Si hoc remedium puellae non prodest, medica alterum ei dabit.

Here puellae is dative singular: to/for the girl.

That is because prodest comes from prodesse, and prodesse takes a dative object:

• alicui prodesse = to help someone, literally to be of use to someone

So:

• hoc remedium puellae non prodest = this remedy does not help the girl

The ending -ae can mean more than one thing in Latin, but here the verb tells you it must be dative.

Because dabit comes from dare, and dare normally takes:

• a direct object = the thing given
• an indirect object = the person receiving it

So in:

• medica alterum ei dabit

we have:

• alterum = the thing being given
• ei = to her, the person receiving it

So ei is dative singular.

Yes. Both refer to the girl.

Latin often does this:

• first mention the noun itself: puellae
• then refer back to it with a pronoun: ei

That is perfectly normal and avoids repeating puellae.

Prodest means it helps, it benefits, or it is useful to.

It is the 3rd person singular present indicative of prodesse.

This verb is a compound of:

• pro-
• esse = to be

But you should learn prodesse as a whole verb with its own meaning:

• prodesse alicui = to help someone / to be good for someone

So prodest does not take a direct object in the accusative the way English help does. It takes a dative instead.

Because hoc remedium is the subject of prodest, not its object.

The idea is:

• this remedy is doing the helping
• or, in this sentence, this remedy is not helping

So it must be nominative singular.

Both words are nominative singular neuter:

• hoc = this
• remedium = remedy

They agree with each other in gender, number, and case.

Alterum is an adjective being used substantively, meaning it stands on its own as if a noun were understood.

Literally, it means:

• the other one
• another one

Here the omitted noun is clearly remedium:

• alterum = another remedy or the other one

Its form is neuter accusative singular, because it is the direct object of dabit and refers to remedium, which is neuter.

Strictly speaking, alter often has the sense the other of two.

However, in many sentences like this, especially in learner material, it is natural to translate it as another:

• the doctor will give her another

So the idea is that if one remedy does not work, a different one will be given.

Dabit is future indicative active, 3rd person singular, from dare.

So it means:

• she will give

That fits the sentence well:

• If this remedy does not help the girl, the doctor will give her another.

Latin can do that in a straightforward condition.

Here:

• si ... non prodest = if ... does not help
• dabit = will give

English does something similar:

• If this remedy does not help, the doctor will give her another.

So there is nothing strange here. The condition is presented as a real possibility, and the main action is future.

Because medica is feminine.

It means female doctor or woman physician. It is the subject of dabit, so it is nominative singular.

So:

• medica dabit = the female doctor will give

If the doctor were male, you would expect medicus instead.

Non negates prodest:

• non prodest = does not help

In Latin, non usually comes before the word or phrase it is negating. So its position here is exactly what you would expect.

Yes, but in a normal Latin way.

Latin word order is much freer than English word order because the endings show the grammar. This sentence is arranged in a natural, readable Latin order:

• Si introduces the condition immediately.
• hoc remedium comes early as the topic.
• puellae appears before prodest, where the listener can easily connect it with that verb.
• medica begins the main clause as the subject.
• dabit comes at the end, a very common place for the verb in Latin.

So the order is not random, but it is also not as rigid as English.

You know from agreement with remedium.

Since remedium is nominative singular neuter here, hoc must also be nominative singular neuter.

That is one of the most important habits in reading Latin:

• find the noun
• see what adjective or pronoun agrees with it
• match gender, number, and case

So:

• hoc remedium = this remedy as the subject of the clause