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ARCHIMEDES 
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And OB, OP, OQ, ... OZ, 00 is an arithmetical progression 
of n terms; therefore (cf. Prop, 11 and Cor.), 
(n-l)OC 2 :{OP 2 + OQ 2 +... + OC 2 ) 
< OG 2 :{OG.OB + ^{OG-OB) 2 } 
< (n-l)OG 2 :{OB 2 + OP 2 +... + OZ 2 ). 
Compressing the circumscribed and inscribed figures together 
in the usual way, Archimedes proves by exhaustion that 
(sector Oh'G): (area of spiral OBG) 
= OG 2 : {OC.OB + ${OG-OB) 2 }. 
If OB = b, OG = c, and (c—b) = (n—l)h, Archimedes’s 
result is the equivalent of saying that, when It diminishes and 
n increases indefinitely, while c — b remains constant, 
limit of h {b 2 + (b + h) 2 + {b+2h) 2 + ... +{b + n— 2h) 2 } 
= (c-b) {cb + ^{c-b) 2 } 
= l(c 3 -& 3 ); 
that is, with our notation, 
| x 2 dx = i(c 3 — b 2 ). 
Jfc 
In particular, the area included by the first turn and the 
initial line is bounded by the radii vectores 0 and 
the area, therefore, is to the circle with radius 2 na as ^(2na) 2 
to (2nd) 2 , that is to say, it is § of the circle or ^n^ira) 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 na 
and 4TT«, and is to the circle with radius in a in the ratio 
of {'r 2 r 1 + -|(r 2 — 7q) 2 } to r 2 2 , where r x = 2nd and r 2 = 4nd\ 
the ratio is 7:12 (Prop. 25). 
If ll x be the area of the first turn of the spiral bounded by 
the initial line, B 2 the area of the ring added by the second 
complete turn, B 3 that of the ring added by the third turn, 
and so on, then (Prop. 27) 
S s = 2B 2 . R,= iR 2 ,...B„ = (n-l)R 2 . 
Also .R 2 = 6 R x . 
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