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After Beatrice's hopeful prophecy, Dante notices something reflected in Beatrice's eyes, like the image of a double candle in a mirror. He looks into her eyes to see if he's not just imagining it.
When he realizes it's there, he turns around to find it.
He sees a point of light so bright that it nearly blinds him, and its far larger than any star. Around it circles nine rings of flame, the first ring out orbiting the Point the fastest, and the other rings revolving more slowly the farther out they go. The thing is huge.
He notices that the ring with the purest light is the one closest to the Point because, Dante believes, the closest ring is the most similar—of all the rings—to the Point itself.
Dante is perplexed by what this is. Beatrice comes to the rescue. She explains that all of Nature depends on this thing. She further explains that the first ring spins the fastest because it has the most desire for God.
Dante thinks for a moment, then says if the actual universe were like this, he would be happy (because Earth as the smallest "ring" would be the purest and closest to God). So why isn't the universe like this model?
Beatrice smiles knowingly and says, it's okay that you don't understand Dante; no one has ever tried before. So let me explain: in the material universe, the blessedness of a material object depends on how much power it has. In matter, greater power corresponds to greater size. So the bigger an object is, the more power it has, and the more blessed it is. This is why this sphere of the Primum Mobile—as the biggest sphere—is closest to God.
But in this model, where each ring represents an Angelic Intelligence (which is decidedly not material), only power matters. Thus, the one with the most power moves closest to God. When matching this model to the material universe, one sees that the angel with the greatest power and who is closest to God matches the sphere with the greatest power (the Primum Mobile) but yet is the farthest from God. It's what your math teachers would call an inverse relationship.
To illustrate his understanding, Dante compares the dawning of his comprehension to the north wind, Boreas, blowing away clouds from the sky and making it clear.
After Beatrice finishes talking, each of the nine rings grows brighter, their individual sparks all shining brilliantly and Dante can hear their voices singing a hymn towards the fixed Point in the center.
Beatrice speaks again, naming each of the rings for Dante's benefit. From the center outwards, the first ring contains the Seraphim, then the Cherubim, Thrones, Dominions, Virtues, Powers, Principalities, Archangels, and Angels.
Beatrice goes on to say that Dionysius, a scholar, was famous for making this hierarchy known to mortals. Later, a certain Gregory came and disputed Dionysius's findings. But when Gregory died and came to Heaven, he saw he was wrong.
Finally, she tells Dante that it shouldn't surprise him that man knows this heavenly secret because its source was St. Peter himself, who saw it and himself served as Dionysius' source.