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It is conventional to number the three persons of Latin and Greek and many other languages so that the first person is the speaker, the second one is the listener, and the third one is anyone else. Did the ancient grammarians already have this numbering? If yes, was it an explicit numbering (first, second, third) or a more implicit ordering (always giving things in the order facio–facis–facit)?

The numbering is most importantly a convention and it makes life easier if everyone sticks to the same nomenclature. I don't know if the modern convention goes back to the antiquity. In the plural the numbering also describes an order of precedence in Latin and Greek: any group containing me is "we", any other group containing you (sg.) is "you (pl.)", and any other group is "they".

Both Latin and Greek sources are fine, as I believe the two languages to be essentially identical when it comes to persons. Any insight into the history of numbering of persons would be quite interesting.

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    As a side note, in the Sanskrit grammatical tradition the numbering is the opposite -- he/she is "first person", etc. – TKR Aug 10 at 18:27
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    @TKR The Greco-Roman numbering corresponds more readily to usage in contract law (where there may not be a third party, and they are often extraneous if they exist) so maybe it's fortunate it caught on. – C Monsour Aug 10 at 20:20
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This numbering goes back to Greek grammarians. Here is the Τέχνη Γραμματική (Art of Grammar) ascribed to Dionysius Thrax:

πρώσοπα τρία, πρῶτον, δεύτερον, τρίτον· πρῶτον μὲν ἀφ᾽ οὗ ὁ λόγος, δεύτερον δὲ πρὸς ὃν ὁ λόγος, τρίτον δὲ περὶ οὗ ὁ λόγος.
"There are three persons ['faces'], first, second, third. The first is the one from whom the speech [proceeds], the second is the one to whom the speech [is addressed], the third is the one whom the speech is about."

It's debated whether this work is really by Dionysius Thrax, who lived in the second century BC, or is from a later date, but in any case the terminology goes back to antiquity.

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