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Oh, Arthur, Arthur. He's still trying to convince himself that he's not in love with Pet Meagles.
This means that not only is he stressed about the Gowan situation, he's also stressed about pretending not to be stressed about the Gowan situation. Oh yeah. It's like an onion. Layers.
Doyce is on to Arthur's poorly kept secret, and they talk about how much Gowan obviously sucks, how much Pet clearly likes him, and how annoyed Meagles is every time he sees how much Pet likes Gowan.
The next time he visits the Meagleses, Arthur runs into Gowan, who does his totally annoying alpha-male jock thing of constantly giving Arthur and Doyce backhanded compliments.
Then Gowan runs down himself and his crappy paintings – which would be a nice self-deprecating thing to do, except the next thing Gowan does is say that every single other artist is also a talentless hack.
Basically, according to Gowan, everything sort of sucks, and also nothing particularly matters. In art, in literature – and also in government. According to Gowan, the fact that the Circumlocution Office is horrendous, corrupt, and totally ineffectual is kind of the point of it. He's infuriating, especially to a goody-goody like Arthur.
Gowan invites Arthur to meet Mrs. Gowan, his mom.
The Gowans are a poor offshoot of a rich and powerful family, so they live in the poor offshoot of a rich and powerful neighborhood in a gross old house.
Mrs. Gowan is about as stuck-up a person as can be imagined. She's very sad that Gowan is forced to work for a living. By which, hardy-har, she's apparently talking about his lazy painting, which he kind of does and mostly doesn't do. Also, she is sad about the idea of a democratically elected government.
Finally, she takes Arthur aside and tells him she's also sad about the potential marriage between Gowan and Pet. According to Mrs. Gowan, the Meagleses have done their utmost to entrap the desirably high-born Gowan into this marriage.
Arthur is totally scandalized to hear Mrs. Gowan talking about the Meagleses this way (especially since they actually don't want the marriage to happen!) but she pointedly ignores him.
Finally, Gowan takes Arthur home, and Arthur realizes that he was only invited to meet Mrs. Gowan because this is what she would say to him. Meaning Gowan wanted to: 1) make Arthur feel bad by hearing the Meagleses insulted, and 2) warn Arthur to back off Pet since Gowan is already in there.
This seems a little paranoid, but it rings true, because that's how Gowan rolls.