A history of Greek mathematics - Wilbourhall.org

74, ARCHIMEDES
And OB, OP, OQ, . . . OZ,
OG is an arithmetical progression
of n terms; therefore (cf. Prop. 11 and Cor.),
(n- \)0G 2 : (OP 2 + OQ 2 + ... + OG 2 )
< OC 2 :{OC.OB + i(OC-OB) 2 }
< (n-l)OC 2 :(OB 2 + OP 2 +... + OZ 2 ).
Compressing the circumscribed and inscribed figures together
in the usual way, Archimedes proves by exhaustion that
(sector Ob'C) :
(area of spiral OBC)
= 00 2 : {OC.
OB + ^(00-OB) 2 }.
If OB = b, OG = c, and (c— b) = (n— l)h, Archimedes's
result is the equivalent of saying that, when h diminishes and
n increases indefinitely, while c — b remains constant,
that, is,
limit of h{b 2 + (b + h) 2 + (b + 2h) 2 +...+{b + '^2h) 2 }
= {c-b){cb + l;(c-b) 2 }
= §(o 3 -6 3 );
with our notation,
Jb
x 2 dx = l(c
3 — o
In particular, the area included by the first
3
).
turn and the
initial line is bounded by the radii vectores and 2ira\
the area, therefore, is to the circle with radius 2 ita as ^(2ttcl) 2
to (27ra) 2 , that is to say, it is § of the circle or ^ir(2iTa) 2 .
This is separately proved in Prop. 24 by means of Prop. 10
and Corr. 1, 2.
The area of the ring added while the radius vector describes
the second turn is the area bounded by the radii vectores 2 wet,
and lira, and is to the circle with radius Aw a in the ratio
of {^2 r i + 3(
r2~ r i)
2
} t° r 2
,
where r x
= 2na and r 2
= 47ra;
the ratio is 7:12 (Prop. 25).
If R 1
be the area of the first turn of the spiral bounded by
the initial line, R 2
the area of the ring added by the second
complete turn, R z
that of the ring added by the third turn,
and so on, then (Prop. 27)
R 3
== 2R 2)
R A
= 3R 2
, R
Also R 2
= 6R X
.
ro
=
4i? 2
, ... R n
= (n-1)R 2
.