Flatland: A Romance of Many Dimensions

Abbott held the same view in his own name. Writing in The Kernel and the Husk (1886) of the appearances of the risen Christ, he observed that conviction “appears never to have been forced on any entirely unwilling unbeliever”. The Square’s failures are therefore not a schoolmaster’s complaint about dull pupils but a theory of belief, in which a truth from beyond can be offered and even shown, but never compelled. What remains is writing. The Square, unable to persuade a boy or a brother, sets his hope on memoirs that may one day reach some unknown mind, and that is also the hope of the book in the reader’s hands.

Analogy as the Bridge to the Unseen

Analogy is the one instrument in the book that works across a border, and the Sphere, failing with words, turns to it in § 16 to lead the Square through a series he can check for himself. A point has one terminal point, a line two and a square four, so the next figure must have eight; a line has two sides and a square four, so the next must have six. The figure is the Cube, and the counts are right. The method asks nothing of the eye. It takes the proportion of what a creature knows and extends it into what the creature cannot picture.

Then the pupil turns the method on his teacher. In § 19 the Square carries both series one step further, to a figure of sixteen terminal points bounded by eight cubes, which is exactly right for the cube of four dimensions. The Sphere, who had promised that everything followed “in strict accordance with Analogy” (§ 16), now cries “Analogy! Nonsense” and soon wants only to “have done with this trifling”. The teacher has dropped his method at the border of his own experience, and the reader, who knows that the Square’s arithmetic is sound, watches the Sphere become the King of Lineland.

Yet analogy in Flatland is not simply vindicated. Within a page the Square’s argument has turned to rapture, and he is calling for a Fifth, a Sixth, a Seventh and an Eighth Dimension as though counting were seeing. Abbott himself wrote in Philomythus (1891) that “There may be regions of four, five, or fifty dimensions”, but that people rightly decline to act on a mere possibility for which they have no evidence. The book’s own ending settles the matter quietly. At his trial the President asks for the Cube to be shown by some means “other than the enumeration of imaginary sides and angles” (§ 22), and the Square has nothing else to offer. Analogy has given him the form of the higher world, its numbers and its order, but not the sight of it, and the form alone persuades no one who has not seen.

Imprisonment and the Martyr of a New Truth

The book opens in a prison, if the reader begins with the Preface, and it ends in one, and between the two Flatland is a world of confinements. Would-be rebels are kept “in honourable confinement for life” (§ 3), Specimens are fettered in the schools, Women live in apartments built to stop them turning, and the Council sends to the Asylum any Square who claims a revelation. Lineland’s people are confined to their line and the Point to its “non-dimensional Gulf” (§ 20), and every world, seen from above, is a prison whose inmates take its walls for the edge of things.

Against this the Square casts himself as a martyr. He compares himself to Prometheus at the height of his vision and again in his cell, and speaks of “the martyrdom which I endure for the cause of the Truth” (§ 22). The Council’s resolution grades the punishment of a new truth by the shape of the one who brings it, with death for an Isosceles, the scourge for a Triangle, the Asylum for a Square or Pentagon, and trial for anyone higher. Its President is the book’s most chilling figure because he is its most reasonable, for he agrees with his prisoner that the Truth will prevail and so sees no harm in locking him up until it does. It costs him nothing to be right.

Yet Abbott denies the martyr his glory. No one is converted, the new truth is not vindicated within the book, and the prisoner, in his last paragraph, doubts the shape of the Cube and fears that Spaceland was a dream. The only conversion the book records is the prisoner’s own, and it is not the one he sought, for in seven years, the Preface reports, he has changed his views of the Women and of the Isosceles, the two classes he most despised. He went up to Spaceland for a new geometry and has come back, slowly, to a new charity.

Symbolism

Light and Fog. Light in Flatland comes from nowhere. It falls equally by day and by night, indoors and out, “whence we know not” (§ 2), and the question of its origin has been forbidden by law after filling the asylums with those who tried to answer it. The Square claims to know that it comes from the world of Three Dimensions, though the book never shows him learning so. Brightness marks life, since living figures shine more than houses, and in the Preface it is the name Flatlanders give to the dimension of height they cannot recognise. Fog, by contrast, is a blessing, “the Nurse of arts and Parent of sciences” (§ 6), because things fade in it with distance and so betray their shape. Flatland’s knowledge is born of dimness. When the Sphere arrives in a dry season, with no Fog in the room, the Square is reduced to feeling him, and in Spaceland he has to learn a new kind of sight altogether, made of light and shade and perspective.