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KANSAS ACADEMY OF SCIENCE. 



ures are constructed and investigated with reference to the greatest 

 attainable simplicity. I can say from experience that there is hardly 

 a subject in descriptive and advanced plane geometry in which the 

 student takes more real interest than in this method of presenting the 

 construction of plane figures in descriptive geometry, and of intro- 

 ducing propositions of projective geometry. It seems to me the most 

 natural way to higher geometry. 



IV. COLLINEATION. 



As in the previous chapter, I shall start from elementary construc- 

 tions in descriptive geometry and through the study of perspective 

 gradually arrive at the most general expression of collineation. 



A central projection, or a perspective, is determined by the plane of 

 projection (pictorial plane) and the center (eye). Assuming the plane 

 of the paper as the plane of projection and any point in space as the 

 center, it is possible to construct the perspective of any figure in space 

 in this plane. The center can be most easily located by a circle in 

 the plane of projection. The radius of this circle is the distance of 

 the center from the plane and "the center of the circle is the ortho- 

 graphic projection of the center of projection upon the plane of projec- 

 tion. This circle has been introduced into geometry by Professor 

 Fiedler, of Zurich, and he calls it distance circle (distanzkreis).* 



Fig. 5. 



Let II' be the plane of projection and II an arbitrary plane, whose 

 projection upon II' shall be made from a center C. Let S be the line 

 of intersection of II' and II, fig. 5. To obtain the projection of any 

 point P' of any point P in II, connect P with the center C ; then the 

 point of intersection of this connecting line (indefinitely produced) 



♦Darstellende Geometrie, vol. I, 1883. It must be mentioned that Courinery already uses a 

 "cercle a distance" in his Geometrie Perspective, Paris, 1828, 



