406 Mr. atwood’s Theory for the Mensuration 
is proportional to the fine of FI and the cofine of DL jointly- ; 
confeqnently the fine of FI being the fame, fin. f ED is propor- 
tional to the cofine of DL ; this will lead to the reafon why in. 
enumerating (art. 4.) the * conditions which limit the magnitude 
of the obferved arcED, the pofitionof the fecondary KP, in refpect 
of the point of interfeclion Q^and of the fixed fecondary KO,, 
was annexed : for it will appear, that every thing elfe being the 
fame, the magnitude of the arc ED will depend on the pofition 
of the fecondary KP, whether it be on one fide of the fixed fe- 
condary KO, or on the other, the angles PKO,^KO, being equal. 
Having fet off Op — OP draw the fecondary inter feeling 
the parallel FIMU in the point U ; and through B and U draw 
the arc of a great circle BU W ; take UW = BU ; and through 
D and W draw the arm of a great circle DW : then by the 
conflrudion and demonflration in art. 6. the angle fubtended 
by the obferved objects will be meafured by the arc DW, and it 
will be eafy to {hew, that DW is not equal to DE, except in 
two extreme cafes ; that is, when the fixed plane of reflection 
DFB is either coincident with the primitive fecondary KO or 
perpendicular to it. Through the points F and U draw the 
arc jo f a great circle VFU, and from D draw the arc DV 
perpendicular to VFU: fince + the fines of half the arcs 
DE, DW, are in a proportion compounded of the proportions 
of the fine of IF to the fine of FU, and of the cofine of DL 
to the cofine of DV, the fines of IF,. FU, being equal by the 
conftr notion, the fines of half the arcs ED, DW, will be in the 
fame proportion with the cofines of DL and DV , which are 
evidently unequal confeqnently, the fines of half the arcs 
DE, DW, and therefore the arcs themfelves, muff be unequal. 
11. The angles PKO, OKjft, remaining equal, when the 
fixed plane of refledion BFD (fig, 4.) is coincident with the 
* Compare fig,. 2, t Supra, 
fecondary 
