I think the collection of the great library of Alexandria is now in Rome.................in the Vatican city..............

he Younger Seneca - On Tranquillity of the Mind

In fact we do know that the Royal Library is mentioned by Livy because he is later quoted by Seneca (died 65AD) in his dialogue On the Tranquillity of the Mind where he also says that a great number of books were destroyed. It has been asserted that Seneca must have got his knowledge about the destruction of the books from Livy but a close reading of the dialogue does not bear this out. Seneca actually only states that Livy thought the library was "the most distinguished achievement of the good taste and solicitude of kings" and then only so as he can disagree.
The actual number of books destroyed that Seneca gives is matter of some controversy that we will need to briefly address. In ancient manuscripts it is common for large numbers to be expressed as a dot placed above the numeral for each power of ten. Clearly in copying it is easy to make a mistake with the number of dots and errors by a factor of ten are frequent. That may have happened in the case of On the Tranquillity of the Mind. The manuscript from Monte Cassino actually reads 40,000 books but this is usually corrected to 400,000 by editors as other sources such as Orosius give this figure for the number of scrolls destroyed. I have not seen the manuscript, of course, so do not know if this way the number is expressed. However, even if it was given in words the difference between 40,000 and 400,000 is also pretty small. I propose therefore that the number given by Seneca, and indeed all other ancient sources, should be ruled as inadmissible as evidence because we cannot be sure of what it was originally.