The Catholic Church and Muslim world are credited for having saved Greek and Roman writing following the fall of the Western Roman Empire (eventually contributing to the Renaissance). To what extent did the Eastern Roman Empire have copies/preserve those same works?

I decided to look into this a bit, because questions like it come up quite a bit. I am no Byzantinist, and I haven't spent any particular time studying the Arab translation movement and none at all on the reception of these texts in Western Europe.

I apologize that this is a wall of text, but I find it to be extremely well argued, so I'd like to quote extensively on the question from Dimitri Gutas and his book Greek Thought, Arabic Culture as I think it both goes to some of the problems in answering the question and also lends itself to some further questions of my own:

Translation requires originals from which to be made; no matter how favorable all factors conducive to a translation activity may be, it cannot take place unless these archetypal texts are available. This raises immediately the question of the status of Greek manuscripts containing secular works in the eighth century, in areas accessible to the translators, i.e., in the Islamic realm and Byzantium.

Students of Byzantium have devoted a significant amount of attention to this question, though for different reasons. For them, the eighth century is the time when the minuscule hand of Greek writing first came into current and standard use, the time of the devastating upheavals of the iconoclastic controversy, the time of the “dark ages” of Byzantium. In the general dearth of information for the study of this crucial century, manuscripts present one of the few reliable sources, and they have accordingly been studied relatively intensively Although there are individual variations among Byzantinists in emphasis and nuance, there is a general consensus, first, that we know little about the period, and second, that the little we know paints a rather bleak picture about the status of Greek secular manuscripts during the period when these works were busily being translated into Arabic. Understandably, for Byzantinists ignorant of Arabic, the Graeco-Arabic translation movement has not figured in such discussions.

In the first place, all existing Greek manuscripts containing secular works until the middle to the end of the eighth century, i.e., until the beginning of the Graeco-Arabic translation movement, were written in uncials, the Greek majuscule hand. The minuscule came into use by about this time, and it took some considerable time before Greek scribes in Byzantium transcribed manuscripts into minuscule. The distinction between uncial and minuscule manuscripts is important for gauging their number and availability. The old, large hand is neat but cumbersome, and it takes more space than the cursive and smaller minuscule hand. This means that it took longer to transcribe codices in uncials than in minuscule, and that accordingly uncial manuscipts would be more expensive than minuscule. The writing material was inevitably animal skins, i.e. parchment. Papyrus was also used, but primarily in Egypt; outside of Egypt its usefulness was curtailed due its greater perishability in more humid climates.

Due to these circumstances it is understandable that during this period (and in this case, throughout the ninth century as well) there appears to have been no book trade in Byzantium to speak of. Book production was laborious and costly and therefore the acquisition of even a very modest private library of a few dozen books was beyond the means of most, if not all, rich intellectuals.*6

Second, we know relatively little about libraries in the Byzantine world because there is scanty information on the subject. We know next to nothing particularly for the earlier period until the ninth century, which is of relevance for our purposes here. But if one may extrapolate with caution from the situation existing after the ninth century, the major collections of books can be expected to have been in monasteries, in the libraries of high officials of Byzantine government (including the imperial library), and in private collections.47

This picture is in agreement with the general tenor of the eighth century in the history of Byzantium, the so-called “dark age,” and its devastating effect on the survival of classical culture. In particular, this was a period, starting already from the middle of the seventh century, when interest in and production of secular literature had completely disappeared. ** Consequently, no manuscripts of secular content were copied; there was no demand for them, and there were no scholars and scientists demanding them. The period of the iconoclastic controversy is singularly devoid of men of any eminence in sciences and philosophy. Then, after the turn of the ninth century, there is a gradual re-emergence of scholarly activity and its development into what Paul Lemerle called “the first Byzantine humanism.”

...

Evidence of this influence [of the Arabic Translation movement on the 9th century "renaissance" in Byzantium] is scarce and mostly anecdotal in the historical sources. On the Arabic side we have some references to missions sent by caliphs or scholars to Byzantium in search of Greek manuscripts. We also have reports about Greek books acquired after the sacking of a city, most famously Amorium (in 223/838) by alMu‘tasim. Such reports in the historical sources may or may not be true, but ultimately they are worthless because they are not specific enough for our purposes. Obviously, for all those Greek writings which were translated into Arabic, the translators had the Greek originals from which to work, and they got them from somewhere. But unless we know the precise provenance and date of the Greek archetype manuscript, information that a particular caliph sent a scholar to “Byzantium” to get it does not improve our knowledge.

Usefully interpreting such reports means identifying the sources of the Greek manuscripts used by the Arabic translators. In this regard scholarship by Arabists has not been as diligent as that by Byzantinists, though admittedly there is very little information and material to work with. We do not have Arabic manuscripts dating from the translation period, nor have the Greek manuscripts of the ninth and tenth centuries been investigated to ascertain whether they have been used for translation into Arabic.

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In a discussion of where the Arab translator Hunayn ibn Ishaq sought his manuscripts:

Other cities Hunayn mentions in connection with Greek manuscripts are Aleppo and I$trran. s2 It is interesting to note that he does not mention once any city under Byzantine jurisdiction, and especially not Constantinople. Given the tenor of the Ris&, in which Hunayn tries to present himself as an indefatigable philologist and strict critic of other translators, it is difficult to believe that he would not have mentioned the fact had he indeed found a Greek manuscript in Constantinople. The nature of this information is in accord with the little that we know, as mentioned above, about libraries containing Greek manuscripts in the seventh through the ninth centuries. We can only guess, in this case, that I$tnayn looked for those manuscipts in monasteries and churches and in private libraries.53

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Evidence from manuscripts [that the Byzantines were receiving texts from the Arabic world] comes from the fact that after a hiatus of apparently over one hundred and fifty years, Greek secular manuscripts began to be copied again around 800.57 The expert and detailed work accomplished by generations of Byzantine scholars has identified those manuscripts which survive from the first threequarters of the ninth century. These manuscripts have received repeated attention from Byzantinists because, in addition to being the major hard evidence for the ninth-century renaissance, they were for the most part written in the new minuscule hand in the context of a movement, aimed at transcribing the old uncial manuscripts, that is responsible for the preservation of most classical literature.58 A brief look at a listing of them makes it immediately apparent that the vast majority, indeed almost all of them, are scientific and philosophical. The question that forces itself upon us in the context of the subject of this book is, naturally, how these manuscripts correlate with the translation movement going on in Baghdad at the very moment these manuscripts were being copied.

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[Of a table comparing manuscripts translated into Arabic vs. those copied in Byzantine territory]:

The table shows an almost perfect positive correlation between the works translated into Arabic and the first Greek secular manuscripts copied during the first fifty years of the ninth century.

Gutas concludes that, provisionally, this would suggest that either A the Byzantines began copying this precise material because of an existing demand, or B in response to and in imitation of the Arabic copying.

Whether such materials, by this time, had to be reacquired from the Arabs themselves is not something I'm familiar with. But the quotes from Hunayn about not seeking these materials in Byzantine cities is striking to me.

As for my own questions, one of my biggest questions about the translation movement is about why some of the pre-eminent texts of the Greek tradition (e.g. Homer, Herodotus, Thucydides) were not translated. While Hunayn's abilities in Greek were apparently such that he could recite Homer by heart, as Gutas writes:

The extensive Homeric citations in Arabic which do survive are all translations of the passages in the works which cite him, primarily Aristotle and Plutarch’s Placita phihophorum.

Gutas makes the case that the reason behind the translation movement was to secure scientific and astrological texts, and to a lesser extent philosophical texts, as part of a regime building exercise, partly as a way to establish the Abbasids in a Perso-Greek intellectual tradition. Nonetheless the lack of a translation of the literary texts in comparsion with, say Kalila wa-Dimna, is striking to me.