MISCELLANEOUS PAPERS. 



223 



Fig. 3. 



Any orthographic or central projection of this configuration leads 

 immediately to the original proposition. We may conversely use re- 

 lations in a plane to establish propositions in space. Take, for 

 instance, four circles in a plane and construct their external centers 

 of similitude (fig. 3). It is found that they form a complete quadri- 

 lateral, whose six points are 3 by 3 coUinear. The four circles may 

 be considered as projections of spheres in space. Here the collin- 

 earity of the vertices of corresponding common tangent cones still 

 exists, and we have therefore the theorem : 



The six external centers of similitude of any four spheres in space 

 are coplanar. 



Similar propositions hold for the internal centers. 



III. ORTHOGRAPHIC PROJECTION. 



In most text-books on descriptive geometry no attention is paid 

 to certain geometrical principles which, as it Avill appear, form the 

 base for nearly all projective constructions. It is also remarkable 

 how easy these principles and relative propositions may be derived 

 from exact intuition in space. From this point of view the necessity 

 of including certain geometrical propositions in a course on descrip- 

 tive geometry is imperative. I shall illustrate the points in question 

 by the treatment of affinity of figures* (homology with an infinite 

 center). 



♦Fiedler: Geometrie der Lage, voL 1, pp. 1-115. 



