A history of Greek mathematics - Wilbourhall.org

3
36 ARCHIMEDES
that, if we continually double the number of the sides of the
regular polygon inscribed in a circle, segments will ultimately be
left which are together less than any assigned area, Archimedes
has to supplement this (Prop. 6) by proving that, if we increase
the number of the sides of a circumscribed regular polygon
sufficiently, we can make the excess of the area of the polygon
over that of the circle less than any given area. Archimedes
then addresses himself to the problems of finding the surface of
any right cone or cylinder, problems finally solved in Props. 1
(the cylinder) and 14 (the cone).
Circumscribing and inscribing
regular polygons to the bases of the cone and cylinder, he
erects pyramids and prisms respectively on the polygons as
bases and circumscribed or inscribed to the cone and cylinder
respectively. In Props. 7 and 8 he finds the surface of the
pyramids inscribed and circumscribed to the cone, and in
Props. 9 and 10 he proves that the surfaces of the inscribed
and circumscribed pyramids respectively (excluding the base)
are less and greater than the surface of the cone (excluding
the base). Props. 11 and 12 prove the same thing of the
prisms inscribed and circumscribed to the cylinder, and finally
Props. 13 and 14 prove, by the method of exhaustion, that the
surface of the cone or cylinder (excluding the bases) is equal
to the circle the radius of which is a mean proportional
between the ' side ' (i. e. generator) of the cone or cylinder and
the radius or diameter of the base (i.e. is equal to wrs in the
case of the cone and 2irrs in the case of the cylinder, where
r is the radius of the base and s a generator). As Archimedes
here applies the method of exhaustion for the first time, we
will illustrate by the case of the cone (Prop. 14). .
Let A be the base of
the cone, C a straight line equal to its
c
E
radius, D a line equal to a generator of the cone, E a mean
proportional to G, D, and B a circle with radius equal to E.