1. QED
What does that mean?

2. Quod erat demonstrandum.

3. Let us compare this Latin expression with its equivalents in three other languages.

4. Quod erat demonstrandum.
Which was to be proved.
Vilket skulle bevisas.
Chto i trebovalos’ dokazat’.

5. The easiest relation to establish is the object relation.

6. Quod erat demonstrandum.
Which was to be proved.
Vilket skulle bevisas.
Chto i trebovalos’ dokazat’.

7. The next relation is most explicitly expressed in Russian.
Chto i trebovalos’ dokazat’.

8. Also in the Swedish equivalent this meaning is expressed by a separate word.
Vilket skulle bevisas.

9. Quod erat demonstra<nd>um.
In Latin, this meaning is located to a morpheme, namely the suffix forming the gerundivum.
Quod erat demonstra<nd>um.

10. Which was (to be)<INF> proved.
In English, the carrier of the meaning is the infinitive form of the verb be. To express this we use the portmanteau morpheme <INF>.
Which was (to be)<INF> proved.

11. Note that the formants of the passive voice need not be accounted for here; they are always syntactic.

12. The past tense meaning is represented by a morpheme, which in all languages, except Russian, is of the portmanteau type: <PRET>. This predicate has two valency positions, one of which is occupied by the implicit speech act verb «dico».

13. «d.» Quod (erat)<PRET> demonstra<nd>um.
«d.» Which (was)<PRET> (to be)<INF> proved.
«d.» Vilket skulle<PRET> bevisas.
«d.» Chto i trebova<lo>s’ dokazat’.

14. «Dico», in its turn, takes as its object the whole sentence, represented by the predicate corresponding to the finite verbs in Russian and Swedish.

15. «d.» Quod (erat)<PRET> demonstra<nd>um.
«d.» Which (was)<PRET> (to be)<INF> proved.
«d.» Vilket skulle<PRET> bevisas.
«d.» Chto i trebova<lo>s’ dokazat’.

16. The relative pronoun remains to be accounted for
The relative pronoun remains to be accounted for. It triggers the two-place implicit predicate «i.e.». It is sufficient to demonstrate this on one example.

17. «i.e.» Quod erat demonstrandum.
The stippled arrow points to the theorem mentioned in the preceding context.

18. Quod erat demonstrandum?
Now let us turn the relative pronoun into an interrogative one. We can let the question mark (the intonation) represent the speech act verb.
Quod erat demonstrandum?

19. Quod erat demonstrandum?
This means that there are two object relations pointing to the word quod.
Quod erat demonstrandum?

20. Dixit, quod erat demonstrandum.
A similar object position of the pronoun we find in so-called objective subordinate clauses.
Dixit, quod erat demonstrandum.

21. In Latin, there is hardly any difference in expression between the two uses of the pronoun.

22. In English, it is preferable to exchange which for what:
He told us what was to be proved.

23. Han meddelade vad (som) skulle bevisas.
A similar replacement takes place in Swedish. In addition, the passive voice of the objective sentence triggers the insertion of a syntactic som.
Han meddelade vad (som) skulle bevisas.

24. On skazal, chto trebovalos’ dokazat’.
In Russian, finally, the new context makes it necessary to drop the particle i, which is connected with the function of the relative pronoun.
On skazal, chto trebovalos’ dokazat’.

25. In addition, the objective chto has a considerably heavier stress than the relative chto. This is still more evident if we compare the pronoun chto with the homonymous conjunction chto.

26. Ja znaju, chtó ty kurish’.
Ja znaju, (chto) ty kúrish’.
In English: I know what you smoke (are smoking) vs. I know that you smoke (are smoking).

27. The End