Write the girl a flirtatious note.
Place the note at her study station when she's not there or drop it off in person and return to your study area. Say something flattering like, "Just wanted to tell you, you have gorgeous eyes," and initiate a date by adding, "Join me for a coffee break?" or, "Can I buy you lunch?" However, if you wish to let the note-passing last a bit longer, wait to initiate the date and simply ask her to write back.
Approach the girl in person if you are not comfortable with writing a note.
In a hushed voice, say something complimentary or ask her about a book she's reading. See if she would like to get coffee or lunch.
Take your favorite book over to where she is sitting and place it on the table in front of her.
Say that you noticed her from afar and thought she might be interested in reading a fantastic book. Tell her she can have the book on one condition --- she must tell you her favorite book so you can read it. Use this exchange as an ice-breaker. If she seems to warm up to you, ask her out for coffee or for her phone number.
- Approach the girl with confidence but be careful not to come on too strong. Remember that you are a stranger and you have to quickly show the girl you are not creepy. Smile warmly and be polite. It may help to say something like, "I usually don't do this, but..." before you go on to ask her out.