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:

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(Hartshorne, Geometry: Euclid and beyond (1997), p. 3)

  • $\begingroup$ Explicitly? I may be wrong, but I would guess some educator after 1950 (Maybe Piaget?) :-) See mathoverflow.net/questions/135347/… $\endgroup$ Oct 28, 2018 at 21:44
  • $\begingroup$ How did you guess? And why so late? The idea is very obvious, isn't it? $\endgroup$ Oct 28, 2018 at 21:52
  • $\begingroup$ 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. $\endgroup$
    – Conifold
    Oct 28, 2018 at 22:44
  • $\begingroup$ “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. $\endgroup$ Oct 28, 2018 at 23:02
  • $\begingroup$ But Euclid had only congruence, no length. $\endgroup$ Oct 28, 2018 at 23:06

1 Answer 1


Intuitively, Euclid understood this. Probably. (It is very difficult to speculate what a person who lived 2300 years ago, and about whom we really know nothing, understood). Euclid's Elements was rigorously formalized by Hilbert (Foundations of Geometry, end of 19th century, there are translations to many languages, including English). He essentially corrected what Euclid "probably understood but could not express well", and filled many gaps in Euclid.

This includes the notion of length. In the modern approach length is a function on the set of intervals which takes values in real numbers. This is equivalent to what you say. Equivalence relation a set is mathematically the same as a function on the set (which sends an element of the set to its equivalence class).

For a very accessible discussion of this and other questions about Euclid's Elements I recommend the book by R. Hartshorne, Geometry. Euclid and beyond. Hilbert's book is also very readable. But on my opinion, Hartshorne is the best. One can also read Euclid, but this is difficult, and it is recommended to read him with modern comments (by Heath or Hartshorne).

  • $\begingroup$ Euclid, "Common Notion I" ... Things which equal the same thing also equal one another. $\endgroup$ Nov 3, 2018 at 14:46

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