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points of intersection of a certain rectangular hyperbola 
and a certain parabola. It is quite possible, however, that 
such problems were in practice often solved by a mechanical 
method, namely by placing a ruler, by trial, in the position of 
the required line: for it is only necessary to place the ruler 
so that it passes through the given point and then turn it 
round that point as a pivot till the intercept becomes of the 
given length. In Props. 6-9 we have a circle with centre 0, 
a chord AB less than the diameter in it, OM the perpendicular 
from 0 on AB, BT the tangent at B, OT the straight line 
through 0 parallel to AB; D : E is any ratio less or greater, 
as the case may be, than the ratio BM: MO. Props. 6, 7 
(Fig. 2) show that it is possible to draw a straight line OFF 
meeting AB in F and the circle in P such that FP: FB=D: E 
(OP meeting AB in the case where D:E < BM:M0, and 
meeting AB produced when D : E > BM: MO). In Props. 8, 9 
(Fig. 3) it is proved that it is possible to draw a straight line 
OFF meeting AB in F, the circle in P and the tangent at B in 
G, such that FP : BG=D: E (OP meeting AB itself in the case 
where D : E < BM: MO, and meeting AB produced in the 
case where D : E > BM: MO). 
We will illustrate by the constructions in Props. 7, 8, 
as it is these propositions which are actually cited later. 
Prop. 7. If D : E is any ratio > BM: MO, it is required (Fig. 2) 
to draw OP'F' meeting the circle in P' and AB produced in 
F' so that 
F'P': P'B = D : E. 
Draw OT parallel to AB, and let the tangent to the circle at 
B meet OT in T.