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 ).