The measure of Dante’s Universe

At the end of the twenty-seventh canto of Paradiso (Divine Comedy), Dante Alighieri describes his ascension towards the ninth sphere of the Ptolemaic model of the universe: the Primum Movens, which contains and gives motion to all the previous skies (Moon, Mercury, Venus, Sun, Mars, Jupiter, Saturn, and Fixed Stars), that rotate around the center of the Universe, the Earth (Figure 1, left). Beatrice, the woman who leads the poet in his itinerarium in deum, explains that the Universe has a fixed center, the Earth, surrounded by eight rotating spheres or skies with an increasing radius, and all these physical elements are included in a ninth sphere, the Primum Movens, from which the movement arises; and this last sphere is not contained by other skies and has no physical place: it exists in the mind of God. The following are the verses in which Beatrice explains the mentioned architecture of the Ptolemaic system. You can find translations to English, for sure, but if you think that you can read Dante in a translation, you are deluding yourself.

Ma ella che vedea il mio disire,
incominciò, ridendo tanto lieta, 
che Dio parea nel suo volto gioire:
"La natura del mondo, che quieta
il mezzo e tutto l'altro intorno move,
quinci comincia come da sua meta.
E questo cielo non ha altro dove
che la mente divina, in che s'accende
l'amor che 'l volge e la virtù ch'ei piove.
Luce ed amor d'un cerchio lui comprende
si come questo li altri; e quel precinto
colui che 'l cinge solamente intende."

Paradiso, XXVII, vv 103-114

At the beginning of canto XXVIII, Dante can now see what is beyond the Primum Movens, a reality called Empyreal, which has a structure that is symmetric to the one of the Ptolemaic universe, but this time God is the center and nine spheres or skies rotate around him (Figure 1, right; Figure 2). The union of these two points, the Erath and God, and of the two sets of nine spheres that include them, is the complete sum of all that exists, according to Dante: of what has been created and of its Creator. But what is its shape or geometric nature? If we imagine Dante as a bi-dimensional being and if we subtract a spatial dimension from all his surroundings, we can see the spheres as circles and assemble them as the parallels of a sphere of which the Earth is one pole and God is the other. Dante moves on the surface of this sphere (a space with two dimensions) and the circles are the intersection of the sphere itself with a family of parallel planes. If we now add again the third dimension to this model, we must conclude that the universe is a four-dimensional sphere and that the skies are its intersections with a family of three-dimensional spaces.

Figure 1. The Ptolemaic system, on the left. On the right, the structure of the Empyreal. From the paper by Peterson mentioned in below.

I found the idea of the universe of Dante as a hypersphere of a space with four dimensions in a 1979 paper by the physicist Mark A. Peterson, “Dante and the 3-sphere” (link). That said, it is possible to calculate the space (in four dimensions) occupied by such a universe. If we introduce a Euclidean space of dimension 4, we can calculate the volume of the sphere of radius R and center in the origin of the coordinate system, by solving the integral:

$V(4,R)=2^4\int_{0}^{R}\int_{0}^{\sqrt{R^2-x_4^2}}\int_{0}^{\sqrt{R^2-\left(x_3^2+x_4^2\right)}}\int_{0}^{\sqrt{R^2-\left(x_2^2+x_3^2+x_4^2\right)}}dx_1dx_2dx_3dx_4$

With a few passages, we get

$V(4,R)=4\pi\left(\pi\frac{R^4}{4}-\frac{1}{3}\int_{0}^{R}\left(R^2-x_4^2\right)^\frac{3}{2}dx_4-\int_{0}^{R}x_4^2 \sqrt{R^2-x_4^2}dx_4\right)$

The two integrals inside the brackets can be easily solved by integration by series (use the binomial expansion) and we get that

$V(4,R)= \frac{\pi^2}{2}R^4$

What is the value of R? It should be the radius of the largest sphere, the equatorial one: the Primum Movens. But it has no radius, to my knowledge, it is not material. We can use the radius of the sphere of the fixed stars, though, that according to Ptolomeus (Almagest, book I, chapter V) is about 20 thousand times the radius of the Earth, $r_T = 6,371 km$ . Therefore, we conclude that the volume of the four-dimensional hypersphere that includes the Universe, according to Dante Alighieri, is $8\pi^210^{16}r_T^4$ , which is about $1.3\cdot10^{45}m^4$ . It is not clear what the meaning of this analysis might be, if any: Dante did not know geometry with more than three dimensions, to my knowledge. But one implication is that a three-dimensional being can not move from one sky of the Ptolemaic system to the other (because they are in different three-dimensional spaces) unless he finds a way to move along the fourth dimension. I think that this would resonate with Dante’s mind and that this would agree with what he had envisioned: he can accomplish his ascent to God only because of external intervention and help from Beatrice.