THE COLLECTION. BOOK V 
393 
Then LH: AG — (arc LE): (arc AB) 
— (arc LE): (arc DE) 
— (sector LHE): (sector DHE). 
Also LH*; AG* = (sector LHE): (sector ACr-B). 
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): (EDK) > B: Z DHE, 
by the lemmas (1) and (2) respectively. 
That is, 
(sector EDH): {EDK) > Z LHE: Z DHE 
> (sector LjHE) : (sector DHE) 
> (sector EDH): (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 pro 
position 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