A history of Greek mathematics - Wilbourhall.org

176 APOLLONIUS OF PERGA
a given point a straight line which shall tut off segments from
each line (measured from the fixed points) bearing a given
ratio to one another! Thus, let A, B be fixed points on the
two given straight lines A C, BK, and let be the given
point. It is required to draw through a straight line
cutting the given straight lines in points M, N respectively
such that AM is to BN in a given ratio. The two Books of
the treatise discussed the various possible cases of this problem
which arise according to the relative positions of
.
the
given straight lines and points, and also the necessary conditions
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 OB meeting AG
in B', and draw B'N' parallel to BN to meet OM in N'.
the ratio B'N' :
BN,
being equal to the ratio OB' :
OB,
Then
is constant.
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 in such a way that B'N :
given ratio = A, say.
AM
is a