Later, in the fourth construction, he used superposition (moving the triangles on top of each other) to prove that if two sides and their angles are equal, then they are congruent; during these considerations he uses some properties of superposition, but these properties are not described explicitly in the treatise.

Because when we studied Elements at math camp when I was ~16 I remember this standing out to me. I think we were going through it as a group, and the instructor asked if anyone could prove each theorem in turn before giving us the answer if we couldn’t. Unsurprisingly, no one could prove this one. When he showed us how it was done I felt a bit… cheated? because no one had told us we could do that. But I didn’t do anything with this feeling, I think I just assumed that everything was fine, I should have been able to work out that we could do that.

Later I learned that no, it was in fact cheating and we could not do that.

It seems worth noting here that

Elementsisn’t entirely rigorous. I don’t remember many details about that, but https://en.wikipedia.org/wiki/Euclid’s_Elements#Criticism has some. I do remember this bit (or at least something very similar):Because when we studied

Elementsat math camp when I was ~16 I remember this standing out to me. I think we were going through it as a group, and the instructor asked if anyone could prove each theorem in turn before giving us the answer if we couldn’t. Unsurprisingly, no one could prove this one. When he showed us how it was done I felt a bit… cheated? because no one had told us we could do that. But I didn’t do anything with this feeling, I think I just assumed that everything was fine, I should have been able to work out that we could do that.Later I learned that no, it was in fact cheating and we could not do that.

Yeah, and sometimes his case analysis was a little less than exhaustive. I think Byrne fixed that, though.