the Trigonometrical Operation. jgg 
The following folution * of that important problem, being the 
only unexceptionable one that 1 have received, is here given in 
the author’s (Mr, Dalby’s) own words. 
Let CE and CP (Plate X. fig. 2.) reprefen t the equatorial 
and polar femi-diameters of the earth, confidered as a fpheroid 
flattened at the poles; P£ and PN two meridians ; pe and pn 
two correfponding ones (that is, in the fame planes) on a 
fphere, having the fame center C. Let the points a , h , A, B, 
on the fphere and fpheroid have the fame latitudes refpedlively. 
Draw the radii tfC, b C, and the verticals AG, BW. ' 
Then, becaufe the angles AON, BDE, in the fpheroid, are 
always equal to the latitudes of the points A, B, thefe angles 
are therefore refpedtively equal to the angles aCn 9 bCe , in the 
fphere, and confequently the verticals AG, BW, are parallel to 
the radii tfC, bC. 
Let the latitude of B or b be greater than that of Aori?; 
and let it be required to make the horizontal angle PAr on the 1 
fpheroid equal to the angle pab , or what the horizontal angle 
would be on the fphere. 
Becaufe the angle pab is meafured by the inclination of the 
1 / 
planes, aCb , aCp , and AG is the common interfeftion of all 
the planes of the vertical circles at A, and is parallel to aC 9 
and in the fame plane ; therefore, when the horizontal angle 
PAr is equal to the angle pab ? the planes GAr, CA muft be 
parallel to each other ; and confequently Gr, the line where 
the plane GAr interfefts the plane of the meridian EP, is pa- 
* From my correfpondence by letter, and otherwife, with Dr. Maskelyne, I 
had reafon to hope, that he would have favoured me with lome communication 
on this fubjeft. No doubt, he has been prevented by other bullnefs , but he will 
probably give his method of folving fpheroidical triangles to the Royal Society on 
feme future occafion. 
Vol. LXXX. C c 
raiiel 
