A history of Greek mathematics Vol.II from Aristarchus to Diophantus by Heath, Thomas Little, Sir, 1921

THE COLLECTION. BOOK V 393
Then LH:AG = (arc LE) :
— (arc LE) :
= (sector LEE) :
(arc AB)
(arc DE)
(sector DEE).
Also LH 2 :AG 2 = (sector Zi/i?) : (sector AGB).
Therefore the sector LHE is to the sector AGB in the
ratio duplicate of that which the sector LHE has to the
sector
DHE.
Therefore
(sector LHE) :
(sector DHE) = (sector DHE) :
(sector AGB),
Now (1) in the case of the segment less than a semicircle
and (2) in the case of the segment greater than a semicircle
(sector EDH) :
by the lemmas (1) and (2) respectively.
That is,
(sector EDH) :
(EDK) > R:l DHE,
(EDK) > L LHE: L DHE
> (sector LHE) :
*
> (sector EDH) :
(sector DHE)
(sector AGB),
from above.
Therefore the half segment EDK is less than the half
semicircle AGB, whence the semicircle ABC is greater than
the segment DEF.
We have already described the content of Zenodorus's
treatise (pp. 207-13, above) to which, so far as plane figures
are concerned, Pappus added nothing except the above proposition
relating to segments of circles.
Section (2). Comparison of volumes of solids having their
surfaces equal. Case of the sphere.
The portion of Book V dealing with solid figures begins
(p. 350. 20) with the statement that the philosophers who
considered that the creator gave the universe the form of a
sphere because that was the most beautiful of all shapes also
asserted that the sphere is the greatest of all solid figures
394 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 semiregular-
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
that given by Zenodorus.
the same as
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 $ >
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,