318 11. POST-EUCLIDEAN GEOMETRY
1. Hellenistic geometry
Although the Euclidean restrictions set limits to the growth of geometry, there
remained people who attempted to push the limits beyond the achievements of
Archimedes and Apollonius, and they produced some good work over the next few
centuries. We shall look at just a few of them.
1.1. Zenodorus. The astronomer Zenodorus lived in Athens in the century fol-
lowing Apollonius. Although his exact dates are not known, he is mentioned by
Diocles in his book On Burning Mirrors and by Theon of Smyrna. According to
Theon, Zenodorus wrote On Isoperimetric Figures, in which he proved four theo-
rems: (1) If two regular polygons have the same perimeter, the one with the larger
number of sides encloses the larger area; (2) a circle encloses a larger area than any
regular polygon whose perimeter equals its circumference; (3) of all polygons with
a given number of sides and perimeter, the regular polygon is the largest; (4) of all
closed surfaces with a given area, the sphere encloses the largest volume. With the
machinery inherited from Euclidean geometry, Zenodorus could not have hoped for
any result more general than these. Let us examine his proof of the first two, as
reported by Theon.
Referring to Fig. 1, let ΑΒÃ and ÄΕÆ be two regular polygons having the same
perimeter, with ABT having more sides than AEZ. Let Ç and è be the centers
of these polygons, and draw the lines from the centers to two adjacent vertices and
their midpoints, getting triangles ΒÃÇ and ΕÆÈ and the perpendicular bisectors
of their bases Ç Ê and ÈË. Then, since the two polygons have the same perimeter
but ΑΒÃ has more sides, BK is shorter than Ε A. Mark off Ì on Ε A so that MA =
BK. Then if Ñ is the common perimeter, we have Ε A : Ñ :: ÆΕÈΑ : 4 right angles
and Ñ : BK :: 4 right angles : ZBHK. By composition, then Ε A : BK :: ÆΕÈΑ :
ZBHK, and therefore EA : MA :: ÆΕÈΑ : ZBHK. But, Zenodorus claimed, the
ratio EA : MA is larger than the ratio ÆΕÈΑ : ÆÌÈΑ, asking to postpone the
proof until later. Granting that lemma, he said, the ratio ÆΕÈΑ : ZBHK will
be larger than the ratio ÆΕÈΑ : ÆÌÈΑ, and therefore ZBHK is smaller than
ÆÌÈΑ. It then follows that the complementary angles ZHBK and ÆÈÌΑ satisfy
the reverse inequality. Hence, copying ZHBK at Ì so that one side is along MA,
we find that the other side intersects the extension of ËÈ at a point Í beyond
È. Then, since triangles BHK and MNA are congruent by angle-side-angle, it
follows that Ç Ê = ÍΑ > ÈË. But it is obvious that the areas of the two polygons
are \HK • Ñ and ^ÈË · Ñ, and therefore ABT is the larger of the two.
The proof that the ratio Ε A : MA is larger than the ratio ÆΕÈΑ : ÆÌÈΑ was
given by Euclid in his Optics, Proposition 8. But Theon does not cite Euclid in
his quotation of Zenodorus. He gives the proof himself, implying that Zenodorus
did likewise. The proof is shown on the top right in Fig. 1, where the circular arc
ÎÌÍ has been drawn through Ì with è as center. Since the ratio ΑΕÈÌ :
sector ÍÈÌ is larger than the ratio ÄÌÈË : sector ÌÈÎ (the first triangle is
larger than its sector, the second is smaller), it follows, interchanging means, that
ΕÈÌ : ÌÈΑ > sector ÍÈÌ : sector ÌÈÎ. But ΕÈÌ : ÌÈΑ :: EM : MA, since
the two triangles have the same altitude measured from the base line EM A. And
sector ÍÈÌ : sector ÌÈÎ :: ÆΕÈÌ : ÆÌÈΑ. Therefore, EM : MA is larger
than the ratio ÆΕÈÌ : ÆÌÈΑ, and it then follows that Ε A : MA is larger than
ÆΕÈΑ : ÆÌÈΑ.
