the 'Trigonometrical Operation . ipj 
Becaufe AG, ‘lAV, BW, rG, are parallel to aC, AC, the 
angles vWB, AGr, will be equal to the angle aCb, or arc ab , 
therefore the arcs ^B, Ar, will each be equal to the arc ab ; 
that is, they are arcs of great circles of the fame value, in- 
tercepted between the meridians PN, PE, at B and A. 
Draw GR perpendicular to the vertical BW ; then, becaufe 
BW and rO are parallel, it will alfo be perpendicular to ; G ; 
and becaufe the axis PW is the common interfeftion of the 
planes of all the meridians, and BW, rW, are in the plane 
of the meridian PB, therefore GR is in that plane ; and be- 
caufe the angle WBr, made by the vertical and meridian, and 
the angle GRB, are right ones, therefore GR is equal to the 
arc Br nearly, and confequently is nearly equal to what fub- 
tends the difference of the horizontal angles on the fphere and 
fpheroid. 
And if GS be perpendicular to the vertical GA, it will be 
equal to the arc Av nearly, and therefore GR, GS, or the arcs 
Br, Av, will be as the cofines of the latitudes of B and A. 
Draw AK the tangent to the meridian at A, to meet the axis 
CP produced ; alfo draw AH perpendicular to the vertical AG, 
to meet Gr produced ; through H draw KHT, and join AT, 
Then, becaufe the points K, PI, are in the plane of the hori- 
zon of A, the line KHT will be in that plane ; and becaufe 
rH and BT are in the plane of the meridian BP, therefore HT 
is alfo in the fame plane, and is what fubtends the angle TAPI, 
the true difference of the horizontal angles, which, when the 
fpheroid is given, may be determined as follows. 
From the nature of the fpheroid, find the length of the ver- 
tical AG ; alfo the points G and W, where the verticals meet 
the axis : then, becaufe the angle AKG is equal to the latitude 
of A, and AGH is its complement, GK and AK will be given. 
C c 2 Let 
