Yes, you are right. In the second case, the syntactic break is between 'raged' and 'down': [The debate has raged] [down through the ages].
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AngliholicIs it right to understand the bolded words in the above as "appeared"Yes.
Angliholicand "on"The debate has continued over time. I guess the trick is to view time as a kind of line or flow so that the debate in question has come down to us through that flow of time.