upon the Antique Dials. 93 



Demonstration. By Horsley's Projections of the Sphere, B. iv. pr. 10. 

 the angle APZ is the projection of the angle AEZ ; and, as is obvious, 



BEL = n. AEZ = nL. 

 Hence, ES' = cot I secnL = FI = tan BE* = tanD'. 



Again, draw CM parallel to PZ, and equal to CD, and join MH. 

 Then CHP, HCM, and PHN are right angles ; and we have 



P D 2 = D C 2 + C P 2 



= D C 2 + C H 2 + H P 2 

 = C M 2 + C H 2 + H P 2 

 = MH 2 + HP 2 . 

 But we have also N H 2 + H P 2 = P N 2 = P D 2 



= M H 2 + H P 2 ; 

 and hence, finally, N H = H M. 



It follows, therefore, from Horsley's prop. 9. ubi supra, that the arc 

 which measures the angle PNX is projected into PX ; that is, the arc D' 

 is projected into PX. It has also been shewn that the angle AEZ is pro- 

 jected into APZ. Hence X is a point in the curve. Q. E. D. 



XII. 



The specific details of the construction are adapted to cases where the 

 point of contact is not of a latitude either very high or very low. Very 

 slight modifications, however, will adapt it to these cases also. The opera- 

 tions are intentionally rendered as simple as possible in reference to the in- 

 termediate latitudes, which could not have been done had we attempted in 

 the same paragraph to render the description practically applicable to all 

 cases. We proceed to consider these particular states of the problem. 



VOL. XII. PART I. N 



