224 



KANSAS ACADEMY OF SCIENCE. 



Fig. 4. 



The horizontal and vertical projections of any point in the bisecting 

 13lane of the second and fonrth angle coincide. From this it follows 

 that the projections of the piercing point of any line with the bisect- 

 ing plane, which shall be designated by U, coincide, and are conse- 

 quently obtained as the point of intersection of the two projections of 

 the straight line. Let 1' and 1" be the projections of a line 1 and Di 

 their point of intersection. In a similar manner, we designate by Ai, 

 Bi, Ci, . . . . ; ai, bi, ci, . . . . the coinciding projections of points A, B, 0, 



and lines a, b, c, in U. As two points determine a straight 



line and as the projections of a point in U coincide, it follows that the 

 projections of any line in U coincide also. Any plane P with the 

 traces ti and t2 intersects U in a line u, whose projections coincide in 

 ui. Evidently the traces ti, ta and the lines u and ui meet in the same 

 point T of the ground line. The line u is therefore determined if 

 another point is known. To find such a point, assume any straight 

 line 1 in the plane P, figure 4, and construct its intersection L witli 

 U. The straight line connecting T with L, or in the projections T 

 with Li, is the required line. Every line in the plane P intersects the 

 line u ; hence its projections meet in a point of the line ui. From 

 this it is seen that the projections of points, lines and figures in a 

 plane P are related by the two laws : 



1. Corresponding projections of lines meet in points of a fixed 

 straight line (axis of affinity). 



2. Corresponding projections of points are situated in parallel 

 lines. 



