A history of Greek mathematics - Wilbourhall.org

Then D :
E > BM :
> OB : BT,
ON SPIRALS 67
MO,
by hypothesis,
by similar triangles.
Take a straight line P'H' (less than BT) such that D :
= 05 : P'H', and place P'fT' between the circle and OT
1
verging towards
'
B (construction assumed).
Then F'P' : P'B = OP' : P'H'
Prop. 8. If D :
= OB :
= D:E.
P'H'
£
E is any given ratio < BM: MO, it is required
to draw OFPG meeting AB in F, the circle in P, and the
tangent at B to the circle in G so that
FP : BG = D : E.
If OT is parallel to AB and meets the tangent at B in 7 1 ,
#¥: MO = OB : BT, by similar triangles,
D:E BT.
Describe a circle through the three points 0, T, C and let OB
produced meet this circle in K.
'
Then, since BC > BT, and OK is perpendicular to OT, it is
possible to place QG [between the circle TKC and BO] equal to
BK and verging towards
'
(construction assumed).