Leonardo and the Last Supper by Ross King

Author:Ross King
Language: eng
Format: epub, mobi
Publisher: Walker Books
Published: 2012-07-15T16:00:00+00:00

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).

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