- THE EARLIEST GREEK GEOMETRY 295
FIGURE 16. The exterior angle theorem.Euclid deals with this issue in the Elements by stating as the last of his assump-
tions that "two straight lines do not enclose an area." Oddly, however, he seems
unaware of the need for this assumption when proving the main lemma (Book 1,
Proposition 16) needed to prove the existence of parallel lines.^17 This proposition
asserts that an exterior angle of a triangle is larger than either of the opposite in-
terior angles. Euclid's proof is based on Fig. 16, in which a triangle ABT is given
with side BT extended to Ä, forming the exterior angle .4ÃÄ. He wishes to prove
that this angle is larger than the angle at A. To do so, he bisects AT at E, draws
AE, and extends it to Æ so that EZ — AE. When ÆÃ is joined, it is seen that the
triangles ABE and TZE are congruent by the side-angle-side criterion. It follows
that the angle at A equals ZETZ, which is smaller than ZETA, being only a part
of it.
In the proof Euclid assumes that the points Ε and Æ are on the same side of
line BT. But that is obvious only for triangles small enough to see. It needs to
be proved. To be sure, Euclid could have proved it by arguing that if Ε and Æ
were on opposite sides of BT, then EZ would have to intersect either BT or its
extension in some point H, and then the line Β Ç passing through à and the line
BEH would enclose an area. But he did not do that. In fact, the only place where
Euclid invokes the assumption that two lines cannot enclose an area is in the proof
of the side-angle-side criterion for congruence (Book 1, Proposition 4).^18
Granting that Aristotle was right about this point, we still must wonder why
he considered the existence of parallel lines to be in need of proof. Why would he
have doubts about something that is so clear on an intuitive level? One possible
reason is that parallelism involves the infinite: Parallelism involves the concept that
two finite line segments will never meet, no matter how far they are extended. If
geometry is interpreted physically (say, by regarding a straight line as the path of
a light ray), we really have no assurance whatever that parallel lines exist—how
could anyone assert with confidence what will happen if two apparently parallel
lines are extended to a length of hundreds of light years?(^17) In standard editions of Euclid, there are 14 assumptions, but three of them, concerned with
adding equals to equals, doubling equals, and halving equals, are not found in some manuscripts.
Gray [1989, p. 46) notes that the fourteenth assumption may be an interpolation by the Muslim
mathematician al-Nayrizi, the result of speculation on the foundations of geometry. That would
explain its absence from the proof of Proposition 16.
(^18) This proof also uses some terms and some hidden assumptions that are visually obvious but
which mathematicians nowadays do not allow.