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Its librarians not only ejected the Fourth Doctor but suspended his lending privileges, because, according to him, "they didn't trust me after I misshelved the Dead Sea Scrolls". Nevertheless, he still had one of their books, Ptolemy's Treatise on the Structure, Position and Medicinal Nature of Celestial Bodies out on loan some 2093 years and four months after it was due back. (PROSE: Eye of Heaven)
The library was set on fire in the 5th century during the Mim invasion of the city. The First Doctor, Susan, Barbara Wright and Ian Chesterton spent some time in the library during their three week stay in the city. Ian was particularly interested, becoming friends with Hypatia, one of its scholars. (AUDIO: The Library of Alexandria)
By the time of his sixth incarnation, the Doctor had saved several scrolls from the fire, (AUDIO: The Carrionite Curse) including two plays by Aristophanes. (PROSE: Timewyrm: Apocalypse) The Eighth Doctor told Tamsin Drew that he couldn't save the Halcyon from the Ice Warriors any more than he could prevent the fire of the Library of Alexandria. (AUDIO: The Resurrection of Mars)