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PAPPUS OF ALEXANDRIA
which have their surfaces equal; this, however, they had not
proved, nor could it be proved without a long investigation.
Pappus himself does not attempt to prove that the sphere is
greater than all solids with^ the same surface, but only that
the sphere is greater than any of the five regular solids having
the same surface (chap. 19) and also greater than either a cone
or a cylinder of equal surface (chap. 20).
Section (3). Digression on the semi-regular solids
of Archimedes.
He begins (chap. 19) with an account of the thirteen semi
regular solids discovered by Archimedes, which are contained
by polygons all equilateral and all equiangular but not all
similar (see pp. 98-101, above), and he shows how to determine
the number of solid angles and the number of edges which
they have respectively ; he then gives them the go-by for his
present purpose because they are not completely regular; still
less does he compare the sphere with any irregular solid
having an equal surface.
The sphere is greater than any of the regular solids which
has its surface equal to that of the sphere.
The proof that the sphere is greater than any of the regular
solids with surface equal to that of the sphere is the same as
that giveniby Zenodorus. Let P be any one of the regular solids,
S the sphere with surface equal to that of P, To prove that
S>P. Inscribe in the solid a sphere s, and suppose that r is its
radius. Then the surface of P is greater than the surface of s,
and accordingly, if R is the radius of S, R > r. But the
volume of S is equal to the cone with base equal to the surface
of S, and therefore of P, and height equal to R; and the volume
of P is equal to the cone with base equal to the surface of P
and height equal to r. Therefore, since R>r, volume of IS >
volume of P.
Section (4). Propositions on the lines of Archimedes,
‘ On the Sphere and Cylinder ’.
For the fact that the volume of a sphere is equal to the cone
with base equal to the surface, and height equal to the radius,
