A history of Greek mathematics - Wilbourhall.org

34 ARCHIMEDES
Therefore AX: AG = (AA'-$AG) :
(%AA'-AG)
= (4AA'-3AG):(6AA'-4AG);
whence AX:XG = (4AA'- 3AG) :
(2AA'-AG)
= (AG + ±A'G):\AG + 2A'G),
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 themselves.
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 semicircle.
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 examine
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 § of the volume of the sphere,
(4) that the surface of the circumscribing cylinder including
its bases is also § 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-