ON THE SPHERE AND CYLINDER, I 
35 
i'-AG) 
iA'-lAG)] 
A A'-AG) 
+ 2 A'G), 
in Prop. 9 for the 
¿sphere the ratio 
the hemisphere in 
. (Props. 4, 5) and 
n the same course, 
at the end of the 
iropositions tliem- 
3S finds the centre 
■ a plane through 
gravity of a semi- 
he order in which 
•, i, n. 
shortly stated in 
des tells us that 
ling them for the 
ay be able to ex- 
The results are 
es that of a great 
any segment of a 
ch is equal to the 
segment to a point 
the volume of a 
eight equal to the 
ne of the sphere, 
cylinder including 
ere. It is worthy 
these propositions 
33 and 34 respec- 
tively), this was not the order of their discovery; for Archi 
medes 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 tvhich 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 extremi 
ties 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 he 
made to exceed any assigned magnitude of the same kind; 
the axiom is of course practically equivalent to Eucl. Y, 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 com 
pressing 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