J74 INVESTIGATION 6f CERTAIli THEOREMS 



To determine length of the meridian, comprehended between the northern 

 LuJe by com- ^"^ Touthem extremities of the illand, amounting nearly to 

 paring an arc of nine degrees. 



Ihat'^f a "Itllel ^^ '^ P'^'"* '^^"^ ^^^^ ^^^^ already been faid, that the refult 

 in fame lat. deduced from this coraparifon would poITefs every advantage, 

 and would be entitled to more credit, than any determination 

 bf the figure of the earth that is yet known. 



34. On the fuppofition that, in a furvey of a country, the 

 meafurement is made along a feries of triangular planes, all 

 given in polition and magnitude, there is yet another method 

 of determining the ligure of the earth, more general than any 

 of the former. On the fuppofition juft mentioned, it is evi- 

 ' dent, that the length of a firaight line or chord, drawn from a 

 given angle of any one of thefe triangles, to a given angle of 

 any other of them, may be found by trigonometrical calculation. 

 Let the latitudes be obferved at the extremities of this chord, 

 and alfo the difference of longitude ; then, from the nature of 

 aa ellipfoid, the length of this fame chord may be expreffed, in 

 terms of the axes a and b, together with the latitudes of the 

 extremities of the chord, and the difference of longitude be- 

 tween them ; and this expreflion being put equal to the length 

 of the chord meafured, will give an equation, in which all the 

 quantities are known, except « and Z;. Further, if a ==& -f- ^* 

 and if the faid expreflion be reduced into a feries, with the 

 powers of c afcending, that feries will converge very rapidly, 

 becaufe c is fmall in refpe6l of a; then, for a firfl approxima- 

 tion, we may rejed all the terras that involve the powers of c 

 higher than the firfl, by which means we fliall have a iimple 

 equation of the form ma -^-nc =:l, where m and n are functions 

 <of the latitudes and difference of longitude, and I is the length 

 of the chord. 



Now, if a fimilar equation be derived from the meafurement^ 

 of any other chord, thcfe two equations will give a and c in the 

 fame manner as in § 6 ; and thus, from the meafurement of 

 any two chords, the figure of the earth will be determined. 



33. The length of the chords, thus meafured, fliould be 

 great, fo that they may, if pofTible, fubtend angles of feveral 

 degrees, and their pofition will be moft favourable when one. 

 of them is in the plane of the meridian, and the other nearly 

 at right angles to it. The numerical computation will be 

 found lefs laborious than might be imagined ; but the com* 



plet© 



