ON THE ANTIQUE HOUR-LINES. 63 



In this view the projections of the parallels are circles, and 

 equal arcs of the parallels are represented by equal arcs of their 

 projections : the construction of the lines which intercept one- 

 sixth part of each semidiurnal arc, is therefore performed by 

 dividing the projection of each semidiurnal arc into six equal 

 parts, and connecting each point of division with its corre- 

 sponding points on the projections of the other semidiurnal 

 arcs. 



In figure 1st, these hectemorial hour-lines are seen to con- 

 verge at that point of the meridian which is marked 66° 30'. 

 This is the point of contact of the horizon, and greatest unseen 

 parallel ; it is also the point where the^mid-day part of the me- 

 ridian cuts the horizon. At this point the semidiurnal arc is 

 indefinitely small, and therefore the lines which divide it into 

 six parts must be indefinitely near to each other, or, in other 

 words, must converge. 



At this point of convergence each hectemorial line is incli- 

 ned at a considerable angle to the meridian. As the line pro- 

 ceeds, the inclination becomes less, till it is nearly as small as 

 the inclination of the astronomical hour-line, which this hecte- 

 morial line cuts at the equator ; and the hectemorial line on 

 this projection is asymptotic with that astronomical hour-line. 

 For the distance of their intersection from P, measured on the 

 plane of projection, is infinite, being equal to the distance of 

 the intersection of the equator and plane of projection, two 

 planes parallel to each other; and however far the projection is 

 extended, the two lines approach indefinitely, but do not 

 meet. 



Take, for example, the third hectemorial line HS (in the 

 figure on the margin) which cuts the ninth astronomical hour- 

 line at the equator ; the projection of a great circle which in- 

 tersects the ninth astronomical hour-line at the equator, is a 



straight 



