ON SPIRALS 
71 
et the tangent 
in, if we bisect 
le spiral in R, 
irai. But by 
easily proved) 
point of PQ 
Dirai, which is 
y the tangent 
s obtuse on the 
e, of the radius 
Let OF meet the spiral in Q'. 
Then we have, alternando, since PO = QO, 
FQ-.QO = PQ:OU 
< (arc PQ) : (arc ASP), by hypothesis and a fortiori. 
Componendo, FO :Q0 < (arc A$Q) : (arc ASP) 
< OQ' : OP. 
But QO = OP ; therefore FO < OQ' ; which is impossible. 
Ion about the 
the subtangent 
point of inter 
com 0 to OP). 
turn and then 
ete turn before 
he deals with 
end of the first 
and any subse- 
te point of the 
as illustrative 
it P of the first 
e first turn, PT 
then it is to be 
with centre 0, 
Therefore OT is not greater than the arc ASP. 
SP. 
t < OT. 
ì ratio PO :0T, 
idicular from 0 
ae OQF meeting 
at 
II. Next suppose, if possible, that OT < arc ASP. 
Measure OV along OT such that OF is greater than OTbut 
less than the arc ASP. 
ThAi the ratio PO : OF is less than the ratio PO : OT, i.e. 
than the ratio of %PS to the perpendicular from 0 on PS; 
therefore it is possible (Prop. 8) to draw a straight line OF'RG 
meeting PS, the circle PSA, and the tangent to the circle at P 
in F', R, G respectively, and such that 
F'R : GP = PO : OV.