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00^ ^) an acute angle smaller than the parallel angle, from which 

 ensues that OP (or perhai)S the prolongation of PO over O) must 

 meet the piclui*e plane; the point of intersection P' is the central 

 projection of P. However, for P outside the cone the acute angle 

 between OP and 00^ is greater than the parallel angle; so now 

 OP is divergent with respect to t, from which ensues that points 

 outside the cone possess no projections at all ; points on the cone on 

 the contrary do, but these projections lie at intinity. 



From the fact that points outside the parallel cone are not projectible 

 we need not infer that these points cannot be determined by Central 

 Projection ; if such a point is regarded let us say as the point of 

 intersection of two straight lines, and if these can be projected by 

 Central Projection, their point of intersection will also be determined 

 in this indirect way. 



3. Let a straight line / be at right angles to t in a point D 

 of T. As the line 00^ is also perpendicular to t, it is possible to 

 bring a plane through / and 00 -^j the trace of which connects O^ 

 with D. This plane will intersect the cone jc in two generatrices, 

 Pi, Ih'^ we now assume that / cuts these two lines in two points 

 Pj and P^, and — to fix our thoughts — that P^ lies between 

 B and P^. The line / possesses two points at infinity, V^^, V^^, 

 which both lie inside the parallel cone; let us suppose that Fj^ 

 lies under the picture plane and V^^ above it, then the succession 

 of the points on / is this: T^i^, D, P^, P^, V^^. 



The projecting ray Ol\^ cuts r in a point F'l of ^ lying between 

 D and 0^ ; we shall call it the first vanishing point of /. In like 

 manner the ray OV^^ prolonged over will cut the line e in a 

 point Y\ lying in such a w^ay that O^ lies between V\ and V\; 

 we shall call V\ the second vanishing point of /. The point 0^ 

 does not lie in the middle between V\ and V\, on the contrary 

 it is closer to V\; if namely we let down the perpendicular OS owt 

 of to I a quadrangle is formed with three right angles, namely 

 at O^, D and S, and from this ensues that the fourth ./ SOO^ is 

 acute. Kow OS is the bisectrix of / F^^^^Fa^, and therefore the 

 perpendicular in O on (AS' the bisectrix of ^ V\OV\; this perpendicular 

 must be placed, as /^SOO^ is acute, between 00^, and OV\y and 

 from this ensues ^ V\00^ < X V\00,. If we now let the 

 rectangular triangle V\00^ rotate about the side 00^ till it lies on 

 the triangle V\00i, thcji we can immediately find that 0^ V\<:^Oi V\. 



1) By OOi we understand tlie straight line prolonged at both ends uulimiledly, 

 by OP however a semi-ray starling from 0. 



27* 



