[Ennui.]

It's possible that she was being friendly with you, or that she was flirting. It's hard to tell from something like this. I'm inclined to believe there was some type of motive behind it, whether friendly or flirting, because if she just wanted to look at more books she'd probably have politely asked you to step away and wouldn't have continued to come closer.
I would approach her about it and shoot your shot! There's nothing wrong with trying to enter a relationship with her as long as you are polite and are okay with taking no for an answer if she isn't actually interested. I'd approach her when she seems to be in a relatively good mood and in a more private area if possible, and let her know that you have noticed her for some time and would like to ask her out.
Alternatively, you can ask her directly what the actions she's taking means. Let her know that you noticed her looking at you and touched arms in the library. Ask her if she was being friendly or if she was flirting with you. Be direct with her about this and let her know that you aren't expecting anything from the conversation, but you just want to know what she means by these things so that you don't get the wrong idea.
I know it's scary to approach her about this, but at least then you will have a better idea of what is going on and there won't be any room for misinterpretation! It could be the start of something great, or you'll know that she doesn't like you like that and you can stay friends with her. It may be awkward for a little while after if that's the case, but I bet she'd come back around.

Best of luck!
Dez