Consider that the question does not concern the origin of the ideas of equivalence relation and equivalence class. It exactly concerns **the origin of the terms** "equivalence relation" and "equivalence class". It seems that the terms weren't in use at least until 1903 where Russell writes:



> 
> Peano has defined a process which he calls definition by abstraction,
>  of which, as he shows, frequent use is made in Mathematics. This
>  process is as follows: when there is any relation which is transitive,
>  symmetrical and (within its field) reflexive, then, if this relation
>  holds between u and v, we define a new entity Ø (u), which is to be
>  identical with Ø (v).
> 
> 
> Relations which possess these properties are an important kind, and
>  it is worthwhile to note that similarity is one of this kind of
>  relations.
> 
> 
> 


**UPDATE**: Thanks to the suggestions given in the comments and a very informative answer of Francois Ziegler, suddenly the following piece from Russell's Introduction to Mathematical Philosophy came into a new light: 



> 
> The question “what is a number?” is one which has been often asked,
>  but has only been correctly answered in our own time. The answer was
>  given by Frege in 1884, in his Grundlagen der Arithmetik. Although his
>  book is quite short, not difficult, and of the very highest
>  importance, it attracted almost no attention, and the definition of
>  number which it contains remained practically unknown until it was
>  rediscovered by the present author in 1901.
> 
> 
> 


So, perhaps that was Frege himself who used the terms! Although, I couldn't find anything in Frege's writings yet. But now, considering Eugen Netto's paper (see Francois' update below) a very more important question now is: Is he indeed Russell who should be credited with rediscovering Frege in 1901?! 


**PS.** I am well aware that asking a new question inside another question is not a good idea. However, up until this post I had the feeling that apart from the origin of the terms I know "everything" about the history of the notions of equivalence relation and equivalence class (part of which has been published [here](http://link.springer.com/article/10.1007/s10649-008-9173-x#)). But, this new information caught me by surprise, and I could not help myself to add the new question. You may just ignore it. 



Who introduced the terms "equivalence relation" and "equivalence class"?
Von Neumann uses "equivalence class" in [*Zur Prüferschen Theorie der idealen Zahlen*, Acta Sci. Math. (Szeged) **2** (1926) 193-227](http://pub.acta.hu/acta/showCustomerVolume.action?id=5089&dataObjectType=volume), p. 197 (viewable after free registration):



> 
> Wir nennen $R$ und $S$ *äquivalent*, in Zeichen: $R\sim S$, wenn (...)
> 
> 
> Satz 2. Es ist stets $R\sim R$. Aus $R\sim S$ folgt $S\sim R$. Aus $R\sim S$ und $S\sim T$ folgt $R\sim T$.
> 
> 
> (...)
> 
> 
> Infolge des Satzes 2. zerfällt die Menge der Folgen realer Zahlen in paarweise elementefremde Klassen untereinander äquivalenter Folgen.
> 
> 
> (...)
> 
> 
> Definition 4. Eine **Aequivalenzklasse**, die lauter Fundamentalfolgen enthält, nennen wir eine *ideale Zahl*.
> 
> 
> 


This seems like an early example, in that he sees fit to add here the footnote: "We define the ideal number as the corresponding set of fundamental sequences, itself; naturally one could also regard it as an ideal element attached to this set."


I note also that Hasse's book *Höhere Algebra* (where Henry Cohn found the earliest occurrence of "equivalence relation" so far) appears to have a [1926 edition too](http://books.google.com/books?id=2UdtAAAAMAAJ&q=%C3%A4quivalenzrelation).




---


**Update 1:** One might also quote a paper by Eugen Netto, [*Über die arithmetisch-algebraischen Tendenzen Leopold Kronecker's*, in: Mathematical Papers read at the International Mathematical Congress (Chicago, 1893), Macmillan 1896, pp. 243-252](http://www.mathunion.org/ICM/ICM1893/), who writes:



> 
> "Jede wissenschaftliche Forschung geht darauf aus, *Aequivalenzen* festzustellen und deren *Invarianten* zu ermitteln (...)."
> 
> 
> Jede Abstraction, z. B. die von gewissen Verschiedenheiten, welche eine Anzahl von Objecten darbietet, statuirt eine Aequivalenz; alle Objecte, die einander bis auf jene Verschiedenheiten gleichen, gehören zu einer **Aequivalenzclasse**, sind unter einander aequivalent, und der aus der Abstraction hervorgehende Begriff bildet die "Invariante der Aequivalenz."
> 
> 
> 




---


**Update 2:** Finally(?) one should probably also quote Vol. 2 of Weber's [*Lehrbuch der Algebra* (1896)](http://archive.org/stream/lehrbuchderalgeb02weberich#page/n573). Literally he uses neither "equivalence relation" nor "equivalence class", but see how close he gets in §152 "Aequivalenz":



> 
> 1. *Zwei ganze oder gebrochene Functionale $\varphi$, $\psi$ im Körper $\Omega$ heissen **äquivalent**, wenn (...)*
> 2. *Zwei Functionale, die mit einem dritten äquivalent sind, sind auch unter einander äquivalent.*
> 
> 
> Theilt man hiernach alle Functionale des Körpers $\Omega$ in ***Classen*** ein, indem man zwei Functionale in dieselbe oder in verschiedene Classen wirft, je nachdem sie äquivalent sind oder nicht, so ergiebt sich (...)
> 
> 
> 


Same for Dedekind's [*Ueber die Theorie der algebraischen Zahlen* (1879)](http://archive.org/stream/vorlesungenberz02dedegoog#page/n559), §175:



> 
> Wir wollen nun zwei Ideale $\mathfrak a$, $\mathfrak a'$ ***äquivalent*** nennen, wenn (...)
> 
> 
> Zugleich ergiebt sich hieraus, dass (...) je zwei Ideale $\mathfrak a'$, $\mathfrak a''$, die mit einem dritten Ideal $\mathfrak a$ äquivalent sind, stets auch miteinander äquivalent sein müssen. Auf diesem Satze beruht die Möglichkeit, alle Ideale in *Idealclassen* einzutheilen; (...) der Inbegriff $A$ *aller* mit $\mathfrak a$ äquivalenten Ideale $\mathfrak a$, $\mathfrak a'$, $\mathfrak a''$ (...) nennen wir eine *Idealclasse* oder kürzer eine ***Classe***
> 
> 
> 


(Of course, calling *class* a family of objects related by some *equivalence* is a custom that can be traced to much older work — cf. Dirichlet [[1863, p. 172](http://archive.org/stream/vorlesungenberz03dedegoog#page/n191)] or Eisenstein [[1847, p. 118](http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=GDZPPN002145928)] or Gauss [[1801, §223](http://archive.org/stream/werkecarlf01gausrich#page/n231)].)


