26 INVESTIGATION of certain THEOREMS 



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

 the chord meafured will not be affected by them ; the amplitude 

 of the arch indeed may be affected by fuch deflections, if they 

 happen at its extremities ; but the effect: 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 minutes, 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 fhall 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 this paper. The 

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

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

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

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



- rt* cof tp a cofp 1 _. 



as in § 4. r zz z zz — 7™, according to 



the method of reduction followed in the preceding articles of 

 this paper. Then, becaufe /i — ~^r = 1 +| fin 9* near- 

 ly, we have 3»oi coftp (1 -j-^finp 1 ) = a cof<p-f <?un<p 2 cof<p, 



or if we divide by cof <p, ~ zz a + c fin <p\ Let ^ = /, then 



/ — a + c fin <p 2 . 



32. Again, 



