A history of Greek mathematics - Wilbourhall.org

558 APPENDIX
the small increases of r and S are in that same ratio.
Therefore
what we call the tangent of the angle OPT is r/a,
i.e. OT/r — r/a; and OT = r 2 /a, or rS, that is, the arc of a
circle of radius r subtended by the angle 0.
To prove this result Archimedes would doubtless begin by
an analysis of the following sort. Having drawn OT perpendicular
to OP and of length equal to the arc ASP, he had to
prove that the straight line joining P to T is the tangent
at P. He would evidently take the line of trying to show
that, if any radius vector to the spiral is drawn, as OQ', on
either side of OP, Q' is always on the side of TP towards 0,
or, if OQ' meets TP in F, OQ' is always less than OF. Suppose
g
p^s^ "7a'
BVco^^ ^^^W'G'
Tax
S/^^AT
^-
u
o
that in the above figure OB! is any radius vector between OP
and OS on the backward ' '
side of OP, and that OR' meets the
circle with radius OP in R, the tangent to it at P in G, the
spiral in R f ,
and TP in F'. We have to prove that R, R' lie
on opposite side's of F' , i.e. that RR' > RF' ; and again, supposing
that any radius vector 0Q' on the ' forward ' side of
OP meets the circle with radius OP in
TP produced in F, we have to prove that QQ' < QF.
Archimedes then had to prove that
(1) F'R:R0 < RR':R0, and
(2) FQ:Q0>QQ':Q0.
Now (1)
is equivalent to
Q, the spiral in Q' and
F'R.RO < (arcEP): (arc ASP), since R0 = P0.