Pater dicit diem natalem filiae omnibus annis gratiorem fieri.

After dicit, Latin uses an indirect statement.

So the sentence is structured like this:

• Pater dicit = Father says
• diem natalem filiae omnibus annis gratiorem fieri = that the daughter's birthday is becoming more pleasing every year

In Latin, an indirect statement normally uses:

• an accusative noun for the subject of the reported statement
• an infinitive for the verb

So here:

• diem natalem is the subject of the reported statement
• fieri is the infinitive verb

Because in an indirect statement, the subject is put in the accusative.

If this were a direct statement, you would expect something like:

• dies natalis ... fit or dies natalis ... est

But after dicit, Latin changes the subject into the accusative:

• dies natalis → diem natalem

So diem natalem is not the direct object of dicit in the ordinary English sense; it is the subject of the infinitive fieri inside the indirect statement.

Yes.

The basic expression is dies natalis, literally birth day.

In this sentence it appears in the accusative singular:

• dies natalis → diem natalem

Both words change because:

• dies becomes diem
• natalis becomes natalem

So diem natalem is just the accusative form of the normal phrase meaning birthday.

Here filiae is best understood as genitive singular:

• filiae = of the daughter

So:

• diem natalem filiae = the daughter's birthday

A learner may notice that filiae could also be dative singular in another context, but here the meaning and structure point to the genitive.

Latin often leaves out possessive words like his, her, or their when the possessor is obvious from context.

Since the sentence begins with Pater, it is natural to understand filiae as his daughter unless context suggests otherwise.

So Latin can simply say:

• diem natalem filiae

where English may prefer:

• his daughter's birthday

Latin does not always need to spell that out.

Because gratiorem agrees with diem natalem.

Since diem natalem is:

• masculine
• singular
• accusative

the adjective describing it must also be:

• masculine
• singular
• accusative

So:

• gratiorem = more pleasing / dearer / more welcome
• it matches diem natalem

This is normal adjective agreement in Latin.

Gratiorem is the comparative form of gratus.

Depending on context, gratus can mean things like:

• pleasing
• welcome
• dear
• agreeable

So gratiorem means:

• more pleasing
• more welcome
• dearer

In this sentence it describes the birthday as becoming more appreciated or more delightful over time.

Because Latin comparatives do not always need to state the second thing being compared.

Here gratiorem means something like:

• more pleasing than before
• increasingly pleasing

The idea of comparison is supplied by the context, especially with omnibus annis.

So the sentence does not need to say explicitly:

• more pleasing than last year

That comparison is understood.

Because fieri means to become or to come to be, while esse means to be.

So:

• gratiorem esse = to be more pleasing
• gratiorem fieri = to become more pleasing

The sentence is expressing a change over time, not just a static description.

Also, fieri is an important verb to recognize because it often functions as the passive infinitive of facere, but it also commonly has the sense to happen or to become. Here the meaning is clearly to become.

Omnibus annis is an ablative plural phrase.

Literally it means:

• in all years

But idiomatically it means:

• every year
• year by year

This is a common Latin way to express repeated time.

So omnibus annis gratiorem fieri gives the sense:

• to become more pleasing every year

Because Latin word order is much freer than English word order.

English depends heavily on position:

• subject before verb
• object after verb

Latin depends much more on endings.

So in this sentence, the endings tell you the roles:

• Pater = nominative, subject of dicit
• diem natalem = accusative, subject of the infinitive in indirect statement
• filiae = genitive, dependent on diem natalem
• omnibus annis = ablative phrase
• gratiorem = adjective agreeing with diem natalem
• fieri = infinitive

Latin can therefore place words in an order that feels natural for emphasis or style, not just for basic grammar. Here it puts fieri at the end, which is very common for an infinitive in an indirect statement.

A learner might wonder that, since filiae can sometimes be dative. But here that is not the best reading.

Why not?

• diem natalem filiae is a very natural way to say the daughter's birthday
• if Latin wanted more pleasing to the daughter, you would usually expect the structure to make that clearer
• the context strongly supports filiae as genitive, not dative

So the normal understanding is:

• the daughter's birthday not
• the birthday to the daughter