Legendo et meditando discipula argumentum libri paulatim intellegit.

They are gerunds, not participles.

A gerund is a verbal noun: it names an action, but it still feels verbal in meaning.

• legendo comes from legere = to read
• meditando comes from meditari = to reflect, think over

So here they mean something like:

• by reading
• by reflecting

A participle would act more like an adjective and agree with a noun. These forms do not do that here.

They are in the ablative because Latin often uses the ablative to express means or method.

So:

• legendo et meditando = by reading and reflecting
• or through reading and reflection

That is, the student comes to understand the book's content by means of these activities.

Yes, in English that can sometimes sound natural, but by reading and reflecting is the most direct grammatical match.

The Latin ablative here mainly suggests means/process: the understanding happens through reading and reflection.

So the safest starting translation is:

• by reading and reflecting

But in smoother English, depending on context, someone might say:

• while reading and reflecting

We know from the case ending and from the verb.

• discipula is nominative singular, the normal case for the subject
• intellegit is third person singular, so it matches discipula
• argumentum is accusative singular, so it is the direct object, not the subject

So even though discipula is not the first word, it is still the subject.

This is a key Latin habit: word order is flexible, but endings tell you the grammar.

Because discipula is the feminine form.

• discipula = female student
• discipulus = male student

So the sentence is specifically talking about a girl / young woman student, not a male student.

It is a noun phrase made of:

• argumentum = accusative singular
• libri = genitive singular

Their roles are:

• argumentum is the direct object of intellegit
• libri means of the book

So:

• argumentum libri = the subject matter of the book
• or the plot/content/outline of the book

Literally, it is the argument/content of the book.

Not exactly.

In this sentence, argumentum does not mean a quarrel or debate. In Latin, it can mean things like:

• subject matter
• theme
• content
• outline
• plot
• proof or evidence in other contexts

So here argumentum libri is better understood as what the book is about, not an English-style argument between people.

paulatim is an adverb meaning:

• gradually
• little by little

It modifies the verb intellegit.

So the sense is:

• the student gradually understands
• the understanding happens step by step, not all at once

Its position before the verb is natural Latin and gives a little emphasis to the slow process.

intellegit is:

• present tense
• active voice
• indicative mood
• third person singular

It comes from intellegere = to understand

So it means:

• she understands
• or in English, sometimes she is understanding

In normal translation here, she understands is best.

No. Latin word order is much freer than English word order.

This sentence begins with Legendo et meditando in order to foreground how the student comes to understand. Then Latin gives:

• the subject: discipula
• the object: argumentum libri
• the adverb: paulatim
• the verb: intellegit

A more English-like order would be something like:

• Discipula paulatim argumentum libri legendo et meditando intellegit

That would still mean essentially the same thing.

So the original order is not random; it is a matter of emphasis and style, not strict necessity.

Because Latin has no articles.

So Latin does not have separate words for:

• the
• a / an

That means:

• discipula could mean the student or a student
• libri could mean of the book or of a book

English has to choose whichever sounds best from the context. In this sentence, the student and the book are natural choices.