Definite article used in beginning of a sentence

Question

I am going through some German Math notes. The notes begin with this sentence:

Die im vorangehenden Kapitel behandelten Regeln und Sätze über Grenzwerte haben alle vorausgesetzt, dass wir die jeweiligen Grenzwerte schon “besitzen".

Could someone explain in details how the 'die' is used in the beginning?

Answer 1

One characteristic of German has been called Klammerstruktur: the phenomenon that words belonging together can be separated and occur at a vast distance from each other, which often poses problems for language learners.

The phenomenon of verbal bracketing is encountered at an early stage for learners.

Ich habe unterschrieben.

Ich habe in der vergangenen Woche nach reiflicher Überlegung den Vertrag für die neue Wohnung unterschrieben.

Habe unterschrieben forms a unit, yet in the second example, the listener has to wait a very long time for the participle unterschrieben to arrive. Before the utterance has been completed, the listener has to guess at what activity the utterance is about, or has to defer interpretation until it has been completed. Basically, the entire sentences intervenes between auxiliary and participle.

Your example shows a similar nominal bracketing, with the article and accompanying noun forming a unit that is separated by what is basically a sentence, paraphrasable by a relative clause.

die im vorangehenden Kapitel behandelten Regeln

die Regeln, die im vorangehenden Kapitel behandelt wurden

The correct strategy for interpreting such phrases is to keep scanning to the right after the article for a combination of adjective and noun that fits.

die … behandelten Regeln (weak adjective, plural noun)

ein im vorangegangenen Kapitel behandelter Ansatz (strong adjective, masculine noun)

aus natürlichen Quellen stammendes Wasser (strong adjective, noun that doesn't require an article)

The phenomenon is so prevalent in academic writing that someone dedicated a chapter of a book to describing an algorithm for decoding such phrases; see Tips for translating participial constructions. (The book is The key to German translation by Cecil Pollard.)

Answer 2

Let's look at a simplified version of the sentence:

Die Regeln und Sätze, ...

Here it is easy to see that Die is simply the article of Regeln and Sätze.

Now let's add an adjective (which is a particple, but this is not important at this point):

Die behandelten Regeln und Sätze, ..

Now let's add a bit more:

Die im vorangehenden Kapitel behandelten Regeln und Sätze, ...

Die is still the article of Regeln and Sätze, it is just that the attributive phrase im vorangehenden Kapitel behandelten separates them.

If the adjective is a participle, it can be accompanied by objects and adverbials. This way one can construct quite large attributive phrases.

An even longer example:

Die den Studenten gestern abend in feierlicher Sitzung überreichte Urkunde..

Answer 3

These are the parts of the sentence:

• Die im vorangehenden Kapitel behandelten Regeln und Sätze über Grenzwerte
  This whole phrase is the subject of the sentence. It is a nominal group (aka "nominal phrase") and it is in nominative case.

• haben ... vorausgesetzt
  This is the predicate (as defined in German grammar, which is another definition as in English grammar where is it just called "the verb", but a German predicate can have multiple verbs on different places in the sentence, but the bigger difference is, that objects do belong to the predicate in English grammar, while objects are not considered to be part of the predicate in German)

• alle
  This indefinite pronoun is a determiner that semantically also belongs to the subject, but stands here at position 3 of the sentence, which is the alternative place for subjects (if they are not at position 1). You could include it into the nominal group that is the subject, but then if replaces the already existing determiner die which is the topic of your question:

  Alle im vorangehenden Kapitel behandelten Regeln und Sätze über Grenzwerte

• dass wir die jeweiligen Grenzwerte schon “besitzen"
  This is a subjunctive object clause. The whole clause is here instead of the accusative object that the verb "voraussetzen" normally requires.

---

Let's dive deeper into the nominal group that is the subject. Each nominal group has a core (usually a noun, this is where the term nominal comes from) and some accompanists which can be determiners and attributes.

The core of the nominal group in your sentence is not a single noun but an enumeration:

Regeln und Sätze

To have enumerations where the textbook requires a noun is always possible

But nominal groups also need a determiner. This is very often an article (definite or indefinite)

der Kern, ein Kern
^{the core, a core}

But it can also be a demonstrative pronoun, ...

dieser Kern, jener Kern
^{this core, that core}

... a possessive pronoun, ...

mein Kern, sein Kern
^{my core, his core}

... or other kinds of words (numerals, indefinite pronouns, etc.)

drei Kerne, einige Kerne, wessen Kern?, ...
^{three cores, some cores, who's core?, ...}

In you sentence it is a definite article:

die Regeln und Sätze
^{the rules and clauses}

But the core can also have attributes. Attributes are often adjectives:

der rote Apfel
^{the red apple}

but also very often objects that can contain other nominal groups. (Here we have prepositional objects, but also genitive objects are very common.)

der Apfel in meiner linken Hand
^{the apple in my left hand}

die Regeln und Sätze über Grenzwerte
^{the rules and clauses about limit values}

Rule of thumb: Adjectives are left of the core (between determiner and core), objects that are used as attributes are right of the core. Of course you can have both of them:

der rote Apfel in meiner linken Hand
^{the red apple in my left hand}

Instead of adjectives you also can have participles:

der gefärbte Apfel
^{the colored apple}

die behandelten Regeln und Sätze
^{the treated rules and clauses = the rules and clauses treated}

But adjectives and particles can have their own attributes:

der rot gefärbte Apfel
^{the red colored apple}

die neulich behandelten Regeln und Sätze
^{the recently treated rules and clauses = the rules and clauses treated recently}

In the two examples above the words rot/red and neulich/recently describe properties of the participles gefärbt/colored and behandelt/treated which by themselves describe properties of the cores. So they are attributes of attributes.

But attributes of attributes are just attributes and can also be something else but adjectives, for example another prepositional object, consisting of a preposition and a nominal group:

der von der Sonne gefärbte Apfel
^{the by the sun colored apple = the apple colored by the sun}

die im vorangehenden Kapitel behandelten Regeln und Sätze
^{the in the previous chapter treated rules and clauses = the rules and clauses treated in the previous chapter}

(Note, that in German attributes of adjectives and participles are always left of them, even if they are objects.)

And you can put this all together to a very nice and elaborated nominal group:

der von der Sonne gefärbte Apfel in meiner linken Hand
^{the by the sun colored apple in my left hand = the apple colored by the sun in my left hand}

die im vorangehenden Kapitel behandelten Regeln und Sätze über Grenzwerte
^{the in the previous chapter treated rules and clauses about limit values = the rules and clauses about limit values treated in the previous chapter}

And of course you can use this nice nominal phrase as a subject or object in a sentence:

• as the subject

  Der von der Sonne gefärbte Apfel in meiner linken Hand schmeckt gut.
  ^{The by the sun colored apple in my left hand tastes well. = The apple colored by the sun in my left hand tastes well.}

  Die im vorangehenden Kapitel behandelten Regeln und Sätze über Grenzwerte sind kompliziert.
  ^{The in the previous chapter treated rules and clauses about limit values are complicated. = The rules and clauses about limit values treated in the previous chapter are complicated.}

• as an object (here: accusative object)

  Ich betrachte den von der Sonne gefärbte Apfel in meiner linken Hand.
  I watch the by the sun colored apple in my left hand. = I watch the apple colored by the sun in my left hand.

  Wir analysieren die im vorangehenden Kapitel behandelten Regeln und Sätze über Grenzwerte.
  ^{We analyze the in the previous chapter treated rules and clauses about limit values. = We analyze The rules and clauses about limit values treated in the previous chapter.}