Dimension and distance

Use in the book

The survey of mathematical definitions of soul distinguishes figure as the limit of extension from extension itself. Severus and Speusippus are then located in this mathematical class, before a more unified causal account is considered. The vocabulary belongs to an argument about the status of soul, not simply a measurement of a visible object.

Places to read

1. I.49.32.5Figure, extension, their form, and their unifying cause are distinguished.

Translation in this edition

“Extension” best fits the passage’s contrast with figure; “distance” is appropriate only where separated positions are intended.

Use in the book

Dimensionality controls what each world's inhabitants can move through, perceive and express. Lineland's King takes length for the entirety of Space. The Square takes his plane for true Space. The Sphere initially treats the three-dimensional world as the final conceivable enlargement. Each locally adequate description becomes false when converted into a declaration about the whole of possible existence.

The debate in §162345 brings a real conceptual difficulty into the comedy. The Square observes that a dimension implies direction and measurement, and offers to accept the Sphere's assertion if he can measure the alleged height or indicate its direction. The request is not answered by adding length to height and breadth, because Flatland already distributes these names among two extents. Nor does calling brightness height establish a measurable third axis. The visitor admits that he cannot supply the demanded indication on the pupil's present terms.

The revised text adds a further complication: Flatlanders possess an infinitesimal height although they cannot apprehend it as a dimension. Their visible existence requires more than a purely abstract mathematical plane. This physical qualification does not give them free three-dimensional movement, and it should not erase the story's governing restriction. The narrative is investigating an inhabited model, with bodies and sight, rather than supplying a perfectly self-contained set of axioms.

Once transported, the Square learns enough to extend the question. A fourth dimension would stand to Spaceland as Spaceland stands to Flatland. The analogy challenges the Sphere's claim that the concept is inconceivable, but no fourth-dimensional realm becomes an object of waking observation. Higher dimensionality remains a possibility reasoned about and, in the later dream, illustrated. The distinction between conceivable, mathematically describable and physically verified remains crucial.

The word also acquires ethical and political force without ceasing to be spatial. Limited Dimensionality becomes what the imprisoned narrator wants a future race of rebels to refuse. That phrase concerns a restriction upon thought enforced by inherited assumptions and by law. Yet an additional dimension is not itself additional goodness. The Sphere's moral correction of omnividence prevents the book's ladder of spatial powers from becoming a simple ladder of spiritual worth.

Places to read

1. §162345“Two Dimensions by four names” — Verbal plurality does not create spatial independence
2. §162345“Dimension implies direction and measurement” — The Square states a test
3. §192345“a Fourth unrecognized Dimension” — The revised argument extrapolates the earlier difficulty
4. §202“the Abyss of No Dimensions” — The lower limit is embodied in Pointland
5. §2223“limited Dimensionality” — A geometrical restriction becomes an intellectual challenge

In this edition

The notes identify additions to §§16 and 192345678910 made for the revised edition of 1884. These respond to objections about visibility and dimensional thickness. They are Abbott's revisions; the modern annotations explaining them should remain distinct from the narrated argument.

Dimension

An independent measurable extent or direction. The debate distinguishes adding a dimension from adding another name for an extent already familiar to the observer. Printed form: Dimension (§ 162345). See Dimension.

Figure

A geometrical form and, in this inhabited geometry, often a person. The Sphere corrects the Square's use of Figure for the prospective Cube, requiring the more specific dimensional term Solid. Printed form: Figure (§ 162345). See Dimension.

Infinitesimal

Extremely small, here too small for the Flatlander to recognize and measure as a third dimension. The revised discussion distinguishes the body's slight height from the brightness through which the Square perceives it. Printed form: infinitesimal (§ 162345). See Dimension.

Parallelogram

A quadrilateral with opposite sides parallel. The educated Square insists that a visible Woman is scientifically a very thin Parallelogram, not a perfectly breadthless abstract line. Printed form: Parallelogram (§ 162345). See Dimension.

Solid

A three-dimensional body in the Sphere's usage. This differs from the Flatland use for a plane figure enclosed on its sides, a verbal disagreement central to the lesson. Printed form: Solid (§ 162345). See Dimension.

Dimension implies direction and measurement (contextual phrase)

§162345. “Dimension implies direction and measurement.”

The Square asks for a criterion rather than another prestigious assertion. If the new height is a dimension, he wants it measured or its direction indicated. On the pupil's present terms, the Sphere admits that he can do neither. The demand is therefore more serious than the surrounding comedy of an eye in the stomach may suggest. Yet it is not the last word: a subject confined to a plane may lack the practical access required to satisfy a test that presupposes movement outside it. The passage stages a genuine tension between intelligible evidence and the enlargement of the framework within which evidence is intelligible. Neither credulity nor simple contempt for the request captures its force.

See also. Dimension · Evidence and demonstration