From Wikipedia on cardinal numbers:
The oldest definition of the cardinality of a set $X$ (implicit in Cantor and explicit in Frege and Principia Mathematica) is as the class $[X]$ of all sets that are equinumerous with $X$.
My question concerns Euclidean geometry, esp. Euclid's Elements in which there are no lengths (of line segments) that may be equal or not (as numbers) but only line segments themselves which may be equal (tacitly: with respect to length) or not.
Who was the first to state explicitly that the length of a line segment $l$ may be considered the class $[l]$ of all line segments that are equal to $l$ (in the sense of Euclid)? And who can be assumed to have known or considered this implicitly? Euclid himself?
Probably not the first but a very important author did state it like this:
(Hartshorne, Geometry: Euclid and beyond (1997), p. 3)