What is the etymology of the term "reflexive" in the context of binary relations?

Question

A binary relation $R$ over a set $A$ is called reflexive if the following is true: $$\forall a \in A. aRa$$

Why are relations called these "reflexive?"

Answer 1

I'm confident the term comes from grammar, where reflexive pronouns are things like "self" in English, and reflexive verbs are verbs whose object is the same as its subject, e.g. Él se lavó (Spanish - He washes himself).

Romance languages in particular have verbs which are only reflexive, e.g. "to sit down" in French is s'asseoir. The reflexive pronoun is always explicitly present. In English, I sit down, you sit down, one sits down; in French, I sit myself down, one sits oneself down, you tell someone "sit (yourself) down!", and so on. My guess is the term "reflexif" has been used in French discussions of French grammar for centuries.

Many languages have reflexive verbs, by any other name, and it's not just an Indo-European thing either (e.g. Hebrew). See https://en.wikipedia.org/wiki/Reflexive_verb.

Answer 2

This is form latin reflexio "back-bending"; the relation "bends" $a$ back to itself (and by the way, "itself" is a reflexive pronoun).