Learn Hardcore Japanese: tosyokan de hon wo mitukemasita.

The particle を marks the direct object of a transitive verb. Since 見つける is transitive, 本を見つけました means found a book. Using が would mark 本 as the subject or emphasize it, which doesn’t fit with this verb’s direct‐object requirement.

Why is the verb 見つけました in the past tense?

The past tense form 見つけました shows that the action of finding the book is completed. For a present/future or habitual sense you’d use 見つけます, and for an ongoing search you might say 探しています.

What’s the difference between 見つける and 探す?

見つける means to find something you weren’t aware of or had lost. 探す means to search for something actively. Typically you 探す first, and once you locate it you 見つける.

Can I switch the order to 本を図書館で見つけました?

Yes. Japanese word order is relatively flexible. 本を図書館で見つけました still clearly means I found a book at the library and sounds natural.

Could I say 図書館に本を見つけました instead?

No. に marks a point of existence or direction, not the site of an action. To express “found at,” you need で. If you say 図書館に本があります, it means there is a book in the library, not I found a book at the library.

Why is there no subject in this sentence?

Japanese often omits the subject when it’s understood from context. Here the implied subject is I, so the full idea is (I) found a book at the library.

How would I say this casually instead of politely?

The plain past form of 見つける is 見つけた, so in casual speech you’d say 図書館で本を見つけた.