Jf j 6 Gen. Roy’s Account of 
Let a and b on the fphere have the fame latitudes and difference 
of longitude as A and B on the fpheroid, and find the angles 
pah, pba , and the arc ab , or angle aCb ; then becaufe AG is 
given, and the angle AGH equal to the angle aCb, AH will 
be given ; with AH and AK, and the included angle HAK. 
(equal to the fpherical angle bap) find the angle AHK, and 
alfo KH ; then, becaufe the triangles KHG, KTW, are in the 
fame plane (that of the meridian BP) and GH is parallel to 
WT, thefe triangles will be fimilar. Hence GK : HK :: 
WG : TH ; now HA, HT, and the included angle AHT (the 
complement of AHK) being given, the angle TAH, the differ- 
ence of the horizontal angles, will be given. 
Example. Let the fpheroid be that of M. Bouguer; and let 
the latitude of A be 49 0 40', of B 50°, and their difference of 
longitude c° 3c 7 . 
From the nature of the fpheroid, the radii of curvature of 
the meridian at the equator and the pole, will be 3465507 and 
3524069 fathoms nearly; their difference is 58562 fathoms, 
the length of the evolute of the meridian ; and the vertical 
AG = 3465507 4-Jh.x 585624-4 x 58562 x fine 49 0 4o'V + T 4 r 
x 58562 x fine 49^ 40')“ = 3509769.5 fathoms; alfo OG = T l 
x 58562 +_ 4 T x 58562 x fine 49° 40 1 2 = 40307.66 fathoms; 
and DW zz JL x 58562 + Jt- x 58562 X fine 50°* = 40397.23 
fathoms. Now, the angles GOC, WDC, being = the lati- 
tudes of A and B, we get CG = 30726 .i 6, and CW =.30946.08 
fathoms, their difference being 219.92 fathoms = GW. 
The fides pa, pb, being equal to 40° zo' and 40° refpe&ively, 
and the included angle = 3c/, will give the angle pab — 43 0 
■ - • ■ • * 
5i / .48".2,, the angle pba — 135 0 45' i 6 "a, and ab, or the 
angle aQb, = 2/ > 
Now 
