Questions: Flirting

I'm not talking about clubs, in settings where people are looking to "hook up" etc, I'm referring to everyday settings, in general social settings with acquaintances and friends.
Guys, why do you look at a girl, to the point of turning your head to look at her, yet don't approach her to talk?
What are you expecting or hoping for the girl to do?
What would stop you from approaching her?
Consider the context of a girl you already know, but not too well.
Any insight on this would be appreciated.

Before I say anything else: I don't believe that astrology has EVERYTHING to do with love or personalities, but I do believe they are a good base to start from.

He is an Aries, I am an Aquarius.

I really need some advice on this. Please be honest and please read through all of this.

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I have been single for about a year now, and have had no interest in finding someone to 'fill the void' as one might say it. I was quite content being single.

That is... until this guy came along. Now he has me questioning whether or not I'd like to let him win me over, or just tell him that I have no interest. I won't lie, I never gave off the vibes of "I'm interested". I just tend to be friendly to everyone, so it was no different with this guy.

His interest in me couldn't have been any more obvious. How so? He came into class one day with a beautiful, fully bloomed tulip. Apparently, he had picked it from his garden. When he handed it to me, he told me that he had seen the flower earlier and that it had reminded him of me, but that my beauty was unmatched by anything he had ever seen before. I was shocked to say the least--a little confused as well, but flattered beyond words. I accepted his gift. Well, after multiple declines to go out on a date with him, he wore me down enough to agree to one. To be honest, it wasn't half bad.

He opened doors for me, pulled out (and pushed in) chairs for me, and slow danced with me. I won't lie--he is like a transplant from the Shakespearian times. He is very, VERY traditional. His good-bye to me? A kneel on the ground and a kiss on the hand.

He is also one of the smartest people I've ever met. He is fluent, and self taught, in 5 different languages. It takes him less than 2 months to learn a new language. It's crazy!

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I am confused. I don't know what my feelings are for him yet. Should I give him more chances? Should I not?