PIERO DELLA FRANCESCA

(b. Borgo San-Sepolcro [now Sansepolcro], Italy, in the 1410s, d. Borgo San-Sepolcro, 12 October 1492),

mathematics, perspective, painting.

For the original article on Piero della Francesca see DSB, vol. 5.

Piero is one of the few persons who figures prominently both in the history of mathematics and art. While he has received much attention as a painter, his contributions to mathematics are not so well known and will be the main topic of this addendum (for Piero and mathematics in general, see the 2005 book by J.V. Field and its references; for Piero and perspective, see also the 2007 book by Kirsti Andersen and its references).

Although many scholars have worked on Piero during the decades since the original DSB article and thrown much light on his work, basic knowledge about him is still lacking. Thus, his exact year of birth is unknown, just as it is unclear how and where he learned mathematics and painting. Likewise, scholars have no dates for when he composed his three books on mathematics. Of these, his Trattato d’abaco (referred to as Trattato) is supposed to be the earliest. It belongs to the so-called abacus books, which, despite their name, presented not the abacus but mathematics on an elementary level for future merchants, bank clerks, artisans, and artists. The books dealt primarily with arithmetic, but often included some algebra and practical geometry as well—which is also true of Piero’s Trattato. In addition, Piero treated some more advanced geometrical objects, such as the regular polyhedra. He returned to these solids in the last of his books Libellus de quinque corporibus regularibus (referred to as Libellus), in which he also included five semi-regular, also called Archimedean, polyhedra (Field, 1997). These solids are called Archimedean because Pappus of Alexandria in his Collection stated that Archimedes had found thirteen convex polyhedra that have regular polygons of more than one kind as faces. It is unlikely that Piero knew about Pappus’s claim, and he may actually himself have rediscovered

some of the Archimedean polyhedra. In Trattato he described two of these, one—later called a cuboctahedron—having eight equilateral triangles and six squares as faces, obtained by cutting off the corners of a cube through the midpoints of the edges. The second was a truncated tetrahedron that Piero constructed by cutting off the vertices of a tetrahedron through points situated one third of the edge length from the corners. In Libellus he came back to the truncated tetrahedron and then added the truncated versions of the remaining four regular polyhedra.

As to the rest of the contents of Trattato and Libellus, Piero benefited a great deal from earlier treatises—the material for which mainly dates back to Euclid, alKhwarizmi, and Leonardo da Pisa (also called Fibonacci) (on Piero and Euclid, see Folkerts, 1996; and Piero and the algebra tradition, see Giusti, 1993).

While able to draw on a tradition for the themes of the Trattato and Libellus, Piero was pioneering new ground when he composed his third book De prospectiva pingendi (On the perspective of painting, referred to as De prospectiva), which is the first work devoted entirely to perspective. Most of the book consists of descriptions of various constructions in a style that tends to become tedious, but is useful because it guides the reader through the entire construction. Piero not only wanted to explain the how of perspective constructions, but the why as well. In other words, he aimed to provide a scientific foundation for his subject. This was a task that he had to start from scratch, and he did get started. Taken stepwise, Piero’s mathematical reasoning was sound, but he did not always provide all the arguments that were needed for his conclusions. For instance, in his “proof” of theorem 24 in the first book, he proved with great accuracy that some triangles are similar, but overlooked the fact that the similarity in itself did not prove his original statement.

De prospectiva is divided into three parts; in the first Piero discussed the problem of constructing perspective images of figures situated in a horizontal plane applying a total of three different methods. In the second part, Piero threw some three-dimensional figures into perspective. The third part of De prospectiva is devoted to a method that applies to both plane and solid figures. This method is based on making a plan and an elevation of a configuration consisting of the eye point, the picture plane, and an object to be drawn in perspective. The method then provides what corresponds to the horizontal and vertical coordinates of the perspective images of special selected points in the object. There are no traces of Piero’s predecessors having applied this method, but still he takes it for granted that his readers knew the technique of constructing plans and elevations of objects. The essential novelty was that Piero applied this technique to perspective, and he may well have been the first to do so. He himself stated that he found this perspective method less abstract and at the same time more powerful than the constructions presented in the two first parts of the book. In his own words, it was “easy to demonstrate and understand” and advantageous to apply “to more difficult solids” (Piero, 1974, p. 129). And he did present impressive examples, among them how to construct the perspective image of a tilted cube in which none of the edges are horizontal or parallel to the picture plane, and a trompe l’oeil giving the impression that a bowl pops up from a table when seen from the intended eye point. What made the example with the tilted cube difficult was to construct its plan and elevation; for this Piero applied a plan, an elevation, and a third projection perpendicular to the two others to describe a rotation of a cube having its sides parallel to the plans of reference.

De prospectiva did in general not receive the appreciation it deserved. Presumably, its mathematical arguments were too difficult for practitioners to follow and its long and detailed descriptions of how to perform constructions deterred mathematicians from studying it carefully.

Piero was admired by Luca Pacioli and Giorgio Vasari, but by the end of the sixteenth century he seems to have been forgotten both as a mathematician and as a painter. Thus, it took four hundred years after his death before his name appeared as the author of a printed book. Although his work was not mentioned often, Piero was not completely without influence. Presumably through Pacioli’s praise of De prospectiva, Daniele Barbaro became aware of the work and copied longer passages from it almost verbatim without revealing his source. Before Barbaro, he himself had behaved similarly by including an Italian version of Libellus as a third section of his own De divina proportione(1509) with out referring to Piero. Through these publications, and presumably also through manuscripts that applied Piero’s material, some of his ideas became part of textbooks. For instance, it is striking that in his books on geometry and proportions, the German painter and mathematician Albrecht Dürer applied methods that are similar to Piero's, including the technique of involving a plan and an elevation to describe movements of bodies in space.

SUPPLEMENTARY BIBLIOGRAPHY

WORKS BY PIERO DELLA FRANCESCA

Trattato d’abaco. Biblioteca Medicea Laurenziana, Florence, Codex Ashburnham 280, ed. Gino Arrighi, Pisa: Domus Galilæana, 1970.

De prospectiva pingendi, Biblioteca Palatina, Parma, Ms. no. 1576, ed. Giusta Nicco Fasola. Florence: Sansoni, 1942. Later editions 1974 and 1984. Written presumably before 1482 in Italian—but with the Latin title. Also published in German, French, and Latin translations.

Libellus de quinque corporibus regularibus, Biblioteca apostolica Vaticana, Codex Urbinas 632. In “L’opera de corporibus regularibus… usurpati da fra Luca Pacioli,” ed. Girolamo Mancini. Atti della Reale Accademia dei Lincei, Memorie della classe di scienze morali, storiche e filologiche 5 (14, 1916): 441–580. Published as Libellus de quinque corporibus regularibus. Rome: Accademia dei Lincei, 1915. Also edited as Libellus de quinque corporibus regularibus; corredato della versione volgare di Luca Pacioli, 3 vols. Florence: Giunti, 1995. Latin version of a manuscript that Piero presumably composed in Italian and is now lost.

OTHER SOURCES

Andersen, Kirsti. The Geometry of an Art: The History of the Mathematical Theory of Perspective From Alberti to Monge. New York: Springer, 2007.

Banker, James R. The Culture of San Sepolcro during the Youth of Piero della Francesca. Ann Arbor: University of Michigan Press, 2003.

———. “Contributi alla cronologia della vita e delle opere di

Piero della Francesca.” Arte Cristiana 92 (2004): 248–258.

Dabell, Frank, and J. V. Field. “Piero della Francesca.” In The Dictionary of Art, edited by Jane Turner. New York: Grove, 1996.

Davis, Magaret Daly. Piero della Francesca’s Mathematical Treatises. Ravenna: Longo, 1977.

Field, J. V. “Rediscovering the Archimedean Polyhedra: Piero della Francesca, Luca Pacioli, Leonardo da Vinci, Albrecht Dürer, Daniele Barbaro, and Johannes Kepler.” Archive for History of Exact Sciences 50 (1997): 241–289.

———. Piero della Francesca. A Mathematician’s Art. New Haven, CT: Yale University Press, 2005.

Folkerts, Menso, “Piero della Francesca and Euclid.” In Piero della Francesca tra arte e scienza, edited by Marisa Dalai Emiliani & Valter Curzi, Venice: Marsilio, 1996.

Giusti, Enrico. “Fonti medievali dell’Algebra di Piero della Francesca.” Bollettino di Storia delle Scienze Matematiche 13 (1993): 199–250.

Jayawardene, S. A. “The Trattato d’abaco of Piero della Francesca.” In Cultural Aspects of the Italian Renaissance: Essays in Honour of Paul Oskar Kristeller, edited by Cecil H. Clough. Manchester, U.K.: Manchester University Press, 1976.

Kirsti Andersen

Piero della Francesca

Piero della Francesca (ca. 1415-1492), painter, mathematician, and theorist, was one of the most influential Italian artists of the early Renaissance.

Piero della Francesca was the son of Benedetto dei Franceschi, a shoemaker and tanner in Borgo San Sepolcro near Arezzo. Piero was called "della Francesca, " according to Giorgio Vasari, because he was raised by his mother, who had been widowed before his birth.

Piero was mentioned in a document of Sept. 7, 1439, as "being with" Domenico Veneziano when Domenico was painting in S. Egidio and S. Maria Nuova, Florence. During the 1440s Piero was in San Sepolcro and Ferrara. In 1451 he executed a frescoed portrait of Sigismondo Malatesta in the Tempio Malatestiano, Rimini. On April 12, 1459, he was paid for work (lost) done in the Vatican, Rome. He decorated the choir of S. Francesco, Arezzo, between 1452 and 1466. From 1467 until his death he remained in San Sepolcro except for brief periods in Bastia (1468), Urbino (1469), and Rimini (1482). He served on the town council of San Sepolcro and was a member of the Company of St. Bartholomew. His will is dated July 5, 1487. Vasari relates that the artist was blind and had to be led about by a boy during his last years. Piero died on Oct. 12, 1492.

The Baptism of Christ is a good introduction to Piero's style. Within an arched frame the baptism is taking place in a landscape strikingly similar to the countryside around San Sepolcro. The static, dour figures arranged in regular geometric patterns across the panel's surface are lit by the limpid, luminous Umbrian light. To the left, a trio of angels restricts the view into the distance; to the right, space opens up to reveal a disrobing figure and, further back, a group of bearded patriarchs before a distant landscape. In this picture Piero showed a concern for rational, measurable space, luminosity, and pearly colors. No documents are associated with this picture. Some scholars consider it Piero's earliest work, about 1440-1445; others date it in the early 1450s.

The legend of the True Cross in the choir of S. Francesco, Arezzo, is Piero's most extensive fresco cycle. The scenes are filled with impassive, static figures that impart a soothing quietness to the various episodes. Even in the battle scenes there is a sense of order and quiet rather than confusion and noise. His interest in clearly articulated, rational space can be seen in the Meeting of Solomon with the Queen of Sheba; his interest in the effects of light in the Dream of Constantine, the first realistic nocturnal scene in Italian art. These murals were painted between 1452, the date of the death of Bicci di Lorenzo, the artist first commissioned to paint them, and Dec. 20, 1466, when a document referred to them as complete.

Other noteworthy works from Piero's mature period are the Flagellation, a panel; the Madonna del Parto, a fresco in the cemetery chapel at Monterchi; and the Resurrection, a detached fresco. In the Resurrection the life-size Risen Christ steps wearily from his sarcophagus while looking directly out at the beholder. Piero depicts him as a manly figure with the awesome vigor of a Byzantine Pantocrator. Behind Christ a mauve Umbrian landscape is lit by the moist, pearly light of dawn, and in the foreground four soldiers are sleeping.

Piero was aware of Flemish painting, as can be seen in his Nativity. He probably knew the art of Rogier van der Weyden and may have met him, for Rogier was in Ferrara the same time (ca. 1450) as Piero. He would certainly have known Justus of Ghent, who served the court at Urbino. The Flemish qualities in Piero's work include his use of the oil paint technique and some iconographic types, such as the Madonna del Parto and the Madonna of Humility in the Nativity.

In 1465 Piero executed a diptych portrait of Federigo da Montefeltro, the Duke of Urbino, and his wife. The artist's major work for Federigo was the altarpiece, Madonna with Saints and Donor, which dates in the late 1470s.

Piero wrote three treatises. De prospectiva pingendi, written before 1482, deals with linear perspective and is still the definitive work on the subject. The other treatises are concerned with painting and business mathematics.

Further Reading

All modern scholarship concerning Piero della Francesca derives from the pioneering work of Roberto Longhi. Longhi's works, which date from 1914 to 1942, are in Italian, but the fruits of his discoveries have been incorporated in English-language works. Kenneth Clark's readable monograph is Piero della Francesca (1951; 2d ed. 1969). Also useful is Piero Bianconi, All the Paintings of Piero della Francesca (trans. 1962). Bernard Berenson wrote interesting essays on Piero in his Central Italian Painters of the Renaissance (1897; 2d rev. ed. 1909) and Piero della Francesca; or, The Ineloquent in Art (1954). □

Francesca, Piero Della (or Piero Dei Franceschi)

also known as Petrus Borgensis

(b. Borgo San Sepolcro [now Sansepolcro], Italy, between 1410 and 1420; d. Sansepolcro, 12 October 1492)

mathematics.

Vasari’s reason (in Lives of the Artists) for the adoption of the feminine form “Francesca” in Piero’s name has been invalidated by the authentication of his father as “dei Franceschi” and his mother as Romana di Perino da Monterchi. Of Piero’s early life–as the wide uncertainty of his birthdate reveals—nothing is known until 7 September 1439, when he was an associate of Domenico Veneziano at Florence. He is not named by Alberti in the famous dedication to his colleagues in Dellla pittura (1436), but Alberti’s influence is revealed in the clearest manner in the architectural studies from Piero’s workshop at Urbino.

Piero’s value to science lay in his pioneering efforts to explore the nature of space and to construct it by his sophisticated study of linear perspective and masterly juxtaposition of color masses. Although his major work on the mathematics of painting, De prospettiva pingendi, was written only after his career as an artist was at an end, it can hardly be doubted that the diminution in the successive members of the black and white pavement in the Flagellation at Urbino must have been achieved by such complex calculations as are subsequently displayed in the prospettiva. These represent a synthesis of the two operational diagrams that he probably learned from Alberti.

But a more strikingly original contribution of Piero’s was his measuring of the distances between successive surfaces of a human head and transferring the plane sections thus obtained into a contoured plan. Luca Pacioli, whose influence in spreading the study of mathematics in the early cinquecento is well known, testified to the assistance of his fellow townsman but later paid him the dubious compliment of including (unacknowledged) in his De divina proportione a large part of Piero’s last work, De quinque corporibus regolaribus.

BIBLIOGRAPHY

I. Original Works. The De prospettiva pingendi was written in the vernacular; a transcript exists in the Palatina at Parma and forms the basis of the definitive text edited by G. Fasola (Florence, 1942). There is a transcript of the (contemporary) Latin trans. in the Ambrosian Library at Milan. Piero’s De quinque corporibus regolaribus exists in a transcript (with figs. by him) in the Vatican Library (Urbinas 632). Excerpts from the original works (in English) are available in E. G. Holt, ed., A Documentary History of Art, I (new York, 1957), 256–257.

II. Secondary Literature. A detailed and analytical study of piero’s life and work is Roberto Longhi, Piero della Francesca (London, 1930), Leonard Penlock, trans. The Introduction by Sir Kenneth Clark to the Phaidon review of his pictures, Piero della Francesca (London, 1951), is both readable and scholarly.

William P. D. Wightman

Piero della Francesca (1415–1492)

Italian painter and a master of the early Renaissance. The son of a shoemaker, who died before he was born, he grew up in the small village of Borgo San Sepolcro near the Tuscan town of Arezzo. He moved to Florence to train as a painter and helped older painters with the decoration of several churches in that city. He worked for a noble patron, Sigismondo Malatesta, for whom he painted a famous portrait, as well as for the pope in Rome. He spent much of his adulthood in the town of San Sepolcro and Arezzo, where he was hired to paint the choir of the church of San Francesco, where he painted a famous fresco cycle known as the The Legend of the True Cross, which was inspired by traditional legends surrounding the cross on which Christ was crucified. The orderly arrangement of figures give these pictures a sense of calm rationality, a new sensibility that made a break with traditions of Gothic painting and its direct appeal to the emotions. The Flagellation, a renowned work of the early Renaissance, presents three mysterious figures in the foreground over whom art historians have been arguing for five centuries.

For the Duke of Urbino, Federigo da Montefeltro, Piero completed a double portrait of the duke and his wife, as well as a famous altarpiece, Madonna with Saints and Donor. Piero's great skill and knowledge of mathematics and linear geometry allowed him to construct masterful paintings with the use of foreshortening and perspective, which gives a three-dimensional appearance to the flat surface of a painting. He wrote a treatise, De Prospectiva Pingendi, on the art of perspective, and works on mathematics, including Treatise on the Abacus, in which he covered the subjects of geometry, algebra, and the problems of perspective.

See Also: Montefeltro, Federigo da

Piero della Francesca (1415–92) ( Piero dei Francheschi) Italian painter. He was familiar with the innovations made by Masaccio, Donatello, Filippo Lippi and others. He drew their achievements together to create a deeply reflective style. His most important work is the fresco series depicting the Legend of the True Cross (c.1465) for the choir of San Francesco, Arezzo.

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