A history of Greek mathematics - Wilbourhall.org

ON SPIRALS 73
are in arithmetical progression. Draw arcs of circles with
radii OB, OP, OQ ...
as shown; this produces a figure circumscribed
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 OzG and OBp' '; therefore, by increasing
the number of
divisions of the angle BOO, we can make the
difference between the areas of the circumscribed and inscribed
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 + OQ 2 + ... + OC 2 ),
and (whole sector Ob'C) : (inscribed figure)
= (n-l)0C*:(0B 2 + 0P* + 0Q 2 +... + 0Z 2 ).