(Recall that a rational number is one that is equivalent to the ratio of two integers.)
The Pythagoreans are credited with the proof of the existence of irrational numbers.
A separate, more general and circuitous ancient Greek doctrine of proportionality for geometric magnitude was developed in Book V of Euclid's Elements in order to allow proofs involving incommensurable lengths, thus avoiding arguments which applied only to a historically restricted definition of number.
Euclid's notion of commensurability is anticipated in passing in the discussion between Socrates and the slave boy in Plato's dialogue entitled Meno, in which Socrates uses the boy's own inherent capabilities to solve a complex geometric problem through the Socratic Method.
He develops a proof which is, for all intents and purposes, very Euclidean in nature and speaks to the concept of incommensurability.
Then the subgroup of the real numbers R generated by a is commensurable with the subgroup generated by b if and only if the real numbers a and b are commensurable, in the sense that a/b is rational.
Thus the group-theoretic notion of commensurability generalizes the concept for real numbers.
Two groups G1 and G2 are (abstractly) commensurable if there are subgroups H1 ⊂ G1 and H2 ⊂ G2 of finite index such that H1 is isomorphic to H2.
Depending on the type of space under consideration, one might want to use homotopy equivalences or diffeomorphisms instead of homeomorphisms in the definition.