A history of Greek mathematics Vol.II from Aristarchus to Diophantus by Heath, Thomas Little, Sir, 1921

THE GONIGS, BOOK VII 175
As we have said, 'Book VIII is lost. The nature of its
contents can only be conjectured from Apollonius's own
remark that it contained determinate conic problems for
which Book VII was useful, particularly in determining
limits of possibility. Unfortunately, the lemmas of Pappus
do not enable us to form any clearer idea. But it is probable
enough that the Book contained a number of problems having
for their object the finding of conjugate diameters in a given
conic such that certain functions of their lengths have given
values. It was on this assumption that Halley attempted
a restoration of the Book.
If it be thought that the above account of the Gonics is
disproportionately long for a work of this kind, it must be
remembered that the treatise is a great classic which deserves
to be more known than it is. What militates against its
being read in its original form is the great extent of the
exposition (it contains 387 separate propositions), due partly
to the Greek habit of proving particular cases of a general
proposition separately from the proposition itself, but more to
the cumbrousness of the enunciations of complicated propositions
in general terms (without the help of letters to denote
particular points) and to the elaborateness of the Euclidean
form, to which Apollonius adheres throughout.
Other works by Apollonius.
Pappus mentions and gives a short indication of the contents
of six other works of Apollonius which formed part of the
Treasury of Analysis. 1 Three of these should be mentioned
in close connexion with the Conies.
(a) On the Gutting-off of a Ratio (Xoyou a7roTOfirj),
two Books.
This work alone of the six mentioned has survived, and
that only in the Arabic ; it was published in a Latin translation
by Edmund Halley in 1706.
+
It deals with the general
problem, Given two straight '
lines, parallel to one another or
intersecting, and a fixed point on each line, to draw through
1
Pappus, vii, pp. 640-8, 660-72.
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' :
Then
OB, 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