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The native speakers of Aramaic were known as Arameans, and settled in an area around 3500 BC where modern-day Syria is. The area was then known as Aram, and is considered by linguists to be the hub and home of Aramaic. The Arameans later began to move to other areas, namely Mesopotamia, both voluntarily and involuntarily. Along with their belongings, they took the Aramaic language and alphabet with them when they left.
As it spread it became more varied, thus the different dialects still around today. The main division is between dialects found on either side of the Euphrates River. The dialects can also be categorized based on time period, falling under either Old, Middle, or Modern Aramaic. Around 500 BC, after Darius I conquered Mesopotamia, the Aramaic of that region and time period became the official language used by different regions of the empire to communicate with each other. By the 3rd century BCE, the Aramaic spoken in modern-day Syria and Egypt was largely replaced by Greek.
end partial quote.
So, because the educated of this area started to speak and be educated in Greek this also makes sense why the first language the Bible would have been translated into out of Aramaic would have been Greek.