The Florentine architect and engineer Filippo Brunelleschi was the first to carry out a series of experiments leading to a mathematical theory of perspective.

Brunelleschi began his career as an architect, which seems to have led him to his perspective studies.

His most stunning accomplishment in architecture is the dome which crowns the Cathedral in Florence, a work that occupied him intermittently from 1417 to 1434.

The technical difficulties involved in erecting the new dome underscored an important aspect of his talents: he was a daring innovator, with a solid knowledge of mathematics and mechanics.

Perfectly in keeping with such interests are the experiments Brunelleschi performed on the subject of mathematical perspective.

Writing a century later, Brunelleschi's biographer, Antonio Manetti, describes one of these experiments, in which Brunelleschi painted the Baptistry in Florence (right), which stands directly before the famous cathedral that Brunelleschi would later dome. 

In 1415, Brunelleschi painted his picture of the Baptistry on the surface of a small mirror, right on top of its own reflection.  Unfortunately, this work has since been lost: it seems to have been intended to be used only in this experiment, not to be preserved.

To demonstrate the fact that his painting was indeed an exact replica that could fool the eye, Brunelleschi drilled a small hole in the mirror and then stood directly in front of the Baptistry, looking through the peephole to see the real building.  He then held up a second, clean mirror in front of his painted panel.  The second mirror blocked the view of the real building, but now reflected his painted version on the original mirror.  

By moving the second mirror in and out of the way, Brunelleschi could check whether his painting was indeed an exact copy of the three-dimensional, octagonal building on the two- dimensional surface of his original mirror.

Once he had verified the accuracy of his painted mirror, it became possible for Brunelleschi to analyze the structure by which three dimensions was translated into two dimensions. 

As Brunelleschi found, there was a mathematical system.  It centered around the central vanishing point, inside the yellow circle in the graphic at the right, where all lines that were perpendicular to Brunelleschi's painted mirror (often called "the picture plane") would converge, like railroad tracks in the distance.

This point determined the horizon line, and was exactly opposite to Brunelleschi's own position standing in front of the Baptistry. 

The horizon line contained other vanishing points as well, where lines determined by structures that were not exactly perpendicular to Brunelleschi's mirror would converge.

For example, those lines determined by the oblique sides of the octagonal Baptistry converge at different vanishing points, which still lay on the horizon line (inside the yellow circles to the left and right of the central vanishing point).

What is clear from Manetti's description is that Brunelleschi used his painted mirror for careful calculations.  His final result showed a logical, rational, mathematic system by which three- dimensional space could be rendered on any two-dimensional surface.

Once Brunelleschi devised and publicized this system of horizon lines and vanishing points, any artist could use it to create convincing spaces in their paintings, without using mirrors.