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

APPENDIX 557
'
obscurity ' of Archimedes ; while, as regards Vieta, he has
shown that the statement quoted is based on an entire misapprehension,
and that, so far from suspecting a fallacy in
Archimedes's proofs, Vieta made a special study of the treatise
On Spirals and had the greatest admiration for that work.
But, as in
many cases in Greek geometry where the analysis
is omitted or even (as Wallis was tempted to suppose) of
set purpose hidden, the reading of the completed synthetical
proof leaves a certain impression of mystery; for there is
nothing in it to show ivhy Archimedes should have taken
precisely this line of argument, or how he evolved it. It is
a fact that, as Pappus said, the subtangent-property can be
established, by purely plane ' ' methods, without recourse to
a 'solid' vevcris (whether actually solved or merely assumed
capable of being solved). If, then, Archimedes chose the more
difficult method which we actually find him employing, it is
scarcely possible to assign any reason except his definite
predilection for the form of proof by reductio ad abswrdum
based ultimately on his famous Lemma or Axiom.
'
'
It seems worth while to re-examine the whole question of
the discovery and proof of the property, and to see how
Archimedes's argument compares with an easier ' plane ' proof
suggested by the figures of some of the very propositions
proved by Archimedes in the treatise.
In the first place, we may be sure that the property was
not discovered by the steps leading to the proof as it stands.
I cannot but think that Archimedes divined the result by an
argument corresponding to our use of the differential calculus
for determining tangents. He must have considered the
instantaneous direction of the motion of the point P describing
the spiral, using for this purpose the parallelogram of
velocities. The motion of P is compounded of two motions,
one along OP and the other at right angles to it. Comparing
the distances traversed in an instant of time in the two directions,
we see that, corresponding to a small increase in the
radius vector r, we have a small distance traversed perpendicularly
to it, a tiny arc of a circle of radius r subtended by
the angle representing the simultaneous small increase of the
angle (AOP). Now r has a constant ratio to which we call
a (when 6 is the circular measure of the angle 0).
Consequently
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
^-
u
g
p^s^ "7a'
BVco^^ ^^^W'G'
Tax
S/^^AT
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
o
Q, the spiral in Q' and
F'R.RO < (arcEP): (arc ASP), since R0 = P0.