That's the great lie--Gibbon, wanting to claim Romanitas for the 'Enlightenment' against Christianity, and all the Muslim apologists now, all want us to forget that there was a Christian civilization that was never overrun by Germanic barbarians, from which the Muslims took most of their learning, from which they copied such civilized ways as they adopted, and which in the end they destroyed.
The name 'Byzantine Empire' is itself part of the lie.
It was the Roman Empire, its capital moved to Constantinople, New Rome, by Constantine, and it preserved classical learning, mathematics included, until its fall. The Caliph of Damascus once offered a great sum of money to the Emperor to let one Leo the Mathematician visit his court--the Emperor declined. Its scholars fleeing the advance of the Muslims, and settling in the most Roman regions of Italy--not Rome itself, but the Veneto--were the trigger for the Rennaisance.
In practical Roman fashion, the few advances in mathematics made during the Christian era in the Roman Empire were related to engineering: Anthemius of Tralles, one of the architects of Justinian's Hagia Sophia, wrote treatises on conic sections, giving some advances on Apollonius and Archimedes, including describing the focal properties of the parabola, and developing the method of drawing ellipses with two pins and a loops of string.
Of course, no one goes on about the advances of 'Byzantine mathematics' (and rightly so, just as with the Muslims, there really weren't very many), and its history is largely forgotten except among us Orthodox, and quirky academic 'Byzantinists' (a lot of whom eventually convert to Orthodoxy).
That was my point, though I didn't make it well. Indeed they were Rome, in law, tradition, and structure, just not religion. They twice reconquered most of the empire. When Rome fell alot of the intellectual capital and important families went to Byzantium and integrated as the cultures were basically the same. And like Rome their architectural achievements were second to none.
Granted, Europe went backwards in alot of areas, especially trade and stability with the fall of the Western Empire, but the Germanics were Romanized enough that they didn't fall back across the board into primitive barbarian tribes either. The Franks, Goths, and others were already on the move into what would eventually be the Holy Roman Empire in what was a desire to put civilization back together with a successor state to Rome. The Byzantines honored a number of those attempts if I recall.
Through all that, the Byzantines were there. So Europe may have went dark in some respects after the fall of Rome, but Western civilization did not die either in my opinion. And Europe kept open ties to the Byzantines, so they had access to knowledge and culture. Worse off in many ways due to the lack of order and trade than when they were under Rome, but not totally out of the loop either.
As you said, civilization eventually transferred back to the Western part of the Empire, or rather the tiny remnants of it that still flourished around Rome, with the advance of the Muslims and a more stable, Christianized Europe. The thread of Western Civilization was never really lost, despite what the Muslim apologists want to believe.