34 
ARCHIMEDES 
Therefore AX: AG = (AA'-fAG): (|AA'-AG) 
= (4AA'-3A(?):(6AA , -4A(7); 
whence AZ:X£ = (4 AA’- 3 AG): (2 AA'-AG) 
— (At? + 4 A/Cr): (AC + 2 A/C), 
which is the result obtained by Archimedes in Prop. 9 for the 
sphere and in Prop. 10 for the spheroid. 
In the case of the hemi-spheroid or hemisphere the ratio 
AX :XG becomes 5 : 3, a result obtained for the hemisphere in 
Prop, 6. 
The cases of the paraboloid of revolution (Props. 4, 5) and 
the hyperboloid of revolution (Prop. 11) follow the same course, 
and it is unnecessary to reproduce them. 
For the cases of the two solids dealt with at the end of the 
treatise the reader must be referred to the propositions them 
selves. Incidentally, in Prop. 13, Archimedes finds the centre 
of gravity of the half of a cylinder cut by a plane through 
the axis, or, in other words, the centre of gravity of a semi 
circle. 
We will now take the other treatises in the order in which 
they appear in the editions. 
On the Sphere and Cylinder, I, II. 
The main results obtained in Book I are shortly stated in 
a prefatory letter to Dositheus. Archimedes tells us that 
they are new, and that he is now publishing them for the 
first time, in order that mathematicians may be able to ex 
amine the proofs and judge of their value. The results are 
(1) that the surface of a sphere is four times that of a great 
circle of the sphere, (2) that the surface of any segment of a 
sphere is equal to a circle the radius of which is equal to the 
straight line drawn from the vertex of the segment to a point 
on the circumference of the base, (3) that the volume of a 
cylinder circumscribing a sphere and with height equal to the 
diameter of the sphere is f of the volume of the sphere, 
(4) that the surface of the circumscribing cylinder including 
its bases is also f of the surface of the sphere. It is worthy 
of note that, while the first and third of these propositions 
appear in the book in this order (Props. 33 and 34 respec- 
tively^ 
medes 
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