What does Alexandrian Library mean in the Bible?

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Watson's Biblical & Theological Dictionary - Alexandrian Library
This celebrated collection of books was first founded by Ptolemy Soter, for the use of the academy, or society of learned men, which he had founded at Alexandria. Beside the books which he procured, his son, Ptolemy Philadelphus, added many more, and left in this library at his death a hundred thousand volumes; and the succeeding princes of this race enlarged it still more, till at length the books lodged in it amounted to the number of seven hundred thousand volumes. The method by which they are said to have collected these books was this: they seized all the books that were brought by the Greeks or other foreigners into Egypt, and sent them to the academy, or museum, where they were transcribed by persons employed for that purpose. The transcripts were then delivered to the proprietors, and the originals laid up in the library. Ptolemy Euergetes, for instance, borrowed of the Athenians the works of Sophocles, Euripides, and AEschylus, and only returned them the copies, which he caused to be transcribed in as beautiful a manner as possible; the originals he retained for his own library, presenting the Athenians with fifteen talents for the exchange, that is, with three thousand pounds sterling and upwards. As the museum was at first in the quarter of the city called Bruchion, the library was placed there; but when the number of books amounted to four hundred thousand volumes, another library, within the Serapeum, was erected by way of supplement to it, and, on that account, called the daughter of the former. The books lodged in this increased to the number of three hundred thousand volumes; and these two made up the number of seven hundred thousand volumes, of which the royal libraries of the Ptolemies were said to consist. In the war which Julius Caesar waged with the inhabitants of Alexandria, the library of Bruchion was accidentally, but unfortunately, burnt. But the library in Serapeum still remained, and there Cleopatra deposited the two hundred thousand volumes of the Pergamean library with which she was presented by Marc Antony. These, and others added to them from time to time, rendered the new library more numerous and considerable than the former; and though it was plundered more than once during the revolutions which happened in the Roman empire, yet it was as frequently supplied with the same number of books, and continued, for many ages, to be of great fame and use, till it was burnt by the Saracens, A.D. 642. Abulpharagius, in his history of the tenth dynasty, gives the following account of this catastrophe: John Philoponus, surnamed the Grammarian, a famous peripatetic philosopher, being at Alexandria when the city was taken by the Saracens, was admitted to familiar intercourse with Amrou, the Arabian general, and presumed to solicit a gift inestimable in his opinion, but contemptible in that of the barbarians; and this was the royal library. Amrou was inclined to gratify his wish, but his rigid integrity scrupled to alienate the least object without the consent of the caliph. He accordingly wrote to Omar, whose well known answer was dictated by the ignorance of a fanatic: "If these writings of the Greeks agree with the Koran, or book of God, they are useless, and need not be preserved; if they disagree, they are pernicious, and ought to be destroyed." The sentence of destruction was executed with blind obedience: the volumes of paper or parchment were distributed to the four thousand baths of the city; and such was their number, that six months were barely sufficient for the consumption of this precious fuel.

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Alexandrian - ) Of or pertaining to Alexandria in Egypt; as, the Alexandrian Library
On - Tradition makes Plato and other Greek philosophers study in Heliopolis; later, the foundation of the Alexandrian Library, on the one hand, deprived Heliopolis of the glory of learning, and, on the other, the old traditions of royal descent from the Sun-god had little weight with the Ptolemys