...ancient Latin and Greek poetry. It seems, rhyme was introduced into Latin poetry in late antiquity/the early middle ages, when the breakdown of the system of vowel quantity made the old ways of doing poetry impossible.

Chinese poetry, on the other hand, did rhyme; the rhymes in the poems collected in the Shijing are the most important source for the reconstruction of the pronunciation of Old Chinese (which was at least as different from modern Mandarin or Cantonese as Archaic Latin is from modern French).