The Imaginary in Geometry 



41 



We saw in fig. 9 that the row of blue points was projective to 

 the row of corresponding black points, in fact, that they were 

 similar, the infinite points corresponding. When we chose our 

 coordinates, however, so that they become infinite on a line z 

 then it is the intersections of blue line and black line with z that 

 correspond. Call these PJ', PJ. Then P x , P 2 , P x completely 

 establish the correspondence of the two rows, giving, as they do, 

 three pairs of coordinated points. It is therefore, possible, given 

 two elements P x , P 2 of a line and given on P 1 P 2 the part iV of a 



third element to choose the other part P z " anywhere whatever on 

 P X "P 2 ". After that all is fixed. For example, given P x , P 2 and 

 with P 3 ' not on P X P 2 ' fig. 35 shows how to find P". It replaces 

 fig. 11. If Poo is changed while P x , P 2 and P z ' are fixed P./' in 



41 



