50 
ARCHIMEDES 
m 
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', V' are the surface and 
volume of the lesser, then V: V' < S 2 : S' 2 but > S?: S'% 
(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) prac 
tically 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 pro 
position 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. tt) is < 3~ hut > 3-|y. 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