Equivalence Relation - Euclidean Relations

Euclidean Relations

Euclid's The Elements includes the following "Common Notion 1":

Things which equal the same thing also equal one another.

Nowadays, the property described by Common Notion 1 is called Euclidean (replacing "equal" by "are in relation with"). The following theorem connects Euclidean relations and equivalence relations:

Theorem. If a relation is Euclidean and reflexive, it is also symmetric and transitive.

Proof:

• (aRc ∧ bRc) → aRb = (aRa ∧ bRa) → aRb = bRa → aRb. Hence R is symmetric.
• (aRc ∧ bRc) → aRb = (aRc ∧ cRb) → aRb. Hence R is transitive.

Hence an equivalence relation is a relation that is Euclidean and reflexive. The Elements mentions neither symmetry nor reflexivity, and Euclid probably would have deemed the reflexivity of equality too obvious to warrant explicit mention.