Prope focum coquus stat, nam focus totam culinam calidam facit.

Because prope is a preposition that takes the accusative case.

• prope focum = near the hearth
• focus = the hearth as the subject of facit

So the same noun appears in two different forms because it is doing two different jobs:

• focum: object of the preposition prope
• focus: subject of the verb facit

This is a very common feature of Latin: noun endings change depending on function.

Coquus is nominative singular.

You can tell because it is the subject of stat:

• coquus stat = the cook stands / is standing

In this sentence, the cook is the one doing the standing, so coquus must be nominative.

Because stat is more specific than est.

• est = is
• stat = stands or is standing

So coquus stat gives a more physical picture: the cook is not just existing near the hearth, but actually standing there.

Latin word order is much more flexible than English word order because the endings show what each word is doing.

So all of these would be understandable Latin:

• Prope focum coquus stat
• Coquus prope focum stat
• Coquus stat prope focum

The version here begins with prope focum to foreground the location: near the hearth.

Nam means for, because, or you see. It introduces an explanation.

So:

• coquus stat ... nam ...
• the cook stands ... because / for ...

The second clause explains the first: the cook is near the hearth because the hearth makes the whole kitchen warm.

Because they are all feminine singular accusative forms.

• culinam is the direct object of facit
• totam agrees with culinam
• calidam also agrees with culinam

So:

• totam culinam = the whole kitchen
• calidam = warm, describing the kitchen

All three forms match because they refer to the same feminine singular accusative noun.

Because after facit, Latin often uses a predicate accusative or object complement.

In English we say:

• The hearth makes the whole kitchen warm

Here, warm describes the object (the kitchen), not the subject (the hearth).

Latin does the same thing:

• focus totam culinam calidam facit

So calidam agrees with culinam, not with focus.

A helpful way to break it up is:

• focus = subject
• facit = verb
• totam culinam = direct object
• calidam = what the object is made to be

So literally:

• the hearth makes the whole kitchen warm

This is a very common Latin pattern:

• [subject] + [object] + [adjective agreeing with object] + facit

Because Latin has no articles.

There is no separate word for:

• the
• a
• an

So coquus can mean:

• the cook
• a cook

And focus can mean:

• the hearth
• a hearth

You decide from the context which English article makes best sense.

Yes, it is related, but the meaning in Latin is different.

In Latin, focus means hearth, fireplace, or domestic fire.

The English word focus originally developed from this idea, but in modern English it usually means center of attention or point of concentration. So in this sentence, you should think of a real hearth, not mental concentration.

Here are the main noun forms in the sentence:

• focus, focum: 2nd declension masculine
• coquus: 2nd declension masculine
• culina, culinam: 1st declension feminine

That is why you see:

• -us in nominative singular for coquus and focus
• -um in accusative singular for focum
• -a / -am with culina

Yes, when prope is used as a preposition meaning near, it takes the accusative.

So you get:

• prope focum = near the hearth
• prope villam = near the house
• prope viam = near the road

This is just something to memorize with the preposition.