Dante uses the idea of contrapasso, or symbolic retribution. Explain how punishment fits the sin in the cases of Paolo and Francesca, Pietro della Vigne, Jason, Ulysses, and Brunetto Latini.
Dante finds himself in front of the gate of Hell. What does the inscription say?
Let's start with your part two of your question first; that is, the inscription above the Gates of Hell. As Virgil and Dante approach the gate, they read its Latin inscription:
Per me si va ne la città dolente,
per me si va ne l'etterno dolore,
per me si va tra la perduta gente.
Giustizia mosse il mio alto fattore:
fecemi la divina podestate,
la somma sapienza e 'l primo amore.
Dinanzi a me non fuor cose create
se non etterne, e io etterno duro.
Lasciate ogne speranza, voi ch'entrate
This may be translated as:
THROUGH me you pass into the city of woe:
Through me you pass into eternal pain:
Through me among the people lost for aye.
Justice the founder of my fabric mov'd:
To rear me was the task of power divine,
Supremest wisdom, and primeval love.
Before me things create were none, save things
Eternal, and eternal I endure.
"All hope abandon ye who enter here."
All the condemned souls you are inquiring about, Francesca and Paolo, Pietro della Vigne, Jason, Ulysses, and Brunetto Latini have no more chances. Unlike Virgil and Dante, their hell is permanent.
Every soul condemned to the nine circles has a punishment that symbolically matches their sins. As Virgil and Dante descend into the Second Circle, in Canto V, they encounter Francesca and Paolo, who had been lovers when they were alive. Their crime is lust and adultery. Both were married to other people. Francesca had been wedded to a very old and crippled man; she fell in love with her husband's young and handsome brother. One day, the two were outside, reading the Arthurian legend of another pair of adulterers, Lancelot and Guinevere. Francesca and Paolo soon begin kissing. Francesca's husband finds out and has the unfaithful lovers killed. In Hell, their punishment is to be buffeted by winds that forever keep them apart.
Dante, himself enthralled with Beatrice, is overcome with compassion for the pair and faints in response. Part of Dante's journey will be to accept the justice of God's punishment for sin.
It is in Circle Seven, Canto XIII that Dante and Virgil find Pietro della Vigne. Like Dante, Pietro della Vigne was a poet and had the ear of Frederick II. As the sinner speaks to Dante, his rhetoric is as tangled as the trees in which he is forever snarled. Della Vigna was accused (later it appears that these accusations were false) of betraying his king: " "I am he that held both keys of Frederick's heart/ To lock and to unlock; and well I knew/ To turn them with so exquisite an art," he tells the travelers.
In sorrow and shame, Pietro della Vigne hung himself, a mortal sin. Suicides, in the Catholic faith, cannot enter Heaven. In Latin, "Vigne" means "vines." Symbolically, trees in Canto XIII are tangled and thorny. Moreover, in added punishment, "the souls will not be reunited with their bodies at the Last Judgment but will instead hang their retrieved corpses on the trees." However, since Dante dictates della Vigne's sin as suicide rather than being a traitor, this punishment is symbolically appropriate.
Jason, the great man in mythology, who rescued the Golden Fleece, is found in Circle Eight, Canto XVIII. He is among the condemned in a "pouch" known as the "malebolge." In this pouch, demons chase the sinners from one side to the other with whips. Jason is there because he abandoned his wife, Medea, and their two children in order to seek fame and fortune. Symbolically, Jason will never find peace and he will be no better treated than any of the other sinners; just another soul for the demon's scourge.
We also find Ulysses, another great man of the myths, in Circle Eight, Canto XXVI. Ulysses sin is fraud. His symbolic punishment is to be engulfed in a flame, along with another fraudulent soul, Diomedes. Ulysses's many frauds included "the stratagem of the wooden horse; luring Achilles; and stealing the Palladium." His flame may burn and be seen forever, but he is still in hell.
Finally, you ask about the symbolism of the punishment for Brunetto Latini. He is found in the Circle Seven, Canto XV. Latini is cast among the sodomites and must live in the scorching desert in an area called the "Abominable Sands." Sodomites had committed crimes against nature, and therefore against God. There does not seem to be any historical evidence for the married-with-children Latini to be condemned here; he may "either posit a substitute vice for the sexual one--linguistic perversion, unnatural political affiliations, a quasi-Manichean heresy--or emphasize a symbolic form of sodomy over the literal act."