Chaucer from Prentice to Poet: The Metaphor of Love in Dream Visions and Troilus and Criseyde

Appendix B: The Golden Proportion

Appendix B The Golden Proportion The proportion known successively as the cut (in the ancient world), the divine proportion (Middle Ages), the golden section (after Kepler), and phi (twentieth century) was probably discovered ca. 400 b.c. The word proportion signifies an equivalence of two ratios, a ÷ b = x ÷ y. Though “division into extreme and mean ratio,” as phi is defined, indicates a specific point on a line, creating two segments that are in a golden ratio, neither of these two segments can be precisely measured. Nor can their ratio to each other be precisely shown with whole numbers (that is, shown with numbers that are integers unto themselves, like 4/5 or 11/25, both of which resolve in terminating decimals, 0.8 and 0.44, respectively). Phi is a nonterminating decimal, called by mathematicians “irrational,” meaning not amenable to whole-number ratio . It is commonly indicated to the fifth decimal place, 1.61803 . . . , with the three dots signifying an infinite number of decimal places. Phi can, however, be easily plotted geometrically. In figure B.1, point C, constructed from the diagonal across two equal squares, is the golden cut of line AB. The algebraic equation for ϕ, discovered long after it was identified by Pythagorean geometry, depends on the square root of 5, another irrational number whose decimal equivalent is 2.23606. Since fourteenth-century mathematicians in Europe were incapable of working with irrational numbers , they normally substituted the rational convergent 20/9 (2.22222 . . . in modern mathematics) for the square root of 5. When a line is divided geometrically at its precise golden cut, and only then, a remarkable relationship exists among its three magnitudes—the shorter segment, the longer segment, and the whole line. The magnitude of the shorter segment, multiplied by ϕ, produces the magnitude of the longer segment, and this longer segment, multiplied by ϕ, produces the length of the whole line. It follows that if the original whole line is extended by a length equal to the longer segment, this extension becomes the shorter segment of a new phi relationship in which the original whole line is now the longer segment. In turn this longer segment can also be folded out to create yet a third demonstration. And so on to the threshold of infinity. For a proportion to be called phi, it must satisfy two equations. The first is an additive equation: a + b = c. The second requires equality between two ratios, for which a simple equation from Euclid is the most convenient demonstration . The Middle Ages learned its Euclid from Nicomachus of Gerasa, whose tenth mean provides the equation that proves a divine proportion among three magnitudes a, b, and c (where a > b and b > c): a - c b —— = — a - b c In every age since the divine proportion was discovered, mathematicians have been captivated by both its elegance and its occurrence everywhere in nature. Phi is the only ratio whose square is itself plus 1 and whose reciprocal is itself minus 1. It governs the fifth Platonic solid, a dodecahedron, thought to be the first manifestation of matter when the Creator converted virtue into essence.1 The Parthenon was apparently constructed according to this proportion, as were the sculptures of Phidias, the first letter of whose name gave Mark Barr the thought in the early twentieth century to use ϕ as its mathematical symbol. The two parts of Horace’s Ars poetica (lines 1–294, discussing the craft of poetry, and lines 295–476, on the interaction between poet and critic) are separated from each other at the poem’s precise golden cut (Le Grelle 1949, 142–56; Duckworth 1962, 76–77). In this instance, mathematics underscores Horace’s thematic point, that the craft of poetry, 1 / Appendix B Figure B.1. Geometric construction of the divine porportion the middle term, is the means by which the lesser extreme, poets and critics, achieve the greater extreme, literature. Virgil’s Georgics I, said by the editor of the Loeb edition to be “perhaps the most carefully finished production of Roman literature” (Fairclough 1978, x), divides its poetic lines into five sections: 37.5 161 55 204.5 51.5 The poet’s aim was evidently to demonstrate mathematically the natural harmony of the universe. The line totals 161 and 204.5 leap out from sections 2 and 4, which discuss technical farming matters, as the total number of days in a year...