How did Renaissance painters learn to fake depth?

Renaissance painters learned to fake depth by turning seeing into geometry. The key rule is simple: an object looks N times smaller when it is N times further from your eye. Follow that rule strictly and parallel lines, like the edges of a road or a ceiling, seem to run together toward points in the distance, usually on the horizon. That system is called linear perspective.

Before it, painters often sized figures by importance rather than distance, and medieval artists, though they knew distant things look smaller, had no systematic method. The breakthrough is credited to Filippo Brunelleschi, the Florentine architect behind the cathedral's great dome, who experimented between 1415 and 1420. According to his early biographers, he painted the Baptistery of Florence, drilled a hole in the panel, and had people look through it from the back at the real building. Then he raised a mirror: the viewer could flip between the painting's reflection and reality, and see how closely they matched.

Within years, nearly every ambitious artist in Italy was using geometric perspective, and around 1435 Leon Battista Alberti wrote down the mathematics, using nothing fancier than Euclid's similar triangles. The method ruled Western painting until the late 19th century. The twist: Brunelleschi's famous panel is lost, and he isn't known to have painted anything else at all.