Trying to explain better why in English the 'h' is used after some 'r's, but not others (when transliterating from the Greek rho), the following patterns usually apply:
1. H is used when there is an initial 'rho': Rhodos, Rhegion, Rhea, Rhetor, Rhapsode, Rhino, Rheumatism, etc.
2. H is used when there is a double 'rho': Pyrrhos, Hemorrhoid, Arrhythmia, Gonorrhea
So when we don't have these two conditions met, we wind up without the 'h', like when we have Korinthos, Kyrene, Alexandreia, Hierosolyma, Tarsos, Pergamon, Trapezous, Demetrias, Thermon, Taras, Brettia, Syrakousai, Trinakrie, etc.
The only exceptions I know of is when you compound words. You take the stem that means "flow" ("rhe-") and put a prefix in front of it and it still will yield the 'h' in English: diarrhea.
We could leave it out for sure, as the rough breathing in these initial and double 'rho's is automatic in virtually all cases, but it doesn't hurt anything to leave it in (even if it seems superfluous to some) and it is familiar to a larger part of our audience.
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