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

ISOPERIMETRIC FIGURES. ZENODORUS 207
themselves, so that, while they got a reputation for greater
honesty, they in fact took more than their share of the
produce. 1 Several remarks by ancient authors show the
prevalence of the same misconception. Thucydides estimates
the size of Sicily according to the time required for circumnavigating
it. 2 About 130 B.C. Polybius observed that there
were people who could not understand that camps of the same
periphery might have different capacities. 3 Quintilian has a
similar remark, and Cantor thinks he may have had in his
mind the calculations of Pliny, who compares the size of
different parts of the earth by adding their lengths to their
breadths. 4
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
Zenodorus wrote, at some date between (say)
200 B.C. and
A.D. 90, a treatise Trepi lo-ofxirpcov o-^fiaTcov, On isometric
figures. A number of propositions from it are preserved in
the commentary of Theon of Alexandria on Book I of
Ptolemy's Syntaxis ; and they are reproduced in Latin in the
third volume of Hultsch's edition of Pappus, for the purpose
of comparison with Pappus's own exposition of the same
propositions at the beginning of his Book V, where he appears
to have followed Zenodorus pretty closely while making some
changes in detail. 5 From the closeness with which the style
of Zenodorus follows that of Euclid and Archimedes we may
judge that his date was not much later than that of Archimedes,
whom he mentions as the author of the proposition
(Measurement of a Circle, Prop. 1) that the area of a circle is
half that of the rectangle contained by the perimeter of the
circle and its radius. The important propositions proved by
Zenodorus and Pappus include the following: (1) Of all
regular 'polygons of equal perimeter, that is the greatest in
area which has the most angles. (2) A circle is greater than
any regular polygon of equal contour. (3) Of all polygons of
the same number of sides and equal perimeter the equilateral
and equiangular polygon is the greatest in area. Pappus
added the further proposition that Of all segments of a circle
having the same circumference the semicircle is the greatest in
1
Proclus on Eucl. I, p. 403. 5 sq.
2
Thuc. vi. 1.
3 Polybius, ix. 21.
4 Pliny, Hist. nat. vi. 208.
5
Pappus, v, p. 308 sq.
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.