Virgil and Arnaut, for
Adalinda Gasparini

Adalinda Gasparini wrote a fine paper
on the French troubadour Arnaut, for an international Dante conference held in

*Tan m’abellis vostre cortes deman…*

Formazione amorosa come *peregrinatio
& periclitatio*

She quotes the first *sestina* (sixtain) of Arnaut, an
intricate form of a poem invented by himself, and explains the *cledisat* or framework – returning
permutations and shifting rings of rime words – via an elegant algorithm
in form of an Archimedean spiral. Arnaut may well have influenced Dante who
mentions him toward the end of Purgatorio 26, where the troubadour announces
himself in Provençal

*Ieu sui Arnaut* (I am Arnaut)

in perfect symmetry to an earlier
Italian announcement

*Io son Virgilio* (I am Virgil)

at the beginning of Purgatorio 7. The
Divina Commedia has 100 canti:

34 (Inferno) 33
(Purgatorio) 33 (Paradiso)

Virgil and Arnaut mark another
partition of the one hundred canti:

40 20 40

40 canti 1-40 (Inferno 1-34, Purgatorio 1-6)

20 canti 41-60 (Purgatorio 7-26)

40 canti 61-100 (Purgatorio 27-33, Paradiso 1-33)

*Io son Virgilio* seven lines into canto 41

*Ieu sui Arnaut* seven lines before the end of canto 60

The partition 40 20 40 generates
musical proportions:

40 / 20 or 2/1
octave

40 + 20 = 60 60 / 40 or
3/2 quint

40 + 40 = 80 80 / 60 or 4/3
quart

4/3 times 3/2 equals 2/1

quart and quint combined yield an octave

A similar number game leads from the
partition 40 20 40 to the so-called *Sacred
Triangle 3-4-5*

40 + 20 = 60 40 + 40 = 80 40 + 20 + 40 = 100

60-80-100 being a multiple
of

The Sacred Triangle

This triples generate a sequence of ever
rounder polygons of 12 20 28 36 … sides (of two or three different lengths per
polygon, multiples of the square roots of 2 and 5 or 2 times 5 that can be
approximated by additive number columns). The slowly rounding polygons
approximate the circle of radius 5 25 125 625 ... in the ever finer grid 10 by
10, 50 by 50, 250 by 250, 1250 by 1250 …

Let us imagine a circle of radius 50
and diameter 100 in the grid 100 by 100, according to the number of canti in
the Divina Commedia. The crossing horizontal and vertical axes mark the center
of the circle, while their ends provide four points of the circumference. The
Sacred Triangle

Let us imagine an ellipse defined by
the partition 40 20 40 as marked by Virgil and Arnaut

This ellipse has remarkable numbers.
The long horizontal axis measures 100 units and the slightly shorter vertical
one practically 98 units (49 plus 49). The side of the inscribed rhomb, nearly
a square, measures exactly 70 units, and the side of the narrow vertical rhomb confined
by the foci (focal points) held by Virgil and Arnaut measures exactly 50 units.
Finally, the circumference of the round ellipse measures practically 311 units.
From all the ellipses of the long axis 100 only this one has such interesting
numbers, integers and near integers. (In mathematical terms, the ellipse
visualizes a whole number solution of the equation aa minus bb equals bb minus
cc, namely, 70x70 minus 50x50 equals 50x50 minus 10x10.)

The medieval saying *deus est sphaera *means that God is present
in the perfect form of the circle. A sphere on paper, reduced by one dimension,
is a circle, still a perfect form. Let us regard the imaginary circle of
diameter 100 as the circle of divine perfection and knowledge and truth, while
the ellipse given by the partition 40 20 40 may symbolize the Divina Commedia
revolving around Virgil, whom Dante owns his style (Inferno 1:85-87), and
Arnaut who would have provided the seed of inspiration, so to say. The full
circle stands for divine perfection, all embracing knowledge and absolute
truth, whereas the ellipse, a round one almost filling the circle, represents
the best of human work that comes close to perfection and truth, but only in
way of better and better approximations, always leaving a gap that can’t be
bridged, not even by the most perfect work of art and most elegant scientific
theory, a principal gap holding surprises for the future, new ideas and
theories to be found and developed by coming generations. We can only approach
the truth, never really reach it. If you look at a disk from an angle, the
circle is reduced to an ellipse. One dimension of the circle is preserved, the
other reduced by a factor between one (full circle) and zero (line). The word
ellipse goes back to the Greek and means wanting, standing behind; something is
missing, left out, as in the elliptic way of speaking. Our human knowledge is
elliptic: adequate in one dimension, wanting in the other (or another)
dimension.

The ellipse defined by the 100 canti of
the Divina Commedia and the partition 40 20 40 marked by Virgil and Arnaut
conveys a philosophical message that goes along with my cosmological
interpretation of the Divina Commedia. The long poem has 14,233 lines, one line
short of the cosmological number 14,234 in Dante’s model universe. One line is
missing, the line of the divine messenger that would complete the epic, as each
canto closes on a beautiful line of poetic power. Only that the one line to be
delivered by the divine messenger would transcend all human wisdom. It is that
single line we are working on, generation for generation.

December 2, 2012,

Franz Gnaedinger

Postscript. An example of symbolic
geometry in an Italian painting, Baptism of Christ, by Piero della Francesca.
The drawings may speak for themselves