Curious about "curious"

I am interested in learning more about the word curious because it is a word that has two interesting and distinct meanings, and I would like to know how these separate meanings came about. The root of the word appears to be "cure," which doesn't seem to relate at all to its modern meanings, so I'm also interested in archaic meanings of curious.
I also don't often use the word curious in its meaning of peculiar or strange. When I do use it, it is most often in the form of curiosity, the noun. It is a word that really applies to me in the sense of my eagerness to learn, and I associate curiosity with a certain amount of imagination and wonder. To me, curiosity isn't a negative concept (i.e. curiosity killed the cat) as I think curiosity is always an important aspect of a student's education and an adult's life. In science, curiosity is what drives new experiments and what produces wondrous new results, and that is something I will take with me in my future.