The Shuowen Jiezi, compiled around 100 AD, contain entries for some 10,000 Chinese characters. The Kangxi Dictionary, compiled almost two millennia later, contain over 47,000 characters. Of the characters in the Kangxi Dictionary, as well as encoded in Unicode (say), I'm aware that a large number of them are variant forms and don't necessarily represent "new" characters.

But glossing over this fact, is there existing scholarship on how many characters were 'invented' throughout Chinese history? Were more characters invented during certain periods of time, or has the development of characters been relatively stable?

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The evolution of the Chinese characters is a very long journey, I don't think there is a clear indication of how many characters were invented in each period and are still in use or not. The table below shows the long journey, and the source article is an excellent reading material.

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