From the very beginning of the play, we know that Septimus isn't your ordinary tutor.
THOMASINA: Septimus, what is carnal embrace?
SEPTIMUS: Carnal embrace is the practice of throwing one's arms around a side of beef. (1.1)
Septimus's response is absurdly literal: as he goes on to explain, "carnal" really does refer to meat. His answer tells us a lot about him right off the bat. First, he has a sense of humor that tends towards the ridiculous, but is grounded in his academic knowledge. Second, even though he's supposed to be teaching Thomasina, he's not above messing with her. And third, as the conversation develops and Septimus gives an answer that's another kind of literal (we won't quote the whole passage here – it's in "Quotes"), we learn that he doesn't have much in the way of prudish sexual hang-ups. By establishing the character of Septimus so clearly right from the beginning, Stoppard also tells us a lot about the kind of play this is going to be – one with a weird sense of humor, plenty of wordplay, and some sexy talk.
Since we've already gotten to know Septimus through how he talks to Thomasina, we're primed to appreciate his mockery of Mr. Chater, who's less his match in verbal wit than Thomasina is. Witness how Septimus neatly derails Chater from his adultery-avenging rampage:
CHATER: I have heard of your admiration, sir! You insulted my wife in the gazebo yesterday evening!SEPTIMUS: You are mistaken. I made love to your wife in the gazebo. She asked me to meet her there, I have her note somewhere, I dare say I could find it for you, and if someone is putting it about that I did not turn up, by God, sir, it is a slander. (1.1)
Does Septimus really believe that Chater's worried that Septimus might have stood Mrs. Chater up for an appointment to have sex with her? Hardly. But by purposely misunderstanding Chater – just as he did with Thomasina's question that opened the play – Septimus breaks the expected story. When Chater rushes into the schoolroom demanding satisfaction from Septimus, he's acting out a part in his own imagined narrative, in which the next step would be that Septimus will either deny the whole thing or agree to a duel. When Septimus admits of his tryst with Mrs. Chater, behaving as if he thinks the real sin would have been standing her up, he departs from the available options in Chater's Choose Your Own Adventure story. By going completely off script in his response to Chater, Septimus gains the upper hand over him, at least temporarily.
Of course, Chater soon gets a sharp kick in the pants from Captain Brice, and the duel does get scheduled. Septimus's decision to sleep in the boathouse rather than risk a bullet may seem cowardly, but only if you buy the idea that a duel cleanses your honor. From a more pragmatic viewpoint, Septimus's refusal to pay his debt with a gunfight shows just how smart he is.
Septimus may be less smart when it comes to Thomasina. He totally misses what she's trying to do with the iterated algorithm thingamagum:
THOMASINA: No marks?! Did you not like my rabbit equation?
SEPTIMUS: I saw no resemblance to a rabbit.
THOMASINA: It eats its own progeny.
SEPTIMUS: (Pause) I did not see that. (2.7)
Here, Septimus's tendency to take things literally blinds him to Thomasina's inventive projects: she says "rabbit," and he thinks, "picture with two long ears, fluffy tail," not "equation that devours its young." His disconnect here suggests that Thomasina's strength is the imagination that Septimus lacks – while he thinks outside the box in managing his love affairs, he's not so good at applying that originality to mathematics. Septimus's failure illustrates the play's larger suggestion that cleverness will only get you so far in both poetry and mathematics: to make really great achievements, you need genius. Septimus's lack of genius in this regard might give us a clue to figuring out what it is that geniuses do differently.
While Septimus doesn't seem to change much over the course of the play, the ending does suggest that he's learned a few things from his time in Arcadia. For one, he gives Thomasina's essay an A grade "in blind faith" (2.7), showing that he takes her ideas seriously even if he doesn't understand them – a far cry from his dismissive tone earlier in the play. Smooching Thomasina may be another example of a change, or perhaps just another case of Lady Croom substitution syndrome. His refusal to join Thomasina in her room, however, may show that he thinks more of her than Mrs. Chater – or just that he's avoiding doing something very, very stupid. Since we know that Thomasina dies soon after the end of the play, their maybe-sort-of romance is left unresolved – and open to interpretation. (Click over to "What's Up with the Ending?" for more on the play's conclusion.)