94 



Improvements in Fundamental Ideas 



cut circle in S, and then SH to cut MM in B, it is evident, P 

 and B are equidistant from the centre of the circle, and there- 

 fore that B is determinahle. The point T where MM cuts 

 NN is also. And the angle HC right to S being equal, LC 

 right to S is equal NN right to MM; and therefore a circle 

 can pass through TBH and Q. Hence the point H where 

 this circle cuts the given one is determinable, and hence HB 

 parallel to MM, and point B, line HQ, and point C. 



Given the points P and Q, in given straight lines MM NN, a 

 point C and an angle ' right' can be found such that E and 

 F being any two points making PE • QF equal a given magni- 

 tude m • n, the angle CE right to F shall be equal angle 6 right. 



It is evident that the determination of the point C involves 

 the solution of the following problem : — Given the base PQ, 

 the difference of the angles at the base, and the rectangle 

 under the sides of the triangle PQC to construct the triangle. 

 It may also be remarked that when MM and NN coincide, 

 the difference of the angles at the base is zero ; and when 

 moreover m . n is negative and greater than one-fourth of 

 PQ . QP, then we have in an implicit manner the theorem 

 which Chasles establishes concerning homographic divisions 

 on the same line whose double points are imaginary. See 

 Chesles' Geometric Superieure, page 118. 



Given a pair of homographic divisions on two straight lines 

 MM NN, and likewise a second pair of other homographic 



