Magistra dicit artem interpretandi non solum legendo, sed etiam audiendo disci.

In English, after says, we usually use that + a finite verb: The teacher says that the art ... is learned.

Latin very often uses a different construction after verbs like dicit: the accusative-and-infinitive construction, often abbreviated ACI. In this pattern:

• the subject of the reported statement goes into the accusative
• the verb of the reported statement goes into the infinitive

So in magistra dicit artem ... disci:

• artem = the subject of the reported idea, but in the accusative
• disci = the infinitive

If you turned the reported idea into a direct statement, it would be:

• Ars interpretandi non solum legendo, sed etiam audiendo discitur.

There, ars is nominative because it is now the normal subject of a finite verb.

Because it is part of that same indirect statement after dicit.

Latin often says:

• dicit + accusative + infinitive

So instead of she says that the art is learned, Latin literally has something closer to:

• she says the art to be learned

Of course, in natural English we translate that as she says that the art is learned.

Also, disci is specifically the present passive infinitive of disco.

• discere = to learn
• disci = to be learned

That passive matters here, because the meaning is not the art learns, but the art is learned.

Interpretandi is a gerund in the genitive singular.

A gerund is a verbal noun: a form that still feels verbal, but functions like a noun. Here it means of interpreting or of translating/explaining, depending on context.

So:

• ars interpretandi = the art of interpreting

Since the whole phrase is inside an accusative-and-infinitive construction, ars becomes artem, but interpretandi stays genitive:

• artem interpretandi = the art of interpreting

It is active in meaning: interpreting, not being interpreted.

That can be confusing because the verb behind it is deponent: interpretor, interpretari. Deponent verbs often look passive in form, but they have active meanings.

So even though the form may look passive-like to a learner, interpretandi means:

• of interpreting
• not of being interpreted

They are gerunds too, this time in the ablative singular.

• legendo = by reading
• audiendo = by listening

The ablative gerund often expresses means or method: how something is done.

So here they explain how the art is learned:

• non solum legendo, sed etiam audiendo disci
• to be learned not only by reading, but also by listening

Because Latin can use the ablative of means without a preposition.

English usually needs by:

• by reading
• by listening

But Latin can simply use the ablative form:

• legendo
• audiendo

So the absence of a preposition is normal. The case ending itself shows the idea of means or method.

It is a very common correlative pair meaning:

• not only ... but also

Here it links the two methods:

• non solum legendo
• sed etiam audiendo

So the sentence contrasts and adds:

• not only by reading
• but also by listening

Latin word order is flexible, so these paired expressions do not always sit in exactly the same places they would in English, but the pair non solum ... sed etiam is easy to recognize once you know it.

They go with disci.

They tell you how the art is learned, not how the teacher speaks.

So the structure is:

• magistra dicit = the teacher says
• artem interpretandi ... disci = that the art of interpreting is learned
• non solum legendo, sed etiam audiendo = not only by reading, but also by listening

In other words, they describe the process of learning, not the act of saying.

Because Latin word order is much freer than English word order.

Placing the infinitive at the end is very natural in Latin, especially in an indirect statement. It lets Latin build up the thought and then finish with the key verbal idea.

So this order is normal:

• Magistra dicit artem interpretandi non solum legendo, sed etiam audiendo disci.

English usually wants the verb earlier, but Latin often prefers to save it for the end.

The sentence does not explicitly say.

Because disci is passive, the focus is on what is learned:

• artem interpretandi = the art of interpreting

The learner is left unspecified, in a general sense: people, students, one, anyone, and so on. That is perfectly normal. The point of the sentence is the method, not the identity of the learner.

Not if it wanted this meaning.

• discere = to learn (active)
• disci = to be learned (passive)

If you used discere with artem as the accusative subject of the indirect statement, the sense would become wrong, as if the art were doing the learning.

What Latin wants here is:

• the art is learned

So disci is the correct form.

Yes, very much so.

Latin often uses:

• a noun
• followed by a genitive gerund

to express the activity of doing something.

So:

• ars interpretandi = the art of interpreting
• ars scribendi = the art of writing
• ars dicendi = the art of speaking

In your sentence, because of the accusative-and-infinitive construction, ars becomes artem, but the basic idea is the same.