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Names of cities and small islands (small enough to have only one town) and a tiny number of other words such as domus are a special case. Here the preposition is omitted and the prepositional case is used.
Edit: I should of course have said locative case. I was probably thinking of the prepositional case in Russian.
Romae is locative case. It's used with names of cities, small islands, and a very small number of other words including domus. Instead of saying in Roma (ablative case) we use the locative and say Romae.
With these words where we use the locative case there's another peculiarity with movement towards them. Ususally this is signified with ad plus the accusative case, but with these words we leave the ad out. So if we're going to Rome we simply say Romam. When we say ad Romam this means near Rome.
Because definite articles in romance languages evolved from Latin demonstratives due to dropping declensions and become more analytical in morphology. That's the "purpose" for why they exist in romance languages and not in Latin. If you're asking for the "reason" instead, then no one can say, because language evolution sometimes happens in ways that are inexplicable. Slavic languages (with the notable exceptions of Macedonian and Bulgarian) don't have articles either, though they sometimes use the word "one" to imply indefiniteness. The reason for this, just as in Latin, can only be speculated. The usual conjecture is that the more synthetic the language is in its morphology, the more declensions there are for its demonstratives and the more freedom there is to not use any demonstrative when the definiteness of a noun can be inferred from the rest of the sentence.
TL;DR: complex morphology in Latin and other synthetic languages tends to not have articles. Languages that are more analytical tend to have articles.
First of all you need to pay attention to the verb in the Latin sentence:
I live (the 1st person, singular) https://www.duolingo.com/skill/la/Greetings/tips-and-notes
Hence the subject is a "virtual" I/ego. This is clearly evidenced by the form of the verb, and the pronoun itself in Latin is usually omitted.
The most common structure in Latin is SOV (subject - object - verb).
That is, Romae is not the subject and not in the nominative. And what is it then? It's the locative case of "Roma"! :)
This is so helpful. I wish the sentences matched the information in the tips. That is, not just listing the word Romae in a table and then expecting you to read and memorize the table (not), but actually saying, hey, there's a case called Locative and it goes like this...