Final answer:
The historical name for the eastern portion of the Roman Empire is the Byzantine Empire, a pivotal center for Orthodox Christianity and Greek culture which contributed significantly to the European intellectual landscape post its fall.
Step-by-step explanation:
The historians' name for the eastern portion of the Roman Empire from the fourth century until its downfall to the Ottomans in 1453 is the Byzantine Empire. Famous as a center for Orthodox Christianity and Greek-based culture, it was strategically significant due to its capital, Constantinople, controlling key straits and serving as the gateway between Europe and Asia.
The Byzantine Empire was a continuation of the Roman tradition in the east and it played a vital role in preserving Roman cultural practices and the spread of Christianity. After the fall of Constantinople, many Greek scholars fled to Italy and contributed to the Renaissance, signifying a significant transformation of western Europe influenced by the intellectual heritage of the Byzantine civilization.