Fraternities with non-Greek letter names weren't unheard of in the 1800s. Rainbow and The Mystical 7 come to mind, and I think there were others. We were, of course, initially founded as the Sinfonia Fraternity, though we started using the Greek letters very early on. (We didn't officially change our name until 1948, though.)
Greek-letter societies arose in a liberal arts context for students who would have studied Greek and Latin. Triangle, FarmHouse and Sinfonia all share the background that they would have been founded for students in specialized degree programs -- engineering, agricultural or music -- for which Greek would not have typically been part of the curriculum at the time. (Indeed, that was an early objection to Greek letters in our name -- Greek wasn't part of a music student's course of study.) I doubt there was too much pressure on Triangle, FarmHouse or Acacia to change their names.
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