doing or saying something that could be considered rude, but being polite, careful, and lightly saying something that could offend but needs to be said....."I tactfully told him he needed to lose weight" its for being polite yet saying something rude that needs to be said. hope that helps.

in your example, the king could not tactfully get the book from his physician, so he killed him to bluntly take it as a last resort, as it seems.....nice