A history of Greek mathematics - Wilbourhall.org

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.
(EDK) > R:l DHE,
That is,
(sector EDH) :
(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