problems. Just like a picture, mathemati-
cal concepts need to be created, invented,
and discovered, much like the French
painter Claude Monet, and others, invent-
ed Impressionism.How did early paintersintegrate
mathematics and painting?
Early painters had a problem: how to rep-
resent the three-dimensional world on a
two-dimensional canvas; or, how to give
paintings depth and perspective. Although
perspective does follow mathematical
guidelines, many early painters were able
to incorporate perspective (or depth per-
ception) intuitively.
The idea of perspective flourished
around the time of the ancient Greeks,
who used a form of perspective to design
certain architectural structures and even
theatrical stage settings. But it is unclear whether they actually understood the math-
ematics behind perspective. One of the first to create the impression of depth by using
certain rules was Italian painter, sculptor, and architect Giotto di Bondone (c.
1267–1337). But those rules were of his own devising and were probably not based on
mathematics. Whatever the case, he clearly worked out a way to represent depth in
space, and he came close to understanding linear perspective.
But it took until the Renaissance before painters explored the science behind per-
spective. In the early 1400s, sculptor Filippo Brunelleschi (1377–1446) made the first
correct formulation of linear perspective using mirrors. He understood that there was
a single vanishing point to which all parallel lines in a plane (or on his canvas) con-
verge; he also understood scales, calculating the relationship between the actual
length of an object and its length in the picture, depending on its distance in the
painting. Writer and mathematician Leone Battista Alberti (1404–1472) was the first
person to write the rules of perspective in 1435, using the principles of geometry and
the science of optics.
By 1450 artist and mathematician Piero della Francesca (1412–1492) had written
an even more extensive mathematical work on perspective, including concepts of art,
arithmetic, algebra, and geometry. The writings of mathematician Luca Pacioli
(c. 1447–c. 1517; or Friar Luca dal Borgo; who was also known as the father of account-
374 ing, see below) delved more into perspective, including in his book De Divina propor-
Giotto di Bondone developed the concept of linear
perspective in his art to depict three-dimensional
space realistically in his two-dimensional paintings.
Library of Congress.