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 meridian, 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 hee- 
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 
