Something over 99% of books written in antiquity have been lost, so the usual form of the question is the other way round -- 'how did such-and-such a text survive?' If a book was a core text in the Byzantine educational system, that generally meant survival, but that only applies to a handful of literary classics: anything outside that core was prone to be lost eventually. Here's an excerpt of an answer I wrote last month, about Herodotus --

Ancient books were transmitted by an ongoing tradition of copying and re-publishing books. In antiquity this was done by booksellers who got scribes to make new copies; there was a severe decline in book culture in the Greek-speaking parts of the Mediterranean between the 6th and 9th centuries, and when it revived starting in the mid-800s, it was thanks to a mix of secular and ecclesiastical scholarship. Book culture in Constantinople was more or less continuous from then until 1453.

After that point, the text of Herodotus survived in copies that had mostly been previously transported to Italy, and disseminated from there. Manuscripts made in that era -- after the development of the printing press -- are often very important too. In the west, different manuscript collections had different levels of stability and permanence. For example, one collection now held at the Vatican, the Palatine collection, came from a collection that was captured in the Thirty Years War from the Electoral Palatinate in Heidelberg in the 1500s-1600s. This in turn had been swiped from the monastery of Lorsch when the monasteries were dissolved in the 1560s. And Lorsch in turn got its collection from a range of sources.

And here's a longer piece I wrote last year that gave more context. There are printed accounts out there too, but for my money they underplay the importance of Greek scribes and scholars in the renaissance.

Nearly all Greek literature that survives, survives in Greek, but there are exceptions: these are the lucky texts, remember. For a few, we do rely on translations into several other languages -- especially Latin and Armenian, but also Syriac and Arabic.

Apollonius of Perga's Cutting-off of a ratio is one of them. Only an Arabic translation survives. For the Conics, we're luckier: books 1-4 are extant in Greek, though there's only a single manuscript. It's at the Vatican, Vat. gr. 206: unfortunately it isn't one of the ones that they've digitised so far. But for books 5-7 we have to rely on an Arabic version; and book 8 is lost altogether.