A history of Greek mathematics Vol.II from Aristarchus to Diophantus by Heath, Thomas Little, Sir, 1921

ON THE SPHERE AND CYLINDER, I 35
tively), this was not the order of their discovery ; for Archimedes
tells us in The Method that
'
from the theorem that a sphere is four times as great as the
cone with a great circle of the sphere as base and with height
equal to the radius of the sphere I conceived the notion that
the surface of any sphere is four times as great as a great
circle in it ; for, judging from the fact that any circle is equal
to a triangle with base equal to the circumference and height
equal to the radius of the circle, I apprehended that, in like
manner, any sphere is equal to a cone with base equal to the
surface of the sphere and height equal to the radius '.
Book I begins with definitions (of ' concave in the same
direction as applied '
to curves or broken lines and surfaces, of
a ' solid sector ' and a ' solid rhombus ') followed by five
Assumptions, all of importance. Of all lines ivhich have the
same extremities the straight line is the least, and, if there are
two curved or bent lines in a plane having the same extremities
and concave in the same direction, but one is wholly
included by, or partly included by and partly common with,
the other, then that which is included is the lesser of the two.
Similarly with plane surfaces and surfaces concave in the
'
same direction. Lastly, Assumption 5 is the famous Axiom
of Archimedes ', which however was, according to Archimedes
himself, used by earlier geometers (Eudoxus in particular), to
the effect that Of unequal magnitudes the greater exceeds
the less by such a magnitude as, when added to itself, can be
made to exceed any assigned magnitude of the same kind
;
the axiom is of course practically equivalent to Eucl. V, Def. 4,
and is closely connected with the theorem of Eucl. X. 1.
As, in applying the method of exhaustion, Archimedes uses
both circumscribed and inscribed figures with a view to compressing
them into coalescence with the curvilinear figure to
be measured, he has to begin with propositions showing that,
given two unequal magnitudes, then, however near the ratio
of the greater to the less is to 1, it is possible to find two
straight lines such that the greater is to the less in a still less
ratio ( > 1), and to circumscribe and inscribe similar polygons to
a circle or sector such that the perimeter or the area of the
circumscribed polygon is to that of the inner in a ratio less
than the given ratio (Props. 2-6): also, just as Euclid proves
D 2
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
c
E
the cone, C a straight line equal to its
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.