A history of Greek mathematics - Wilbourhall.org

50 ARCHIMEDES
To return to Archimedes.
Book II of our treatise contains
further problems : To find a sphere equal to a given cone or
cylinder (Prop. 1), solved by reduction to the finding of two
mean proportionals; to cut a sphere by a plane into two
segments having their surfaces in a given ratio (Prop. 3),
which is easy (by means of I. 42, 43) ;
given two segments of
spheres, to find a third segment of a sphere similar to one
of the given segments and having its surface equal to that of
the other (Prop. 6)
;
the same problem with volume substituted
for surface (Prop. 5), which is again reduced to the finding
of two mean proportionals; from a given sphere to* cut off
a segment having a given ratio to the cone with the same
base and equal height (Prop. 7). The Book concludes with
two interesting theorems. If a sphere be cut by a plane into
two segments, the greater of which has its surface equal to S
and its volume equal to V, while S', Y f are the surface and
volume of the lesser, then V: V < S 2 : S' 2 but > S*:S'i
(Prop. 8) : and, of all segments of spheres which have their
surfaces equal, the hemisphere is the greatest in volume
(Prop. 9).
Measurement of a Circle.
The book on the Measurement of a Circle consists of three
propositions only, and is not in its original form, having lost
(as the treatise On the Sphere and Cylinder also has) practically
all trace of the Doric dialect in which Archimedes
wrote ; it may be only a fragment of a larger treatise. The
three propositions which survive prove (1) that the area of
a circle is equal to that of a right-angled triangle in which
the perpendicular is equal to the radius, and the base to the
circumference, of the circle, (2) that the area of a circle is to
the square on its diameter as 11 to 14 (the text of this proposition
is, however, unsatisfactory, and it cannot have been
placed by Archimedes before Prop. 3, on which it depends),
(3) that the ratio of the circumference of any circle to its
diameter (i.e. n) is < 3y but > 3-^f.
Prop. 1 is proved by
the method of exhaustion in Archimedes's usual form : he
approximates to the area of the circle in both directions
(a) by inscribing successive regular polygons with a number of