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I have to agree with Ashton, that I was kind of offended and irritated that the book seemed so sexist, and we were reading it at a women's college!  But it wasn't until I realized that it was first published in 1884, and read that the A. Square didn't really believe in the degradation of women, that I began to fully enjoy the original insight that Abbott gave into his story.  In fact, after reading the book, I was amazed at how many correlations it had with Brecht's Galileo!  Clearly there are parallels here, such as:

1.) Both Galileo and A. Square fight against the main power of the time (the Catholic Church, and the government of Flatland)  to try and persuade people of the truth, but to little success at the beginning.

2.) In the beginning of the play, Galileo tries to teach Andrea (the newer generation) his ideas, because children seem to be much more open minded.  The same is with A. Square, his first attempt at teaching the Gospel of the Third Dimension is to his Grandson.  Fresh blood, whose mind hasn't been confined to the standards and morals of society yet.  I think this small action speaks a lot about the characters, and proves that they want to lay the foundation of these new ideas with youngsters who can bring about a new era.

There are  Many other parallels but I just thought these two were particularly important to recognize.  Any other parallels that are really important?