Nested prepositional phrases

Question

I'm trying to learn me some Latin recently, using Euler's works as my training material, since some of them already have English translations, so I can compare my attempts with theirs, and use them as a learning aid.

One of them (E26) starts with the following sentence:

Notum est hanc quantitatem aⁿ+1 semper habere divisores, quoties n sit numerus impar, vel per imparem praeter unitatem divisibilis.

which has been translated by Jordan Bell as:

It is known that the quantity aⁿ+1 always has divisors whenever n is an odd mber or is divisible by an odd number aside from unity.

which seems to be correct. But now I'm trying to unpack the structure of this sentence from this. The thing that I have trouble with is the nesting of prepositional phrases at the end of this sentence, because they seem to be nested one within another, and I was taught that in Latin, prepositional phrases can only be attached to verbs, never to nouns. But when I try to apply this guideline here, it doesn't seem to make any sense :q Here's how I understand the structure of this sentence, and please correct me if I'm wrong:

From what I see, it starts with an indirect statement (accusative + infinitive), after a "verb of knowing" (well, in a way): "Notum est" = "It is known". Known what? Well, whatever the rest of the sentence tells us, so "it" refers to the indirect statement that follows (which I presume is the subject of the sentence, the aforementioned "it", which is then linked to the adjectival complement – the participle "notum" that describes it).

Now there's a bunch of words in accusative, followed by an infinitive "habere", so I presume this is the "accusative + infinitive" construction that is being reported as "well known".

But then there's that other clause that seems to be telling us something more about when the number of this form has divisors (i.e. attached adverbially to the infinitive verb "habere", in the same way as "semper"). So when does it have divisors? Whenever ("quoties") the exponent n has certain properties described in what follows.

So whenever n is a number ("numerus") of that particular type. Of what type? Well, I see two descriptions that tell that, presented as alternatives with "vel":

1. a simple adjective "impar" (so, when the number is odd), or…
2. a compound adjective "divisibilis" with this nested prepositional phrase that further describes it.
   Divisible by what?
   By an odd number ("imparem"), but not just by any odd number, but only by those that are not 1 ("unitatem").

So the way I see it is that the prepositional phrase "praeter unitatem" ("except unity") tells us something more about what type of odd ("imparem") is meant, so it must describe the adjective "imparem". And this adjective (used substantively, I guess) is in turn the object of the external preposition "per", making a compound phrase "per imparem praeter unitatem" ("by an odd (number) other than unity") which describes the adjective "divisibilis" (telling us what it should be divisible by). And finally, the entire expression "divisible by an odd number other than unity" is used adjectivally to describe the noun "number" ("numerus"), as one of the alternatives (the other one is "impar").

Is this interpretation correct?

My uncertainty comes from that guideline that says that prepositional phrases must be bound to verbs, not nouns. But I'm equally uncertain about the validity of that guideline, because if prepositional phrases in Latin can also act adverbially (as they usually do in other languages), then surely enough they could describe verbs (as all adverbs do), but adverbs can also describe adjectives and other adverbs, so if that's the case in Latin as well, then there shouldn't be anything wrong with describing adjectives or adverbs with prepositional phrases as well. Is my thinking correct?

I dug through a couple of books about Latin grammar, and despite them being rather volumous (one having over 700 pages!), I couldn't find much (if anything at all) about the construction and usage of prepositional phrases in Latin, let alone whether can they be nested in the way I described above :q So any references to online resources or books about that particular subject would be appreciated.

Answer 1

Your analysis seems entirely correct.

In short, I would say praepositional phrases can indeed be used to modify nouns and adjectives, just as in English and other Indo-European languages, especially when the noun or adjective has a somewhat praedicative 'feel' to it, like divisibilis.

It is true that Latin has a preference for adjectives to modify nouns, and, if that isn't possible, for nouns in the genitive; but neither of those usually specifies the kind of relation between one thing and the other, which praepositional phrases can.

populus Romanus

The adjective is readily available, and so this is preferred over populus Romae. Further, there is no ambiguity. A praepositional phrase would really not add anything. So I think this is what you will normally see.

iter fluminale

This is possible. It is a journey related to a river. But it can be ambiguous, because the relation between journey and river is ambiguous.

iter ad flumen

iter praeter flumen

These two phrases could be used to resolve any ambiguity: is it a journey to a river, or along it?

Notice how the noun iter has a somewhat praedicative aspect: like the verb ire, it easily attracts adverbial phrases related to motion.

This can also apply to modifying nouns, although [numerum] imparem does not seem praedicative at all. But the notion "except" in praeter is so universal that I think it can be attached to almost anything.