( 392 ) 



It is clear tliat the central projection /' of / coincides with the 

 trace e of the projecting plane 01 = e, and at the same time that 

 / is determined by its point of intersection and by one of the two 

 vanishing points ; the second will be found by letting down the 

 perpendicular OS out of O to /, and by setting otf at the other side 

 of OS an angle equal to the parallel angle, formed by OS and 

 the only existing ray parallel to /. But further can be remarked 

 that / is also determined by its two vanishing points, or what comes 

 to the same by its two points at infinity ; to find / we should but 

 have to bisect the angle formed by the two projecting parallel rays 

 of /, to mark on the bisecting line a segment OS corresponding 

 to Vj /^ Vi^ V^^ as parallel angle, and to erect the perpendicular 

 in >S' on OS. 



The line / is divided into four segments by its two points at infinity, 

 its point of intersection and its two points F^, y-^^ (whose projections 

 lie at infinity), and /' in like manner by its points at infiuity P\^, 

 P\^, the point D, and tlie two vanishing i)oints ]"\, V\ of /; the 

 connection between these diflerent segments of / and /' is as follows. 

 To the infinite segment V^^D corresponds the finite segment V\D, 

 and to the finite segment BP^ the infinite segment DP\^; to the 

 points between P, and P^ no projections correspond, because the 

 projecting rays of these points are divergent with respect to r ; 

 to the infinite segment P^V^^ on the contrary a segment of /' 

 again corresponds, namely the infinite segment P\^ V\. There now 

 remain on /' only the points between the two vanishing points, to 

 vi^hich also belongs 0, ; to these no points of / correspond, their 

 projecting rays being divergent with respect to /. 



§ 4. If a line lit is to cut the surface of the parallel cone 

 in two points, the length of DO^ may not exceed a certain upper 

 limit, so that the results just found do not hold for all lines i t. 

 Let us again suppose through 00^ an arbitrary plane e, and let us 

 now first regard 00, itself. If we let down out of O, on to 7;, 

 the perpendicular 0/f, then because p^ is parallel to e, the angle 

 7'0^P\^ is the parallel angle beh:)nging to(>i7', and therefore angle 

 7\),0 is smaller than this parallel angle, because 0,0 cuts the 

 line j>.j (namely in ^^>); and 0(),P' ., being equal to 90^, the paral- 

 lel angle TC^.'p,^ > 45°, and angle 7Y>i(><45°. If in « we move /, 

 first coinciding with 00„ in such a way that it remains in D per- 

 pendicular to e, namely towards the side of r\^ (therefore from 

 P\^), then the perpendicular D2' on y>, becomes continually greater, 

 and so (see N". 1) tiie parallel angle TDP\^ continually smaller; 

 as soon as the perpendicular DT has attained such a length that the 



