On 1609 Cardinal Federico Borromeo founded the Biblioteca Ambrosiana in Milan. Depending on how "public library" is defined, the Ambrosiana was possibly the the second public library in Europe, after the Bodleian at Oxford. However, the Ambrosiana was preceded in Italy by the library at the Domincan convent of San Marco (1444) and the Biblioteca Medicea Laurenziana (1571), both of which were characterized as "public" libraries when they were founded. Thus, there may be some uncertainly as to which library was actually the first "public" library in Europe.
To build up the Ambrosiana's collections Cardinal Borromeo's agents scoured Western Europe, and even Greece and Syria for books and manuscripts. In 1606 they acquired the complete manuscripts of the Benedictine monastery of Bobbio, founded in 614, and the library of the Paduan Vincenzo Pinelli, whose more than 800 manuscripts filled 70 cases when they were sent to Milan, and included the famous extremely early illuminated miniatures of the Iliad, the Ilias Ambrosiana.
"During Cardinal Borromeo's sojourns in Rome, 1585–95 and 1597–1601, he envisioned developing this library in Milan as one open to scholars and that would serve as a bulwark of Catholic scholarship against the treatises issuing from Protestant presses. To house the cardinal's 15,000 manuscripts and twice that many printed books, Construction began in 1603 under designs and direction of Lelio Buzzi and Francesco Maria Richini. When its first reading room, the Sala Fredericiana, opened to the public, December 8, 1609, it was, after the Bodleian Library in Oxford, the second public library in Europe. One innovation was that its books were housed in cases ranged along the walls, rather than chained to reading tables, a practice seen still today in the Laurentian Library of Florence. A printing press was attached to the library, and a school for instruction in the classical languages.
"Cardinal Borromeo gave his collection of paintings and drawings to the library too. Shortly after the cardinal's death his library acquired twelve manuscripts of Leonardo da Vinci, including the Codex Atlanticus. . . ." (Wikipedia article on the Biblioteca Ambroisiana).