Isosceles triangle

Isosceles triangle

In Brief

Ruthlessly mocking Bloom's romantic stereotype of Spaniards, Italians, and other Mediterranean peoples as hotblooded types, Stephen lobs some counter-examples in Eumaeus: "we have the impetuosity of Dante and the isosceles triangle, Miss Portinari, he fell in love with and Leonardo and san Tommaso Mastino." The Trinitarian numerology of Dante Alighieri, the serene mathematics of Leonardo da Vinci, and the massive logic of Thomas Aquinas all would suggest that an Italian can be just as cerebral as the next guy.

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Stephen's sardonic picture of Dante falling hotly in love with a triangle refers to the poet's unprecedented and unrepeated abstraction of the experience of love. Dante reconciled the adored lady of troubadour erotic tradition with Christian theology at a high cost: he made her almost entirely a vessel of insight into the disembodied truths of divinity. This transformation is realized most fully in the Paradiso (finished not long before his death in 1321), but the groundwork for it was laid in the Vita Nuova (1295). The relentless Trinitarian echoes in the Divine Comedy have their origin in the obsessive numerology of this early work, where the spiritual exaltation of encountering something utterly new (nuova or nova) in Beatrice overlaps constantly with the number nine (nove).

The remarkable canzone in section 19 of the Vita Nuova presents Beatrice (the "Miss Portinari" that Boccaccio identified as the object of Dante's adoration) as a divinely created miracle ("God intended to make something utterly new") who can lead the mind to heaven. When Beatrice's death is announced in section 28, the reader receives, not an outflowing of emotion, but "three reasons" for not talking about the loss in any detail, and a disquisition on a related number: "since the number nine has appeared many times in what I have already written (which clearly could not happen without a reason), and since in her departure this number seemed to play an important part, it is fitting that I say something here concerning this, inasmuch as it seems to fit in with my plan" (trans. Mark Musa).

Section 29 explains: "Let me begin by saying that if one counts in the Arabian way, her most noble soul departed this life during the first hour of the ninth day of the month, and if one counts the way they do in Syria, she departed in the ninth month of the year, the first month there being Tixryn the First, which for us is October. And according to our own way of reckoning, she departed in that year of our Christian era (that is in the year of Our Lord) in which the perfect number [i.e., ten] had been completed nine times in that century in which she had been placed in this world: she was a Christian of the Thirteenth Century. One reason why this number was in such harmony with her might be this: since, according to Ptolemy and according to Christian truth, there are nine heavens that move, and since, according to widespread astrological opinion, these heavens affect the earth below according to the relations they have to one another, this number was in harmony with her to make it understood that at her birth all nine of the moving heavens were in perfect relationship to one another. But this is just one reason. If anyone thinks more subtly and according to infallible truth, it will be clear that this number was she herself—that is, by analogy. What I mean to say is this: the number three is the root of nine, for, without any other number, multiplied by itself, it gives nine: it is quite clear that three times three is nine. Therefore, if three is the sole factor of nine, and the sole factor of miracles is three, that is, Father, Son, and Holy Spirit, who are Three in One, then this lady was accompanied by the number nine so that it might be understood that she was a nine, or a miracle, whose root, namely that of the miracle, is the miraculous Trinity itself. Perhaps someone more subtle than I could find a still more subtle explanation, but this is the one which I see and which pleases me the most" (trans. Musa).

Stephen's absurd image of Beatrice as a geometrical figure makes sense in the context of this bizarre passage. An "isosceles" triangle is one with two or more equal sides––a category which would include the perfectly balanced equilateral triangle.

Leonardo da Vinci, another Florentine with an interest in divine order, pairs well with Dante's architectonic rationality. In addition to his well-known labors in natural science and engineering, Leonardo indulged a keen interest in geometric shapes and numerical ratios. His Last Supper and Mona Lisa have been shown to make use of antiquity's "golden ratio" of approximately 1.6:1 (based on the irrational number phi which begins 1.6180339887). Leonardo explored the use of linear perspective in paintings like The Adoration of the Magi and was probably the first European artist to employ the technique of anamorphosis. Perhaps most relevant to Stephen's conception of Dante, he also embraced the classical idea that man is the measure of all things, a quintessential creation whose proportions reveal the divine mathematics underlying all of reality.

In the notebook drawing he called Vitruvian Man, or "The proportions of the human body according to Vitruvius," Leonardo illustrated the Roman builder's belief that ideal human proportions should inform works of architecture. In the text above and below the drawing he articulates many of Vitruvius' observations. Among them: the length of a man's outstretched arms are equal to his height; the length of the face from chin to hairline is one tenth of the total height; the length of the open hand from the wrist to the tip of the middle finger is the same; the maximum width of the shoulders is a quarter of the height; the distance from the elbow to the tip of the fingers is a quarter of the height; the base of the penis lies at one half of the height; the distance from the bottom of the feet to the bottom of the knee is one quarter of the height; and "if you open your legs enough that your head is lowered by one-fourteenth of your height and raise your hands enough that your extended fingers touch the line of the top of your head, know that the center of the extended limbs will be the navel, and the space between the legs will be [aha!] an equilateral triangle."

About Thomas Aquinas ("san Tommaso Mastino" means "Saint Thomas the Mastiff," or "The bulldog of Aquin" as Stephen calls him in Scylla and Charybdis), little commentary is needed. The Summa Theologica and Summa Contra Gentiles are among the most hyper-rational texts ever produced. Simply reading their tables of contents yields the impression that all of creation can be comprehended, divided, subdivided, pigeonholed, articulated, clarified, and harmonized by human logic. Thomas would have been associated with Dante in Stephen's thoughts because so many ideas of the Commedia were taken from his writings.

JH 2018
William Blake's etching of Beatrice and Dante before the three beasts that block his ascent toward the sun in Inferno canto 1, most likely an allegory for the three categories of sin that he will face in Hell. Source:
Beatrice and Dante before the three Persons of the Holy Trinity, in a 15th century manuscript by Giovanni di Paolo held in the British Library. Source: Wikimedia Commons.
The nine celestial spheres of Ptolemaic cosmology which Dante maps onto the nine orders of angels contemplating God in the Empyrean, in figure 5 in the 1986 Penguin Paradiso translated by Mark Musa.
Vitruvian Man, ca. 1490, pen and ink with wash on paper, from a notebook of Leonardo da Vinci, held in the Gallerie dell'Accademia, Venice, photographed by Luc Viatour in 2007. Source: Wikimedia Commons.