How do you approach a girl who is studying in the library?

You are going to be an intruder as she's probably studying. Patience and you will have your prize...

If she's deep in concentration, forget it. Wait until it looks like she's bored. Walk by and say something to the effect of "Hey, I'm going to be over here, let me know when you're ready for a break and we'll go (insert something fun here...won't be hard to top studying in the library). Keep it low-key...record store, stroll around campus, or a beer.

Good luck.

A key thing to consider here is that a library generally speaking is not the ideal place to strike up a conversation with someone. Most libraries will obviously be rather quiet, and assuming there are other people in the immediate area you should probably keep it that way.

Most libraries usually will have some sort of cafe or coffee shop close by (especially if this is a library on a campus) that would be a little more suitable.

As mentioned above, wait until she seems distracted or bored as opposed to focused, then just go over and introduce yourself politely and ask her to join you for a little break.

Nick & Marc seem to be on the right track -- read signals (focusing on studying vs bored, looking around...) and don't be creepy/threatening (opening with a rape joke = bad) or a d&$# in a book. :) You are a good & fun guy, but she doesn't know that yet.

If you are really interested in the answer and care about how you make that woman feel and understanding her reactions, then this is a good guide: A Guy's Guide to Approaching Strange Women Without Getting Maced

From that guide: If you speak, and she responds in a monosyllabic way without looking at you, she’s saying, “I don’t want to be rude, but please leave me alone.” You don’t know why. It could be “Please leave me alone because I am trying to memorize Beowulf.” It could be “Please leave me alone because you are a scary, scary man with breath like a water buffalo.” It could be “Please leave me alone because I am planning my assassination of a major geopolitical figure and I will have to kill you if you are able to recognize me and blow my cover.”

You don't, she is studying after all. She obviously has more important things to be doing.

Quietly, of course.

A pretty girl reading a book in a library is probably above your league. She is above all of our leagues. She is untouchable. So dont waste your time.

The first poster wasn't bad but there's more. I wouldn't talk about what she's reading right away because unless you genuinely cared about her input on that subject it would seem like you're using it to hit on her. Instead you could ask her for her advice on something. Say "excuse me, I only have a moment... [insert question]. An example would be "I could use a female perspective. I'm looking for something new to read. Do you have any ideas?". If you do know something about what she's reading after this could be a good time to throw that in. "hey thanks for your help by the way I saw you were reading..." After this little conversation you could say "this has been fun but I have to go. You seem really interesting is there a way we could stay in touch". If she likes you, she might give her number and if she's not sure or is seeing someone she might give her facebook info and you may be able to put more time in there. Nice and Easy. Hope that helps.

Having been that girl, I'd say your best bet is to walk up say something like "Hey I'm Dan, I'm sitting over there, let me know if you want a break and maybe we can go get a cup of coffee or if you need help with history" (or whatever you know a lot about and are in the library for). Then walk away. She'll probably say thanks and then in half an hour when she's bored of what she's doing, she'll come over. I wouldn't wait for any kind of conversation but you'll have piqued her interest.