Book Cover design by Brian Barth, Smithsonian Books

Math and the Mona Lisa
Chapter One

A Life Well Spent

"The water you touch in a river is the last of that which has passed and the first of which is coming. Thus it is with time present. Life, if well spent, is long " — Leonardo da Vinci

Late medieval and early Renaissance Italy witnessed many changes, including a revival of the mercantile economy, the emergence of a vernacular literature, and the first serious efforts to recover the classical tradition of learning. Feudalism, with the landed nobility controlling the lives and destinies of the populace, began to lose its grip. The Holy Roman Empire and the Roman Catholic Church increasingly failed to provide social and political stability. National monarchies, especially those of France and England, rose in importance, and in Italy, the city-state became the preferred form of political organization. One city-state, Firenze (Florence), located in north central Italy, took the lead in projecting the new indefatigable spirit of humanism, a return to the classical ideal of man being the measure of all things, and it became the incontestable intellectual capital of Renaissance Europe. The city's preeminence was displayed in literature—with Petrarch, Dante, and Boccaccio—but most prominently in painting, sculpture, and architecture. The brilliant painter Giotto appeared early in this remarkable period. The next hundred years gave rise to the artist and architects Masaccio, Alberti and Brunelleschi; then, toward the end of the fifteenth century, the matchless trio of Leonardo, Michelangelo, and Raphael burst onto the scene.

            An explosive catalyst for the change was the invention by Johannes Gutenberg of the printed book in 1455.1 Before the print revolution, Europe's libraries contained 30,000 volumes. Within fifty years the number of books had increased to three million. The Renaissance also saw the European voyages of discovery, resulting in dramatic expansion in the size of the known world. The Protestant Reformation ignited further intellectual commotion, with an attendant eruption of various dissident sects. Finally, the Renaissance artist, who saw the need to describe nature in the way it really presented itself and not in some idealized or ecclesiastically dictated way, was instrumental in the launching of modern science.

            The changing intellectual milieu of the Renaissance spread quickly to Rome, Milan, and Venice. One ingredient for its accelerated development in Italy came with the conquest of Constantinople by the Ottoman Turks in 1453. A number of significant Byzantine scholars migrated to Italy at the invitation of the Italian humanists, among them, Theodore Gaza, John Argyropoulos, and the most influential of all, Demetrius Chalcondyles. These scholars brought with them the first serious efforts to recover the classical tradition of learning and afforded Italian humanists access to the classic Greek texts and manuscripts preserved in Constantinople.

            Any discussion on the ascent of civilization must necessarily include the rise of the university. Toward the end of the eleventh century the first of the studia generalia, precursors of universities,2 had appeared in Bologna. In the twelfth century others were begun up in Paris, Oxford, Modena, and Parma, and in the thirteenth century in Cambridge, Padua, Siena, Perugia and Palermo. The universities did not give rise to the Renaissance, but they would come to benefit significantly from it. While the Italian universities were the first to be founded in Europe they were the last to be liberated from the scholastic tradition grounded in the works of Aristotle. Their doctrine was salutary for the rebirth of rigorous intellectual discourse in the manner of the ancient philosophers, but it focused mostly on theological issues in a doctrinaire way. Thus the early emergence of universities in Italy with their scholastic tradition has to be regarded as a red herring for the development of science in Italy.3

         Not very much is known about Leonardo da Vinci's personal life. According to his earliest biographer, Giorgio Vasari, he was an uncommonly handsome man, well-built, full of charm and grace, but he left no definitive image of himself. One reasonable candidate for a Leonardo likeness is a red chalk drawing, found in Turin in the mid-nineteenth century and believed by many to be a self-portrait of Leonardo in his old age. There is a mesmerizing quality in the eyes, simultaneously exuding wisdom, sadness, and acute intelligence that only a truly insightful psychologist-artist could capture (plate 1). Another possible likeness, also from his mature years, is a colored chalk profile portrait thought to be by one of his pupils; David Alan Brown, of the National Gallery of Art, Washington, D.C., makes a compelling case for the subject of this work.

            Leonardo lived his sixty-seven years in a time of frequent wars and political and social upheaval, but also in a period of artistic and intellectual ferment unrivaled since the Golden Age of Greece. He embodied the Renaissance spirit. In his own time he was known as Il Fiorentino (the Florentine), although by the late sixteenth century Giorgio Vasari was already referring to him as "Leonardo da Vinci."

            It is not known exactly where Leonardo was born, but convincing arguments have been offered by a number of biographers that he was born on April 15, 1452, in the Tuscan village of Anchiano near the town of Vinci, on the outskirts of Florence. He was the illegitimate son of Ser Piero di Antonio da Vinci and a peasant girl named Caterina. The young couple never married, the boy living the first five years of his life with his mother, grandmother, and a peasant from Anchiano, whom the mother eventually did marry. Meanwhile, Ser Piero married Donna Albieri, a woman of his own station, and only when he found that his wife was infertile did he seek and gain guardianship of Leonardo.

            During the next ten years the boy lived in his father's family home in Vinci, never gaining formal adoption or the benefit of the respected family name. There have been speculations by a number of authors—including Sigmund Freud—that in the home of his mother and maternal grandmother, and later in the home of his stepmother and step-grandmother, the boy perhaps received attention bordering on the worshipful. These factors have been offered as possible ingredients for his unusual psyche, his exquisite sensitivity, superhuman drive, surpassing intelligence, and probable homosexuality—although this is all conjecture.

            Ser Piero eventually married two more times, fathering twelve other children, but none exhibited similar gifts, nor left the slightest mark on civilization. A rare reference to Leonardo's siblings dates to 1504, when Ser Piero died at the age seventy-seven, when the siblings launched a successful conspiracy to cut Leonardo off from any share of his father's estate.

            Had Leonardo been born legitimate he most likely would have been groomed to become a notary—just as his father, grandfather, and great grandfather had been. That option was not open to children born out of wedlock. He had no formal schooling, although he had some private tutoring, what we might call "home schooling." His unconventional education did not include the study of Latin or Greek, and he seems to have felt inadequate in never being able to read the works of the classical authors in their original languages. With a lack of formal education he did not benefit from many of the great classical ideas, but neither was he saddled with the bad ones. Leonardo, untutored by a university education, was thus also uncontaminated by it.

            The choice of a vocation for the young boy presented his father with a quandary. From the beginning Leonardo displayed certain talents for art and music, and accordingly Ser Piero’s decision was simplified. When Leonardo was fifteen, he moved with his paternal family to Florence, and two years later, was apprenticed to the painter's workshop of Andrea del Verrocchio, a talented goldsmith, sculptor, and painter. Verrocchio—whose name means "true eye"—was one of Florence’s most influential artists.            Leonardo flourished in Verrocchio’s workshop, developing the skills that would serve him throughout life. He mastered the techniques for grinding rare-earth elements to create colors, making brushes, even casting bronze, as well as the latest principles of perspective and composition. He also learned from the master the techniques of chiaroscuro, using light and shade in pictorial representation, and sfumato, the blending of chalk strokes to make a seamless, smoky shadow, the latter a procedure that Verrocchio developed himself. Most importantly, he learned the importance of understanding anatomy, so that he could build the body from the inside out. It was clear from the start, however, that Leonardo wanted to build on what was known, and experimentation with all of the elements of art became his modus operandi. In his early days in the workshop Verrocchio assigned Leonardo one of the two kneeling angels in the Baptism of Christ (Florence, Uffizi). Although masters often gave a student a secondary figure to paint, this proved to be a strategic mistake, because here that single angel becomes the painting's visual focus. Verrocchio, overwhelmed by Leonardo's mastery, never picked up a brush again.

            Leonardo acquired his wonder and passion for nature in his childhood in the verdant hills of his native Tuscany, observing nature as both an artist and a scientist. He had a "mysterious cave" in the hills that inspired his life-long passion for geology.6 And indeed the peregrinations convinced him that the earth was much older than the contemporary view—an early triumph of observation over orthodoxy. His earliest known landscape drawing, a pen and ink study from 1473 (Florence, Uffizi), made when he was twenty-one years old and living in Florence, depicts the Arno Valley seen from a hilltop. The scene is rich with artistic devices—impeccable perspective, draftsmanship, and shading. But also present is the degradation in the quality of light with distance seen through the eyes of the physicist, and the rock formations seen through the eyes of the geologist. In his codices he classified rocks and pondered their origin, identifying sedimentary rocks—long before the invention of geology and its classifications. In the striations and strata in rocks—some presenting horizontal, others oblique configurations—he wondered about the possibility of "uplifting" as a mechanism for their formation, anticipating the development four centuries later of the theory of plate tectonics with its attendant process of uplifting as the established explanation for mountain chains. In Leonardo's drawings and in several of his greatest paintings, geologically interesting topographies become commanding backdrops.

            If science—searching out nature's secrets—was a noble cause for Leonardo, so was technology—the building of inventions to make life more efficient, more comfortable, more interesting. The conviction, however, that nature's own inventions were by far the most beautiful ("nothing is lacking, and nothing is superfluous") meant that we should begin by trying to emulate nature's own creations. "If birds can fly, so should humans be able to fly," he wrote, a belief that fueled his life-long interest in creating machines that could carry man aloft. Some of his mental inventions may have been produced, but most remained just theories. Among the designs that fill the notebooks one can see that he prefigured, among other devices the bicycle, the automobile, the tank, the collapsible bridge, the parachute, the underwater diving mask, the flamethrower, and the submarine. To satisfy his scientific curiosity he performed ballistic experiments in order to determine projectile trajectories and painstaking dissections to understand anatomical structure. With seamless facility he functioned as anatomist, botanist, geometer, physicist, architect, mechanical engineer, hydraulic engineer, civil engineer, and even aeronautical engineer. His drawings of anatomical and botanical subjects are especially unmatched for their virtuosity and insight, which in some cases would not be surpassed for two or three centuries.

            Leonardo's drawings reveal a negative slope (top left to bottom right slant) in the shading. Most scholars believe this is evidence that he was left-handed or possibly ambidextrous. He is also famous for his use of mirror text; the right-to-left direction in the handwriting is more natural for a born left-hander; since pulling a pen rather than pushing it avoids smearing the freshly laid ink. Leonardo annotated his sketch of an old man using his characteristic mirror text. The left image is a digitally produced reflection of the original at the right, and reveals the familiar letters of Latin script. The symbols identifying various subdivisions in the right image are also in the mirror text, as clarified in the reflected image on the left (fig. 1.1).

            Leonardo never married, and he may never have had a sexual relationship with a woman; there is no evidence that he had any children. His only interest in women seems to have been as subjects for commissioned portraits—three of which became pivotal masterpieces of Western art. That he may have been a homosexual was long suspected by art historians. Relationships between masters and young apprentices may have been quite common in Leonardo’s day, and on April 9, 1476, Leonardo was denounced anonymously to the Ufficiali di Notte in Florence, the official in charge of public morals, leading to sodomy charges against him. These charges were eventually dropped for lack of evidence, but the toll on Leonardo made the period excruciatingly painful for him. Commissions from wealthy patrons dried up, although this was perhaps also due to his growing reputation for not finishing works on time.

            In 1481 Lorenzo de' Medici (the "Magnificent”), the visionary patron of Florentine artists, received a request from the Pope Sixtus IV to send Florence’s leading painters to collaborate on the murals in the newly erected Sistine Chapel. Lorenzo complied, submitting a team that included Botticelli, Ghirlandaio, Perugino, Piero di Cosimo, and Cosimo Rosselli, but glaringly omitted Leonardo’s name. In July of the same year Leonardo had to settle for painting the Adoration of the Magi for the monastery of San Donato a Scopeto near Florence, which required him "to supply all his materials and complete the commission within 24 or at the most 30 months"4 or else forego the commission. True to form, he became sidetracked, did not complete the work on time, and failed to receive the compensation. But the painting, although unfinished, was so magnificent and revolutionary in its conception that Leonardo’s contemporaries––including his rivals, back from decorating the Sistine walls––stood in front of it transfixed and awestruck.

            By 1482 Leonardo, dispirited from the humiliation of his last few years in Florence, moved to Milan. It was known among artists that its ruler, Duke Ludovico Sforza, wanted to erect a massive bronze equestrian monument in memory of his father. Leonardo wrote to the duke, applying for a position of court engineer, a capacity in which he could design buildings, sanitation systems, portable bridges, and a variety of terrible weapons; only as an afterthought did he mention that he also was an artist.

            Leonardo was hired and moved to Milan, where he would spend the next sixteen years. The Sforzas had launched a program of importing artists and intellectuals to transform the wealthy city from the Lombard hinterland to one that would rival Florence as a cultural capital. Among Leonardo’s contemporaries in Milan were the architect Bramante, who would later design Saint Peter’s Basilica in Rome, and the mathematician Luca Pacioli, whose De divina proportione Leonardo would illustrate.

            Leonardo envisioned the colossal equestrian monument as a possible crowning glory for his artistic endeavors. Once he received Ludovico’s final approval to proceed, he moved quickly to finalize his design from myriad possibilities he had contemplated. He toiled feverishly, sculpting a full-size model out of clay, in preparation for casting the final product in bronze. But as fate would unfold, the French chose just that moment to lay siege to Milan. Ludovico, desperate to stave off the better-armed enemy, had the metals allocated for the bronze horse cast instead into cannons. The Milanese were unable to stop the French, whose troops entered the city with relative impunity. When they came upon Leonardo’s clay model, the Gascon archers among them immediately identified it as an object for target practice. Exceedingly disheartened by the ignominious end to his dream, Leonardo could do nothing. He understood the desperation for survival of the Sforza, their dynasty rapidly drawing to an end. Ludovico, disguised as a Swiss pikeman, tried to flee, but was arrested by the French, who placed him in a dungeon, where he lived out the last a decade of his life. Meanwhile, Leonardo, with his assistant Andrea Salai and his mathematician friend Luca Pacioli, moved first to Mantua, and from there on to Venice for several months. It must have been in this period that the seeds of De divina proportione were sown. The book, published in 1509 in Venice, reveals Leonardo's lifelong preoccupation with geometric shapes and patterns—the informal doodling, finally found its way into print.

            In his Milanese period from 1482 to 1500, Leonardo was at his inventive peak, especially regarding science and engineering. When the French defeated Milan and the residents of the city overthrew Ludovico, he was forced to abandon the city with approximately six hundred florins. He was not paid for the last two years of his work, and the equestrian statue would never be finished. He had produced a few paintings, including his Lady with theErmine (Portrait of Cecilia Gallerani), the second of only three secular portraits of women, and the Last Supper, which Kenneth Clark called "the keystone of European art." In addition, he accumulated countless sketches of his mental inventions, in notebooks or on unbound sheets.

            Leonardo returned for a short time to Florence, where writers, poets, artists and architects received him with deference, but did not accord him the adulation of the younger and exceedingly talented sculptor and painter, Michelangelo. He came very close to taking a court engineer position in Constantinople, similar to his post in Milan, and to painting a portrait of the Ottoman sultan Beyezid II. In his application he contemplated preliminary designs for a bridge over the Golden Horn and a pontoon bridge across the far more expansive Bosporus. As events unfolded, however, the position fell through and Leonardo went to work in Urbino, under the patronage of the city's new leader, Il Principe, Cesare Borgia.

            Leonardo saw in Borgia a potential unifier of the scattered city-states of Italy. Also in Urbino he met and befriended Niccolò Machiavelli, author of The Prince, a book modeled largely on the career of Cesare Borgia. Within five years, however, disillusioned by the unbridled tyranny of the Borgia family, Leonardo was ready to leave Urbino. Through Machiavelli Leonardo received a commission from Florence, where he and Michelangelo were each to paint a mural on facing walls of a hall in the Palazzo Vecchio. The massive mural by Leonardo was to celebrate the Battle of Anghiari, in which Florence defeated Milan in 1440. Although some thoughts began to take shape in his notebooks, the commission was withdrawn. The prospect of a double mural in the same room by the two titans of the Renaissance evaporated as quickly as it had appeared.

            Leonardo had even briefer stays in his next few towns. In 1506, upon an invitation from Milan's French occupiers, he returned to Milan, where for a time Louis XII became his protector and patron, according him the title Painter and Engineer of the King. It was at this time that Leonardo also met two young men who were to have a lasting effect on him. The first was Marcantonio della Torre, the brilliant young professor of anatomy at the University of Pavia, who gave Leonardo some direction for his future anatomical studies, and the second was the aspiring young artist Francesco Melzi, son of Leonardo's landlord, Girolamo Melzi. His relationship with young Melzi, who would become his apprentice and closest disciple, has been described as one of mutual devotion, approaching that of father and son. He was only six years in Milan the second time, for when the French were evicted after their defeat in the Battle of Pavia, a Sforza heir, Maximiliano, took over the city and saw in its future little need for art, learning, or universal genius. Leonardo, accepting a commission from Pope Leo X, went to Rome in 1513. But his preoccupation with producing a new type of varnish instead of a new painting was another personal downfall, and resulted in the pope shifting his focus on the talents of the young Raphael. Soon afterward Leonardo suffered a stroke that left him partially paralyzed in his right arm, effectively ending his career as a painter.

            In 1516 the King of France, Francis I, befriended Leonardo and invited the aging artist and his two assistants to France. Leonardo and his tiny entourage journeyed to Amboise, about sixty miles south of Paris. He brought chests, bulging with manuscripts, his collection of books, the scale drawing for Virgin and Child with Saint Anne, Saint John the Baptist (now thought to be Bacchus), and the one painting with which he would not part, the Mona Lisa, created a decade earlier in Florence. The group moved into the manor house at Cloux, a short walk from the Chateau d'Amboise, the king's own abode.

            Leonardo died three years later, on May 2, 1519. He was in the care of a genuinely devoted king, who rushed down from Paris upon having heard of Leonardo's impending death. A generous benefactor who had asked for little in return, Francis I inherited the Mona Lisa, and Leonardo's manuscripts and books were bequeathed to Melzi.



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