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Say that the Chinese ambassador Gan Ying (Old Chinese *kˤam ʔraŋ) arrives in Rome in the year 97 AD, in the reign of Emperor Nerva. This leads to at least some degree of contact, enough for a few big Chinese names to be transliterated into Latin.
The thing is that Old Chinese, the type of Chinese spoken in the first century AD, was a very... strange-sounding language. Really. Its consonant clusters are on par with Georgian.
Here's a few guesses at how the Romans might write a few major Chinese names:
Gan Ying
Modern Mandarin Chinese:
Old Chinese:
Hypothetical Roman transliteration: Camranus (I feel ˤ and ʔ would be ignored)
Confucius
Modern Mandarin Chinese: 孔子 Kŏng zĭ
Old Chinese: 孔子 *Kʰˤoŋʔ tsəʔ
Hypothetical Roman transliteration: Chontseus (Greek Kʰ was written "ch" by Romans)
Mencius
Modern Mandarin Chinese: 孟子 Mèng zĭ
Old Chinese: 孟子 *mˤraŋ-s tsəʔ
Hypothetical Roman transliteration: Merantseus
Han Dynasty
Modern Mandarin Chinese: 漢朝 Hàn cháo
Old Chinese: 漢國 *n̥ˤar-s C.qʷˤek
Hypothetical Roman transliteration: Narsquecia
Emperor of China
Modern Mandarin Chinese: 皇帝 Huáng dì
Old Chinese: 皇帝 *Gʷˤaŋ tˤek-s
Hypothetical Roman transliteration: Quantex
Luoyang (capital of the Han dynasty)
Modern Mandarin Chinese: 洛陽 Luò yáng
Old Chinese: 洛陽 *kə.rˤak laŋ
Hypothetical Roman transliteration: Craclagnum
Qin Shi Huang (first Chinese emperor)
Modern Mandarin Chinese: 秦始皇 Qín shĭ huáng
Old Chinese: 秦始皇 *dzin l̥əʔ Gʷˤaŋ
Hypothetical Roman transliteration: Tsinlequanus
In a hypothetical world, the Roman Emperor might have sent an envoy to Cetrananum (Chang'an) and Craclagnum to ask his good friend the Grand Quantex of Narsquecia for the books of Chontseus and Merantseus.