Quo diutius aurifex aurum polit, eo nitidius monile fit.

It is a very common Latin pattern called a correlative comparative. It corresponds to the English pattern the more ... the more ..., the longer ... the shinier ..., and so on.

So in this sentence:

• quo diutius = the longer
• eo nitidius = the more brightly / the shinier

This construction shows that one thing changes in proportion to another.

Both quo and eo are ablative singular neuter forms.

In this construction, the ablative has the sense of by how much and by that much:

• quo = by which amount
• eo = by that amount

So the pattern is literally something like:

• by how much longer the goldsmith polishes the gold, by that much brighter the necklace becomes

That sounds awkward in English, but it explains why Latin uses these forms.

Because the quo ... eo ... construction normally uses comparatives.

Diutius is the comparative adverb of diu, meaning for a long time. So:

• diu = for a long time
• diutius = for a longer time, longer

Here it modifies polit, so it tells you how long the goldsmith polishes.

Here it is best understood as a comparative adjective, agreeing with monile.

Why?

• monile is neuter singular nominative
• nitidius can be either:
  • a comparative adverb, or
  • a neuter singular comparative adjective

Because it goes with fit and describes what the necklace becomes, it works as a predicate adjective:

• monile fit nitidius = the necklace becomes shinier

So even though the form looks like an adverb, in this sentence it is naturally taken as an adjective with monile.

In Quo diutius aurifex aurum polit:

• aurifex is the subject: the goldsmith
• aurum is the direct object: the gold
• polit is the verb: polishes

So the structure is:

• aurifex = who is doing the action
• aurum = what receives the action

In eo nitidius monile fit:

• monile is the subject
• fit is the verb
• nitidius is the predicate adjective describing the subject

So the necklace is the thing that becomes shinier.

Because fit means becomes, comes to be, or is made, while est just means is.

That matters here because the sentence describes a change:

• not the necklace is shiny
• but the necklace becomes shinier

So fit is exactly the right verb for a change of state.

Fit is the 3rd person singular present indicative of fio.

Fio is an important verb meaning:

• become
• come to be
• sometimes be made

So monile fit means the necklace becomes.

Polit is 3rd person singular present active indicative of polio, polire, meaning to polish or to smooth.

So:

• polio = I polish
• polit = he/she polishes

Here the subject is aurifex, so the goldsmith polishes.

Aurifex is a 3rd-declension noun, nominative singular here.

It means goldsmith and is built from:

• aurum = gold
• -fex = maker/doer, related to facio

So aurifex literally means something like gold-worker or gold-maker, though the normal English meaning is goldsmith.

Not really. They are related in meaning, but they do different jobs in the sentence:

• aurifex = the person working
• aurum = the material being worked on

Latin often allows this kind of pairing. In fact, it can sound neat and natural because the words are closely connected.

Latin word order is much more flexible than English word order because Latin uses endings to show grammatical roles.

This sentence puts the paired comparison in prominent positions:

• quo diutius first
• eo nitidius later, answering it

That arrangement makes the proportional relationship very clear. English usually relies more on fixed word order, but Latin can move words around for emphasis and style.

Because Latin expresses that idea with its own idiom: quo ... eo ...

Languages often package the same idea differently. English uses the article the in a special adverbial way here, but Latin uses two correlated ablatives instead:

• quo = by which amount
• eo = by that amount

So this is just the normal Latin way of saying what English says with the ... the ....