It is perhaps slightly misleading to call it "the first public library" but nevertheless its use was not the prerogative of a private group.
ɪɾ ɪz pɚhˈæps slˈaɪtli mɪslˈiːdɪŋ tə kˈɔːl ɪt "ðə fˈɜːst pˈʌblɪk lˈaɪbɹɛɹi" bˌʌt nˌɛvɚðəlˈɛs ɪts jˈuːs wʌz nˌɑːt ðə pɹɪɹˈɑːɡətˌɪv əvə pɹˈaɪvət ɡɹˈuːp.