APPENDIX 
557 
‘ obscurity ’ of Archimedes; while, as regards Vieta, he has 
shown that the statement quoted is based on an entire mis 
apprehension, and that, so far from suspecting a fallacy in 
Archimedes’s proofs, Yieta 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 analy 
sis 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 sub tangent-property can be 
established by purely ‘ plane ’ methods, without recourse to 
a ‘solid’ vtvcns (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 absurdum 
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 describ 
ing 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 direc 
tions, we see that, corresponding to a small increase in the 
radius vector r, we have a small distance traversed perpen 
dicularly to it, a tiny arc of a circle of radius r subtended by 
the angle representing the simultaneous small increase of the 
angle 0 (AOP). Now r has a constant ratio to 6 which we call 
a (when 6 is the circular measure of the angle 6). Consequently