Here's an attempt. In 1757, on a whim, some chinese Explorers head west to discover what is out there. They make their way through Turkey etc. selling silk.
Eventually, they make their way to the libraries in Rome and read about Ancient Rome.
Assuming they are still alive, they return to China with this information and it becomes popular.
Ove 10 years, this republican idea kicks in as the Nobles think they shouldnt be run by the Manchus. By 1789 they overthrow the Qing Dynasty and declare The Chinese Republic.