Poetical cosmology in Dante Alighieri’s Divina Commedia / © 1999-2003 by Franz Gnaedinger, Zurich, fg(a)seshat.ch, fgn(a)bluemail.ch / www.seshat.ch


Dante-Referat Vilnius






Poetical Cosmology in Dante Alighieri's Divina Commedia



Professor Wilhelm Pötters from the University of Würzburg has a fascinating thesis: that it was the medieval DEUS EST SPHAERA which inspired Giacomo da Lentini's invention of the sonnet, Francesco Petrarca's Canzoniere and Dante Alighieri's Divina Commedia.


Imagine a circle whose diameter measures 14 units. Using the value 22/7 for π, the area of the circle can be transposed into a rectangle that measures 14 by 11 units. Wilhelm Pötters believes that this transformation of a circle into a rectangle of practically the same area was the origin of the sonnet, which comprises 14 lines of 11 spoken syllables (endecasillabi).


Who was Francesco Petrarca's Laura? By examining both Laura's attributes and the metric of the Canzoniere, Wilhelm Pötters came to a surprising conclusion:


     LAURA, an unreachable woman - laverta - Donna Veritas - the inaccessable truth, symbolized by the number π


Wilhelm Pötters found similar (though more complex) ideas in Dante Alighieri's Divina Commedia including these (and a few more) key numbers: 3 and 9 (totality); 10 (perfect number), 10, 100, 1000, 10000, 100000 (powers of 10); 7 or 14 and 11 (numbers of the sonnet); un cinquecento diece e cinque = 515 (divine messenger); 61 (according to Gerhard Goebel the number of the 60 virgins plus Beatrice); and finally 1510010500000 (un cinque cento diece cinque x 100,000). Wilhelm Pötters multiplies 515 by 61 and obtains 31415. Then he divides the product by a power of 10 and obtains an excellent value for π, namely 31415/10000 = 3.1415.


There are 14,233 lines in the Divina Commedia. According to Pötters, this number can be seen as the diameter of the cosmos, while the number 1,510,010,500,000 represents the volume of a sphere enveloping the universe. The corresponding diameter measures 14,234.0050... units, a little more than the number of lines, this indicating that the divine messenger comes from a zone beyond the visible universe, namely from the Empireo.


How could Dante have deduced the value 3.1415 for π? He probably used an ancient method which I reconstructed from Egyptian and Babylonian mathematics. It may have survived in northern Africa and been brought to Europe by Leonardo Fibonacci. If we know that π is smaller than 4 but a little greater than 3 we can draw up this number sequence:


  4  (plus 3)  7  10  13  16  19  22  25

  1  (plus 1)  2   3   4   5   6   7   8


If you know that π is more than 3 but a little less than 22/7 you will obtain this number sequence:


  3  (plus 22)  25  47  69  91  113  135  157  179  201

  1  (plus  7)   8  15  22  29   36   43   50   57   64


  223  245  267  289  311  333  355  377

   71   78   85   92   99  106  113  120


If you know that π is less than 315/100 but a little more than 311/99 you can start another π-sequence by using this generator:


  63  (plus 311)   5 x 63 (plus 5 x 311)   315 (plus 1555)

  20  (plus  99)   5 x 20 (plus 5 x  99)   100 (plus  495)


  315  (plus 1555)  1870  3425  4980  6535  8090  9645

  100  (plus  495)   595  1090  1585  2080  2575  3070


  11200  12755  14310  15865  17420  18975  20530  22085 

   3565   4060   4555   5050   5545   6040   6535   7030


  23640  25195  26750  28305  29860  31415

   7525   8020   8515   9010   9505  10000


The numbers of the above generator play an important role in Dante's poetical cosmology. His model is basically a Ptolemaic one, consisting of 9 + 1 = 10 heavenly spheres and making use of cosmic miles, a poetical mile, and a common mile. Here are the cosmic miles:


    small cosmic mile   =  1 a

    medium cosmic mile  =  2 a

    large cosmic mile   =  3 a


  EARTH, center of the universe


    diameter                 99 medium cosmic miles

    circumference           311 medium cosmic miles


  MOON, heavenly sphere I


    radius of sphere I     7117 small cosmic miles

    diameter moon            63 small cosmic miles

    circumference           198 small cosmic miles


    (the moon would appear in the correct angle / size)


  MERCURY, heavenly sphere II


    radius of sphere II   10065 small cosmic miles


  VENUS, heavenly sphere III


    radius of sphere III  12327 small cosmic miles


  SUN, heavenly sphere IV


    radius of sphere IV    7117 medium cosmic miles

    diameter sun             63 medium cosmic miles

    circumference           198 medium cosmic miles


    (the sun would appear in the correct angle / size)


  MARS, heavenly sphere V


    radius of sphere V    15914 small cosmic miles


  JUPITER, heavenly sphere VI


    radius of sphere VI   17433 small cosmic miles


  SATURN, heavenly sphere VII


    radius of sphere VII  18830 small cosmic miles


  FIXED STARS, heavenly sphere VIII


    radius of sphere VIII 20130 small cosmic miles


  PRIMUM MOBILE, an invisible sun, heavenly sphere IX


    radius of sphere IX    7117 large cosmic miles

    diameter of PM           63 large cosmic mile

    circumference           198 large cosmic miles


  EMPIREO, beyond divine sphere X, home of the divine messenger


    radius of sphere X    22506 small cosmic miles


The surfaces of the spheres remain in the following chain of proportions:


     1 : 2 : 3 : 4 : 5 : 6 : 7 : 8 : 9 : 10


All rounding errors are small or even tiny. The diameters of sphere I (moon), IV (sun) and IX (Primum Mobile) measure exactly 14,234 small, medium and large cosmic miles.


The poem counts 14233 endecasillabi. Add 1, and you obtain the cosmological number 14234. This proves to be very significant.


When Luzifer fell from the heavens, he crashed onto the north pole, pressed and propelled the ice masses of the artic region into the very center of the earth, thus creating a large funnel, namely the Inferno. He is stuck in the ice lake at its bottom. The south pole and hemisphere fled from him and thus formed the mountain of the Antipurgatorio and Earthly Paradise. Here is a schematic cross-section of the disfigured planet:  Dante 1 / Dante 2 / Dante 3 / Dante 4 / Dante 5 / Dante 6




            o   o --------- o    o

         o         o     o          o

        o           o   o            o

        A-----X-----B C D-----Y------E     Equator

        o            o o             o


         o                          o    Antipurgatorio


           o                      o


             o                 o


                o           o    Earthly Paradise





Original diameter of the Earth:AE = 99 medium cosmic miles. Radii of the hemispheres of the Inferno: XA = XB = YD = YE = 869/9 small cosmic miles. Radius of the hemisphere of the ground of the ice lake: CB = CD = 22/9 small cosmic miles. Radii of the Antipurgatorio and Earthly Paradise: AE = EA  = 99 medium cosmic miles


Luzifer is represented by point C in the (former) center of the Earth. Draw nine arcs around C and you obtain the spheres of the Inferno. Their radii measure 11, 22, 33, 44, 55, 66, 77, 88, 99 small cosmic miles. If the radius measures 99 small cosmic miles, you obtain the equator. If the radius measures 88 small cosmic miles, you obtain roughly the polar circle. In between is found the inhabited world as then known. Strangely enough, it would correspond to sphere 1 of the Inferno, probably signifying that our world is affected by Luzifer.


Now for the Antipurgatorio and Earthly Paradise. Imagine nine horizontal rings whose radii measure 11, 22, 33, 44, 55, 66, 77, 88, 99 small cosmic miles Lay the nine rings on the mountain. The largest ring will cover the equator. The nine rings and the peak of the mountain define 9+1 = 10 regions. Let the road begin somewhere at the equator and go around the mountain, in such a way that you ascend by one ring every time you complete a circuit of the Antipurgatorio. You will slowly reach the peak. The road will first be steep, but near the top almost horizontal.


  1 small cosmic mile   =  83 / 54 pi  poetic miles

  1 medium cosmic mile  =  83 / 27 pi  poetic miles

  1 large cosmic mile   =  83 / 18 pi  poetic miles


  54 pi / 83  small cosmic miles   =  1 poetic mile

  27 pi / 83  medium cosmic miles  =  1 poetic mile

  18 pi / 83  large cosmic miles   =  1 poetic mile


  1 poetic mile  =  108 common miles


  1 poetic mile         =  1.021963... medium cosmic miles

  1 medium cosmic mile  =  0.978508... poetic miles


  radius of a circle          1 small cosmic miles

  diameter                    1 medium cosmic miles

  circumference exactly     332 common miles

  diameter of the Earth      99 medium cosmic miles

  circumference exactly  32,868 common miles (99x332)


Now for the Malebolge. Sphere 8 of the Inferno is defined by the rings KL and MN of the funnel:



                        K   O   L

                          M P N



Diameter Earth AE = 198 small cosmic miles. Radii funnel XA = XB = YD = YB = 869/18 small cosmic miles. Malebolge: upper ring KL, lower ring MN. // Radius CK = radius CL = 22 small cosmic miles. The upper ring of the Malebolge is defined by the radii OK and OL, the circumference of the ring measures exactly 22 poetic miles. // Radius CM = radius CN = 11 small cosmic miles. The lower ring of the Malebolge is defined by the radii PM and PN, the circumference of the ring measures exactly 11 poetic miles. Dante and Virgil enter the steep part of the funnel called Malebolge. The Malebolge volge 22 miglia, Dante and Virgil see the moon below or beneath their feet. Later on, when they leave the morose region, the Malbolge volge 11 miglia. The 22 and 11 miles are poetic miles and correspond to 2,376 and 1,188 common miles (when we, in the real world, cover one common mile, Dante and Virgil in their dreamlike world cover one poetic mile or 108 common miles).


Strange: Dante and Virgil see the moon below their feet? Might the rings of the Inferno be an anti-heaven? the moon below a counterpart of the lunar sphere above? How about comparing the spheres of the Inferno with the heavenly ones? Let me try:





     outer radius of Inferno 1   99 small cosmic miles (equator)

     inner radius of Inferno 1   88 small cosmic miles


Sphere 1 of the Inferno covers, or better, affects the region from the equator till about the polar circle. While Heaven 9 is ruled by the Primum Mobile, Inferno 1 is ruled by another primum mobile, namely lonza leone lupo; panther lion wolf; voluptuousness, pride / arrogance and greed / avarice.  Dante 2a





     outer radius of Inferno 2   88 small cosmic miles

     inner radius of Inferno 2   77 small cosmic miles


The second sphere of the Inferno begins at about the polar circle and reaches almost the top of the disfigured planet. Instead of stars shadowy flocks of birds pass by.  Dante 2b





     outer radius of Inferno 3   77 small cosmic miles

     inner radius of Inferno 3   66 small cosmic miles


Sphere 3 of the Inferno covers the pole of the disfigured planet (a ring, no longer a point). While Heaven 7 is ruled by Saturn, mythological founder of the Golden Age, Inferno 3       is ruled by Cerberus.  Dante 2c




     outer radius of Inferno 4   66 small cosmic miles

     inner radius of Inferno 4   55 small cosmic miles


Sphere 4 of the Inferno lies on the upper region on the inside of the funnel. While Heaven 6 is ruled by Jupiter, Inferno 4 is introduced by the enigmatic pape satan - a bitter depiction of the pope who banned Dante and reigns over Italy as if he were the pagan god Jupiter.  Dante 2d





     outer radius of Inferno 5   55 small cosmic miles

     inner radius of Inferno 5   44 small cosmic miles


Heaven 5 is ruled by the war god Mars while Inferno 5 is the place of the garrison Dis    (dis-agree, dis-accord, dis-sension).  Dante 2e





     outer radius of Inferno 6   44 small cosmic miles

     inner radius of Inferno 6   33 small cosmic miles


Heaven 4 is ruled by the Sun, whereas in sphere 6 of the Inferno are mentioned sarcophagi and a blind prison - yet also a ray of hope: a sweet ray and an all-seeing eye (Beatrice?)  Dante 2f





     outer radius of Inferno 7   33 small cosmic miles

     inner radius of Inferno 7   22 small cosmic miles


Heaven 3 is ruled by Venus; when passing Inferno 7, we read that the fall of Luzifer made tremble and shake the universe as if it were feeling love.  Dante 2g





     outer radius of Inferno 8   22 small cosmic miles

     inner radius of Inferno 8   11 small cosmic miles


Inferno 8 is called Malebolge:


     circumference of the upper ring   22 poetic miles

     circumference of the lower ring   11 poetic miles


The walls of the funnel are by now becoming very steep. - Heaven 2 is ruled by Mercurius. The Roman god Mercurius was an alter ego of the Greek god Hermes who played an important role in Homer's Odyssey. Hermes was called Argeiphontes, meaning an Argus-eyed, most watchful killer of snakes and/or dragons, which may be understood in a political sense. Hermes might well be the alter ego of Homer. And if Mercurius were an alter ego of Dante? The Malebolge is by far the most important sphere of the Inferno. Yet it isn't ruled by the Argeiphontes Hermes or by Mercurius, but by a dragon or a snake.  Dante 2h



INFERNO 9 (bottom of Inferno, ice lake/sea) - HEAVEN 1 - MOON


     outer radius of Inferno 9   11 small cosmic miles

     inner radius of Inferno 9    0 (Luzifer)


Dante and Virgil, passing the Malebolge, see the moon below them. Heaven 1 is ruled by the moon. Many a moon shone down into Inferno 9. The cool light of the moon is suiting to the cold place.  Dante 2i



All in all I speculate that the Inferno in the funnel of the disfigured planet can be understood as Luzifer's anti-heaven.  Dante 2j


Beginning, middle and conclusion of the Divina Commedia:


  Line 1         Nel mezzo del cammin di nostra vita

  line 2         mi ritrovai per una selva oscura,

  line 3         ché la diritta via era smarrita.


  line 7,116       Questo triforme amor qua giu di sotto

  line 7,117     si piange: or vo' che tu de l'altro intende,

  line 7,118     che corre al ben con ordine corrotto.


  line 14,230      A l'alta fantasia qui manco possà

  line 14,231    ma giù volgeva il mio disio e il velle,

  line 14,232    sì come rota ch'igualmente è mossa,

  line 14,233    l'amor che move il sole e l'altre stelle.


  line 14,234  missing line of the divine messenger?


When Luzifer fell from God's grace and crashed on our planet, pressing the artic into the center of the Earth, thus creating the funnel of the Inferno as a demonic anti-heaven, the universe shook and trembled. Sin and pain came into the world, the harmony of the spheres was disturbed. However, there is hope: a divine messenger will come and restore the harmony whereupon the heavenly spheres will move again come rota ch'è igualmente mossa. - How will the messenger perform this wonder? By means of a divine message. God created the world by his word, and another powerful word will hopefully save us. What word or line? A l'alta fantasia qui manca possa - we lack the fantasy to imagine the powerful divine word, the single line that will restore the cosmological number 2 x 7,117 = 14,234 and the former harmony of the heavenly spheres, and free the world from sin and pain ...


There are 100 Canti in the Divina Commedia (a number found in the generator of the π-sequence), and every Canto ends in a line of often touching and moving poetical power. The divine messenger may end the Divina Commedia in the same way: by adding one single line with infinitely more poetical power, thus restoring the cosmological number and the former harmony of the heavenly spheres. We know the name of the messenger: a number belonging to an excellent π-value. However, all our values for the number of the circle are approximate values - a tiny part is always missing, and calculating the minute rest turns out to be more difficult than all previous steps.


In one of his letters professor Wilhelm Pötters told me that the very center of the Divina Commedia is the word tu = you. Virgil addresses Dante, while Dante writes for his reader, for me, for you ... In the middle of the poem Dante gives a diagnosis of what goes wrong: all we do is motivated by love, but by a wrong love in a triple form, questo triforme amor, or then by a true love gone wrong. You may remember the cross-section of the disfigured planet Earth: does it not resemble a heart? a core as mentioned in a sonnet by Giacomo da Lentini? L'AMOR CHE MOVE IL SOLA E L'ALTRA STELLE - love, the Primum Mobile of the universe ...


Wilhlem Pötter's thesis on the origin of the sonnet and early Italian poetry has far reaching consequences and may really provide a missing link between antiquity and the Italian Renaissance.






GERHARD GOEBEL, Poeta Faber Habilitation, Heidelberg 1971 (and several letters written to me).  WILHELM PÖTTERS, La natura e l'origine del sonetto, Una nuova teoria, Firenze, Leo S. Olschki Editore, 1982; Wer war Laura? Versuch einer Identifizierung der namentlich bekannten Unbekannten in Petrarcas Liebesdichtung, Miscellania Mediaevalia, Veröffentlichungen des Thomas-Instituts der Universität zu Köln, Herausgegeben von Albert Zimmermann; Mensura, Mass, Zahl, Zahlensymbolik im Mittelalter, Band 16/2; Walter de Gruyter Berlin New York 1984; "Un cinquecento dieci e cinque" (Dante, Purgatorio 33, 43), Der Name des Boten Gottes als Problem der Wortbildung und der poetischen Kosmologie, Latinitas et Romanitas, Festschrift für Hans Dieter Bock zum 65. Geburtstag, herausgegeben von Annegret Bollee und Johannes Kramer, Romanistischer Verlag Bonn 1997; Nascita del sonetto, Metrica e matematica al tempo di Federico II; Presentazione di Furio Brugnolo, Longo Editore Ravenna 1998;(and several letters written to me).




Dante-Referat Vilnius


Im frühen Oktober 2003 bekam ich freundlicherweise die Einladung, ein Gastreferat in einem Dante-Kurs an einem Seminar der Universität Vilnius in Litauen zu halten, Reise und Herberge würden mir von einer Stiftung bezahlt. Ich freue mich sehr über diesen Auftrag, da ich allerdings viel Arbeit habe und die Reise nach Vilnius schon sehr weit ist, mache ich einen andern Vorschlag: ich schreibe ein Referat in sehr einfacher Form und schicke es auf einer CD nach Vilnius, beziehunjgsweise an die Stiftung, welche die CD weiterleiten mag. Das Referat finden Sie über diesen link:  Dante-Referat Vilnius