O o 
.e spiral 
(Fig. 2) 
F, and 
OT, the 
i.e. as G 
increases 
BM-.MO, 
he spiral 
s always 
beyond 
Pappus’s 
actually 
isition in 
PO:OV 
, we can 
7' where 
mding to 
is proves 
PM: MO, 
APPENDIX 
561 
approaches that ratio without limit as R approaches P. But 
the proof does not enable us to say that RF' '.{chord PR), 
which is > RF' : PG, is also always less than PM : MO. At 
first sight, therefore, it would seem that the proof must fail. 
Not so, however; Archimedes is nevertheless able to prove 
that, if PV and not FT is the tangent at P to the spiral, an 
absurdity follows. For his proof establishes that, if PFis the 
tangent and OF' is drawn as in the proposition, then 
F'O : RO < OR: OP, 
or F'O < OR', ‘ which is impossible Why this is impossible 
does not appear in Props. 18 and 20, but it follows from the 
argument in Prop. 13, which proves that a tangent to the spiral 
cannot meet the curve again, and in fact that the spiral is 
everywhere concave towards the origin. 
Similar remarks apply to the proof by Archimedes of the 
impossibility of the other alternative supposition (that the tan 
gent at P meets OT at a point U nearer to 0 than T is). 
Archimedes’s proof is therefore in both parts perfectly valid, 
in spite of any appearances to the contrary. The only draw 
back that can be urged seems to be that, if we assume the 
tangent to cut OT at a point very near to 'T on either side, 
Archimedes’s construction brings us perilously near to infini 
tesimals, and the proof may appear to hang, as it were, on 
a thread, albeit a thread strong enough to carry it. But it is 
remarkable that he should have elaborated such a difficult 
proof by means of Props. 7, 8 (including the ‘ solid ’ u evens of 
Prop. 8), when the figures of Props. 6 and 7 (or 9) themselves 
suggest the direct proof above given, which is independent of 
any vevens• 
P. Tannery, 1 in a paper on Pappus’s criticism of the proof as 
unnecessarily involving ‘ solid ’ methods, has given another 
proof of the subtangent-property based on ‘ plane ’ methods 
only ; but I prefer the method which I have given above 
because it corresponds more closely to the preliminary proposi 
tions actually given by Archimedes. 
1 Tannery, Mémoires scientifiques, i, 1912, pp. 800 -16.