Dante's poem is a masterwork in medieval numerology, with profoundly deep intellectual and structural similarities related to where a particular sin or virtue is addressed. The overall theme of the poem involves becoming one with God, or one with the Trinity, or three in one. The poem thus uses multiples of three and one in significant ways.
As others have noted, Dante's masterpiece contains three different epics of the after world—Inferno, Purgatorio, and Paradiso, comprising one complete poem. The number three mystically represents the Christian trinity, yet one also represents the unity of God. Each individual poem contain thirty-three cantos, with one prologue. Christ was thirty-three when he died, and this was considered the perfect age, or the age one would be in the afterlife (don't overthink that). One hundred was a perfect number (10 x 10), so his trilogy is complete. Cantos that are multiples of ten often have instructional matter that relates to understanding the organization of these worlds. Within the epics, cantos corresponding to a multiple of three seem especially relevant, and the cantos have a correspondence in other epics (lust vs proper love, for instance). One can therefore read the entire work and see the fullness of Dante's understanding of human metaphysical thought.
The terza rima rhyme scheme (aba bcb cdc...) maps itself out across tercets. The interlocking rhymes create the sensation of footsteps walking through the cantos, much like Dante the pilgrim journeys through the after life. This rhyme scheme propels one until the last line of each canto, at which point one is invited to pause and make sure one understands the wisdom contained in the canto.
The Holy Trinity—God the Father, the Son, and the Holy Spirit—is emphasized again and again in the structure and poetic form of the Divine Comedy. The poem has three books, one each for Hell, Purgatory and Heaven, and each of these realms has nine (that is 3x3) circles, so the total number of areas described is 27 (3x3x3). There are three types of sinners in Hell, each occupying three circles. The verse form is terza-rima, a form invented by Dante specifically for this poem, in which three-line stanzas (tercets) are interwoven using the rhyme scheme aba bcb cdc etc.
In the deepest pit of Hell, Satan has three faces, mirroring the Triune God in Heaven and showing the symmetry and order of the universe (Hell, and even Satan himself, is under God's control and therefore reflects his orderly nature). There are numerous other threes—the three creatures which bar Dante's path to paradise, the triple-headed Cerberus who guards the gates of Hell—and they are often mirrored by a similar group of three in the other books.
The number three, which was thought to have divine significance, structures The Divine Comedy in a repeated way. Three beasts—the lion, the leopard, and the she-wolf—block Dante's path to paradise. The dog Cerberus has three heads. There are three kinds of sin in hell—lack of self control, violent behavior, and lying or fraud—and each type occupies three levels of hell. Further, Satan himself has three faces in a grotesque parody of the trinity. After life, there are three states of being: torment, preparation for heaven, and heaven.
Paradise is set up in threes, with nine levels, just like hell, and the poem itself is written in terza rima, three-line stanzas. All of this reinforces a main theme of the poem, which is that the universe is orderly and completely under divine control. People can try to rebel against God, but in the end everyone falls into his proper place in a carefully conceived cosmos in which justice is enacted.
The sequence of the poem, which has Dante first going through hell, then purgatory, and only then ascending up the mountain to heaven, reinforces another theme: that one needs to be humbled and aware of one's sin—and sin in general—to be able to truly perceive and receive the gifts of paradise. Dante has lessons to learn before he can be reunited with Beatrice and approach what is truly holy.
Number symbolism is widely used by Dante in The Divine Comedy, both in terms of the poem's structure and content. The number three is particularly important in this regard. The poem is written in terza rima, an Italian form of poetry written in stanzas consisting of three lines. In regard to the content of the poem, it's notable that Hell is presented as having nine circles (three times three), and at the very lowest level, Satan has three faces. At the very beginning of the poem, when Dante finds himself lost in a dark wood, he's scared by three fearsome beasts: a lion, a leopard, and a wolf. And of course The Divine Comedy itself consists of three books: Inferno, Purgatorio, and Paradiso.
There are numerous other examples of such number symbolism throughout the poem. But why was Dante so seemingly obsessed with the number three? The answer lies in the importance of the number three in the Christian tradition. Three is the number of the Holy Trinity, the three persons of the Godhead according to orthodox teaching: Father, Son, and Holy Spirit. For Dante as with most of his contemporaries, it was the divine love of the supreme Godhead that makes everything happen, that sustains the entire cosmos, that in the final, unforgettable words of the poem, "moves the sun and the other stars."
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