Leonardo and the Last Supper (32 page)

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.”
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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.
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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.
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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.”
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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, 1.618
²
= 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 appropriate 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.”
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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.”
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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.
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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.”
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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.
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