64 ON THE ANTIQUE HOUR-LINES. 



straight line parallel to the ninth astronomical hour-line on 

 this projection j but the third hectemorial line is continually 

 approaching to the ninth astronomical hour-line ; the distance 

 between them at the horizon being HD — arc. 45° X tan. polar 

 distance, and afterwards it is [arc 45° — \ semidiurn. arc) X tan. 

 polar distance of the star ; \ semid. arc 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 j it follows, that the projections of the hectemorial 

 lines are not projections of great circles. If a 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 arc 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 



hectemorial 



