Hilaritas matris tristitiam puellae minuit.

The subject is hilaritas.

You can tell because hilaritas is in the nominative singular, the case usually used for the subject of a finite verb. So:

• hilaritas = cheerfulness / merriment → subject
• minuit = lessens / diminishes

So the basic core is:

• hilaritas minuit = the cheerfulness lessens

The direct object is tristitiam.

It is in the accusative singular, which is the case typically used for the direct object. So:

• tristitiam = sadness → direct object

That gives us:

• hilaritas ... tristitiam minuit = the cheerfulness lessens the sadness

Both matris and puellae are genitive singular forms.

The genitive often expresses possession or a close relationship, and here both words are best understood as of phrases:

• matris = of the mother
• puellae = of the girl

So the sentence breaks down like this:

• hilaritas matris = the cheerfulness of the mother
• tristitiam puellae = the sadness of the girl

Because Latin commonly places a genitive next to the noun it modifies, and that is what happens here:

• hilaritas matris = the mother’s cheerfulness
• tristitiam puellae = the girl’s sadness

So even though English might rearrange things differently, the pairings are clear from both word order and case endings.

On its own, puellae could indeed be:

• genitive singular = of the girl
• dative singular = to/for the girl
• nominative plural = girls

But in this sentence, puellae is best taken as genitive singular because it naturally depends on tristitiam:

• tristitiam puellae = the sadness of the girl

Also, the verb minuit does not need a dative here. The structure sadness of the girl makes much better sense than sadness to/for the girl.

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

In English, word order does most of the work:

• The mother’s cheerfulness lessens the girl’s sadness

In Latin, the cases tell you who is doing what:

• hilaritas = subject
• tristitiam = object
• matris, puellae = genitives

So Latin does not need to keep a strict subject–verb–object order. The sentence could be rearranged in several ways and still mean the same basic thing, as long as the forms stay the same.

Latin has no definite article and no indefinite article.

So Latin does not have separate words for:

• the
• a / an

Whether you translate a noun as the mother, a mother, or just mother depends on context.

That is why:

• hilaritas matris can be translated as the mother’s cheerfulness or a mother’s cheerfulness
• tristitiam puellae can be translated as the girl’s sadness or a girl’s sadness

They are:

• hilaritas, hilaritatis — cheerfulness, merriment
• mater, matris — mother
• tristitia, tristitiae — sadness
• puella, puellae — girl
• minuo, minuere, minui, minutum — to lessen, diminish

The forms in the sentence are:

• hilaritas = nominative singular
• matris = genitive singular
• tristitiam = accusative singular
• puellae = genitive singular
• minuit = 3rd person singular

Because mater is a third-declension noun.

Its principal forms are:

• mater = nominative singular
• matris = genitive singular

So when Latin wants to say of the mother, it uses matris.

This is something learners often have to memorize: third-declension nouns often change their stem between nominative and the other cases. Here the working stem is matr-.

Because tristitia is a first-declension noun, and -am is the normal accusative singular ending for that declension.

So:

• tristitia = nominative singular
• tristitiam = accusative singular

Since tristitiam is the direct object of minuit, the accusative is exactly what we expect.

Minuit is 3rd person singular active indicative from minuo.

In many contexts it can mean either:

• he/she/it lessens — present
• he/she/it lessened — perfect

That happens because the present and perfect 3rd singular forms are spelled the same in this verb.

So context must decide whether it is:

• lessens or
• lessened

If your given translation shows a present meaning, then read it as lessens here.

Yes. In fact, that is often the most natural English translation.

So you can translate:

• hilaritas matris as the mother’s cheerfulness
• tristitiam puellae as the girl’s sadness

Both are equivalent to the of translation:

• the cheerfulness of the mother
• the sadness of the girl

A natural analysis is:

• hilaritas matris = the mother’s cheerfulness
• tristitiam puellae = the girl’s sadness
• minuit = lessens / diminishes

So the whole sentence means:

• The mother’s cheerfulness lessens the girl’s sadness.

That is probably the smoothest English version, even though the Latin word order is not the same as the English one.