c6 INFESTlGJ-tlON of certain THEOREMS 



tions, when the redudlions are all properly made, the length of 

 the chord me;ifured will not be affe(fled by them ; the amplitude 

 of the arch indeed may be afFedted by fuch defledlions, if they 

 happen at its extremities ; but the effedl of this error will be 

 rendered the lefs, the greater the arch that is meafured. We 

 may fuppofe, therefore, that the chord of a large arch of a pa- 

 rallel of latitude is meafured, and the amplitude of the arch itfelf 

 at the fame time accurately afcertained. This laft may be done, 

 either by meafuring the convergency of the meridians, if it be 

 in a high latitude, or by any other method of afcertaining dif- 

 ferences of longitude which admits of great accuracy. The 

 chord being thus given in fathoms, and the arch fubtended by 

 it being given in degrees and minixtes, the radius of the parallel 

 itfelf becomes known. 



31. Now, if we would compare the radius of a parallel thus 

 found, with a large arch of the meridian, we fliall have by that 

 means a determination of the figure of the earth, not lefs to be 

 relied on than that given in the beginning of tliis paper. The 

 inveftigation is eafy by help of the tlieorems in § 5. and 6. 

 Let FO be the radius of a parallel to the equator, which pafles 

 through F, the latitude of which is (p, and is fuppofed known ; 

 and let FO found by the method juft defcribed be = r, then,- 



n* cof C! a cotip' ,. 



•IS in ^ A. r = - '=: — ^ =, accordmg to 



' a 



the method of redudion followed in the preceding articles of 

 this paper. Then, becaufe /i - ^fi"^' = i -|-^ fin <p' near- 

 ly, we have r = fl cpf 9 (1 + I fin(p') = a cof (p 4- c fin <p' cof <p, 

 or if we divide by cof ?>, ■^ = a + c{m(p\ Let ^ = /, then 



/ zz rt -f- £• fin(p^ 



32. Again, 



