The main difference between the Roman Empire and the Byzantine Empire concerned the official religions they practiced. Whereas the Roman Empire was officially pagan up for most of its existence, the Byzantine Empire was Christian. The Byzantine Empire was the significant remnant of the Roman Empire that survived in southeastern Europe for a thousand years after the official fall of Rome in 476 CE. As noted, a key difference with Rome was that the Byzantine Empire was always Christian rather than pagan. This hardwired into Byzantium a lack of cultural openness to the kind of religious diversity that had helped classical Rome to expand and thrive.
Another important difference was the relative weakness of Byzantium vis-à-vis the Roman Republic's power in its heyday. While powerful in some ways, Byzantium did not function as a hegemonic cultural, political, and military superpower in the same way as did the classical Roman Empire. This had the downside of leaving western Europe vulnerable to attacks, particularly from Viking marauders, that would not have occurred under the Roman Empire, but this also created an upside in which the western Europeans were forced to create their own vibrant and flexible cultural, political, and military institutions and infrastructures in order to survive.
Byzantium remained crucially important, however, because it controlled Constantinople, the gateway to the Mediterranean as well the gateway to overland passages to Asia. This was a source of access to vital trade routes with the East that this remnant of the Roman empire safeguarded for western Europe. Unfortunately, however, unlike Rome in its heyday, Byzantium ultimately lacked military might to keep this territory from Muslim conquest.
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The Three-Fifths Compromise solved a dispute between "3.states with and without slavery" This compromise "solved" the issue that took place when the slave states wanted to count their slaves as part of the state population in order to gain more seats in Congress.
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