The best I can do is hazard an attempt without dictionary. A 'Deus Ex Machina' is a plot device using divine intervention; that is to say, it is aid unlooked for that comes without precedent; an intervention without plot set-up in order to make the story work. Perhaps my definition is narrow, but if it's accurate, Tolkien doesn't use it. The Eagles are written deeply into his entire mythos and play their role effectively.
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