64 ON THE ANTIQUE HOUR-LINES. 
straight line parallel to the ninth astronomical hour-line on 
this projection; but the third hectemorial line is continually 
approaching to the ninth astronomical hour-line ; the distance- 
between them at the horizon being HD = are. 45° x tan. polar 
distance, and afterwards it is (arc 45° —'} semidiurn. arc) X tan. 
polar distance of the star ;\ + semid. are increases, but never at- 
tains to be 45°, so that the distance never becomes equal to. 
nothing, and tan. polar distance increases indefinitely. 
All great circles are seen under the form of straight lines in 
this projection of the sphere ; and therefore the projection of 
one great circle cannot be an asymptot to the projection of 
another ; it follows, that the projections of the hectemorial 
lines are not projections of great circles.» Ifa straight line be 
drawn through the point H, (in the figure on the margin,) cut- 
ting off a given aliquot, one-half, for example, from a semi- 
diurnal are on the projection, it will cut off a smaller aliquot. 
from the meridional extremity of the other semidiurnal arcs, 
in proportion as they are nearer to the point H; and in order 
that a straight line drawn from H may cut off the same aliquot 
part from several concentric arcs 
which are included. between the 
versed sine HN and sine HO of the 
outer arc, it is necessary that the 
chords of these arcs be parallel to 
each other; which happens ‘only 
in the case where all the arcs are of 
90°, then H coincides with P, and 
then the straight line which cuts off 
the same aliquot from every arc, is a line passing through P 
the centre ; and this sole case is a central projection of a sphere 
so placed, that each semidiurnal arc is 90°, the poles of the 
equator being in the horizon. In ‘this position alone are the 
ey hectemorial 
