What do we know about the organizational structure of the Library of Alexandria?

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In the Wikipedia article you have copied there is a reference to the Pinakes.

Searching I have found and article on this website Library Philosophy and Practice (LPP) that explains the organization of the books, manuscripts,... of this vast library. I copy some fragments:

About Zenodotus, the first librarian

Zenodotus, the Great Library's first librarian, introduced a rudimentary organization system whereby texts were assigned to different rooms based on their subject matter. Zenodotus first inventoried the Library’s holdings, which he then organized into three major categories. [...] Within each of these divisions, Zenodotus organized the works alphabetically by the first letter of the name of their author [...] Library staff under Zenodotus attached a small dangling tag to the end of each scroll, which contained information on each work’s author, title, and subject so that materials could be easily returned to the area in which they had been classified

About Callimachus of Cyrene.

Callimachus produced the pinakes [...] Callimachus listed works alphabetically by author and genre. He did what modern librarians would call “adding metadata” -- writing a short biographical note on each author, which prefaced that author’s entry within his catalogue. [...] In addition, Callimachus noted the first words of the work, and the total number of lines in the document. Later librarians were to make marginal notations in the pinakes, which provided even more information on the nature of the catalogued document [...] By consulting the pinakes, a library patron could find out if the library contained a work by a particular author, how it was categorized, and where it might be found.

But...

While the pinakes is very similar to what modern librarians would refer to as a library catalogue, it did not cover the entirety of the holdings of the Great Library. It dealt only with the largest and most often used portion of the collection. However, Zenodotus’ organizational principle did cover the entire Library.

According to biblicalarchaeology.org gives more details about the "meta-data" done by Callimachus.

The Pinakes identified each volume by its title, then recorded the name and birthplace of the author, the name of the author’s father and teachers, the place and nature of the author’s education, any nickname or pseudonym applied to the author, a short biography (including a list of the author’s works and a comment on their authenticity), the first line of the work specified, a brief digest of the volume, the source from which the book was acquired (such as the city where it was bought or the ship or traveler from which it was confiscated), the name of the former owner, the name of the scholar who edited or corrected the text, whether the book contained a single work or numerous distinct works, and the total number of lines in each work.

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Strabo, who was in Egypt from 24 to 19 BC and gave us detailed accounts of the Museum, claims the inspiration for the Library's organization - at least initially - was Aristotle (emphasis mine):

From Scepsis came the Socratic philosophers Erastus and Coriscus and Neleus the son of Coriscus, this last a man who not only was a pupil of Aristotle and Theophrastus, but also inherited the library of Theophrastus, which included that of Aristotle. At any rate, Aristotle bequeathed his own library to Theophrastus, to whom he also left his school; and he is the first man, so far as I know, to have collected books and to have taught the kings in Egypt how to arrange a library.

Source: Strab. 13.1.54.

The story seems likely; Aristotle was the father of taxonomy and his personal library was one of the earliest acquisitions of the Library. The early librarians probably modeled their own approaches upon Aristotle's. However, we don't have any details on how Aristotle organized his library, and to what degree the early librarians adopted his methods.

What we do know, is that Zenodotus of Ephesus, the first librarian according to the Suda, arranged books in different rooms according to their subject matter. Then, within each room, he arranged the books alphabetically (by their author's name). His system may seem rudimentary by today's standards, but for the time it was revolutionary. In fact, it's the earliest known instance of alphabetic organization.

Callimachus of Cyrene furthered Zenodotus system in his magnum opus, the Tables of people who distinguished themselves in all branches of learning. Pinakes, as the work is better known, is a bio-bibliographical survey of the more significant Greek writings, divided in 120 books. According to Wikipedia:

Callimachus' system divided works into six genres and five sections of prose. These were rhetoric, law, epic, tragedy, comedy, lyric poetry, history, medicine, mathematics, natural science and miscellanies. Each category was alphabetized by author.

Unfortunately only a few fragments of the Pinakes survive, but from those fragments we know they included biographical data, including the author's other works, and general notes for each work (e.g. its extend). Callimachus' Pinakes were the key to the vast collection of Library, and were constantly updated by subsequent librarians and grammarians, as the collection grew.

Sources:

1 Πίνακες των εν πάση παιδεία διαλαμψάντων.

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