Jesus spoke Aramaic or Hebrew doesn't really matter since all the New Testament writers wrote in Greek because that was the common and most widespread language of the Eastern Roman Empire and most of the early Christians were native Greek-speakers.
In the 16th century, Erasmus assembled a Greek New Testamrnt based on the majority Greek texts. It's true that there were changes made from the Greek to the Latin, but, Erasmus didn't rely on the Latin in assembling his Greek New Testament and neither did translators of the Old Testament into English. But, then again, there was more than one version of the Bible translated from Greek into Latin over several centuries and they weren't all corrupted. The way to tell obviously being the outliers. If translators had two Latin texts that read differently, if one did not match the majority of Greek, Latin, Syrian, Arabic and other early text, it was obviously a corruption and therefore discarded.
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