If you do not consider the ancient, famous libraries to be public, Wikipedia puts ~1600 as a first date of modern public libraries. While copyright has a first

If you do not consider the ancient, famous libraries to be public, Wikipedia puts ~1600 as a first date of modern public libraries. While copyright has a first occurance date in the 1700s, in some locales much later. Either way it seems public libraries were around at least a century before copyright, but by other measure, in some places, much longer.