In the pre-lexical stage of native speaker language acquisition patterns would have been learnt (or perhaps we could say we would have become familiar with them). The patterns of genitives are composed of έων, εως, -ας, -ης, -ος, -ους, -ου and -ων elements - with various stress patterns, but not in all possible combinations.
In other words, if we had learnt Greek natively, we would have patterns like "τῆς blah-ῆς bláh-ης". Which we would have recognised when we had come across passages like:
Matthew 7:11 wrote:διὰ τῆς στενῆς πύλης
A pattern like "τῆς blah-ῆς bláh-ας" as in:
Luke 13:24 wrote:διὰ τῆς στενῆς θύρας
Acts 24:2 wrote:διὰ τῆς σῆς προνοίας
when heard as a mature native speaker, would have been processed in a similar to the way I would listen to Americans saying /fæst/ for "fast". For somebody who would have first recognised the "τῆς blah-ῆς bláh-ης" pattern, they may at some stage of their development (perhaps 3 y.o.)
corrected the "τῆς blah-ῆς bláh-ας" pattern to "τῆς blah-ῆς bláh-ης", and for other early learners it would have been vice-versa.
Another genitive pattern that would have been learnt in the pre-lexical stage would have been "τῆς bláh-εως", that would allow recognition of phrases like:
Romans 2:23 wrote:διὰ τῆς παραβάσεως
Galatians 3:26 wrote:διὰ τῆς πίστεως
At an early stage (perhaps <3 years old) those would be recognised individual patterns, rather than a "genitive" pattern as the "δια του" that you mentioned implies. At some point around 3 years of age, there is a cognitive break through in language learning, and certain of the patterns would have been recognised as being grammatically equivalent.
When that would have happened, would have been that there would have been a leveling effect, similar to what we have in our children saying "maked" as an over extension of a common rule. According to the direction that Greek took to develop into Modern Greek, the basic unit that others were made to conform to was "τῆς blah-ῆς" / "τῆς bláh-ης" for some forms of feminine genitives. That, however, would not have been universal across all children.
It would have taken longer for a recognition of the grammatical equivalence of genitives across genders and that would lead to a confusion in genders (more often where natural and grammatical gender are different, less when they are the same) up until the ages of 7, 8 or 9 - depending on the individual child's cognitive development. The way that that happened, would depend on the order that the child recognised that some of the things that where up until then thought to be singularities in the language were actually "grammatically" equivalent - i.e. that they fulfilled the same syntactic function in similar sentence patterns.
Learning in that way, the δια could be followed by a diverse, but finite set of patterns, and in listening to or reading Greek, we would have (sub-consciously) recognised a pattern, and consciously paid attention to the lexical values of the lexically significant words nested in one or other of those patterns.
Looking at it in that way, the "δια του" that you mentioned could be seen as a short-hand way of referring to all those possible genitive patterns that a mature native speaker would have had - some beginning with the article and some not.
We have not learnt Greek like that, and you are not now either. You are learning to actively recognise the grammatical case with an English name in the foreground of your thinking. We start with a common patter and diversify, rather than start with all patterns and recognise equivalences.
I agree with Barry and Paul's sentiments...
Barry Hofstetter wrote:I've never seen it done before, but I agree with Paul -- the more we can use Greek to do Greek the better off we are. Paul, I think the rationale for using τοῦ is that the masculine or or possibly neuter singular would be the default form if one is simply expressing which case the preposition takes.