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]