I would like to elaborate a bit on what Julio wrote.
Gauss does not use equivalence classes in DA for
equality of integers modulo n. (Naturally, Gauss does
not have the general notion of equivalence relation.)
On the other hand, in DA Gauss considers a more
sophisticated equivalence relation, viz. the equivalence
of binary quadratic forms. Here he does use equivalence
classes for this kind of equivalence [section 223]. But
Gauss immediately goes on to consider a representative
from each equivalence class. Since he then proceeds to
use only the representatives and not the equivalence
classes, it is clear that the equivalence classes play
no essential role for him. They are simply part of a
classification scheme (and he takes a biological
approach here, following LInneaus: classes are broken
down into orders, which in turn are subdivided into
genera [Section 228]).
Dedekind follows Gauss in using representatives rather
than equivalence classes. Thus Dedekind partitions the
universe of sets according to cardinality and then
chooses a *representative* for each cardinal [1888,
section 34].
I also have a request to make about Gauss's article
"Theoria residuorum biquadraticum. Commentitio secunda"
[1832]. Does anyone know of a translation of this Latin
article into any of English, French, German, or Italian?
I would be extremely interested if anyone does.
Cordially,
Greg Moore