Often I read that "X's quality Y was the greater by virtue of Z", which makes perfect sense as a result of me being used to this form of expression, but not when I am now trying to understand what is that "the" referring to. If we drop the "the", what precisely happens to the meaning? Which is the the's noun?
In particular, I can't figure out which one to use:
"Because A is much greater than B, C is also greater."
"Because A is much greater than B, C is also the greater."
It seemed quite clear to me at the time of writing, but now that I re-read the question and the answers, I can see that this is quite confusing, actually. Because I didn't provide examples.
Example: from here (emphasis mine :)
The excitement was the greater because constitutional, or quasi-constitutional, issues were intertwined to an almost Tudor degree with the personal and dynastic; indeed the wider significance is the more pressing because of the passions with which recent events have been fuelled.
This has the form:
A was the greater because B, C, D, E, blah blah
Pretty awful piece of pretentious writing if you ask me, but it does use language correctly.
Example: from Mrs Daffodil Digresses (emphasis mine)
[...] husband [...] refused to revisit the scene [...] and remained inexorable, when Alphonse F., entering while the discourse continued, volunteered to spare his friend’s feelings by visiting the chateau and obtaining the required papers.
The marquis thanked him cordially, adding, that the relief was the greater, inasmuch as he would have been compelled to enter their favourite sitting-room, in which their last, as well as so many happier, hours were passed.
Example: from The British Magazine (emphasis mine)
[...] they broke open the doors, and were confounded at finding the habitations abandoned. Their astonishment was the greater, as they could not comprehend in what manner the French escaped; and when they did, they could not make use of their canoes pursue the fugitives, because the were still encumbered with ice, which prevented that kind of navigation.
So, this usage is usually to be found in literary works of archaic variety. I'm leaning to treat it as one of those language patterns that doesn't have an analytic explanation, and is to be learnt and used as a unit, by feel. I suppose that it can be interchanged with "greater still" in all of those examples, and feels about right. So my rule for its appropriateness would be: it's appropriate to use iff substitution with "greater still" preserves the meaning.