The vialis case is found in Eskimo languages. This is another name for prolative case. Usage: movement using a surface or way. Example: by way of or through the house.
The dative case (also called the third case) is a grammatical case generally used to indicate the noun to whom something is given.
In this case, the noun or pronoun's case is determined by the preposition, NOT by its function in the sentence.
Here, the subject, Ich, is in the nominative case, the direct object, das Buch, is in the accusative case, and zum Verleger is in the dative case, since zu always requires the dative (zum is a contraction of zu + dem).