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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 AB ’produced in F, and 
the tangent at B in G; and draw BPH, BGT meeting 0T, the 
parallel through 0 to AB, in H, T. 
Then PF: BG > FG: BG, since PF > FG, 
> 0G : GT, by parallels, 
> OB: BT, a fortiori, 
> BM: MO; 
and obviously, as P moves away from B towards 0T, i.e. as G 
moves away from B along BT, the ratio 0G:GT increases 
continually, while, as shown, PF: BG is always > BM: MO, 
and, a fortiori, 
PF: (arc PB) > BM: MO. 
That is, (4) is always satisfied for any point Q' of the spiral 
‘ forward ’ of P, so that (2) is also satisfied, and QQ' is always 
less than QF. 
It will be observed that no vevens, 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' where 
0T < 0V'< 0V, 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: MO,