by Ross King
The sole Roman architectural treatise to survive antiquity, The Ten Books on Architecture was hugely influential in the fifteenth century. Vitruvius’s treatise was an eclectic combination of the philosophical and the practical, describing everything from the phases of the moon to how to construct catapults and battering rams. One chapter is entitled “On the Primordial Substance According to the Physicists”; the next is called “Bricks.” Such a book was bound to appeal to Leonardo. “Enquire at the stationers about Vitruvius,” he wrote in one of his ubiquitous book-hunter notes.20 He must have acquired a copy soon after the first printed edition appeared in 1486.
One chapter of The Ten Books on Architecture is called “On Symmetry: In Temples and the Human Body.” Vitruvius seems to have spent a great deal of time measuring people’s faces and bodies. He saw in the human body a combination of ratios and proportions, of fractions and modules, of subtle interrelationships between the different parts. For instance, the distance from the hairline to the tip of the chin was the same, he claimed, as the length of the hand from the base of the palm to the tip of the middle finger. This measure equaled a tenth of a person’s height, while the distance from the chin to the crown was an eighth. “If we take the height of the face itself,” he continued, “the distance from the bottom of the chin to the under side of the nostrils is one third of it; the nose from the under side of the nostrils to a line between the eyebrows is the same.”21
Because he took nothing on authority, not even on that of Vitruvius, Leonardo began conducting similar experiments of his own. By about 1490 he was systematically measuring the heads, torsos, and limbs of a number of young men, including a pair that he called Trezzo and Caravaggio, after their hometowns in Lombardy. One of these two men was probably the model for Vitruvian Man.22 He was therefore able to come up with an infinite series of refinements and additions to Vitruvius’s measurements, declaring, for example, that the space between the mouth and the base of the nose is one seventh of the face, while the distance from the mouth to the tip of the chin is “a fourth part of the face and equal to the width of the mouth.” Meanwhile the palm of the hand, he discovered, “goes twice into the length of the foot without the toes,” and the distance between the nipples and the top of the head was a quarter of a person’s height.23
What was the point of all these ratios and proportions? For Vitruvius, all temples should be built according to strict proportions, which he defined as “a correspondence among the measures of the members of an entire work.” The best example of proportionality could be found, Vitruvius pointed out, in the human body, because it was “designed by nature.”24 Since the correspondences among the parts of the human body reflected the order of nature, exact bodily proportions were worth studying as the model for how the various parts of Roman temples could harmonize with each other and reflect this same order and beauty.
Fifteenth-century architects such as Alberti and Francesco di Giorgio were mesmerized by this idea of harmonizing architecture with the proportions of the human body. Not coincidentally, Leonardo’s inch-by-inch studies of Trezzo and Caravaggio corresponded closely with his architectural ambitions, such as his design for the domed crossing of Milan’s cathedral. He was also interested in using these measurements, together with perspective, to place painting on a firm scientific footing, though his proportional hairsplitting certainly exceeded the demands of normal artistic practice. He believed proportion was to be found everywhere in nature, even speculating that there must be a discoverable proportional relationship between the circumference of a tree’s trunk and the length of its branches.25
Pacioli, too, concerned himself with proportion, as the full title of his treatise suggests. Besides giving instructions in the conduct of business, the Summa de arithmetica had attempted to apply the laws of mathematics and proportion to art and architecture. Pacioli pursued these studies even more intensively in Milan. Soon after his arrival he began composing a book on which he collaborated with Leonardo, who provided the illustrations at the same time as he worked on The Last Supper. Pacioli’s book was to be called De divina proportione (On Divine Proportion). There was certainly much in its pages to stimulate Leonardo. Pacioli was interested not merely in measurements such as the distance between the lips and the chin; he concerned himself with nothing less than the proportions of God and the universe.26
The collaboration between the two men was evidently a happy one. Pacioli admired Leonardo’s artistic genius every bit as much as Leonardo admired the friar’s facility with mathematics and geometry. Pacioli called Leonardo “the prince among all human beings,” and he later remembered their collaboration on On Divine Proportion with much nostalgia, writing of “that happy time when we were together in the most admirable city of Milan.”27 Leonardo had little or no input into the content of the treatise, but he was asked by Pacioli to contribute drawings of sixty polyhedra.
Composed in the course of a year or two following his arrival in Milan, Pacioli’s On Divine Proportion is yet another of his fat, mind-numbing disquisitions, this time on geometry and proportion rather than accounting. It is composed in poor Italian and brings to mind the comment supposedly made by Samuel Johnson about a manuscript being both good and original: “But the part that is good is not original, and the part that is original is not good.” He drew freely on the mathematical thought of Plato, Euclid, and Leonardo da Pisa (known to later centuries as Fibonacci). He also took from his old teacher Piero della Francesca—so liberally, in fact, that he was accused of plagiarizing Piero’s De quinque corporibis regularibus.
On Divine Proportion concerned itself with one proportion in particular. In Book 6 of The Elements, Euclid had demonstrated how to divide a line so the ratio of the shorter section to the longer one equaled that of the longer section to the line’s entire length. Euclid called this process dividing a line “in mean and extreme ratio,” and the ratio was expressed in an irrational number that begins 1.61803 and continues to infinity. This ratio would manifest itself in a wide number of phenomena, including in the ascending series of numbers described by Leonardo da Pisa (whose work Pacioli knew well) and now famously known as the “Fibonacci sequence”: 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, and so forth. In this series, each number is the sum of the previous two; moreover, after the first few in the series, each number divided by the previous one yields a ratio that approximates (but only approximates) 1.61803. For example, 21 ÷ 13 = 1.61538.
Pacioli christened dividing a line “in mean and extreme ratio” (as it was known for many centuries) with a much more evocative name: he called it “divine proportion.” For Pacioli the proportion was divine because of its intimate connection to (and here the Franciscan emerges) the nature of God. The mathematical properties of this ratio—the fact that, for example, 1.6182 = 2.618—he regarded as divine rather than coincidental. Among the arguments he advanced to prove his point is that both God and divine proportion are irrational, by which he meant they are both beyond reason and not expressible as the ratio of two integers. Pacioli’s number crunching had clearly moved well beyond bookkeeping and progressed into the lofty realms of ontology.
Besides Euclid, Pacioli also borrowed heavily from Plato’s Timaeus, a work that dealt with such weighty matters as the origin of time, the sun, and the “soul of the world.” Pacioli was attracted in particular to the section dealing with polyhedra. It is difficult to overestimate the importance of these polyhedra for Plato. They were not just geometrical fancies: they formed, he believed, the building blocks of the physical world. In his description of the universe, the four elements (earth, air, fire, and water) are solid bodies expressed by four distinct polyhedra: respectively, the cube, the octahedron (diamond), the tetrahedron (pyramid), and the icosahedron (soccer ball). Plato was extremely vague about his reasoning but stated that the tetrahedron, made from four equilateral triangles, was “the substance of fire” (presumably because of its flame-like shape), while the globular icosahedron made it the polyhedron appropriat
e for water. Such analogies may seem odd to us, but in some respects they were the ancient Greek equivalents of the ball-and-stick molecular models used in, for example, Watson and Crick’s double helix.
In fact, Plato had his own equivalent of a “molecule of life”: a fifth polyhedra, the dodecahedron. To the dodecahedron he assigned a particularly vital role. Composed of twelve interlocking pentagons, this figure had the cosmological function of encompassing and structuring all the others: a kind of geometrical Higgs boson. “God used it for the universe,” Plato asserted with no undue explanation, “in embellishing it with designs.”28
Significantly for Pacioli, this bit of cosmic origami could only be constructed by means of divine proportion, which inhered in the relationship between the sides of the pentagons and their diagonals. Moreover, the dodecahedron encompasses the other four bodies—quite literally, because they can fit inside it (sometimes simultaneously, as in the case of the tetrahedron and the cube). The dodecahedron was therefore a perfect metaphor for the all-encompassing quintessence, though for Pacioli it was, of course, more than a metaphor: the dodecahedron partook of the divine itself. b Divine proportion as described by Pacioli was an attempt to offer a Christianized version of Plato’s account of the Demiurge creating a dodecahedron-shaped universe. It was, in many respects, a geometer’s or an accountant’s attempt to prove the existence of God.
Leonardo’s task of illustrating these geometric figures was not an easy one. Among the sixty drawings he needed to provide was the rhombicuboctahedron, whose twenty-six sides (involving eighteen squares and eight equilateral triangles) must have involved him in considerable mental funambulism. However, Leonardo clearly relished the job. One of his notebooks has careful drawings of the five basic solids accompanied by a rhyme: “The sweet fruit, so attractive and refined / Have already drawn philosophers to seek / Our origins, to nourish the mind.”29 For the manuscripts he produced drawings of these “sweet fruit,” done in ink touched up with watercolor, showing the bodies both in solid form and—in masterpieces of perspective and three-dimensional geometry—in see-through skeletal form.
Leonardo’s dodecahedron
Pacioli was evidently delighted with Leonardo’s results. He later wrote that the artist had created “supreme and very graceful figures,” ones that (in the familiar comparison) not even ancient artists such as Apelles, Myron, and Polykleitos could have surpassed.30
Leonardo’s participation in this project and his friendship and close collaboration with Pacioli raises the question of whether he used divine proportion in The Last Supper. After all, Pacioli recommended his treatise to “clear-sighted and inquiring human minds,” and he promised that anyone who studied “philosophy, perspective, painting, sculpture, architecture, music and other mathematical disciplines” would find in his work “a very delicate, subtle and admirable teaching and will delight in diverse questions touching upon a very secret science.”31 Did Leonardo, then, make any use of Pacioli’s “secret science”?
Since the middle of the nineteenth century, numerous claims have been advanced about the artistic application of divine proportion, or what has become known variously as the “golden section,” the “golden ratio” or—in honor of Phidias, who supposedly employed it in the construction of the Parthenon—the Greek letter phi (ϕ). (In fact, the architect of the Parthenon was not Phidias: he was in charge, rather, of its decorative scheme. Early sources variously credit Iktinos, Kallikrates, and Karpion.)
The golden section is found without question in nature: in pineapples, sunflowers, mollusk shells, and the spiral shape of galaxies such as the Milky Way. Various writers have claimed the golden section can also be found in everything from the Egyptian pyramids to Greek vases and Gregorian chants. But the golden section is a modern obsession. The name was invented only in the nineteenth century. The majority of the claims for art and architecture do not withstand close scrutiny, and a growing literature has comprehensively debunked most of these assertions. The dimensions of the Parthenon, for example, by no means readily support the widespread theory that Phidias (or, rather, Iktinos, Kallikrates, and Karpion) knew or used divine proportion. Theories about the pyramids are difficult to prove because they are based on modern mathematical systems rather than the ones used by the ancient Egyptians.32
Pacioli’s arrival in Milan served to stimulate even more Leonardo’s keen interest in mathematics and geometry. He began filling his notebooks with multiplications and square roots. He praised the “supreme certainty of mathematics.”33 Many hours were spent taking numbers to the power of three or four, dividing and subdividing geometrical figures, and involving himself in the age-old problem of “finding a square equal to a circle.”34 So intense did these preoccupations become that they, rather than some catastrophic mishap or disconsolate realization about the impossibility of his designs, may have been responsible for temporarily curtailing Leonardo’s interest in flying machines. Coincidentally or not, Leonardo’s ambitions for flight began to languish the moment Pacioli arrived in Milan.
Yet the evidence for Leonardo’s interest in or use of divine proportion is scanty indeed. Nowhere in his notebooks did he mention or describe divine proportion: the concept can be found neither in his numerous comments on proportion nor in his even more voluminous notes on mathematics and geometry. Indeed, the complete absence of any reference to divine proportion is one of the surprises of Leonardo’s writings. If he believed divine proportion was indeed the key to beautiful design, something that had a universal applicability, why did he not mention it in his lengthy discussion of proportion in his treatise on painting?
Nor can Leonardo’s paintings easily be adduced as evidence for experimentation with divine proportion. Geometric shapes can certainly be found in his paintings, as in the case of the equilateral triangle formed by Christ’s head and arms in The Last Supper. Moreover, rectangles created via the golden section may be imposed on some of the various faces in his paintings. The superimposition is, however, usually arbitrary and unconvincing. For example, a painting sometimes offered as proof is Leonardo’s unfinished St. Jerome Praying in the Wilderness, painted sometime in the 1480s. According to David Bergamini’s Mathematics, published in Time-Life’s wonderful “Science Library” series, the golden rectangle “fits so neatly around St. Jerome that some experts believe Leonardo purposely painted the figure to conform to these proportions.”35 However, the visual evidence suggests, on the contrary, that the “experts” have purposely arranged the rectangle to conform (albeit imperfectly) to the figure of St. Jerome, whose arm extends well beyond this rectangular confinement and whose head is inconveniently below the upper perimeter. The theory is further troubled by the fact that Leonardo painted St. Jerome Praying in the Wilderness a decade before he met Pacioli and learned about divine proportion. The exercise recalls the efforts of the mathematician who, in the course of some doubtlessly pleasing research conducted during the 1940s, claimed to have discovered the golden section in the chest and waist measurements of Hollywood star Veronica Lake.36
The reality is that the brilliance and appeal of Leonardo’s paintings have little to do with rectangles or measuring sticks and everything to do with astounding powers of observation and unsurpassed understanding of light, movement, and anatomy. St. Jerome Praying in the Wilderness takes its power from the profound contrast between the dazed and sinewy old hermit—in whom we see Leonardo’s fascination with the structural members of the human body—and the lithe torsion of the lion curled on the ground before him. This beast was no doubt drawn from life, modeled by one of the captive lions in the enclosure behind Florence’s Palazzo Vecchio, literally a few yards from his father’s house. Leonardo may even have dissected one of these lions at some point: “I have seen in the lion tribe,” one of his notes reads, “that the sense of smell is connected with part of the substance of the brain which comes down the nostrils, which form a spacious receptacle for the sense of smell.”37 That kind of curiosity and dedication—a willingne
ss to study the fierce creatures padding around their pens and then to peer inside their sectioned skulls—was the secret of Leonardo’s artistic genius, not a dedication to drawing rectangles.
Leonardo’s St. Jerome Praying in the Wilderness with rectangle imposed, exploring the supposed relationship of the painting to the golden section
One reason why Leonardo did not use divine proportion in his works (or advocate it in his writings) is that Pacioli himself did not actually promote it as a design template for painters and architects. That is, Pacioli did not champion divine proportion as the key to perfect design: indeed, the thought never seems to have occurred to him. He advocated instead the Vitruvian system: one based not on the irrational mathematical constant 1.61803 but rather on simple ratios (Vitruvius believed 10 to be the “perfect number”).38
Leonardo trusted empirical study far more than abstract concepts. He was more inclined to believe his own eyes and a set of calipers than one of Plato’s pronouncements on the shape or proportions of the universe. Like Niccolò Machiavelli, he was interested in “things as they are in a real truth, rather than as they are imagined.”39 He could have known from his measurements of Trezzo and Caravaggio that one widespread modern notion about the golden section—that if you divide your height by the distance from your navel to the floor you get 1.61803—was demonstrably incorrect. Indeed, the height versus navel measurements of his Vitruvian Man (sometimes held out as bodily evidence of the golden section) yield a ratio of approximately 1.512, a good deal short of this universal ideal—and a perfect example of the sort of fudging required by so many theories about the artistic use of the golden section. Leonardo’s more empirical approach to human proportion led him to state ratios in other terms, using round numbers. Typical of his method was his claim that a man’s height equals eight times the length of his foot, or nine times the distance between his chin and the top of his forehead.40 Or, as we have seen in the case of The Last Supper, he was interested in the application of tonal intervals to painting and architecture, using ratios such as 1:2, 2:3, and 3:4 to organize the architectural components of the room where Christ and the apostles sit.
