68 ON THE ANTIQUE HOUR-LINES. 



equator and meridian. This projection, which is drawn at 

 figure 3d, contains some of the hectemorial lines from the point 

 of their contact with the greatest wholly unseen to the points 

 of their contact with the greatest wholly seen parallel. It forms 

 a polar dial for the latitude 66° 30, when placed parallel to the 

 plane of the sixth astronomical hour-line. 



The lines thus drawn in the above projections are curved; 

 consequently they are not the projections of great circles ; nei- 

 ther are they the projections of small circles, for if they were, 

 they must necessarily touch the horizon at its intersection with 

 the mid-day portion of the meridiau, because in that point the 

 hectemorial lines converge, and do not go on the other side of 

 the horizon. Now, if small circles so placed, be drawn on the 

 sphere or projected on a plane, it will be found that their 

 course deviates entirely from the course of the lines bounding 

 the hectemoria. The hectemorial lines, therefore, do not coin- 

 cide with small circles on the sphere, nor with conic sections 

 on the central projection. 



The projections above exhibited shew that each pair of liec^- 

 temorial lines for a given meridian and latitude, is an equicru- 

 ral curved line ; but this is only one branch of an entire curve, 

 because the diameter whose extremity has traced a pair of 

 these lines on the surface of the sphere, has still to complete 

 its revolution, which is done when it has arrived by progressive 

 and continuous motion at the point from which it set out. In 

 order to accomplish this with the same kind of motion with 

 which they described the first branch, the extremities of the 

 diameter leave the two parallels that touch the horizon, and 

 proceed to cut off the same aliquot part from the semidiurnal 

 arcs belonging to this second point of departure, as they had 

 done from the semidiurnal arcs of the first point of departure; 

 the second point of departure is to be considered as the mid- 

 day point of a horizon on the opposite side of the equator to 



the 



