176 
APOLLONIUS OF PERGA 
a given point a straight line which shall cut off segments from 
each line {measured from the fixed points) hearing a given 
ratio to one another.’ Thus, let A, B be fixed points on the 
two given straight lines AC, BK, and let 0 be the given 
point. It is required to draw through 0 a straight line 
cutting the given straight lines in points M, N respectively 
o 
such that AM is to BN in a given ratio. The two Books of 
the treatise discussed the various possible cases of this pro 
blem which arise according to the relative positions of the 
given straight lines and points, and also the necessary condi 
tions and limits of possibility in cases where a solution is not 
always possible. The first Book begins by supposing the 
given lines to be parallel, and discusses the different cases 
which arise; Apollonius then passes to the cases in which the 
straight lines intersect, but one of the given points, A or B, is 
at the intersection of the two lines. Book II proceeds to the 
general case shown in the above figure, and first proves that 
the general case can be reduced to the case in Book I where 
one of the given points, A or B, is at the intersection of the 
two lines. The reduction is easy. For join OR meeting AO 
in B', and draw B'N' parallel to BN to meet OM in N'. Then 
the ratio B'N': BN, being equal to the ratio OB': OB, is con 
stant. Since, therefore, BN: AM is a given ratio, the ratio 
B'N': AM is also given. 
Apollonius proceeds in all cases by the orthodox method of 
analysis and synthesis. Suppose the problem solved and 
OMN drawn through 0 in such a way that B'N': AM is a 
given ratio = A, say.