# Lengths as equivalence classes

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 is simple:

Who was the first to state explicitly that the length of a line segment $$l$$ is 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)

• Explicitly? I may be wrong, but I would guess some educator after 1950 (Maybe Piaget?) :-) See mathoverflow.net/questions/135347/… – Francois Ziegler Oct 28 '18 at 21:44
• How did you guess? And why so late? The idea is very obvious, isn't it? – Hans-Peter Stricker Oct 28 '18 at 21:52
• This is the first time I saw length defined as an equivalence class, could you provide a reference where it is done? I suspect that not only nobody treated it this way "implicitly", but even whoever did it explicitly had to maintain some serious effort to think of it this way. Dedekind cuts do come to mind, as well as Hilbert's construction of real numbers out of geometry axioms using segments. – Conifold Oct 28 '18 at 22:44
• “Obvious” to us, maybe. E.g. Hilbert in Grundlagen der Geometrie (1899, translation) still defines length as $\sqrt{(x_1-x_2)^2+(y_1-y_2)^2+(z_1-z_2)^2}$ and congruence of segments as equality of their length, not vice versa. – Francois Ziegler Oct 28 '18 at 23:02
• But Euclid had only congruence, no length. – Hans-Peter Stricker Oct 28 '18 at 23:06