The premise is that the author, who describes himself as A. Square, is living in a universe of only two dimensions. He describes various aspects of such a world, such as gravitation, how you would be able to tell a circle from a square, if you were looking at what, to us, would be the edge of two silhouettes, and so on. Quite imaginative work. Although the subtitle includes "Romance," there's no real love in the book.
Here's definition 3 of romance, from the Free Online Dictionary: a. A long medieval narrative in prose or verse that tells of the adventures and heroic exploits of chivalric heroes: an Arthurian romance. b. A long fictitious tale of heroes and extraordinary or mysterious events, usually set in a distant time or place. c. The class of literature constituted by such tales.
This book is a romance in the second sense.
Perhaps the best part of the book is an experience of a three-dimensional world, after which the protagonist tries, and fails, to convince his fellow Flatlanders that such a thing is possible. (He also tries, and fails, to persuade his three-dimensional mentor, clearly an intelligent being, that a four-dimensional world might be possible.) Although Abbott almost certainly couldn't have known it, Einstein's physics considered time to be a fourth dimension, and string theory suggests that the universe may have 10 or more dimensions.
This edition had a good introduction by Alan Lightman. Lightman compared Abbott to Lewis Carroll, who, like Abbott, wrote about mathematics. (Carroll is, of course, famous for his fantastic writing.) Lightman suggests, plausibly, that Abbott was writing some social satire, especially about the sexes in Flatland. (Women were straight lines, men geometric shapes.) Lightman also said that Abbott was a pioneer in thinking, and probably the book has been important in the development of scientific thought. A 1920 article on Einstein's physics referred to the book, he says.
For me, the importance of the second part of Flatland lies not in its literal geometrical and dimensional discussion, but in its more shrouded warning of too much complacency in the scientific enterprise -- and, by extension, all of life. (p. xii)
Scientists, around the time Abbott wrote Flatland, and for a number of years afterward, thought that science knew about everything there was to know. They were wrong. They are now, if they think such.Thanks for reading! Live in as many dimensions as you can.