416 
PAPPUS OF ALEXANDRIA 
solution of a certain quadratic equation.) Pappus observes 
that the problem is always possible (requires no 8i.opL(rfj,6s), 
and proves that it has only one solution. 
(5) Lemmas on the treatise ‘ On contacts’ hy Apollonius. 
These lemmas are all pretty obvious except two, which are 
important, one belonging to Book I of the treatise, and the other 
to Book II. The two lemmas in question have already been set 
out a propos of the treatise of Apollonius (see pp. 182-5, above). 
As, however, there are several cases of the first (Props. 105, 
107, 108, 109), one case (Prop. 108, pp. 836-8), different from 
that before given, may be put down here: Given a circle and 
two points D, E within it, to draiu straight lines through D, E 
to a point A on the circumference in such a ivay that, if they 
meet the circle again in B, C, BC shall he parallel to BE. 
We proceed by analysis. Suppose the problem solved and 
DA,EA drawn (‘inflected’) to A in such a way that, if AD, 
AE meet the circle again in B, C, 
BC is parallel to DE. 
Draw the tangent at B meeting 
ED produced in F. 
Then Z FBD = Z AGB = LAED\ 
therefore A, E, B, F are concyclic, 
and consequently 
FT) DF= AD DD 
But the rectangle AD. DB is given, since it depends only 
on the position of D in relation to the circle, and the circle 
is given. 
Therefore the rectangle FD. DE is given. 
And DE is given; therefore FD is given, and therefore F. 
If follows that the tangent FB is given in position, and 
therefore B is given. Therefore BDA is given and conse 
quently AE also. 
To solve the problem, therefore, we merely take F on ED 
produced such that FD . DE — the given rectangle made by 
the segments of any chord through D, draw the tangent FB, 
join BD and produce it to A, and lastly draw AE through to 
C; BC is then parallel to DE.