A history of Greek mathematics - Wilbourhall.org

560 APPENDIX
Next, for the point Q' on the ' forward ' side of the spiral
from P, suppose that in the figure of Prop. 9 or Prop. 7 (Fig. 2)
any radius OP of the circle meets A B produced in F, and
the tangent at B in G ;
Fig. 2.
parallel through to AB, in H, T.
and draw BPH, BGT meetmg 0T, the
Then PF:BG> FG: BG, since PF > FG,
> 0G :
GT, by parallels,
> OB :BT, a fortiori,
> BM:M0;
and obviously, as P moves away from B towards 0T, i.e.
moves away from B along BT, the ratio OG.GT increases
continually, while, as shown, PF:BG is always > BM:M0,
and, a fortiori,
That is,
PF:(8lycPB) > BM-.MO.
as G
(4) is always satisfied for any point Q' of the spiral
1
forward ' of P, so that (2) is also satisfied, and QQ' is always
less than QF.
It, will be observed that no vtvcri?, and nothing beyond
'
plane ' methods, is required in the above proof, and Pappus's
criticism of Archimedes's proof is therefore justified.
Let us now consider for a moment what Archimedes actually
does. In Prop. 8, which he uses to prove our proposition in
the ' backward' case (R', R, F'), he shows that, if P0 : 0V
is any ratio whatever less than P0 : 0T or PM : MO, we can
find points F' , G corresponding to any ratio P0 : 0V f where
0T < 0V < OF, i.e. we can find a point F' corresponding to
a ratio still nearer to P0 : 0T than P0 : OF is. This proves
that the ratio RF' : PG,
while it is always less than PM:M0,