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or simply the Greek Empire.
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That's because there was a Greek sort of veneer over
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what would have been local differences.
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You'd have Egyptians speaking ancient Egyptian languages
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but the elites in the cities would be speaking Greek,
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so the culture, the elite culture would still be Greek
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and the same way in Syria.
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The Ptolemies are easier to keep straight
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because they tended to all be named Ptolemy.
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They might have a nickname
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ike Ptolemy Philadelphus is a very famous one,
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but then they would give them numbers.