Ai! Lá polin saca i quettar!
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This was the preposition for “to(wards)” in Sindarin for relationships of both space and time, and was derived from the root √NA of similar meaning (PE17/146-147). Before consonants na used vocalic mutation and before vowels it was reduced to n’ (PE17/147). In the phrase ai na vedui Dúnadan “Ah! at last, Dúnadan!” it was used with the sense “at (a point of time or place)” (PE17/16). It is not clear whether this is a loose translation or an actual variant meaning.
Conceptual Development: The word G. ar {“at, by, beside” >>} “at” appeared in the Gnomish Lexicon of the 1910s, with the meaning “at, towards” when combined with the dative (GL/19). The word ar was glossed “at, to” in Gnomish Lexicon Slips modifying that document (PE13/110).
Neo-Sindarin: The Gnomish precursor ar indicates that a simultaneous meaning of “at” and “to” is possible, but Gnomish distinguished the two by using a dative inflection that was not a feature of Sindarin. For Sindarin, I would assume basic sense of na is “to”, and the sense “at” is actually “arriving at” implied when the modified noun is a terminus in space or time: “to last > (arriving) at [the] last” or “to destination > (arriving) at [the] destination”. Compare compare Q. na in notes from the early 1950s that was used “with sense ‘to’, but this at nearest means ‘towards, to a position near, alongside’ (PE21/79)”.
See also an “to, for” used of more purely dative relationships.