the great library at Alexandria was destroyed by budget cuts, not fire...,

io9 | Though it seems fitting that the destruction of so mythic an institution as the Great Library of Alexandria must have required some cataclysmic event . . . in reality, the fortunes of the Great Library waxed and waned with those of Alexandria itself. Much of its downfall was gradual, often bureaucratic, and by comparison to our cultural imaginings, somewhat petty. For example, the Roman Emperor Marcus Aurelius Antoninus suspended the revenues of the Mouseion, abolishing the members’ stipends and expelling all foreign scholars. Alexandria was also the site of numerous persecutions and military actions, which, though few were reported to have done any great harm to the Mouseion or the Serapeum, could not help but have damaged them. At the very least, what institution could hope to attract and keep scholars of the first eminence when its city was continually the site of battle and strife?

What's interesting here is Phillips' emphasis on how the decline of the library rested as much on its reputation as a learning center as it did on the number of books in its collection. What made the Museum and its daughter branch great were its scholars. And when the Emperor abolished their stipends, and forbade foreign scholars from coming to the library, he effectively shut down operations. Those scrolls and books were nothing without people to care for them, study them, and share what they learned far and wide.

The last historical references to the library's contents meeting their final end come in stories about the events of 639 CE, when Arab troops under the rule of Caliph Omar conquered Alexandria.

Luciano Canfora has written one of the most complete histories of the library, based on primary source material — documents written by people who knew and worked in the library. In The Vanished Library, he describes what the library at Alexandria had been reduced to by the time of its ultimate destruction in 639:

The Serapeum had been destroyed in the attack on the pagan temples in 391. The last famous figure associated with the Museum had been Theon, father of the celebrated Hypatia who studied geometry and musicology and whom the Christians, convinced in their ignorance that she was a heretic, barbarously murdered in 415 . . . Naturally, the city's books had changed, too; and not only in their content. The delicate scrolls of old had gone. Their last remnants had been cast out as refuse or buried in the sand, and they had been replaced by more substantial parchment, elegantly made and bound into thick codices - and crawling with errors, for Greek was increasingly a forgotten language. The texts now consisted chiefly of patristic writings, Acts of Councils, and 'sacred literature' in general.

This was not Ptolemy's great collection, nor was it the center of scholarship in what was then the modern world. It was a broken-down remnant of its former self, neglected for centuries. The collection was mostly stocked with materials that reflected what Judeo-Christian bureaucrats would have considered important; these materials did not reflect the Greek ideal of universal knowledge that had birthed the library in the first place.

In the end, it was only this diminished version of the library that was burned on the orders of Caliph Omar when Emir Amrou Ibn el-Ass took the city.