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Does the Latin verb nosco come from Greek, or did the shared root (cf. γιγνώσκω) end up in Greek and Latin separately? According to Wiktionary, it seems to be the latter case, as the free dictionary traces the ancestry of nosco back to proto-Italic, not to Greek. This came as a surprise to me.

If indeed the two words evolved independently, then what would explain the shared stem vowel ō which comes from the PIE root *ǵneh₃-? Would the change have come from an ablaut early on and then ended up in the two languages?

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Latin nōscō and Greek gignōskō are cognates, but neither is directly derived from the other. They both come from the Proto-Indo-European root *ǵneh₃- "to know", plus the inchoative *-sḱ- marking the present tense.

To quote Wtrmute's answer here:

*h3 is called the "o-colouring laryngeal," which means that it "colours" (i.e., changes) a neighbouring e into *o or *ō: the former from *h3o and the latter from *oh3. See *h3ewis > ovis and *deh3[r/n]m̥ > dōnum.

*ǵneh₃-sḱ- thus became *gnō-sk-, which became Latin gnōscō, later nōscō.

Occasionally in PIE a present form would include reduplication, which usually indicated the perfect (much more common in Greek than Latin). That was the case here. From the reduplicated form *ǵi-ǵneh₃-sḱ- came *gi-gnō-sk-, and thus Greek gignōskō.

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    Just as an aside, reduplicated perfects were somewhat more common in old Latin; e.g., fefeci becomes Classical feci. – Anonym Oct 1 '17 at 20:22

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