How Maths Put It All in Perspective

By Roderick Conway Morris

FLORENCE 15 December 2001

Uffizi, Florence
Perspective drawing of a chalice by Paolo Uccello, c.1430

If Renaissance Italians had not been so preoccupied with mathematics, the history of Western art might have been very different.

Every city in this patchwork of states had its own money, weights and measures, and converting them quickly and accurately was essential for trade. And as barrels, bales and other containers also varied in size, this involved not only arithmetical but also geometrical formulas to assess their volumes.

A large part of the Italian school curriculum was taken up with instructing pupils in such practical mathematics, and a command of them was expected in an educated person, while specialists busily investigated the upper and outer realms of the subject for intellectual, astrological and mystical motives. Not all artists were academically educated, but many were to some extent, and as the status of artists increased they moved more and more in cultivated circles.

By the early 15th century, the intuitive perspective developed by Giotto, Lorenzetti and their contemporaries (and independently by painters in the Netherlands) had reached the limits of its applicability, and the technique was not going to progress until it was put on a sounder, more scientific basis. Indeed, it is emblematic that having gained the reputation of being the finest artist of his age, Giotto was entrusted with the task of designing a campanile for Florence's cathedral. But lacking the mathematical background to make the necessary calculations, the artist came up with a project that was structurally unsafe and that had subsequently to be seriously modified.

The breakthrough came with a generation of artist-intellectuals born in the late 14th and early 15th century. The architect, painter and sculptor Brunelleschi was credited, in the words of the humanist Cristoforo Landino, as 'the inventor or the rediscoverer of perspective' (for, as Landino was fully aware, its theoretical and practical roots went back to antiquity), while the architect, artist and polymath Alberti and the artist and mathematician Piero della Francesca were no less important in its elaboration and diffusion. But it was two less-formally schooled friends of Brunelleschi, the sculptor Donatello and the painter Masaccio, who created the first surviving monuments to the new science.

Masaccio was born 600 years ago, and the anniversary has been marked by the unveiling of the recently completed restoration of his 'Trinity' in the Santa Maria Novella church - the first of a ground-breaking series of frescoes and panels painted over just four years before his premature death at the age of about 27 - and a fascinating exhibition, 'In the Footsteps of Masaccio: The Invention of Perspective' at the Uffizi.

The show was organized by Filippo Camerota of Venice's University Institute of Architecture, with the collaboration of Paolo Galluzzi, the director of Florence's History of Science Museum, which has reconstructed a wonderful lineup of working models of the contraptions used by 15th- and 16th-century artists to apply the new theories of perspective. (The show continues until Jan. 20.)

By the 14th century, 'perspective' was already being classified by scholars as 'practical geometry,' of the kind described in the ninth-century Islamic treatise by Al Farabi (available in Latin) as a method of measuring 'the height of trees and walls, the broadness of rivers, the depth of valleys, the height of mountains.' But the challenge to Brunelleschi and his colleagues became to translate these techniques of measurement and other mathematical formulas into a representational system capable of creating a seemingly three-dimensional image on a flat or low-relief surface.

One of the simplest aids for reducing a 3-D object to an image of a flat surface was a mirror, whose use by artists was attested by ancient sources. One of Brunelleschi's earliest demonstrations employing a mirror is plausibly reconstructed in the exhibition. The device was placed at some distance from the Baptistery beside Florence's cathedral, and consisted of a painting of the building facing away from the viewer, but reflected back to him by a mirror.

The distance between the mirror and the painting was measured out in proportion to the distance between the observer and the Baptistery, so that when the observer peered through a small aperture at the Baptistery and then slid the mirror in front of the building, the reflected painting appeared to be exactly the same size as the reality. Such mechanical devices, a diverting range of which can be seen here, were the first of many that could help artists construct perspective schemes. Some of these were almost certainly used as teaching aides, but it is difficult to assess how much they were employed day-to-day by professional artists once the basic rules of linear perspective had been established and tested.

On the other hand, much useful evidence has come to light of the complex underdrawing and scoring carried out by Masaccio and later Renaissance artists to measure out the perspective schemes of their works.

Paolo Uccello was particularly noted for his tireless devotion to the study of perspective, and the remains of one of his ruined frescoes on display here with its dense underlying grid of bisecting lines, exposed when the work was removed from the wall, confirms that his reputation in this respect was by no means exaggerated.

But Uccello's paintings also vividly illustrate that a mastery of perspective was only one step in a complicated process that still involved as much individual vision and intuition as ever in the creation of great works.

The successful perspective rendering of the classical architectural setting of Masaccio's 'Trinity' was a major factor in establishing it as a landmark in the history of painting. Indeed, the conquest of linear perspective in turn had enormous implications for architectural drawing and design, not to mention illusionistic theatrical scene painting.

Terrestrial and celestial mapmaking was radically transformed, and variations of the kinds of mechanical devices used by artists to measure out perspective for purely illusionistic purposes were soon proving useful for making surveys from a distance of enemy fortifications and calculating ranges for artillery.

Other gentler art forms, too, enjoyed an extraordinary flowering as a result of the discovery - notably trompe l'oeil pictures in intarsia, or inlaid wood paneling, that, sometimes designed by the greatest artists of the age, including Paolo Uccello and Piero della Francesca, reached a breathtaking level of ingenuity and beauty, never subsequently surpassed.

First published: International Herald Tribune