Magistra monet muscas et apes non tangendas esse, ne puellae pungantur.

They are accusative because they are the subject accusatives of an indirect statement after monet.

In English, we say:

• The teacher warns that flies and bees should not be touched.

In Latin, after a verb like monet, the content of the warning can be expressed with:

• accusative + infinitive

So:

• muscas et apes = the things being talked about in the indirect statement
• esse = the infinitive that goes with that statement

Even though muscas et apes are the logical subject of tangendas esse, they appear in the accusative because that is how indirect statement works in Latin.

Tangendas esse means to be touched in the sense of should be touched / must be touched / are to be touched.
Because of the non, here it means:

• not to be touched
• should not be touched
• must not be touched

This is a gerundive + esse construction, often called the passive periphrastic.

So:

• tangendas = a gerundive from tangere (to touch)
• esse = to be

Together, they express necessity or advisability.

So muscas et apes non tangendas esse means:

• that flies and bees are not to be touched
• or more naturally, that flies and bees should not be touched

Because it agrees with muscas et apes.

Both nouns are:

• feminine
• plural
• accusative

So the gerundive has to match them:

• muscas = feminine plural accusative
• apes = feminine plural accusative
• tangendas = feminine plural accusative

This agreement is just like an adjective agreeing with the noun it describes.

Because Latin indirect statement normally needs an infinitive.

After monet, the sentence gives the content of the warning as an indirect statement:

• muscas et apes non tangendas esse

The infinitive is esse.
Without esse, the construction would be incomplete.

So the pattern is:

• monet
  • accusative subject
    • infinitive phrase

Here that becomes:

• monet
  • muscas et apes
    • non tangendas esse

Yes. In this sentence, monet introduces an indirect statement:

• Magistra monet muscas et apes non tangendas esse

That means:

• The teacher warns that flies and bees should not be touched

Latin often uses indirect statement after verbs of saying, thinking, knowing, and also sometimes after verbs like warning.

Because non negates tangendas esse inside the indirect statement.

So:

• non tangendas esse = not to be touched

By contrast, ne introduces a subordinate clause with the subjunctive, usually meaning:

• lest
• so that ... not
• in order that ... not

That is exactly what happens later in the sentence:

• ne puellae pungantur

So the sentence uses both:

• non to negate the indirect statement
• ne to introduce the negative purpose/preventive clause

It is a negative purpose clause or a closely related clause of prevention:

• ne = lest / so that ... not
• pungantur = subjunctive, because purpose clauses use the subjunctive

So the sense is:

• lest the girls be stung
• so that the girls may not be stung

It explains why the warning is given.

Because it is in a clause introduced by ne.

Latin regularly uses:

• ut
  • subjunctive for positive purpose
• ne
  • subjunctive for negative purpose

So:

• ne puellae pungantur = lest the girls be stung

The subjunctive here is not mainly about uncertainty; it is there because the clause expresses purpose or prevention.

Because the meaning is lest the girls be stung, not lest the girls sting.

So:

• pungantur = they may be stung

This is the present passive subjunctive of pungere.

The passive is natural here because the girls are receiving the action.

Because puellae is the subject of pungantur.

In the clause:

• ne puellae pungantur

the girls are the ones who might be stung, so they are the grammatical subject.
That makes them nominative plural:

• puellae = the girls

If it were accusative puellas, it would not fit the grammar of this clause.

Not exactly.

In this sentence, ne does not just mean a simple not. It means something more like:

• lest
• so that ... not
• in order that ... not

So:

• non tangendas esse = that they should not be touched
• ne puellae pungantur = lest the girls be stung

A learner should be careful not to treat non and ne as interchangeable.

Because the gerundive usually adds an idea of necessity, obligation, or propriety.

So:

• tangi would simply mean to be touched
• tangendas esse means to have to be touched / to be to be touched / should be touched

With non, that becomes:

• should not be touched
• must not be touched

So this is stronger than a simple passive infinitive.

Yes. Tangendas esse is a form of the passive periphrastic.

The passive periphrastic is:

• gerundive + form of sum

It expresses necessity or obligation.

For example:

• liber legendus est = the book must be read
• muscas et apes non tangendas esse = that flies and bees must not be touched

Because this one is inside an indirect statement, the form of sum is the infinitive esse.

Latin word order is much more flexible than English word order because the endings show the grammatical relationships.

So Latin can place words for emphasis, rhythm, or clarity.

In this sentence:

• Magistra monet comes first, giving the main action
• muscas et apes comes next, introducing what the warning is about
• non tangendas esse completes the indirect statement
• ne puellae pungantur comes at the end as the purpose/prevention clause

English depends much more heavily on word order, but Latin does not need to follow the same pattern.

Not in the form actually written.

If Latin wanted to say the teacher warns the girls not to touch flies and bees, it would normally use a different construction, with puellas as the object of monet and then wording for not to touch.

But here the sentence says:

• monet muscas et apes non tangendas esse

So the content of the warning is:

• that flies and bees are not to be touched

Then the ne clause explains the reason:

• lest the girls be stung

So the girls are involved in the second clause, not as the direct object of monet.

That is simply how those nouns work in Latin.

• musca, muscae = fly, feminine
• apis, apis = bee, feminine

So in the accusative plural:

• muscas
• apes

Because both are feminine plural, the gerundive that agrees with them is also feminine plural:

• tangendas

Yes. It gives the practical reason or purpose behind the warning.

The overall sense is:

• The teacher warns that flies and bees should not be touched, lest the girls be stung.

So the teacher gives the warning to prevent the stinging of the girls.

That is why the ne clause fits so naturally here.

Pungantur is present passive subjunctive, third person plural.

Breaking it down:

• pung- = stem of pungere (to sting)
• -antur = third person plural passive
• subjunctive because it follows ne

The present subjunctive is normal in a purpose clause after a present main verb like monet.

So the sequence is natural:

• monet = present
• ne ... pungantur = present subjunctive

A useful breakdown is:

• Magistra — nominative singular, subject, the teacher
• monet — present active indicative, warns
• muscas et apes — accusative plural, subject accusatives of indirect statement, flies and bees
• non tangendas esse — gerundive + infinitive, should not be touched
• ne — lest / so that ... not
• puellae — nominative plural, subject of pungantur, the girls
• pungantur — present passive subjunctive, may be stung / be stung

So the structure is:

• main clause: Magistra monet
• indirect statement: muscas et apes non tangendas esse
• purpose/prevention clause: ne puellae pungantur