Medica pulsum pueri tangit et dicit puerum melius valere.

Medica is the feminine form, so it means female doctor.
If the doctor were male, Latin would normally use medicus.

In this sentence, medica is the subject: she is the one doing the actions tangit and dicit.

Medica is in the nominative singular, which is the case normally used for the subject of a sentence.

The verb tangit means she touches or he/she/it touches, and medica matches that as the person performing the action.

Pulsum is the direct object of tangit. It is the thing being touched.

• tangit = touches
• pulsum = the pulse

So medica pulsum tangit means the doctor touches the pulse.

Pueri is genitive singular, meaning of the boy.

So:

• pulsum pueri = the boy’s pulse or the pulse of the boy

This is a very common Latin way to show possession. English often uses ’s, while Latin often uses the genitive case.

This puerum is not the same grammatical role as pueri.

• pueri = of the boy after pulsum
• puerum = the boy in the clause after dicit

After verbs like dicit, Latin often uses an accusative + infinitive construction for reported speech or thought. In that construction, the logical subject of the infinitive goes into the accusative, so puerum is correct.

Because Latin often expresses that-clauses differently from English.

English says:

• She says that the boy is better

Latin usually says:

• She says the boy to be better

That sounds strange in English, but it is normal in Latin. This pattern is called the accusative and infinitive construction, often shortened to ACI.

So:

• dicit puerum melius valere
  literally = she says the boy to be better / to be feeling better
  naturally = she says that the boy is feeling better

This is the indirect statement after dicit.

Its parts are:

• puerum = the subject of the indirect statement, put in the accusative
• valere = the verb of the indirect statement, put in the infinitive
• melius = an adverb modifying valere

So the structure is:

• dicit
  • accusative subject
    • infinitive verb

That is one of the most important sentence patterns in Latin.

Melius means better, while bene means well.

Since the sentence means the boy is better than before, Latin uses the comparative adverb melius.

So:

• bene valere = to be well
• melius valere = to be better / to be feeling better

Valere literally has meanings like to be strong, to be well, or to be healthy.

In this sentence, it means something like:

• to be feeling well
• to be in good health

So melius valere means to be feeling better or to be healthier.

Because it depends on dicit in an indirect statement.

After dicit, Latin does not normally use a separate finite verb the way English does in she says that he is better. Instead, Latin uses:

• an accusative for the subject
• an infinitive for the verb

That is why we get puerum ... valere instead of something like puer melior valet inside the reported statement.

The word order is meaningful, but not as fixed as in English.

Latin relies a lot on word endings, so the sentence could be rearranged more freely than an English sentence. For example, the cases tell you what each noun is doing:

• medica = subject
• pulsum = direct object
• pueri = possession
• puerum = accusative subject of the indirect statement

Still, the given order is natural and clear:

• first the doctor examines the boy
• then she gives her conclusion

So the sentence flows nicely as: The doctor touches the boy’s pulse and says that the boy is feeling better.

Because the boy has two different grammatical roles in the sentence.

1. In pulsum pueri, the boy is the owner of the pulse, so Latin uses the genitive: pueri.
2. In dicit puerum melius valere, the boy is the subject of the indirect statement, so Latin uses the accusative: puerum.

So even though both words come from puer, they have different endings because they do different jobs.