by Larry Witham
45.Mark A. Peterson, “The Geometry of Piero della Francesca,” The Mathematical Intelligencer 19 (Summer 1997): 33-37.
46.Quoted in J. V. Field, “Piero della Francesca’s Mathematics,” in The Cambridge Companion to Piero della Francesca, ed. Jeryldene M. Wood (New York: Cambridge University Press, 2002), 154.
47.Ibid., 155.
48.For the “two cultures” bridged by Piero and Leonardo see James R. Banker, “Three Geniuses and a Franciscan Friar,” lecture, The Frick Collection, New York City, March 20, 2013.
49.Piero, quoted in Field, Piero Della Francesca: A Mathematician’s Art, 30.
50.Plato introduces two kinds of “means,” or ratios, in Timaeus, the first being the so-called golden ratio, or “mean and extreme ratio” as phrased by Euclid. Plato said: “There were two kinds of means, the one exceeding and exceeded by equal parts of its extremes, the other being that kind of mean which exceeds and is exceeded by an equal number.” See Edith Hamilton and Huntington Cairns, eds., The Collected Dialogues of Plato, Including the Letters (New York: Pantheon Books, 1961), 1165. Later, Kepler said: “Geometry harbors two great treasures: One is the Pythagorean theorem, and the other is the golden ratio. The first we compare with a heap of gold, and the second we simply call a priceless jewel.” Quoted in Alfred S. Posamentier and Ingmar Lehmann, The Glorious Golden Ratio (Amherst, NY: Prometheus Books, 2011), 12.
51.As an irrational fraction, the ratio has been called phi and is the basis of a numerical pattern discovered by Leonardo of Pisa called, after his name, the “Fibonacci sequence.” Here, as the numbers are added together in the sequence—0, 1, 1, 2, 3, 5, 8, 13, 21, and so on—and when the larger number is divided by its predecessor, the ratio is roughly 1 to 1.6.
52.There are at least sixteen different ways that the ratio has been derived from geometrical forms. See Posamentier and Lehmann, The Glorious Golden Ratio, 16-38. For all the implications of the golden ratio see Scott Olsen, The Golden Section: Nature’s Greatest Secret (New York: Walker, 2006); and Mario Livio, The Golden Ratio, The Story of Phi, the World’s Most Astonishing Number (New York: Broadway Books, 2002). Enthusiasts and skeptics continue to debate whether the ratio exists in ancient man-made marvels such as the pyramids, famous paintings, and, for example, the waistline ratio of Leonardo da Vinci’s ideal, spread-eagle Vitruvian Man (late 1480s).
53.See Judith V. Field, “Rediscovering the Archimedean Polyhedra: Piero della Francesca, Luca Pacioli, Leonardo da Vinci, Albrecht Dürer, Danielle Barbaro, and Johannes Kepler,” Archive for History of Exact Sciences 50 (September 1997): 241-289.
54.On Piero’s use of the golden ratio to solve problems with the polyhedra see James R. Banker, “Three Geniuses and a Franciscan Friar,” lecture, The Frick Collection, New York City, March 20, 2013.
55.Margaret Daly Davis, “Piero’s Treatises: The Mathematics of Form,” in The Cambridge Companion to Piero della Francesca, 136.
56.For a summary of various treatments of the Flagellation see Carlo Ginzburg, The Enigma of Piero: Piero della Francesca, 48-101, 116-28; Marilyn Aronberg Lavin, Piero della Francesca: The Flagellation (New York: Viking Press, 1972); Creighton Gilbert, “Piero della Francesca’s Flagellation: The Figures in the Foreground,” Art Bulletin 53 (March 1971): 41-51; Ernst H. Gombrich, “The Repentance of Judas in Piero della Francesca’s ‘Flagellation of Christ,’” Journal of the Warburg and Courtauld Institutes 22 (January-June 1959), 172; and Rudolf Wittkower and B. A. R. Carter, “The Perspective of Piero della Francesca’s Flagellation,” Journal of the Warburg and Courtauld Institutes 16 (1953): 292-302.
57.For a Montefeltro biography see Maria Grazia Pernis and Laurie Schneider Adams, Federico da Montefeltro & Sigismondo Malatesta: The Eagle and the Elephant (New York: Peter Lang, 1996), 13-25.
58.Lavin proposes a theory that the court official who commissioned the Flagellation was Ottaviano Ubaldini della Carda, who sought consolation from his son’s death by the plague. See Lavin, Piero della Francesca, 107.
59.For one analysis of mathematics in this work see Wittkower and Carter, “The Perspective of Piero della Francesca’s Flagellation,” 292-302.
60.The measurement is given in Lavin, Piero della Francesca, 101.
61.See John White, The Birth and Rebirth of Pictorial Space (Cambridge, MA: Harvard University Press, 1987 [1957]), 194-95.
62.Lavin, Piero della Francesca, 96. By local tradition, for example, Piero’s painting was about the assassination of Oddantonio.
63.The Jerome interpretation is the unique contribution of John Pope-Hennessy, The Piero della Francesca Trail (London: Thames and Hudson, 1991), 17.
64.Jerome, quoted in Ancient Christian Writers: Letters of St. Jerome, vol. 1, trans. Charles Christopher Mierow (New York: Paulist Press, 1963), 165-66.
65.See Wladyslaw Tatarkiewicz, “The Great Theory of Beauty and its Decline,” Journal of Aesthetics and Art Criticism 31 (1972): 165-80.
66.Pseudo-Dionysius, quoted in Berys Gaut and Dominic Mciver Lopes, eds., The Routledge Companion to Aesthetics (London: Routledge, 2000), 28.
67.Aquinas, quoted in Gaut and Lopes, eds., The Routledge Companion to Aesthetics, 34. It comes from Summa Theologiae, I, 39, 8.
68.Leon Battista Alberti, On the Art of Building in Ten Books, trans. Joseph Rykwert, Neil Leach, and Robert Tavernor (Cambridge, MA: MIT Press, 1988), 302.
69.Leonardo on color in Martin Kemp, The Science of Art: Optical Themes in Western Art from Brunelleschi to Seurat (New Haven, CT: Yale University Press, 1990), 266-69.
70.Valla, quoted in Michael Baxandall, Painting and Experience in Fifteenth-Century Italy (Oxford: Oxford University Press, 1972), 85.
71.Alberti, On Painting, 82.
72.The optimal luminosity of Leonardo’s painting has been studied by John Shearman, cited in Margaret Livingston, Vision and Art: The Biology of Seeing (New York: Abrams, 2002), 115. For luminosity in general, see Livingston, 109-122, especially “Luminance Over the Centuries” (115).
73.The two Piero paintings that show a strong chiaroscuro are Vision of Constantine in Arezzo and the predella panel Stigmatization of Saint Francis in the Perugia polyptych (Saint Anthony Altarpiece). Piero may have influenced other painters in the movement toward extremes in chiaroscuro, as seen finally in Caravaggio and Rembrandt. It has been speculated that a Piero fresco in the Vatican Palace had used dramatic dark and light since, once Raphael had erased it for his own work, Raphael painted Liberation of Saint Peter in a highly chiaroscuro contrast of dark and light.
74.As ever, the dating on the Resurrection is in dispute (scholars have dated the painting in a thirty-year range from 1442 to 1474). The agreed-upon benchmark is the 1441 political takeover of Sansepolcro by Florence. After that, the use of civic buildings changed. In 1442, Florence abolished the Sansepolcro town council and instituted a governing “Captain” over a weaker town assembly. However, the citizens continued to improve the main civic building, the Palazzo dei Conservatori, and may have envisioned Piero’s painting as decorating its entrance hall. Either way, a more generous Florence in 1459 decided that “it is convenient to give and concede the said building to the commune,” in effect giving Sansepolcro back its municipal hall. Piero’s painting may have adorned it well before this time, however, but on the building’s return, the painting surely became a stronger local symbol of independence. Quoted in Michael Baxandall, “Piero della Francesca’s The Resurrection of Christ,” in Michael Baxandall, Words for Pictures: Seven Papers on Renaissance Art and Criticism (New Haven, CT: Yale University Press, 2003), 133.
75.See Baxandall, “Piero della Francesca’s The Resurrection of Christ,” 130 n.18. If there was a political reason behind Piero turning to Siena for arististic models, not Florence, it may have been the fact that rapacious Florence had by now taken over every independent city in Tuscany except Siena,
which was a symbol of resistence to Florence, the “wolf of Tuscany.”
76.Ibid., 150 n.45.
77.See Laurie Schneider, “The Iconography of Piero della Francesca’s Frescos Illustrating the Legend of the True Cross in the Church of San Francesco in Arezzo,” Art Quarterly 32 (1969): 24. She cites how modern iconographers have taken Piero’s barren tree to be symbolic of the Adam-Christ (old-new) dichotomy, and makes the same argument herself. More recently, it has been argued that the green paint simply has fallen off.
CHAPTER 4
1.See Angiola Maria Romanini, Assisi: The Frescoes in the Basilica of St. Francis (New York : Rizzoli, 1998); Elvio Lunghi, Basilica of St Francis at Assisi: The Frescoes by Giotto, his Precursors and Followers, trans. Christopher Evans (London: Thames and Hudson, 1996); and Leonetto Tintori and Millard Meiss, The Painting of the Life of St. Francis in Assisi, with Notes on the Arena Chapel (New York: New York University Press, 1962).
2.The true-cross story has survived in three other known frescos: by Agnolo Gaddi in the choir of Santa Croce Church, Florence, at the end of the fourteenth century; by Cenni di Francesco in 1410 at San Francesco Church in Volterra; and in sinopia by Masolino in the St. Helena chapel, Santo Stefano, Empoli. See Laurie Schneider, “The Iconography of Piero della Francesca’s Frescos Illustrating the Legend of the True Cross in the Church of San Francesco in Arezzo,” Art Quarterly 32 (1969): 23-48.
3.For the story of the Bacci family see Carlo Ginzburg, The Enigma of Piero: Piero della Francesca, new edition, trans. Martin Ryle and Kate Soper (London: Verso, 2002 [1985]), 27-31. A Bacci was in the papal Curia and the family had ties to the humanists in Florence. According to Vasari, Piero used the portraits of Luigi Bacci together with “Carlo and others of his brothers and many Aretines” in the scene where the victors prepare to behead the evil Chosroes. Giorgio Vasari, Lives of the Most Excellent Painters, Sculptors, and Architects, vol. 1, trans. Gaston du C. de Vere (New York: Abrams, 1979), 474.
4.In this sentence, I quote from the old version of Jacobus de Voragine, The Golden Legend; or, Lives of the saints, as Englished by William Caxton (London: J. M. Dent, 1900). For a more concise modern translation see Jacobus de Voragine, The Golden Legend: Readings on the Saints, trans. William Granger Ryan (Princeton, NJ: Princeton University Press, 2012).
5.John Pope-Hennessy, The Piero della Francesca Trail (London: Thames and Hudson, 1991), 48.
6.For a contrast between neoclassical stillness and Impressionist movement see Margaret Livingston, Vision and Art: The Biology of Seeing (New York: Abrams, 2002), 74-76.
7.David Talbot offers these three summaries in Longhi, PDF, 111-12.
8.Kenneth Clark, Piero della Francesca (London: Phaidon, 1951), 28. To be precise, Clark says the first side, the left wall, is “much cooler, dominated by grey and white, green and blue; the [right] wall, although still light, is warmer, with much pink, lilac and grape purple to set off the blue and white.”
9.Ginzburg, The Enigma of Piero. Ginzburg hypothesizes the influence of the Bacci family and other church leaders in changing the frescos.
10.Augustine, Concerning the City of God Against the Pagans, ed. John O’Meara, trans. Henry Bettenson (London: Penguin Books, 1984), 304.
11.Contract quoted in Philip Hendy, Piero della Francesca and the Early Renaissance (New York: Macmillan, 1968), 120.
12.Roberto Bellucci and Cecilia Frosinini, “Piero Della Francesca’s Process: Panel Painting Technique,” in Contributions to the Dublin Congress, 7-11 September 1998: Painting Techniques: History, Materials and Studio Practice, ed. Ashok Roy and Perry Smith (London: International Institute for Conservation of Historic and Artistic Works, 1998), 91.
13.Bert W. Meijer, “Piero and the North,” in Lavin, PFL, 144, 156 n.22.
14.For this theory see Bellucci and Frosinini, “Piero Della Francesca’s Process,” 89-90.
15.For a full review of the circumstance around the commission of the Saint Augustine Altarpiece, related events in Piero’s life, and the fate of the altarpiece, see the exhibition essays in Nathan Silver, Piero della Francesca in America: From Sansepolcro to the East Coast (New York: The Frick Collection, 2013).
16.Vasari, Lives, vol. 1, 472.
17.On the migration of the Saint Augustine Altarpiece elements to the United States see Silver, Piero della Francesca in America. In England, Charles Eastlake acquired the Michael the Archangel in Milan. The painting bore the forged signature of Mantegna, but Eastlake suspected three other artists, including Fra Carnavalle. After the work passed to the National Gallery in 1867, it went under a “School of Della Francesca” label until it was declared a Piero in 1901.
18.For the Saint Anthony Altarpiece see Piero Bianconi, All the Paintings of Piero Della Francesca, trans. Paul Colacicchi (New York: Hawthorn, 1962), 61-63; and Marilyn Aronberg Lavin, Piero della Francesca (New York: Phaidon, 2002), 200-220.
19.Millard Meiss, “Ovum Struthionis: Symbol and Allusion in Piero della Francesca’s Montefeltro Altarpiece,” in Studies in Art and Literature for Belle da Costa Green (Princeton, NJ: Princeton University Press, 1954), 99.
20.Cited in James D. Proctor, ed., Science, Religion, and the Human Experience (New York: Oxford University Press, 2005), 40. Famously, Kenneth Clark, in writing on Piero, said he only belatedly “realized that Virgin is behind the first column.” Kenneth Clark, Piero della Francesca, second rev. edition (London: Phaidon, 1969), 66. This is perhaps another case of Piero’s “spatial games” noted herein, chapter 6 n.27.
CHAPTER 5
1.Pope Pius II was elected on August 19, 1458. Some Piero scholars, including James R. Banker, believe Piero had traveled to Rome once sometime earlier, making this his second visit. By the same token, some Piero scholars believe he was in Florence twice; but, as with Rome, the evidence is at best circumstantial (that is, Piero traveled a great deal).
2.See Carlo Ginzburg, The Enigma of Piero: Piero della Francesca, new edition, trans. Martin Ryle and Kate Soper (London: Verso, 2002 [1985]), 23-24, 27, 45.
3.Pius II, Reject Aeneas, Accept Pius: Selected Letters of Aeneas Sylvius Piccolomini (Pope Pius II), ed. Thomas M. Izbicki, Philip D. W. Krey, and Gerald Christianson (Washington, D.C.: Catholic University of America Press, 2006), 396.
4.See Marshall Clagett, Archimedes in the Middle Ages, vol. 3 (Philadelphia: American Philosophical Society, 1978), 321.
5.James R. Banker, “A Manuscript of the Works of Archimedes in the Hand of Piero della Francesca,” Burlington Magazine 147 (March 2005): 165. It is Banker’s conclusion that Piero’s handwritten copy is an unnamed manuscript now in the Biblioteca Riccardiana in Florence. In this complex detective story of Archimedean manuscripts, Banker meanwhile states two hypotheses of how Francesco’s manuscripts ultimately showed up in the ducal library in Urbino: either a pope sent them there, or Piero carried them there; Banker sides with the second interpretation.
6.For these and more see James Hankins, Plato in the Italian Renaissance, vol. 1 (New York: E. J. Brill, 1990), 3-26; and John Monfasani, “Platonic Paganism in the Fifteenth Century,” in Reconsidering the Renaissance, ed. M. A. di Cesare (Binghamton, NY: Center for Medieval and Early Renaissance Studies, 1992), 45-61.
7.For concise biographies see Dermot Moran, “Nicholas of Cusa and Modern Philosophy,” in The Cambridge Companion to Renaissance Philosophy (Cambridge: Cambridge University Press, 2008), 173-92; Dermot Moran, “Nicholas of Cusa (1401-1464): Platonism at the Dawn of Modernity,” in Platonism at the Origins of Modernity, ed. Douglas Hedley and Sarah Hutton (Dordrecht, the Netherlands: Springer, 2008), 9-29; and Jasper Hopkins, “Nicholas of Cusa,” Dictionary of the Middle Ages, vol. 9, ed. Joseph Strayer (New York: Charles Scribner’s Sons, 1987), 122-25.
8.See Arthur O. Lovejoy, The Great Chain of Being (New York: Harper, 1960 [1936]). Lovejoy emphasizes that the Neoplatonist tradition—mys
tical, scientific, and mathematical—had the central idea of “plenitude,” which means the optimal degree of existing things and divine creativity in all the spaces of the cosmos.
9.See Thomas Whittaker, The Neo-Platonists: A Study in the History of Hellenism (Cambridge: Cambridge University, 1918). The chief Christian Neoplatonist (besides, modestly, Augustine and Boethius) was Dionysius the Areopagite, believed to be a Syrian monk in the patristic period. See Sarah Coakley and Charles M. Stang, eds., Re-thinking Dionysius the Areopagite (Malden, MA: Wiley-Blackwell, 2009).
10.For an English translation of On Learned Ignorance see Nicholas of Cusa, De docta ignorantia, second edition., ed. and trans. Jasper Hopkins (Minneapolis: Arthur J. Banning Press, 1985).
11.Cusa, De docta ignorantia, 69. Kant had rebelled against European rationalism, which purported to prove God, the soul, and much else, and said the mind cannot know the “thing in itself” (ultimate reality), but instead the mind has categories (such as time and space, for example) that organize the flux of physical perceptions.
12.Ibid., 54, 50.
13.Ibid., 61, 52; Cusanus’s De conjecturis (conjecture) and De beryllo quoted in The Cambridge Companion to Renaissance Philosophy, 184. In defining conjectures, Cusanus said “Man is the creator of conceptual beings and of artificial forms that are only likenesses of his intellect, even as God’s creatures are likenesses of the divine intellect” (184). See Moran, Cambridge Companion, 179, on Cusanus’s use of conjecture toward mathematics.
14.For a history of how the Platonist and Christian belief that Ideas in the transcendent realm evolved into a belief in the power or divinity of ideas in the human imagination see Erwin Panofsky, Idea: A Concept in Art Theory (New York: Harper and Row, 1975 [1968, English] [1924]), and Umberto Eco, Art and Beauty in the Middle Ages, trans. Hugh Bredin (New Haven, CT: Yale University Press, 2002 [1959]). See also herein chapter 9, n.36.
