Both Uses

contrary
in
Prince Caspian: The Return to Narnia
(Edited)

"On the contrary, your Majesty," said the Dwarf.
Chpt 10

on the contrary = an expression used to intensify denial of an idea
She did not want to get up; not because she was still tired - on the contrary she was wonderfully rested
Chpt 10 *

Definitions:

(1) (contrary as in: a contrary idea)
different (perhaps opposite or mutually exclusive)

In formal logic, contrary propositions or contrary arguments describe two things that cannot both be true. They might be described as mutually exclusive. For example, a pet cannot be both a cat and a dog. It doesn't need to be either, but if it is one, it cannot be the other. So the statement that a pet is a cat and the statement that the same pet is a dog, are contrary statements.
(2) (contrary as in: a contrary personality)
disagreeable -- typically in reference to someone's personality
(3) (contrary to as in: contrary to)
in opposition to

You could also think of this as meaning, despite, or in spite of, or in conflict with; but since a common meaning of contrary is opposite, you many find it easiest to think of contrary to as in opposition to.
(4) (on the contrary as in: on the contrary)
an expression used to intensify denial of an idea
(5) (to the contrary as in: to the contrary)
with an opposite or different effect; or something with an opposite or different effect
(6) (meaning too rare to warrant focus)