A history of Greek mathematics Vol.II from Aristarchus to Diophantus by Heath, Thomas Little, Sir, 1921

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 f : 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 PFand not PT is the tangent at P to the" spiral, an
absurdity follows. For his proof establishes that, if PVis the
tangent and OF' is drawn as in the proposition, then
F'0:RO < OR': OP,
or F'O < OR', which l 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 tangent
at P meets OT at a point U nearer to
than T is).
Archimedes's proof is therefore in both parts perfectly valid,
in spite of any appearances to the contrary. The only drawback
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 infinitesimals,
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 ' vevcris 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 vevcns.
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 propositions
actually given by Archimedes.
1
Tannery, Memoires scientifques, i, 1912, pp. 800-16.
1523.3