28 
ARCHIMEDES 
Mechanical Theorems, Method (communicated) to Eratosthenes). 
Premising certain propositions in mechanics mostly taken 
from the Plane Equilibriums, and a lemma which forms 
Prop. 1 of On Conoids and Spheroids, Archimedes obtains by 
his mechanical method the following results. The area of any 
segment of a section of a right-angled cone (parabola) is § of 
the triangle with the same base and height (Prop. 1). The 
right cylinder circumscribing a sphere or a spheroid of revolu 
tion and with axis'equal to the diameter or axis of revolution 
of the sphere or spheroid is 1\ times the sphere or spheroid 
respectively (Props. 2, 3). Props. 4, 7, 8,11 find the volume of 
any segment cut off, by a plane at right angles to the axis, 
from any right-angled conoid (paraboloid, of revolution), 
sphere, spheroid, and obtuse-angled conoid (hyperboloid) in 
terms of the cone which has the same base as the segment and 
equal height. In Props. 5, 6, 9, 10 Archimedes uses his method 
to find the centre of gravity of a segment of a paraboloid of 
revolution, a sphere, and a spheroid respectively. Props. 
12-15 and Prop. 16 are concerned with the cubature of two 
special solid figures. (1) Suppose a prism with a square base 
to have a cylinder inscribed in it, the circular bases of the 
cylinder being circles inscribed in the squares which are 
the bases of the prism, and suppose a plane drawn through 
one side of one base of the prism and through that diameter of 
the circle in the opposite base which is parallel to the said 
side. This plane cuts off a solid bounded by two planes and 
by part of the curved surface of the cylinder (a solid shaped 
like a hoof cut off by a plane); and Props. 12-15 prove that 
its volume is one-sixth of the volume of the prism. (2) Sup 
pose a cylinder inscribed in a cube, so that the circular bases 
of the cylinder are circles inscribed in two opposite faces of 
the cube, and suppose another cylinder similarly inscribed 
with reference to two other opposite faces. The two cylinders 
enclose a certain solid which is actually made up of eight 
‘hoofs’ like that of Prop. 12. Prop. 16 proves that the 
volume of this solid is two-thirds of that of the cube. Archi 
medes observes in his preface that a remarkable fact about 
(1912) containing The Method, on the original edition of Heiberg (in 
Hermes, xlii, 1907) with the translation by Zeuthen {Bibliotheca Mathe- 
matica, vii s . 1906/7).