Each shift in point of view is like a shake of a kaleidoscope. One must keep shaking until a pattern which makes sense emerges:

Circumstantial evidence is a very tricky thing -- it may seem to point very straight to one thing, but if you shift your point of view a little, you may find it pointing in an equally uncompromising manner to something entirely different [BOSC].

--- it is one of the curses of a mind with a turn like mine that I must look at everything with reference to my own special subject. You look at these scattered houses, and you are impressed by their beauty. I look at them, and the only thought which comes to me is a feeling of their isolation, and of the impunity with which crime may be committed there [COPP].

When once your point of view is changed, the very thing which was so damning becomes a clue to the truth [THOR].

Sherlock Holmes was a man, however, who when he had an unsolved problem upon his mind would go for days, and even for a week, without rest, turning it over, rearranging his facts, looking at it from every point of view, until he had either fathomed it, or convinced himself that his data were insufficient [TWIS].