Servus mulam prope stabulum ligat, ne in viam exeat.

Because mulam is the direct object of ligat.

• mula = mule in the nominative singular
• mulam = mule in the accusative singular

In the sentence, the slave is tying the mule, so mule receives the action of the verb. That makes it accusative.

Servus is the subject of the main verb ligat.

The nominative case is used for the person or thing doing the action. Here:

• servus = the slave
• ligat = ties

So servus is nominative because the slave is the one doing the tying.

After prope, Latin normally uses the accusative.

So:

• stabulum here is accusative singular
• prope stabulum = near the stable

This is something learners often have to memorize: prope is a preposition that takes the accusative.

Because in can take either the accusative or the ablative, depending on the meaning.

• in + accusative = motion into / onto / toward
• in + ablative = location in / on

Here the idea is that the mule might go out into the road, so motion is involved:

• in viam = into the road / onto the road

If it meant in the road as a location, you would expect in via.

Because exeat is a subjunctive form, and it is used after ne to express negative purpose.

• exit would be indicative: he/she goes out
• exeat is present subjunctive: he/she may go out or, in this structure, so that he/she may not go out

So ne ... exeat means so that it may not go out or lest it go out.

Here ne means so that ... not or lest.

It introduces a negative purpose clause. The idea is:

• the slave ties the mule
• for the purpose of preventing something
• namely, that it go out into the road

So ne in viam exeat means so that it does not go out into the road.

It is in the present subjunctive because the main verb ligat is in a primary tense (present), and Latin normally uses the present subjunctive in a purpose clause after a primary tense.

So the pattern is:

• main verb in present: ligat
• purpose clause introduced by ne
• present subjunctive: exeat

This is part of what is often called the sequence of tenses.

There is no separate subject stated in the subordinate clause, so the understood subject is the most natural one from context: the mule.

So:

• Servus mulam ... ligat = the slave ties the mule
• ne in viam exeat = so that it may not go out into the road

Latin often leaves out pronouns when the meaning is clear.

Exeat comes from exeo, exire, exii/exivi, exitum, meaning go out.

The form exeat is:

• 3rd person singular
• present
• active
• subjunctive

So it means he/she/it may go out, and here with ne it becomes so that he/she/it may not go out.

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

In this sentence:

• Servus is the subject because it is nominative
• mulam is the object because it is accusative
• ligat is the main verb

So even if the order changed, the endings would still tell you a lot. However, the given order is very natural:

• main statement first
• purpose clause after it

That said, Latin authors can move words around for emphasis.

Latin has no definite article like English the and no indefinite article like a/an.

So:

• servus can mean slave or the slave
• mulam can mean a mule or the mule
• stabulum can mean a stable or the stable

You decide from context which English article is best.

It tells you where the slave ties the mule.

So it is an adverbial phrase of place:

• prope = near
• stabulum = stable
• prope stabulum = near the stable

It modifies ligat, telling us where the tying happens.

In classical Latin, a negative purpose clause is normally introduced by ne, not ut non.

So:

• ut = so that
• ne = so that not / lest

That means ne in viam exeat is the standard classical way to say this.

Ut non is generally used in other kinds of clauses, not the normal negative purpose construction.

Because the sentence contains two different kinds of clauses.

1. Servus mulam prope stabulum ligat

   • this is the main statement
   • so the verb is indicative
   • ligat = ties

2. ne in viam exeat

   • this is a purpose clause
   • so the verb is subjunctive
   • exeat = may go out

So the change in mood reflects the difference between a plain statement and a clause expressing intention or purpose.