As James Burke describes in The Day the Universe Changed, perspective wasn't just some new, cool way of drawing. It represented a shift in the way Europeans thought, not only about art, but about the universe. Instead of a kind of diagrammatic painting, in which figures were placed and sized to represent the hierarchy of creation, paintings using perspective depicted how things look to humans. The human viewpoint became important, in art, architecture, and other endeavors.
And though the basics of perspective were derived mathematically, its use in art was intended to create the illusion of human vision. The Brunelleschi experiment, from about 1415, was a mirror image painting of the Baptistery in Florence. This painting, with an eye hole in the middle, was set up facing the Baptistery. Viewers could look through the eye hole into a mirror to see the painting, or, with the mirror removed, look directly at the Baptistery. So effective was Brunelleschi's rendering that viewers claimed not to be able to tell the difference. (Of course, I would think the fact that one of the views had the viewer's eyeball in the middle might be a giveaway, but I wasn't there.)
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