The fifth proposition in the elements is the proves that the angles opposite the Congruent sides in an isosceles triangle are congruent. Euclid proves this by extending the congruent sides of the isosceles triangle, shown above as the blue triangle, to form two large overlapping triangles which he shows are congruent. Then he shows another pair are congruent, namely the two overlapping triangles that share the base of the equilateral triangle above. Then, by some simple arithmetic he explains how this leads to the conclusion that the two base angles of the isosceles triangle are congruent. This proposition has come to be known by commentators as “The Bridge of Asses”. No one knows exactly why it has come to bear this name, most likely because the diagram resembles a bridge truss, but I’ll share a couple of famous anecdotes about it.
First, there was a famous English mathematician and professor who made a point of exacting this Proof from the least capable students. He would ask a low performing student to perform the proof everyday when they came to class, until at last, the “ass crossed the bridge”.
Second, this was a proof that was considered difficult by many students and in some schools became a method of determining whether a student should pursue further studies in mathematics. Students who passed became the asses who were doomed to carry the burden of extended studies in mathematics.
When I first saw this proof, especially since I had seen simpler proofs, it struck me as unduly complex. Those poor Greeks didn't have algebra and couldn't do better. Euclid was the ass. Of course, I am exaggerating a little. Now as I have begun reading Euclid from the beginning it seems perfectly in place with his logical development of ideas. It also doesn’t seem difficult at all, if one truly goes through the argument step by step. Nonetheless, it has been compared to Pappas’s proof and sometimes made out to be a behemoth in comparison. Luckily, in my copy of the Elements Pappas’s proof was featured in the commentary. Since I had often been annoyed by the short presentation of Pappas’s proof and had not seen the full version of Pappas’s proof before I will feature it below:
Suppose that segment AB is congruent to segment BC. Suppose that we consider the triangle as if it were two triangles; namely, triangle ABC and triangle CBA. Now, segment AB is congruent to segment BC, and BC is congruent to AB. Now, "both" triangles have angle ABC in common. Thus, the "two" triangles ABC and CBA are congruent by SAS (see proposition 4) and therefore their corresponding angles are congruent, namely angle A and angle C.
This is more or less the full version of the Pappas proof, which I found in the comment section in my Heath version of The Elements. The commentator is comfortingly critical of the version posed by modern editors that I have found distasteful.
Suppose that segment AB is congruent to segment BC. Suppose that we consider the triangle as if it were two triangles; namely, triangle ABC and triangle CBA. Now, segment AB is congruent to segment BC, and BC is congruent to AB. Now, "both" triangles have angle ABC in common. Thus, the "two" triangles ABC and CBA are congruent by SAS (see proposition 4) and therefore their corresponding angles are congruent, namely angle A and angle C.
This is more or less the full version of the Pappas proof, which I found in the comment section in my Heath version of The Elements. The commentator is comfortingly critical of the version posed by modern editors that I have found distasteful.
I should note that they way that I have proved this theorem in the past is similar to the way Euclid starts his proof of proposition 10, by splitting the triangle into two triangle by constructing a bisector of the vertex angle and showing the two triangle halves are congruent.
