aud Elementary Theorems of Geometry. 



95 



divisions on these same two lines ; to find two points 00', one 

 in each line, such that they shall be corresponding points in the 

 two pairs of homographic divisions. 



ANALYSIS. 



According as the first pair of homographic divisions cut the 

 lines MM NN into proportional segments or not, find a 

 pair of points AA' in these lines, such that AO shall have 

 to A'O' a constant determinable ratio, or find the two points 

 A and A' so that AO . A'O' shall be a constant determinable 

 rectangle. And according as the second pair of homographic 

 divisions cut MM NN into proportional segments or not, 

 find the points B and B in these lines such that BO and B'O' 

 shall have a constant determinable ratio to each other, or 

 that BO . B'O' shall be of a constant determinable magnitude. 



Now since the points AA' are given, if we assume a point 

 c, then (by one of the preceding porisms) we can find a point 

 D and circle DFC such that OD and O'C will cut in some 

 point F in circle DFC. 



And since B & B' are given points, we can find another 

 point G and circle GCE (by one of the preceding porisms) 

 such that OG & O'C will cut in some point E in circle GCE. 



Now since angle GI right to E or = CI right to E or 

 F=DI right to F or O, therefore a circle can pass through 

 GID and O ; but GDI are given points, therefore the point O 

 in which this circle GDI cuts NN is given, and hence OGE 

 and CEO' and point O' in MM. 



Given the lines 1, 2, 3, 4, and the points A, B, C, D, one in 

 each line ; given also the points a', b', c', d'; through a' and b' 



