296 10. EUCLIDEAN GEOMETRYAs Aristotle's discussion of begging the question continues, further evidence
comes to light that this matter of parallel lines was being debated around 350, and
proofs of the existence of parallel lines (Book 1, Proposition 27 of the Elements)
were being proposed, based on the exterior-angle principle. In pointing out that
different false assumptions may lead to the same wrong conclusion, Aristotle notes
in particular that the nonexistence of parallel lines would follow if an internal angle
of a triangle could be greater than an external angle (not adjacent to it), and
also if the angles of a triangle added to more than two right angles.^19 One is
almost tempted to say that the mathematicians who analyzed the matter in this
way foresaw the non-Euclidean geometry of Riemann, but of course that could not
be. Those mathematicians were examining what must be assumed in order to get
parallel lines into their geometry. They were not exploring a geometry without
parallel lines.
2. Euclid
In retrospect the third century BCE looks like the high-water mark of Greek ge-
ometry. Beginning with the Elements of Euclid around 300 BCE, this century saw
the creation of sublime mathematics in the treatises of Archimedes and Apollo-
nius. It is very tempting to regard Greek geometry as essentially finished after
Apollonius, to see everything that came before as leading up to these creations and
everything that came after as "polishing up." And indeed, although there were
some bright spots afterward and some interesting innovations, none had the scope
or the profundity of the work done by these three geometers.
The first of the three major figures from this period is Euclid, who is world
famous for his Elements, which we have in essence already discussed. This work is
so famous, and dominated all teaching in geometry throughout much of the world for
so long, that the man and his work have essentially merged. For centuries people
said not that they were studying geometry, but that they were studying Euclid.
This one work has eclipsed both Euclid's other books and his biography. He did
write other books, and two of them—the Data and Optics—still exist. Others—the
Phenomena, Loci, Conies, and Porisms—are mentioned by Pappus, who quotes
theorems from them.
Euclid is defined for us as the author of the Elements. Apart from his writ-
ings, we know only that he worked at Alexandria in Egypt just after the death of
Alexander the Great. In a possibly spurious passage in Book 7 of his SynagogS,
Pappus gives a brief description of Euclid as the most modest of men, a man who
was precise but not boastful, like (he implies) Apollonius.
2.1. The Elements. As for the Elements themselves, the editions that we now
have came to us through many hands, and some passages seem to have been added
by hands other than Euclid's, especially Theon of Alexandria. We should remember,
of course, that Theon was not interested in preserving an ancient literary artifact
unchanged; he was trying to produce a good, usable treatise on geometry. Some
manuscripts have 15 books, but the last two have since been declared spurious by
the experts, so that the currently standard edition has 13 books, the last of which
looks suspiciously less formal than the first 12, leading some to doubt that Euclid
wrote it. Leaving aside the thorny question of which parts were actually written
(^19) Field and Gray (1987, p. 64) note that this point has been made by many authors since
Aristotle,