ON SPIRALS 
73 
are in arithmetical progression. Draw arcs of circles with 1 
radii OB, OP, OQ ... as shown; this produces a figure circum 
scribed to the spiral and consisting of the sum of small sectors 
of circles, and an inscribed figure of the same kind. As the 
first sector in the circumscribed figure is equal to the second 
sector in the inscribed, it is easily seen that the areas of the 
circumscribed and inscribed figures differ by the difference 
between the sectors OzC and OBp'; therefore, by increasing 
the number of divisions of the angle BOG, we can make the 
. And it was 
re 
L rchimedes), if 
circumference 
the extremity 
3f the ‘second 
, if c n be the 
ith the radius 
lius), the sub- 
th turn = nc n . 
emity, and the 
the initial line 
sircle, the sub- 
KP (measured 
is devoted to 
,nd its several 
radii vectores. 
p. 26). Take 
ng an a^ BC 
escribe a circle, 
equal parts by 
radii in points 
OQ ... OZ, 00 
e property of the 
difference between the areas of the circumscribed and in 
scribed figures as small as we please; we have, therefore, the 
elements necessary for the application of the method of 
exhaustion. 
If there are n radii OB, OP... 00, there are (n— 1) parts of 
the angle BOG. Since the angles of all the small sectors are 
equal, the sectors are as the square on their radii. 
Thus (whole sector Ob'0): (circumscribed figure) 
= (n~l)OC 2 : (OP 2 + 0Q 2 +... + 00 2 ), 
and (whole sector Oh'O): (inscribed figure) 
= (n- 1 )00 2 : (OB 2 + OP 2 + 0Q 2 + ... + OZ 2 ).