Cum puella ad bibliothecam profecta esset, librum in subsellio relictum invenit.

Here cum introduces a subordinate clause that gives the circumstances or time for the main action.

So:

• Cum puella ad bibliothecam profecta esset = when / after the girl had gone to the library
• librum in subsellio relictum invenit = she found a book left on the bench

In this kind of sentence, cum often means something like when, while, or since, depending on context. Here the sense is mainly when/after.

Because cum clauses very often take the subjunctive, especially when they are circumstantial or narrative rather than just giving a simple time fact.

So Latin is not merely saying:

• At that time, she had gone to the library

It is using the clause as background for the main event:

• When/after she had gone to the library, she found...

That is why you get profecta esset rather than something like profecta erat.

Profecta esset is pluperfect subjunctive.

It is formed from:

• the perfect participle: profecta
• plus the imperfect subjunctive of sum: esset

Together they mean:

• had gone
• had set out
• had departed

The pluperfect shows that this action happened before the action of the main verb invenit (found).

Because proficiscor is a deponent verb.

A deponent verb:

• has passive-looking forms
• but an active meaning

So although profecta looks like a passive participle, with a deponent verb it has an active sense:

• profecta esset = she had gone / she had set out

This is very common in Latin and often surprises English speakers at first.

Because it agrees with puella.

Puella is:

• feminine
• singular
• nominative

So the participle must match it:

• profecta = feminine singular nominative

If the subject were masculine, you would expect profectus; if plural feminine, profectae, and so on.

Because ad takes the accusative case when it means to or toward a place.

So:

• ad bibliothecam = to the library

This is normal Latin usage:

• ad urbem = to the city
• ad villam = to the house/farmhouse
• ad scholam = to school

Because in can take either the ablative or the accusative, depending on meaning.

• in + ablative = in / on a place, showing location
• in + accusative = into / onto a place, showing motion toward

Here the book is already located there:

• in subsellio = on the bench or in the seat

So Latin uses the ablative: subsellio.

Subsellium is a neuter noun meaning something like:

• bench
• seat
• sometimes a school bench or low seat

Its exact English translation depends on context. In this sentence, bench is very natural.

So:

• in subsellio = on the bench

Because relictum agrees with librum.

Librum is:

• masculine
• singular
• accusative

So the participle describing it must also be:

• masculine
• singular
• accusative

Hence:

• librum relictum = the book left behind

If the noun were nominative, you would expect relictus instead.

This is a noun phrase with a participle modifying the noun:

• librum = book
• relictum = left
• in subsellio = on the bench

So literally it is:

• a book left on the bench

You can think of relictum as similar to a shortened relative clause:

• librum qui in subsellio relictus erat
• the book which had been left on the bench

Latin often uses participles where English might use a relative clause.

The subject is still puella.

Latin often leaves the subject unstated when it is clear from context. The verb ending in invenit tells you it is third person singular, and the earlier noun puella is the obvious subject.

So the sentence means:

• When the girl had gone to the library, she found a book left on the bench.

Invenit is perfect indicative, meaning she found.

The perfect tense here presents the main action as a completed event in the past. The cum clause uses the pluperfect subjunctive (profecta esset) to show an action that happened before that main event.

So the sequence is:

1. the girl went to the library
2. she found the book

Yes. Latin word order is much freer than English word order because the case endings show how the words relate to one another.

For example, the same basic meaning could still be expressed in other orders, such as:

• Cum ad bibliothecam puella profecta esset, librum in subsellio relictum invenit.
• Librum relictum in subsellio invenit, cum puella ad bibliothecam profecta esset.

The given order is perfectly natural, but not the only possibility.

It can be understood as when in a broad sense, but because the verb is pluperfect (had gone), the action of the cum clause is already complete before invenit.

So in English, depending on style, you might render it as:

• when the girl had gone to the library
• after the girl had gone to the library

Both reflect the idea that her going to the library happened first.

Classical Latin has no articles like English the and a/an.

So:

• puella can mean the girl or a girl
• librum can mean the book or a book
• bibliothecam can mean the library or a library

The translator decides from context which is more natural in English. In this sentence, the girl, the library, and a book are all reasonable choices depending on the intended context.