68
ARCHIMEDES
♦
Let QGO meet the original circle in P and AB in F. Then
OFPG is the straight line required.
For CG. GT = OG . GQ = OG . BK.
But OF: 0G = BT: GT, by parallels,
whence OF. GT = OG . BT.
Therefore CG . GT: OF. GT = OG . BK : OG . BT,
whence CG : OF = BK : BT
= BC:OP
Conoids
l 2 + 2 2 -f-
summat:
(71-1) ('
The san
are a 2 , (
= BC: OP.
the mor<
Therefore OP :OF=BC: CG,
and hence PF: OP = BG : BC,
or PF: BG = OB : BC — D : E.
Archil
itself, tl
radius v
(= the <
Pappus objects to Archimedes’s use of the i/evais assumed in
Prop. 8, 9 in these words:
second c
bounded
the ‘ first
‘ it seems to be a grave error into which geometers fall
whenever any one discovers the solution of a plane problem
by means of conics or linear (higher) curves, or generally
solves it by means of a foreign kind, as is the case e.g. (1) with
the problem in the fifth Book of the Conics of Apollonius
relating to the parabola, and (2) when Archimedes assumes in
his work on the spiral a veva-is of a “solid” character with
reference to a circle; for it is possible without calling in the
aid of anything solid to find the proof of the theorem given by
Archimedes, that is, to prove that the circumference of the
circle arrived at in the first revolution is equal to the straight
line drawn at right angles to the initial line to meet the tangent
to the spiral (i.e. the subtangent).’
describe«
and so o
as radius
sum of
distance)
Props,
spiral coi
through
position,
r = a 6.
than 7t, <
the first,
There is, however, this excuse for Archimedes, that he only
assumes that the problem can be solved and does not assume
the actual solution. Pappus 1 himself gives a solution of the
particular vevais by means of conics. Apollonius wrote two
Books of veva-ei?, and it is quite possible that by Archimedes’s
time there may already have been a collection of such problems
to which tacit reference was permissible.
Prop. 10 repeats the result of the Lemma to Prop. 2 of On
1 Pappus, iv, pp. 298-302.
of a circ
‘ first cir
are two ]
OP
if P, Q
spiral, ai
OP.C
Prop. :
