A history of Greek mathematics - Wilbourhall.org

208 SUCCESSORS OF THE GREAT GEOMETERS
area. Zenodorus's treatise was not confined to propositions
about plane figures, but gave also the theorem that Of all
solid figures the surfaces of which are equal, the sphere is
greatest in solid content.
We will briefly indicate Zenodorus's method of proof. To
begin with (1)
; let ABC, DEFhe equilateral and equiangular
polygons of the same perimeter, DEF having more angles
than ABC. Let G, H be the centres of the circumscribing
circles, GK, HL the perpendiculars from G, H to the sides
AB, DE, so that K, L bisect those sides.
the
AM
Since the perimeters are equal, AB > DE, and AK > DL.
Make KM equal to DL and join GM.
Since AB is the same fraction of the perimeter that the
angle A GB is of four right angles, and DE is the same fraction
of the same perimeter that the angle J)HE is of four right
angles, it follows that
that is, AK :
But
AB:DE=lAGB:lDHE,
LAGK-.L DHL.
MK=
AK :MK > lAGK:l MGK
(this is easily proved in a lemma following by the usual
method of drawing an arc of a circle with G as centre and GM
as radius cutting GA and GK produced.
The proposition is of.
course equivalent to tan a/ tan /S > x/fi, where \tt > a > /?).
Therefore
and consequently
Z MGK > Z DHL,
Z GMK < Z HDL.
Make the angle NMK equal to the angle HDL, so that MN
meets KG produced in N.