by Larry Witham
Where Piero surpasses other abacus books of his day, however, is in his survey of geometric problem-solving. Of about five hundred problems in the Abacus Treatise, Piero made 147 of them about geometry. In this, he does a lot with triangles. Then he gives shorter examples with squares, rectangles, and higher polygons. A typical problem might be: “There is a triangle ABC in which the side AB is 14, BC 15, AC 13; in which I want to put the two largest circles that are possible. I ask what will be their diameters.”47 Most of these problems originated in Euclid’s Elements, as Piero notes. But as an artisan, Piero is doing something new, and this is to combine Greek science with the mind of the workshop artisan. Such a merger of these two Renaissance cultures—the humanist and the artistic—has been most celebrated by history in a figure such as Leonardo da Vinci, but Piero got it started.48 Dry as his Abacus Treatise may have been, it suggested the first hint of a new kind of literature that bridged art and science.
When it came to the mysteries of mathematics and number, however, Piero was mute. Speaking somewhat conventionally, he said that “with God’s help” he hoped to assist merchants in their mathematics, and he perhaps betrayed a bit more of his philosophical outlook when he said he would even venture into algebra, “if it please God.”49 As to the theology of number, however, there are no such effusions, traditional or solemn. When Piero uses the triangular mathematics of the Pythagorean theorem, for instance, he says nothing about the mystical Pythagoras, a figure at the center of a secretive religious fraternity, and yet an inventor of the practical method of calculating triangles.
What Piero’s Abacus Treatise reveals is perhaps a different kind of philosophical interest, and that is the aesthetics of proportion: perhaps divine, but certainly practical and visual. Piero’s extensive use of the rule of three, for example, is a constant demonstration of relating parts and quantities one to another. Medieval and Renaissance definitions of art were heavily weighted toward proportion as the essential element of beauty. Piero fused this aesthetic and mathematical concept into his goals as a painter. He was not unique in this interest, but clearly he would master proportion in ways that led to remarkably beautiful paintings and, late in his life, a unique ability to work with complex visual geometry, such as the so-called Archimedean polyhedra.
Piero’s interest in proportion was further revealed in the ways he handled some of the most famous formulas of ancient geometry, though here again he offered no speculation. His Abacus Treatise, for example, drew upon what would be called the “golden ratio” and “divine proportion” by later writers. Euclid called this mathematical law the “mean and extreme ratio,” and long before him Plato had identified it in his Timaeus creation story as the ratio, or proportion, by which the world was made.50 What Plato and Euclid meant was that any line segment could be divided so that the shorter part’s ratio to the longer is the same ratio as the longer to the entire segment. This is a ratio of 1 to 1.61803 … , which introduces an irrational number (that goes on infinitely) into the proportion.51 As geometers would note across the ages, this ratio could be found in a remarkably large number of geometrical shapes, suggesting that this number—now called phi—was fundamental to nature and even of divine origins.52
Rather than elaborate on a divine ratio, Piero’s Abacus Treatise reveals his interest in the so-called Platonic shapes, to which he dedicates the final 22 pages of his Abacus Treatise (which is 170 pages long). Plato’s dialogue on the natural world, Timaeus, most famously described these shapes, and thus they have been called Platonic. They were later called the “five regular solids” as well. Plato viewed them as the building blocks of the Creation. The shapes are so simple, fundamental, and perfect that they were hypothesized to be the constituent substances of both the world and the ether of the heavens: 1. tetrahedron (fire); 2. cube (earth); 3. octahedron (air); 4. dodecahedron (ether); and 5. icosahedron (water).
Euclid also analyzed the shapes, minus any mystical implications. And then after him the Greek mathematician Archimedes expanded their number, going beyond the five “regular polyhedra” to produce thirteen complex polyhedra. These complex solids have more than one kind of face, typically some combination of triangles, squares, and hexagons. Most of these could be created by trimming off the corners of the five Platonic solids, a visual process to be called “truncating.”53
In Piero’s day, the thirteen complex polyhedra of Archimedes had been essentially lost to history and would not be mapped and named until 1619, when Johannes Kepler inscribed them in one of his works. Between Euclid and Kepler, Piero actually made a small contribution in his Abacus Treatise. By truncating some Platonic solids, he illustrated two of the “lost” thirteen Archimedean polyhedra. And following the Archimedean tradition (of trying to fit solids inside one another based on mathematical calculations), Piero tried his hand in at least two cases in the Abacus Treatise. He showed how his two Archimedean polyhedra fit tightly into a sphere, the two being, as Piero said, “a body with 8 faces, 4 triangular and 4 hexagons,” and then a body with fourteen faces, six square and eight triangular.
That last body was something new in Renaissance geometry until Kepler gave it an exact name: the cuboctahedron. By the end of his career, Piero would mathematically figure out and draw six of the polyhedra that Archimedes had originally formulated, but which had essentially disappeared from history with the inexorable loss and destruction of ancient Greek sources (preserved for a time in later Greco-Roman manuscript copies). Piero would also show his debt to Plato and Euclid by applying the golden ratio to some of his analyses of polyhedra.54 For the time being, the Abacus Treatise was probably helpful to merchants and students and to future mathematicians, as would be seen in the notorious case of one Piero follower, the mathematician Luca Pacioli (of which much more later). But the treatise had little to offer painters. Its obvious forte was to present far more plane and solid geometry than were seen in any other abacus text of Piero’s period.
History would not be especially kind to Piero’s little opus, either; without his name firmly attached to the manuscript, apparently, his authorship would be forgotten. It would thereafter be anonymously cited, copied, or quoted from for centuries. Not until the twentieth century (around 1917, to be precise) would anyone prove that the miraculously surviving Abacus Treatise had been done in the hand of Piero, giving him full credit once and for all.55
Piero’s gift with mathematics would prompt future generations to look back at his works, from the Baptism of Christ onward, to find secret geometries, golden sections, or other such clues. The first painting of Piero that truly lends itself to mathematical analysis, however, is the one that is clearly the most complex geometrically of his (surviving) works. This painting’s historia would also stir expansive debate on what cryptic “story” Piero was trying to tell. The work has been known as The Flagellation of Christ, a small rectangular panel that is just two feet tall.56 It may have been part of an altar or even the decorative side of a wooden chest, called a cassone.
In painting the Flagellation, and probably doing so in Urbino, Piero would have made one of the most arduous journeys in middle Italy. Urbino sits atop the high crest of a mountain range. It is surrounded not just by grassy hills and fertile pockets of valley, but by something akin to a harsh lunar landscape, volcanic in origin. From Sansepolcro, Piero might have gone northwest over the treacherous Rocca Trabaria, a three-thousand-foot elevation of stratified rock and trees on the border between Umbria and the Marches. If traveling from Rimini on the Adriatic coast, Piero could take a road inland at Pesaro, winding through ever-rising rugged hills, finally reaching two steep hills on which Urbino sat like an eagle’s nest.
In these years of Piero’s travels, Urbino had received a new ruler, the soldier and patron Federico da Montefeltro, who was no less than the arch-rival of Sigismondo Malatesta of Rimini.57 A little younger than Sigismondo, Federico was just as ambitious militarily and just as proud of his humanist training and
his spending on the arts. Their provinces shared a border and their families some marriage ties. But they were inveterate enemies. In the form of the mock battles that typified Quattrocento warfare, the two princes promoted border skirmishes to control land and trading routes. They sparred under their dynastic symbols: for Federico the eagle, and for Sigismondo the elephant.
Urbino had recently become a dukedom, elevated to that rank by Pope Eugenius IV, and its first leader was the legitimate heir, the young Oddantonio Montefeltro. A year later, in 1444, Oddantonio was assassinated and his brother Federico—born illegitimate but, like Sigismondo, declared legitimate by Pope Martin V—took control. Piero may have had occasion to visit Urbino earlier in his life, but his painting of the Flagellation of Christ (probably) in Urbino is his first known connection. If he once had worked with the House of Malatesta in Rimini, in the long run Piero would forge a stronger and longer bond of patronage with the House of Montefeltro.
The Flagellation of Christ received its name because of its literal imagery of a Jesus-like figure being whipped. Based on the biblical story, it was not an uncommon image for a late-Gothic work, though Piero was able to present a far more naturalistic rendering and modern setting than seen in the medieval paintings. Federico does not appear as a character in the painting, so he was not its patron. Piero may have done the work for a humanist official in Urbino, or he may have done the small panel—with its exacting mathematical calculations—as a kind of experiment and meditation.58 Whichever was the case, Piero had taken a conventional story and seemingly added mystery upon mystery, both in narrative and mathematically.
Piero’s portrayal of a flagellation story unfolds in two halves. The left side shows a background scene of a man tied to a column. He is being whipped as a royal figure looks on. These are presumably Pontius Pilate and Christ. In the foreground on the right, three other men are posed. They look like real people Piero had in mind, and the dress is both strange and specific, as if Piero had seen it in courts or on the street. One of them wears a dark silk garment emblazoned with the so-called pineapple decoration. This was chic Italian fashion, and it may have passed before Piero’s very eyes. Then again, he could have observed it in a painting by the Flemish master Jan van Eyck, whose works with elaborate costumes were then appearing in Italian courts.
More perplexing, the three men in the foreground seem to be, by one logic at least, discussing the flagellation event taking place behind them. The bearded man is speaking, gesturing to a listening companion. In between them is a youth with blond hair, his eyes strangely lit. No one is looking directly at the viewer, as does one angel in Piero’s Baptism painting, but the Flagellation’s blond youth has virtually the same face as another angel in the Baptism. So, is he, too, supposed to be an angel? As the questions mount up, the plot thickens.
Given the panel’s reasonable size, one can imagine how Piero proceeded in his work. He first did a plan-and-elevation drawing of the picture’s open-air portico, the colonnades, and the receding buildings. He used his compass and straightedge to put in the perspective lines, and in one case he used a cartoon and pouncing to outline a turban on a figure in the foreground.
Like few other paintings in the Renaissance, the Flagellation is the robust work of a mathematician.59 According to the linear perspective Piero uses, the picture represents an outdoor space extending back 250 feet.60 A black stone tile in the foreground seems to be Piero’s unit of measure. The picture is ten by seven of these units. The viewer, in turn, is way off center. From a low vantage point, the beholder looks across an array of complex floor tiles, walls, and façades. The scene incorporates stone and marble in white, purple, red, and pink, and it has two sources of light—and so on, a jungle of detailed visual patterns and special effects for the art sleuth willing to make the effort. Despite its odd angles and cross-cutting lines, the Flagellation is a remarkable work of illusion, an airy outdoor/indoor scene unified by light and color.
For any painter attempting complex perspective, there were calculations to be made. As an illusion, linear perspective works best when the viewer of a painting stands in exactly the right spot, and the early Renaissance seemed to have arrived at a rule of thumb on this matter. Painters drew their perspective lines so that the ideal viewing distance was equal to somewhere between the width of the painting and one and a half times the width. A perspective painting five feet across, for instance, would be the most visually accurate for viewers five to eight feet away.61 For the Flagellation, Piero bucked the trend. His perspective lines put the ideal onlooker much farther away, a viewing distance of two and a half times the width of the picture. Why he did this can only beg more speculation. Perhaps it was because of its original location, or simply because Piero enjoyed a challenging experiment in perspective—or because it simply looked good to him, which is probably the core motive behind most artworks, all things being equal.
Ideal distances notwithstanding, an insatiable curiosity remained: What was the painting about? By this time in his life, Piero was doubtless a circumspect philosopher and a keen observer of the political tumult of his age, from the warring Italian city-states to the continued strife with the Turkish Empire around the Mediterranean. For many future interpreters of Piero, the experience of seeing the ebullient hopes of the Council of Florence (1439), and then the Turkish conquest of Constantinople (1453), must have been something that any artist with a conscience would be compelled to paint about. In the years when Piero mulled or even painted the Flagellation, Pope Pius II had called a special Council of Mantua (1459) to try to launch a crusade against the Turks.
According to one popular view, Piero had all this geopolitical drama in mind while painting the Flagellation: the Turkish infidels had scourged Byzantium as Pontius Pilate had lashed Christ. In the foreground, three key personalities of the day—mystery men, to be sure—may have been political figures close to the events. Over the centuries, there would be thirty-five distinct interpretations of Piero’s historia, either biblical, based on local headlines of the day, or symbolic of international events.62 The most longstanding local tradition had said that Christ’s tribulation symbolized the assassination of Oddantonio, and some of the personalities of Oddantonio’s short reign stood in the front, sad or perplexed. To others, the Flagellation would be an analogy of the conflict of the Latin and Greek Christians with the Turks.
But something else entirely could be at play, if one looks at the work with a totally contrarian interpretation. It might not be the story of Christ at all. More people than Christ had been lashed in the saga of man’s inhumanity to man. Such a lashing was dreamed of by St. Jerome and vividly written down. That story of Jerome had been painted before. His dream relives his intellectual struggle. He is brought before the throne of heaven and whipped for the sin of loving pagan literature more than the Bible.63 The transgression is true, he writes:
I could not do without the library which I had collected for myself at Rome by great care and effort. And so, poor wretch that I was, I used to fast and then read Cicero. After frequent night vigils, after shedding tears which the remembrance of past sins brought forth from my inmost heart, I would take in my hands a volume of [Roman playwright and humorist] Plautus. When I came to myself and began to read a prophet again, I rebelled at the [prophet’s] uncouth style… .64
For this backsliding, Jerome had been called before the judgment seat, and God rebuked him, saying: “Thou liest; thou art a Ciceronian, not a Christian. For where thy treasure is, there will thy heart be also.”
If Piero had been painting the “Flagellation of St. Jerome,” then the topic being discussed by the figures in the foreground could well have been the merits of classical versus biblical or patristic literature, and perhaps the struggle that any Renaissance Christian humanist—like Piero—faced in such an age of intellectual ferment, and this in a culture that still emphasized piety. In the courts where Piero had visited, not only were the Aristotelians and Pl
atonists confronting each other. The topic of the hour was how to Christianize the Greco-Roman past, the recovery of which was turning out to be a central birthright of the Renaissance, for the encounter of antiquity and Christianity was now far exceeding anything seen in the Middle Ages.
In the Flagellation, Piero may have been meditating on his own Jerome-like struggle. He may have been presenting his viewers their own choice between the Bible and the classics. This flagellation story—biblical or otherwise—is, after all, presented by Piero as taking place in a Mediterranean court in classical times. One early Christian theologian, at war with his pagan rivals, had already posed the dilemma: “What has Athens to do with Jerusalem?” The incompatibility of the two was further suggested in the sixth century, when the Christian Emperor Justinian apparently closed the great academy in Athens that carried on in the tradition of Plato and Aristotle. For those familiar with these concerns, Piero’s painting probably roiled with such tension. It was a visual debate over how much the art, literature, mathematics, and philosophy of antiquity could be integrated with Christianity. A small painting, the Flagellation would prove to be one of the biggest, and most beloved, enigmas in the history of art.
Like most of Piero’s paintings, the Flagellation also evoked the question—both Christian and classical—on the experience of visual beauty. Well past the Renaissance, the classical and Christian views of beauty had been remarkably the same, in essence identical. Plato had put the ideals of Beauty and the Good in a transcendent realm, and thus viewed artisans as finally unable to imitate such intellectual perfection. Despite this Platonist skepticism, the Greek tradition nevertheless twinned Plato’s universal Beauty with particular beauties.65 Beauty is known by the pleasure it brings, said the Greeks, but in Plato and Aristotle that experience was defined differently. For Plato it was mental pleasure, while for Aristotle it was “catharsis,” or a physical and emotional release. This physical emphasis was taken further by the pleasure-loving Epicureans. Neither Plato nor Aristotle would ever see the day, of course, when today’s brain science also would use the baseline of pleasure as the effect of visual beauty. Neuroscience, however, would look for this pleasure in something that is happening among the brain’s neurons.