MISCELLANEOUS PAPERS. 



221 



Two triangles, ABC and A'B'C, are homologous, 



1. When the rays AA', BB', CC, joining corresponding points, are 

 concurrent ; or, also, 



2. When the points of intersection of corresponding sides A B and 

 A'B', BC and B'C, CA and C'A' are collinear. 



Each of these two definitions as a hypothesis necessitates the other 

 as a thesis. 



Casey, in his Sequel to Euclid, which is very rich in beautiful ex- 

 amples and propositions, but without an organism, reduces advanced 

 geometry to an incoherent mass of metrical facts. To prove the 

 propositions concerning homologous triangles, he introduces ratios of 

 areas of triangles, and applies the theorem of Menelaos concerning 

 transversals.* The same method is followed in most treatises on 

 plane geometry. The immortal elements of Euclid, which in them- 

 selves are of rare beauty and rigor, lead very soon to sterility when 

 applied to projective properties of figures. 



Fig. 1 



Homologous triangles appear in the simplest manner as plane sec- 

 tions of triangular pyramids. Let V be the vertex of such a pyramid 

 and ABC and A'B'C the intersections of its edges, with two oblique 

 planes, P and P', respectively. Let S be the line of intersection of P 

 and P'. A glance at the figure (fig. 1) shows that AB and A'B', 

 BC and B'C, CA and CA' meet in points of S. 



* Loc, cit., book sixth. 



