The Bactra Review Geometry Civilized

The pathetic state of the subject [geometry] in northern Europe
around 1025 AD, over five centuries after the deposition of the last Roman
emperor, appears from an exchange of letters between two of the most learned
men of the time, Raimbeau (pronounced something like `Rambo') of Cologne and
Raoul of Liège. This correspondence amounted to a geometrical
tournament, in which Raoul, who was much the younger man, attempted to earn his
spurs by defeating Raimbeau. At Raoul's invitation, Raimbeau shot off the most
difficult problem he had run across --- and to which, as it turned out, he did
not know the answer. The problem was one with which we are acquainted: to
prove that the sum of the interior angles of a triangle equals a straight angle
(Euclid I. 32). Raimbeau may have taken it from a book that gave a few simple
Euclidean propositions without demonstrations or, indirectly, from the texts of
the Roman agrimensores....

They began by throwing dust in one another's eyes. It occurred to Raimbeau
that if triangles have interior angles they must also have exterior ones. What
might they be? They asked all their friends. No one knew. No one they
consulted could put his hands on the first book of Euclid's
Elements. Raimbeau ended by taking `interior' to mean `acute' and
`exterior' to mean `obtuse'. With this understanding he found it impossible to
prove anything at all. He would have had trouble even if he had his
definitions right. Neither he nor Raoul had the slightest conception of a
geometrical proof. Raoul did manage to show that the sum of the interior
angles is a straight angle in the special case of a triangle with two equal
sides, formed by drawing the diagonal of a square. But the general case eluded
him. All he could suggest was either to declare the proposition true by
intuition or to draw a triangle on parchment, cut out its angles, and place
them to together to form a straight angle.

Raoul's technique cannot satisfy a geometer.... [I]t is not general: to
prove, or make plausible, the proposition in his way you would have to cut up
every triangle you wanted to analyze. The principal reason for
constructing formal proofs is to demonstrate, once and for all, a property
common to *all* geometric figures defined in the enunciation of the
proposition. [pp. 71--2]

More conclusive proof that the Dark Ages were, in fact, dark, could hardly
be asked for.