of light. TheIl both methods Jhave been repeated a fiafficient number of-ttimesa with all

F f X irnaginable



o (8t#. RoY9s Xccount of a

imaginable care, we fhall thetl, alld not till thens be able to judge to which ttle preference may 6e due. Thlls five or fis long fiatioFlsX in or nearly in the parallel of Greenwich, fuch, for inIlance, as- ttlat of Shooter'srhill Tower, would reach from the eaR quite to the weA of the inand: and as a very confiderable degree of confiRency might be expeEted among the refults for equal portions of the parallel, thts method feems to be as likely as atly to furrliffi data for deterluilling thenature of tlat fplaeroid or figure of the earth.

Cable cf the degrees of the earzb, conJ?rvSed on the bypotbeJ*s sf M, BOUGUER

:13efide the talzle of comparifotl of the are between Green- wich and Perpignan, which I have already endeavoured to explain, this Paper is accompanied with another, which, as well as the former, was ariginally intended folely for my owrl uSe. 07Vith this view it was at firR otlly computed for every five xninutes of the 5XS atld s2d degrees of latitude, that I might thereby be enabled more readily to compare the longitudes and latitudes of the refpeAive Rations in our progrefs tonvards the coaR; aIld more particularly to fit it for opelationsin the fouth parts of England, as likely to be firR carried into execution

ordern howeverS to rerlder it more generally uSeful, it has fance beeIl extetlded to every Eve degrees in the higher and lower parts of the quadrantX and to every fingle deglFee in inter- mediate latitudes. The table not only contains tl-le degrees of the meridiarl and of longitude, but alfo thofe of a great circle erpcndicular to the meridian and likewife fuch as are c)blique to itX for the other feven points of the compaSs. With regard to the con{trudion of the tableX it is only neceSary to make

fome



propoJ*ed trggonomearicat Operatgon. J z Z X

fome remarks on that column of it which colutains the fum of the three equations, or diffierence between the degree of the nzeridiatl and the correEponditlg degree of the great circle perpetldicular to it, which is the moLl troubIeiZome to compute, btl-t muR be found before the degree of longitude catl be obtained. The firR part of M. BOUGUERSS equation confiRs of 7.ths of the diSerence letsveen the degree of the meridian at the equa tor and that at the pole, vExX 545.Iz fathoms, to be conRantly addede Secondly, 4ths of the increlllent of the correfpondillg degree of the meridian above that at the equator to be fub traFted; and, thirdly, 4bths of a tlaird proportional to the exceEs of the degree at the pole above that at the equator as radius, and tlle fille of the correfpollding latil:ude7 to be added Now it will be foundS that this la{} part of the eguation 4Sths if uniformly applied, would have preduced abfurd refults at the 7$th degree of latitude, that is to fay, the degrees of a great circIe would there have become greater than the degree at the pole. The equatiotl I4Tths or 4th would in like man ner have produced abfurd refults between the 79tt1 and 80th degree. Even gths will not gO on further than the 8sths and the higheR equatiotl that will go through the whole qua- drantS uniformly applied, mufl: not excted o7°o parts of the third proportional. M. BOUGUER hitnfelf had found this acl accordingly had applied the equatiotl with a certain modificaF tion or abatement, svhich 1levertllelefs heXlnakes no tnentiOIl of xn his book. Seeitlg, therefore that the degrees of a great circle perpelldicular to the meridian ditrfer moR frorn thoSe of latitude about tlle tropics9 I havev at the zoth degreeR applied the equation <45ths or 38ooo parts, and made the divifor itlcreafe by unity for each degree of the- quadrant above that po;Xat to the poleX where it becomes TS°o parts Bclow the zoth d$gree

t}-l:



zz Gez 009S 08M<+ 4 0

tI1e distiSorS Itl like manne3r) increafes bEv-unit to tlle eqlJator where it becomes s5Z°O prtSo With re@eA to ethe degrees of great circles: obli9uelv fituated- +,o the meridian they vere cEziefly itltellded fbr iacXlttating tlle com-putations ofJ curvature al3d reXa&;onX where itl any particular caS, it might be come lsecefiry to obtain to ;reat precifion, the relative heigIst- o-f a -tery diRallt Ratiotl with regard ;-to thatX occupied by- the ;n{lrxlllaent; :ir ill ftlch CiS7 in firiAneX the atigIe which t;he line tuakes utith tlle tneridian ffiould be attended to-

rbe coziputatiotls $;for thke colwfirudion of} tlwis table have been, as may eaIily l)e judged, iiclentlr 1aborious > but hav ing once begun T was itlduce-d to go tilrough with them, cor*> ceiving that a table of this brt v would be of general utility >

atld - that it tnight be nveryu lLorlg stldeed bef<3re the- refults of iture operations) yet- to be undertaken in diSerent and very renzote parts of the globe would irtlith cbta for any thitlg better ]:t muR leby the united eSorts of tulightened nations collIpiring as it svere together in promoting the cauSe of fci etlce that greatR oltjeEts? fuch as the determinatioti of the tnag*tlitude and figure of the earth can ultimat:ely 1te obtainedO Each Ihoulsl cotltribute to it atld all have it more or leIiX in tlltir power) accordilg to their particular fituatiorl, and that

X Much care has been be:Rowed ia the variouscomputations for the cone itruAion of the two tab1es appertNlling to tilis Paper In tie 1aR, which has becn the moR laheriotlsX the d.iSelaences itm to go otl fo ursiformlya it is hoped, er) error of any cotliqvlence will be f4undr llere st is proper that I ffiollld mentlon a typical crrvtvg in one of the tabIes belongirlgto my Paper on the 13arometer pulDliflled in tlle LXAT£I. volvne of Tr;wrii:ad£ioxls, br I 7 7 7 . Tt is ln Tt5. Vi coiltaining obirvations on heights ricar Carnarvon. Ia the coluXmn of obServed teight of >W1:ercury on Meel X;il1o) AuguR 4> 1775t h 7m P*3W x}Ilead of 27.7 IA read 27eR T4o

portlon



propoSed t{zggonoruetrical per-aZios. 223

portion of its ilsul*\face sartAicTl has alletl to theit^ lIzare; ill the general diRributiotw of thirlgsQ

Under t1An (tl@ilc¢SS atld tntmediately urlder tXne eyei of a

Sovereigrl who Ioves atlcA cIt-erlfhes tEle fciel:lces mucla may be dotle sctithill the lllnits of tIze E;ritin iflltldsa xvhichg rec3ionong frQ!n BERRL1efS iL] CL1ffi1k tO kI10 WCNC11 PArtS Of ELerry in Ire-

atldS coJ<tallo tzea.r X 2 degrees of lotagittlde; ancI coutlting from the Chat}nel otl the :fouttl to tTle Orktle,Y iSlands Q1lly On tlle

north about g degrees of latitude.>? Ellth-tilis gelleltal purpo-f2- in ^7iew5 atld al: the fatne t;ine to Iny the 13eR foundaltiotl for tlle furvey of Ehe BritiS i0alnds I rould propofe as the opera ;tioils pteferSbly to be executed that feriefes of t:rianglle,5 «ould

be extellded alolag di:Serent tmeridiatlsX pa-rtiWularly thefe of the chief QbCervatories? and of rOlC -retmarlrable hills itlnthe cetitral arts of the ifland, tilL they 11 itlto tile fea on tlle foitlth arld

north, alld that, acrofs theSe a g;recll: rlurnber of ferle.Ses, itw the parallels likevvife o:f thb chlief (}lfervatorits> aI1d of re -markable hills or emitlenciesS Ikoulc3 be extended from the eaR to the weR coaIt. T}ff-<> 1t>^skia-hillotl the eaIterrl {kirt o£ ShropIhireX feems to be an eAigil3le poitat) Izeing ;.n ia £entrall- fituatioIl, fufficiently corlfpicuous, and yet 7nLOe 0 hagh as to

3render the frequetat accefsto it-very-inconvelwietO dIf it:ffiOvuld lbe foutld, that lolagitudes rnay be deftertnitled by the ilaIlru ment to the det,ree of accuracy which istexpe&:gdX and that t may likewife be advatltageoufiy appliedS illdepeildently- of

the zellith fedor, in tracirag ally parallel- whofc lattude has been aiready aScertained, or the para11e1 fof any remarliabl:$ pointg whofe meridian is knowlzX but not it-s flatitude, ttlen the ;nurnq ber of theSe paro-llel 0ri+s caranot be too tYluch tnultiplled X fo-r a ;g;reat map7 of ltheX- Crans-*eseicugcd g?ih dHucvcare-£-worlld

folIu.tely



*t wit gU-. g0Y$ Aacauat of 4

abSolutely give tl<e figure of the eartl-, fitace the fllorteR diITancf: frola t11e fulrface to its axis would thereby be oStaitbed.

flthe Brit;aa domitlions ila the EaR-Indies offer a fcene particularly favourable for the meafurement of five degrees of lati tude oll tlle coaR of Choromandel, as has beell lwoticed by Mrv 9ALRYMPLE, F R. S. in his Paper on the Marine Survey of that CoaIl. Tsro degrees of longitude, at each extremity of this arcs ShonId likewiCe be meafured.

wlflhe plains of Bengal, direAly ullder the northern tropic, afford another fituation s^7here it srould be of great confequence to determine the letagths of a degree or two of latitude, alld as many of lollgitude. Thefe two operatiotls could not fail to lJe patronifed by the EaR-lndia Company, wllo lhould defray the expence; flnceX whatever tended fo much to tlae improve- ment of fcience itl general, and fo direScly to that of naviga- tion in particular, muR be thought important to a Body of blerchants whoSe power,- as well as opuletlce Rand at this day unequalled in the mercantile hiRory of the nvorld.

But there is one operation yet to be melltioned that would contribute more than ally other to the determination of the sSgure of the earth, svhich ist the accurate meafurement of fome degrees of lotlgitude on the equator, becauSe thereby¢-the length of its Semi-diameter would be immediately inown, which hitherto has only beerl theoretically co-mputed from the meafured portions of the meridian. The Portuguefe feem to be poSeXed of the moE advantagegous fltuation yet known on the globe for that purpofe: z for h. DE LA CONDAMINE has teld us, that at the mouth of the river of Amazons, llear the fort of Macapa, ilu thrce mirlutes north latitude there are

-S - estenfive



pvofofed tronoe/rical erd/ZS¢ :2,5

txtellfive opell plains where a-t1 operation of this CoS wotxld tlot have been fo difficult as was at firft imagitled, when he Ezro- poiRed the fcheme to the Aca(lemjr of scietlces7 a year before the voyage to 2v,liZo was thought of. E1ad thls beell adoptecl as the fcelle of their opbrationsg inRead of that e]evatecl rallw compreheIaded between the lofty ranges of tE1e AndesX the buEluefs would have been much fooner accon plilhed3 zvith infinitely lefs labour and fatigueS alld Itill mo-re fatisfaacorily7 tillce degrees of lougitude as well as- of latitude would probably have beela meafureEl. Here thell the Portu- p;uefe Ihould determine the le1lgth of two or nao*e degrees of longitude, atld as many of latitude, by way of lproof or cor reScioil of the Peruvian c)bServations. A well colldutited opet ration of this kind could llOt fail to add to tlle celebrity of a

natio-ll who? by their diScovery of the xlew svorld in the sveRy atld having opelled the route by the Cape of Good IIope to the caRerll ficle of the globeX may lJe cortfidered as th-e :60-utlders oIT tnodern navlgatlorl.

With regard to CBperaitions iIl high 1lorthern latitudes, which would doubtlefs be of great importatlceX the Ruffian Empire muI} certainly aiord a variety of ftuationsX more or leEs diffi- cultS for the puirpofe; allel arl Etnpreli s^7ho corn;trlallds fo greauS a proportiotl of that regiotl of the world here it is IEzd ex- ploratory diScos7eries are at plefetlt carrJ7ing orl by her order) can fcarcely le f;.lImpofed to fuffer a reigtl, othersviCe fo b-ril- liant, to expire, vuithout direEting forrlethitlg of this fort to lJe executed ullder the polar circle9 or as llC>lz to the pole as the feverity of the cltitmate vill permit by uTay of collfirmation or corredcion of the L aplanLd tneafretnelltX which yet Rands fingle alld by itSelf7 without aray collaterbil proof of its exa&- nefs. If fuch operatiolls as thoSe I have fuggeRed towards th

VOLe LXS VlIc G g; c1O.fc-



:z6 GX RoYs AcaoatS &a cloWe of this Paper were well executed in diSrent pares af the- worldS little would then remaitl to be done towards the deter millatioll of the magtlitude and figure of the earthg except multiplyingX as llluch as poilble experiments with the fame pendulu3Lns fro;n the equator to very high foutherrl and northerll latitudes that ime judgement might thereby lze lirmesl of the fimilarityX or otherwiCeS of the two polar files of tlae fpheroid.

W 1[ L1:X4 R O Ya t5, 227 s7871

[_w_

E R R A r EAQ

X1n the Secotid Table ibjo1ned to tilis Paper page 2z8 under tH head 44 Degrees of Lellgitude" n the columa of the length of each degree in thoms9 lv 240 fromthe bottom (belng that which anfvsrers to the Latitude of Greenwich) for 38I6I.69 read 38I64I03; and in the next column of slifererltesX 1 >g. froln the bottomt ir 53. I 3 read So.8a > and t.he next Iane below for :t 6. X Z read X S 143*

Ta the Map of the TriangIesX br SxLS tead 8ELSETo

In the Firflc lTableX ;pX %27* tht 170 1l2t Qf tht-ft ¢Ql-aa t onet fox 38;X6X td 38 X 64*Q<,

.wt a >

tomparXton



Gen. RoY's Accoant of a propoJned

Comparifoll of the obferved letlgth of the celeRial arc ofthe meridiall, comprehellded between the parallels of

tr E R R

C E L E S T I A L A R C 4

Comprehended betwxeen the parallels of Gleich and Perpigruan.

DifFerencesof latitudes Afeafured Suppoftngthe eartn to <

or celeRial arcs. Arc. be a fphere. Places of ObEervation. Latitudes.

Deg.Min.&c. 54 Dec- : parts { I 2

° | , o, /'// O Fathoms. Fatho.ns.l Error. Fathoms. Error. Fathollls. Error. Fa

-

[Greenwich Roy obferv 51 284000 } 8 46 481oo 8. 78 534343 0> 535927 535588. S

Greenwich and M= MP -|--diffo265052|o,4475 27235 27350 27331-

2 { Perpignaa 42 4l|542 00 } 8 t9 S7| 8 8 3325; 50649g. 507 108. + 609 ) + So85778 + 207S SOs257- + i 758 3; Point WI and Paris - -|__diff.2 XI 351 8 2, I93Ij I334I7- I33467 + S° X 33934 + 5 7 I34038, + 62I I

3tparisRoyalobServatoV 43 5°14°°} 6 822006.l394 373082. 373641 + 53;9 >,2: 374603 +152 X 374374. +1292 0: 3

Paris and Bourges diff.I 45 9 I9 I. 7525i I06564. I06658e + 94 > :: I07007. + 443 > * I06937. + 373 > O I

4tperplgnan 47 552400 } 423 12!41|4 38691 266518. 266983. F 465 0 267596. + 1078 + 267437- + glS + 2

Bourges arld Rodes diff.2 43 5I 5 2.6309 166cog. I66200. + I9l a I66620. + 420 I665I8. + 509 I

5{Perpignan _ 42 415200 t 39 136 1 656 100509. 100783. + 274, > 100276. + 467 Io0919, + 410 14 _ _ _ _ _ _ _ _ .

s[Semi-diameter of the equator - _ 3486975 } ,ooooo 35°4545 1 I 5 2 350I802 } I8I94 34 Semi-asis between the poles - _ 3486975 3484973 J 9 7 3483608 34

VRatio of the femi-diameters - - I79-047 tO I78 o47 192.483 tO I9I.483 2]

Difference of the legarithms of the ratio - o.OO24323 0 0022622 t.<

Error. Error. Error. Some rEquator _ _ 60484 5 60859-I + 3746 604845 ° ° 60484.5 ° ° 61 other Degrees of tlle 45th _ 60777.6 60859.1 + 8I.S 60993.4 +2I5.8 60957*7 I80*I 64

- chief meridianat the 66° 20' - - 6I194.3 60859§I -335v2 6I342*3 I48w0 6I281.9 + 87.6 6 proper- pole 60859*I 6IS09.3 6I437-I 6

ties. Excefs of the dearee of the meridian at the pole,] above that at the equator - - iy ° ° I024.8 952.6

Degrees of lon- [Eqllator _ _ 60859 I 6II65.8 6III7.9 6 ittlde at rtle 5 43 32 iollth of France _ 44354-4 4412I-3 -233-I 4446I-2 f I06.8 44430v4 + 76.o 4

' l 5I° 28' 40" Greenwich - 379°4 2 38225 9 38x8I.3 3,

vOL. LXXVII.



Gen. RoY's Accoant of a propoJ"ed trigoromerical ()peration.

> ofthe meridiatl, comprehellded between the parallels of Greenwich a1ld Perpigtlaxl, with the meafured and computed lengths of th

tr E R R E S T R I A L A R C.

Coruputed arc OI1 differeIlt hypotbefes.

,ftng the eartn to Suppofxng the Earth to be an oblate ellipSoid. be a fphere.

I 2 3 4 5 6

;ns. l Error. Fathoms. Error. Fathollls, Error. Fathoms. Error. Fathoms. Error. Fathoms. Error. Fathoms. Error.

43 C7s 535927 53ss88. 53So88 53497°* 533942 533860. 35. ,w, 27350. 2733I- 2730I z7294. 2724I. 27230. 08. + 609) + So85778 +207S So8257- Jr i758 go778 jw +12887 So7676. +II77 506701- + 2027 506630- + 131

67. + S° X I33934@ + SI7 I34038, + 62I I33747- + 330 I337I5* + 298 I33454- + 37 I334I2. - 5 4I t 5!;9 < 374603 + Is2I X 374374- + I292 m 374°4°^ + 958 C 37396Iv + 879 ten 373247- + I55 * 3732I8. + I36

58* + 94 > :: I07007. + 443 > * I06937. + 373 > o I06834 + 270 > + I068I0. t 246 F + I06605. + 4I > + I06582. + I8

B3. t 465 0 267596. + 1078 + 267437. + 918 + 267206. + 688 + 267151. + 633 ; 266642. + 124 + 266636- + IIS 30. + I9l Q I66620. + 420 I665I8. + 509 IO6370. + 36I I66334* + 325 I660I7. + 8 I66005. - 4 83. + 274, > 100976 + 467 100919. + 410 1oo836. F 327, 100817. + 308. 100625. + 116, 100631. + 122

>75 }.oO2oo 35°4545 } 19572 3581608 } 18194 3497814 } 1619° 3481221 } 15713 3479949 } 15174 3487958 } 1123 I79.047 tO I78 o47 19^-483 tO I9I.483 2I6.06 to 2I5.06 222.55 to 22I,55 230. to 229 31° 3 tO 309.t o,oo24323 0.0022622 O.0020148 O.0CI9558 o,OOI89=4 O.OOI4018

_ __ __.

Error. Error. Error. Error. Error. Error. Error 9-I + 374.6 60484+5 ° ° 60484+5 ° ° o484.S 0.0 60484.5 6038z.6 - IOJ.g 60484.5 ° 9.1 + 8I.S 60993X4 + 2IS.8 60957*7 + I80.I 60905.g t I28.3 60893 5 + I IS.9 60777.6 o.o 60777*6 o 9§I -335v2 6I342*3 I48w0 6I281.9 + 87.6 6Ii94-3 0,0 6II7344 - 20 9 6Io47-8 _ 46-5 609j7.7 - I6 g.I 6I509.3 6I437-I I332*2 6I308,4 61 I77@0 6I073-0

D.0 I024.8 952.6 847S7 823.9 794S4 588.5

).I 6II65.8 6II17.9 61048.5 sI033@0 6091I.1 60876.2 r.3-233.I 4446I-2 + I06.8 44430v4 + 76sO 44355+7 + IX3 4424Ie7 - I2-7 44250.I - I04.3 442I3*4 - 14I s 38225 9 38x8I.3 3813° I 38II7.7 37978*7 37989.9

LI h



'erpigtlaxl, with the meafured and computed lengths of the correfpotlding terre{Erial arc betwee21 thefe parallels.

R I A L A R C.

b differeIlt hypotbefes.

Suppofing the earth to lle a fpheroid differing from the ellipSoid.

0 be an oblate ellipSoid. More prominent inmiddleLefs prominent in middle latitudes than the ellip- latitudes than the ellipfoid foid. of the fame femi diameters.

4 5 6 7 I 2

i Fathoms. Error. Fathoms. Error. Fathoms. Error. Fathom6. Error. Fathom3. Error. Fathoms. t Error.

P 3 9 533942 533860. 53375I 534997 533737

27294 2724I 2723°@ 27206 27296. 27243*

So7676- + II77 50670I + 2027 506630- + I3I) 506545 + 46] 50770I. + 1202] 506494, _ 55

133715. + 298 133454- + 37 133412. - 5 133360. _ 57 133722. + 3o5 133438. +2

37396Iv + 879 ^ 373247- t I55 * 3732I8* + I36 * 373I85* + I03 @ 373979 + 897 < 373056 -26

I068I0. t 246 F td- I06605. + 4I > + Io6582. + I8 > < I06555 - 9 > t I06815* + 25I t X I06570 + 6 > o

267ISI. + 633 ; 266642. + I24 + 266636. + II8 + 566630. + II2 + 267I64. + 646 + 266486. -32

I66334* + 325 I660I7. + 8 I66005. _ 4 I6599I - I8 I66342§ + 333 I65928. -8I

I008I7. + 308, I00625e + II6, I0063I. + }22J I00639 + I30 I00822v + 3I3 I00558@ +49,

3481221 } 7 3 3474775 }15174 34486915g4 }11236 3485542 }6455 3496934 }15713 3477210 }19530 222.55 to 22I,55 230. to 229 31° 3 tO 3o9.3 540 to 539 222.55 to 221^55 I79.4 to I78,4

O.0CI9558 o,OOI89=4 0.0014018 oooo80so 0.0019558 o.oO24274

Error. Error. Error. Erro1. Brror. Error. 60484.5 6038z.6 - soJ 9 60484.5 °° 60609 0 + I24,5 60484.5 0.0 60484-5 ° ° 60893.5 + I IS 9 60777.6 o.o 60777*6 0.0 60777,6 °-° 60896.5 + x I8.9 60740 o -37-6 6II7344 - 20 9 6Io47.8 - x46.5 609j7*7 - I6e6 6089I.6 + 3°3 7 6I175 7 I8.6 6I20367 + 9'4

6I308.4 61I77@0 6I073-0 60g4,.o 6I30895 6IS06.6

823.9 794 4 588.5 338.o 824 o I022.I

sI033@0 6091I.1 | 60876.2 60834I 6lO33-0 6I029.6

4424Ie7 - I2-7 44250.I - I04.3 442I3*4 - 14In0 44I4l,9 - 2I2.S 4434I.8 - I2.6 44373 5 + I9,I

38II7¢7 37978*7 37989.9 37932-4 3gII7.2 38I6I.7

Table

eratZon. 227



Gen. RoY's Accov?

|' ['able of tlle degrees of the Earth conftrud[ed oll the hypothefis of M. BOUGUERS LVJ I Sq

r

I Degrces of gLCAt Xllcre- - *o - s -

ments Lengths of degrees of the meridian in

fathoms.

4 Diff. 60484.5° } o.o6 60484s6 } o.87 60485^43 l 3.6 6048g.o8 } 9 4I 6O498.49 } :8 6

60548 38 } 36 75 6 698; 3 } 63.86 ! l s4.86

60673 85 } I5 55

60705 62 } I 6.88

60722 5° } x7.52 60740I02 } 18.IS

60758@I7 } I8I75 60776'92 } I9,32

60796.24 } I 6 59

60836 47 } 37 } I.72

60838vI9 } I

6084 64 } I 7

660843;37 } I 74

60846.8+ } *73 60848.58 I74 608So32 } 1-74 608S2 o7 } , 75

So85S57 } 1975 60857.32 *75

* . 2.76 608S9.o8' F 1.76

60862 61 } l 76

60866 I4 } I 30

60867 94 } °.48 6086969 } I 77

6087I.47 } I 78 60873.25 } 1'79 60875°4 } 78

60878 61 } I @79

} 2 69

6c90° 3° T 2z.0<

Degre^* of great circles perpendicu-

and deg. of great

lar to the mcridian. above the

deg. at thf equator.

I _F

,oo

o.o6 o.93 4 58

I 3.99 32.6 I

63688 I Io.63 I 74 49

I 89*35 ao+.go z2X.12 238.oo 255.52 273*67 292v+2 3 I.74

328.33 33 I .60

35 I.97

353*69 355*42 357. IA

358987 360.6 X

362 34 364.O8 365.82 367 57 369.32 37 I.07 372.82

_w

374.58 376434 378.s I

379-87 38 I .64 382.94 383s43 385 I 9 386.97 388.75 39o.s4 392 32 3g4. l l

4I5e80 . A Q _

lIatitude¢.

1 *

c) fl // o ol o

5__ 10- _

lS -

zo _ _

2 -

30 - - 35 - - 40 - -

41 _ _ 42 - -

43 -

44 - - 45 - - 46 - - 47 _ _ 49 - -

Paris 48 So X 4

49 - 50 -

, u

5° 5- 5D IO -

50 15 - 50 20 - 5° 2S -

50 sC -

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5° 55 - ST O -

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51 5- 51 IO -

SI I5 -

5120 -

51 2g - Grecawich 5 1 2g 40

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of the three ecau;tiolls.

545 I %

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5 389*88



Gen. RoY's Accoynt :f a prapaJc;d trigoro?^Zurical Ope^azione

^ . S

pothefis of M. BOUGUER, lvhere the illcrenlellts to the deg;rees of the meridian altove that at th fquares of tile fllaes of tle latitudes.

, " , , f _

| Dii 3=l330 ; | DiS

IC4.40 6065Z.76 88+43 60 x o4* 67 o653,40 88 73160 iossss 606j-5.7+ 8945I ,60 so6.44 6066I.82 90*78 60 I08Z40 6O673.35 9I90 60 o8X7g o692.S9 92s22 60 o7.76 6072z,20 9I35 60

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77.61 60998@44 6s 8o 60c 77.4 I 60999.90 65.62 60< 77@>o 6s00I37 65s46 60c 77 OI 6I002.84 65.29 60c

74.60 61020.64 63.24 60c t.sQ btooR \^ 6T.: 6nr

Degrces of gLCAt circles obliqele to the nzerid;an, the diSel-ences betzzeen the meridional degrees, S

I)erpendiculal to thenl, beiXlg in the ratio of the dquares of the filles of obliquitt. yre^* of great les perpendicu- to the mcridian.

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w - - f - - - f - - f - - l - - - - - -- - - -- - - -

- - "S . v '

-

-

7; Z28

the meridian altove that at the equator folloxv tlle ratio of the 4th power, or Iquareel

s betzzeen the meridional degrees, anrl thofe of great c}rcles dquares of the Elles of obliquity.

- Degrees of longitude.

______

DiS. 6 I029.65

60I09.86 } I 9 4

58968 S3 } I586 20

52984 1 3 } 2 839 °O

+6865.74 }32I9639 I } 687 62

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38283.93 } 6906

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36885 35 } 854.52

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b

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24 Spkfnent Xo M4jo>Gaaeral Royss Accouxt of the Mode prot

poJ2d to befbZlowed termening the relvve Szzgatxon of thc

Royal Orvatorzes + C;reellwich and Paris. See p. I 88*

Read Nov. gf I 7X7* Sr 1f the account of tSwe propoSed trigorlometrical opera 1 tionsfor determining the diSerellce between the merf s;lians of the Royal ObServatories of Cawreensvich atld Parisy I haves at p 2XI. alld 233¢ had occafoon to remark on aIl itlconfifteIlcy buald in the futn of the three eguatioIlss 25 Rated by WI. BOUGUERS br obtainitlg the Iengths of degrees of great circles perpetldicular to tlle meridtan above rlleir coireSpoIldillg degrees of latitude without hktlg been aware of t5le true fource of the error} br svhi-ch I am itz- delzed to the illveRigarlons of the AIlronomer Royals whos l1aViIAg il2tad it OUt, obllgillgly commulliclted the Iiaze to nC aboXt tlle tiine o:E tlle annual receX of tEwe SocCty i t1at ttld of Ju1ze IaR.

At p %8(Q alld ag;ain at p. 3X3 and 3X+* of M Bou- wEs Bool the fubtraEive bratlch of the equationy or l:he part of D<} rl<e gravicetitric arc anfwering to the diSrerwce Joetweexl the rdii of curvature at the equator and giveIl latiw tude,, has earolweoufly beell expreSd in ssords Wtlls inRead of >45t}1S which the aRgebraic ybr?Zala juAly gives for it, and ac COrding to Fl1iCh M. BOUGUER}S tal3le has beetl accurately conzputed lElot fuIped:t;lg atly thirlg of this IBrtn no illtima

T t t X toa



466 tion of it^ apyzea;r;nb aluetlg the erZ42 arad- 1lo tzotice havig been hitElerto talibuX as ir as I ktlosv>* of Ibch a mEake extRtsng in that itzAly celeljrated worl llow fo matay years ir} circulatlotl tlaror3ghout the svol 1d ; liwReaLd thvreire of ,45 thsS

or WS0°6. parts of tie thi^rd roportiorwal9 the 1a{} addtittve mem- 1+tr of the equatioIan l £ilbflitutetl aXaol:}atv twith a certaiLl NlOCStiO) ll()wtStfs AS ated il] tlle beil*e-+trleltrorled pages) aSOUtltilZg 0I11 to 3 7 o )arts of the third proportionalg being the laigiatR tt-aat wou1d apply to the wl jole quadatltS svithouth producitlg abR*rd relts Thus I olStatned approximate de glaees of ,eat ttCls tt3tS of lotgitucle diffel Xg 51lt little from tEaoi of h4. BOUsUtRs atld COlNpEllfatTIlg itl a grent meaire althou;h -not altogntlaerX br tiaeftheN undiScovered cXufb of the rzaiRake of 21Qtl pat of the arc 1DGn for 4 _ ^, =zT<>,

e * ^ ..* - ^ss > e. 7 * * *

to atlnex a Spplernerwrary tablRe) svhele the degrees of great circles alld of*logztucle are ;iccurately computed by the cor- Teded hlberad:ivo bra.nch 4ths- of DG ln{lead oiF atTls as it 110W

fiatlds iw tlne orvgnal tableX From itziieEion it sv1ll appear that tiae maxs3zlutn of correAion amounts xzearly to 5 iitE-lotms at the 7Orh degree of latitudes dinzini>Ihing gradually hou thellce to the pole on one {ide alld the equntor on t-t)e other; where it vani*es. l[>he maximuln of corrediotl ir the- degreesh Qf lollgltudey amounti1wg to about xw fithoms is applicable between the sStll atld sgth dgree of latinde wliem M, 3OUGUERsS degree of the meridian becomes equal to his degme of longitude on the equator From this poillt, it dimilaithes gradually to t}ae pole otl olle fide, and the etator m tlje otherj, where it in like manner difappears.

_ . . ..

Wlt 1 regard to degrwes of great circles fituated obliquely to t-he Ineridian, it is I;Xfficiently obvious, that they are fo little affeded as to xlld¢r it o£ but Iinall importance whether they

..

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467 are correAed or not a 1)t1t brfuch as ale irupulous to :EraRional parts of fithorns theIi rnay fatis ttwemie:lves witll great

cility atad exaEttatX, by attenditlg tx3 atld proporttontng by the cornmon tabular diSrences as it} the bllowing exanaple of the appliCAtiOll Of the corrhiotl to oblique elegrees in theA atitdt 05 i tt11\\7iti ) z ° -9 Sf 49//.

, Origi!llal | StlPI) CorreEed Col reFted | ; u * * talDu,ar coI rezvb

Ollglty. . diSx tt<Ce degres

d:sScI e)ct; tot.

; t ,,_ - 'f v _,,, , , *

- arh, Fath Fatl1> Fath

:t A 1* t 5 te 0 45 ° 2,04.4 -I*65 -p 202*80 bIO7O,1,20

h 33 drS ff 78 24 -o63 { -t } 7 t bIIa S 5

t a a t e t; o t t-r 1 ts a t e r z

C.lIC>ttoSs fitllsltC! | 67 3go } 66.33 _O+5+ t + 65+79 l 6og26 $+

iell iCli3t1, tn Lht 7g ,5 } 44£32 -o 36 { + } 43 96 { 61257 60 l¢atsiti.lde of Greetl- . 6 r,

sViCil. gO o 4 I5.56 0§I2 4 + } 15*44 | 6 ' 44

Here 1t is to obirwred that half of rlae corrediotl is CQll

Rat1t1g to be appl;ed at tlle 4>tll degree of olbliquity atld {itace the diSt¢1Rces betistectl the terms itl the progre«otz eqtlaily

renOvet1: o11 eitler Lide frotn AS are always equal to each other, iS f8}lONv5s ttlAt the correEl;ed diikrences are to le applied with the coutraXy fgnX tl<OS betweell 45° atad the meridian beitg illb cTimitlutio1S; atld theSe betsveen 45° alld the eaR or weft pointS lbettlg in augtuttltatton of ttic lelagth-tilI, ill the firR cai the degree becom-es equa-l to that of latitude X and in the laIt- to that oiT a great circle perpendicular tQ the meridlall

1 s s r * r . s +

s ne 1 aDle or tomparlton p. 2274 1S no wnere aweded 1Zy the al^teration which 11;as beenX the fuhjedb of this difcuflion9< except it1 tllt tWQ laR lines from the bottom} as adverted tod atnnng the trrvza fubjoined to this ipplement.

Tlais illtimatlon of M. 330UGUERsS IniIt-akC in expreR;ng hi irmula iXF w;ords, was accompatwied with the converl;on of

that



mr

i4bb

tlat fcormula by L)r« i^NJAsttNz itlto tllc bllowitlg} adapted .

vx z+S xcvf zlat3coi 4 lats to laatural {ines - 30 -. a ls the exceS of tlle radius of curvature at the pole alJove that at the equator} atld the fign c)f either terln is otaly to be charaged when tlle doubled or quadrupled latitude becotne greater thall go°} alld lefs tllatl 270o.

I etnl:)race this opportutlity of metltionillg anotller circumv

IlCe wholly utlknown to me at ffie tiale my Paper svas cotn poid Frorn svhat hvls been tlaere faid3 at p z I te and Io oll to p zzo. it will probaljly be infbrred that I coll{idered tEle propoSed mode of determitling the diSrttaces of lotlgitude by rlle obServa- tions of t:he pole Rar, made svith a wery accurate inItrument, x ather as netstfi not krlowing that the fame idea) or olle 1learly the {itne tlad lJeire oceurred to rlae Ilev. lMr MICt-IELIJ, AI1 beel

tlXeatecl 011 by 11irn in his very iI]geIliQUS Paper it1 ti1e Pl-iloX phical lnrallfaAions Vol. LVI fE)r the yelr ff766. Tllat t smuR llave read that valuable perfbrtna1ace al)out the titne of ots publicatlon is not to be doubted; but in the lapfe of b

mally yecrrs7 every trace of it had gotle from my retneml)ratlce, ttherwite I +rould have moll cestaItlly refirred to it ill tllc -proper place, atld with the attetltiotl tllat it fo svell deferves. t-Iowevet l wi£hout enteriIg here into particulars, it will ol)vioufly appear:, that the one 11as IlOt been borrowed frcsm tlle vtller.

E R R A r A. T1w pe 95* 1* n7* ir between Gravelit3es and Calais read between WAtttIl and

Gavelilles. -And in 1* 28* for 56ei 4%t o't read 46) 52t Ott-

:irl p. 197, itl the diRatCe ov the parallel of Rodi-'Xs Som that of Dunkirkt fbr b 50t 5I 4lXr rea.d 6) 4O/ SItt +!/t

tll tile laft llne of the note at the bottorn of p. 2I7. inRead Of Iv 49t Stt.8, tt32 I 49 48 *8

{£} pf 2X, l * litze 6+ S;em the topf in£Leact of Tqyths) read X85 ths. In the 'I>;able of Cornpalifbn p %z7. the laSt lirxe but one of the IaR ceIzImn

l)z1t one fr +373*5 read ++37Z.o ln tlle Ilest line below, ir 38 I 64.o as fbr- ulerly corl^edetlf read 38iGzov And in the right-hand coluinz} 1a: liile bllt Ilc tloil} tJit lottel> f6r + I.X? rcad + I;t,6,



-

R

Supplementary Table for the D l

R - - - - | - -

-

10 _ -

u u u

-

6I029.62 } , 6I03i@64 } ;

6 I o38.o2 6I047@82 } II 6 106 30 3 24 6I084.82 6l I I0,54 } ; 6 I t4 I vs 2 } 3 6I I77¢I3 ,

6I I84*8I } 61 I92264 61200v57 } 6 X 204 93 l 61208174 } 6t2I7*00 } 6I225v39 } 6 I 233*89 } 6I242*49 t

6I249@77 } 6125I.I8 } 61259*96 i

R . ,

61 260.69 6l26I*43 } 61262*I6 61262*90 } 6I263z64 } 61264*37 } 6I265.II 6IZ65v84 } 6 I 266*59 } 6T267.p 6 I 268.o6 } 61268.80 }

oloO

* owo4 o.86 - o8s

o.67 -- o.74

-o86 -I,02 -I.33

W._ ,

_ I.34 -I .40 -I.SI

-I.53

-i'54 -I.66 - I 67 - l 77 -I.83 -- X *87 -I.87 -I*94

_ _S

- 1W97 -t*97 -1.95 -I 99

1.99 _ zo t

_ .oz

2.o3 - -z.o3 - 2.04 .-- z.on

-2*o}

-

- - J -

9.80 6¢48 o.s2 5.72 o.88 ,S.7 I

7.68

7.83 7@93 4.36 3@S I

8R6 8 39 80so

8.60 7.28 J.4l 8.78

o73

o.74 o¢73 o74 o74 o.73 o.74 o.73 o.7s o.73 o74 o74

Degrees of great Slpple- Sircles perpendictllar melltary to the meridian cor- correEtion reEted. fior {leg.

Of long.

_|__ __ -

2.02 6.38

Sum ofthe Supple- 3 equat. inenvary Tths of corre&;on c DGbeing for degv t fubtrac- of grcat r

toves circles. _ _B__

545vIt aooo S47oo8 I + o.O7 552.595 + o.82 558 739 -O.go 565*81 I _ o7o 567.7 I 5 -o 83 562.156 - o 99 546 285 _ 1.25

5I8v144 I-73 .

* 5 I0.955 -I ¢78 503@235 _ Iv90

S 494*946 - 2to7 w 49°s392 -2 I l u 486@244 - 2.16 @ 476*984 - 2 30 - 467X2X6 - 2X42

456.972 -2. 58

- 446.252 -2.75 + 436w936 --2.83

43S-o84 -2.85

- 423o488 3.o3

_ _

- 422.504 -3.o6

42. I .5O8 -3.o6

- 420.523 _ 3.07 4x9X534 3x I

- 4I 8s526 -3 I 3 _ XI753I 3tIx

- 4I64528 3.IS

- 4I5e524 -3*l6

414.5I5 --- 3.20

4I3l500 -3.2I

4 2e49I -3.2I o 41X477 -3*2%

Degrees of longitude correEted. p Places.

6 1029o62 DiS:

607199732; 68866 589687 666 } It85 02

Soo84 I X } 2839v 16 46864,4z } 32I9*20 _ _ . f 687 63

"t 1 igs,

e . } 67.83

re s 6S .

Equator

S. of France

Paris

Grl

L3titUiS.

__>_

o 8 S)

O O O

S - - I0 - -

I5 _

20 - -

25 _ _ 30 - - 35 - - 4° - - __

41 -

42

43 _ 43 32 44 - 45 - 46 - 47 - 48 - 48 50 49 - 50 _

so 5 50 IO

50 I5

5O zo

50 25

5° 3° 5° 35 5° 4° 5° 45 so 5o S° S5 SI O d



--- E - - I _ X - - * - w *- -W- ||W-- - -- - - - - - -- 1 *u||S__S--- - - sBa,B- _ _ - W-- - m - w - -- w --

w -- -

-- - - W-| -

SuppGe-

cc

.1

- v - -

Supple- mentary orreEtion for deg. of great circles. _ | -

um of the 3 eqllat. 45ths of BG belng ibtrac-

tive. _0

0 -

DiS.

6127102l°ff,4

6 1 2 7 3 o4 } 0 54

61275.76j °;74

6IZ77269 l - - - J 8v94

5 ^ 1°4 iS } s@1

6134023} 8580

fi } 42fi39

6X9 69 t 27 5

-B--

- w -

-

l |- | S S | - s- - - - | X - l u * * | y |- - y - l - |

u u u

I

- - 2.0 X

-2.02 |

2.o3 .o6

- 2.07 -2.09 -2.04

-2.09 - 2.07

2^o7 - - - Z Zo6 | -2.o8 - B 2. I I -2.09

_

-2.I3 L -2.20

-2.24 - s.27 - 2.30

-2*33 -2.35 - 2.35 - 2.34

,

-2*23 -2.I6 -I@87 -I.27 - oe74 -0.20

i -0.00

- l -

e28

-5 39 -5*47 - 4.89 -3.8 I

-2.I6

+0.00

S1

- - -

for the Degrees of the Earth.

| Degrees of longitude colreEted.

Degrees of great circles perpensliclllar to the meridian cor- reEted.

Places. Latitudes.

Deff.

3835°@97} 9wo7 $} 69Ey

i 'S; 55 f ; ; N

- " i4729@38

24547,73 } 1697'4

5 360 20 } 53 I 6@9 I oooo.oO } 5360^2Q

O S JJ

SI O O

51 5 - 5I I0 - 5I IS - 5t 20 -

5l 25 -

5t 28 4O 5I 3>> 5 35 - 51 40 - 5I 45 SI 50 -

5t 55 - 52 0 0

__-

53 - - 54 _ _ 155 - - 56 _ _ 57 _ _ 58 - -

l

58 42 47.4 59 - - 60 - -

4l In477 4I0.459

4°9 437 4o8.+o8 407.387 4064357 405@60 X 40s.3zo 404.2g i 403.253 402.2 t3

40X.X68 4oo I z8 399 o80

386.325 373-232 359.85 I

3462X87 332.30I

3I8.ZI3 308.o7S

303.963 289.569

-3.22 -3.22

e25 -3e29

3.29 3.3o 3.3o

.3s 3*32 3*32 3*37 3 37

-3.38 3*4o

-3@35 -3@73

3.9o 4.a7

-4*23 -4*39 _4.5 I

- 4@55 -4.69

-

Greenwich

6S - ?land 66 .o

7° 75 - 80 - 85 -w

Pole go o

2I7.320 I 98¢36 x I48.229 8 7 *82 I r 4 I .569 I 0* 309

0.000 1

-

o

Lap

N. B. The degree of the meridian, whofe center anfwers to the lattuds of 43° 32tt is in length 607 I4n54 fathomse



An Account of the Mode Proposed to be Followed in Determining the Relative Situation of the Royal...

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Transcript of An Account of the Mode Proposed to be Followed in Determining the Relative Situation of the Royal...