Flatland: A Romance of Many Dimensions

Plato’s Cave. The nearest classical parallel, and probably a deliberate one in a book by a Senior Classic, is the allegory of the Cave (Republic VII, 514a–517a), and the likeness runs well beyond the shadows of the first page. The prisoner set free is dragged up “a steep and rugged ascent” (515e, in Jowett’s translation), dazzled by the light and slow to see, as the Square, carried off against his will, first knows only darkness and a sight that is not like seeing. The man who goes back down is laughed at, and his fellows would kill him if they could catch him (516e–517a), which is the end of Part II in little. The differences are as telling as the likeness. Plato’s returning philosopher would be put to death; Abbott’s is merely locked up, by a President who professes to agree with him, a sharper and more Victorian irony. And where Plato’s ascent ends in the sun, the Square’s ends in a question that his guide will not answer. The notes add a passage from later in Republic VII (529a–c), where Socrates mocks the idea that gazing at the sky is looking upward, and it is the best gloss in Plato on the book’s riddle of an upward that is not Northward. Plato’s upward, like Abbott’s, is no direction of the senses.

The Meno and the Symposium. The Sphere’s lesson in § 16 has the form, and probably the model, of Socrates’ lesson to the slave boy in the Meno (82b–85b), a run of questions in which the pupil supplies each step. It even ends in the Socratic way, with the teacher telling the pupil that he has answered his own question. But Flatland reverses the dialogue’s hope. Socrates warns that true opinions run away unless they are tied down by reasoning (97d–98a), and the Square, who cannot demonstrate what he saw, watches his own run away. The Symposium matters more, because there the likeness turns into a difference. The Dedication’s hope that a writer initiated into three dimensions may lead his readers on to four, five or six has the shape, probably by design, of the lesser and greater mysteries taught by Diotima, the wise woman who instructs Socrates (209e–210a). Yet Diotima’s ladder has a top, “beauty absolute, separate, simple, and everlasting” (211a–b, Jowett), while the Square rebels against any limit “short of Infinity” (§ 19). Flatland is the Symposium’s ladder with the summit taken away and modesty set in its place, and that is the deepest thing this edition has to say about Abbott and Plato.

Aristotle and Pointland. The Monarch of Pointland reads as a parody of Aristotle’s God, the mind whose only object of thought is its own thinking, in whom thinker and thought are one, and whose contemplation is the pleasantest of all things (Metaphysics Λ.9, 1074b34; Λ.7, 1072b24). The Point has every one of these marks, and the parody is probably deliberate. The twist is Abbott’s own. Aristotle’s self-thinking mind is the summit of being, and Abbott sets it at “the lowest depth of existence” (§ 20). There the Sphere draws the moral that “to be self-contented is to be vile and ignorant”, which is close to Diotima’s warning that the ignorant are satisfied with themselves (Symposium 204a). The Point calls itself “the One, and yet the All in All”, but it is not the One of the third-century Platonist Plotinus, which lies beyond thought; it is thought with nothing to think but itself, and it is very happy.

The infinite sphere. A reader who knows the mystics will hear more in the Sphere than Abbott put there, and all of this is reflection only. The Book of the Twenty-Four Philosophers, whose earliest manuscripts date from the twelfth century, calls God an infinite sphere whose centre is everywhere and whose circumference is nowhere. The theologian Alan of Lille made it an intelligible sphere, a figure of thought rather than of space, and the philosopher Blaise Pascal turned the formula on nature itself. The American essayist Ralph Waldo Emerson begins his essay “Circles” (1841) with the same figure and argues two things the Dedication might have said, that “around every circle another can be drawn” and that “The only sin is limitation.” There is no evidence that Abbott read the essay, yet no Transcendentalist text stands closer to the Square’s rebellion against the limits of his world. The book itself, however, refuses the mystic’s equation of God and sphere, for its Sphere is irritable, fallible and corrected by his own pupil, and his power of seeing everything is shared, he admits, by thieves. The infinite sphere of the mystics is the One the Square imagines above the Sphere in § 19. The book never shows it.