Jason begins his second speech by reinforcing the necessity of winning the verbal battle almost instead of being correct in his actions: "δεῖ μ', ὡς ἔοικε, μὴ κακὸν φῦναι λέγειν" (It is necessary that I, as it seems, to prove myself to speak not badly )" (Μήδεια 522). Later, he refers to his argument with Medea as an "ἅμιλλαν...λογῶν (contest of words)" (Μήδεια 546), the same phrase Gorgias uses to describe the rhetorical battles of the sophists on various topics. These meta-phrases and Jason's rhetorical strategy of numbering causes and disproving Μήδεια point-by-point (also a tactic that Gorgias uses to demonstrate the innocence of Helen), and most particularly his argument that it was not Medea that helped him, but rather Aphrodite, who compelled her to help (one of Gorgias's arguments), shows that Jason is not taking the content of the argument seriously. This would not seem so striking-- after all the Athenians certainly liked rhetoric-- except that Medea is passionately arguing for her own life and those of her children. Going into exile meant not only leaving her homeland but that they would effectively be wandering as beggars ("πτωχοὺς ἀλᾶσθαι παῖδας ἥ τ'ἔσωσά σε" line 517), vulnerable to thieves and murderers and with not guarantee of safe passage or a place to go. This makes Jason seem more horrible than he already does and it actually sounds worse in the Greek than I remember in English.
One thing that Herodotus II pointed out is that although Jason seems awful, the rhetorical skills he displays also make him seem intelligent. He is not doing this because he is stupid; he is doing it because he has no conscience. This is fascinating because Jason was one of the men of the heroic age who traveled on the Argo. From what I remember of my tragedy class, Medea acts like a Sophoclean hero (as described by Bernard Knox). The Sophoclean hero, as Knox describes him  is a relic of the heroic ages who still obeys by the old codes and cannot function in a rapidly modernizing world where the values of honor and oaths no longer hold the same power. This certainly seems to be true of Medea. The odd thing is that Jason, the Argonaut, demonstrates none of these heroic tenancies; he is thoroughly a "modern man" placing the value of conspicuousness  (ἐπίσημος) rather than everlasting κλέος (glory won in the traditional heroic fashion).
- In this translation, I take the neuter adjective to be functioning as an adverb. It could also function (as Mastronarde suggests) as an alternative phrasing for δεινὸς λέγειν (skilled at speaking) (Mastronarde258).
- I use the male pronoun even though Antigone also demonstrates the pattern of the Sophoclean hero. I remember this from reading Bernard Knox' article "The Ajax of Sophocles" (JSTOR) but I am sure it can also be found in The Heroic Temper.
- This is "notableness" but in the sense of something being remarkable in a worldly way. The word's primary meaning is the mark that is stamped on coinage.(see LSJ entry).