Line

Use in the book

The report on Chrysippus says that lines, like bodies and other continua, are divisible without limit. It also denies that endless divisibility requires an already completed collection of infinitely many constituents. The line is a useful example because the argument can address division without relying on the physical thickness of a drawn mark.

Places to read

1. I.14.1eUnlimited divisibility is distinguished from composition out of an actual infinite collection.

Translation in this edition

Retain “line” in the geometrical list; a drawn stroke is not the definition being asserted.

Use in the book

Section 12 gives the straight line a levelling function. A Triangle, Square, Pentagon or Circle presents the same basic kind of visible object. The inhabitants cannot inspect one another as the reader inspects the diagrams. Their elaborate hierarchy consequently requires indirect techniques to recover distinctions that immediate appearance does not disclose.

Section 323 then interrupts the account with the short declaration that Women are Straight Lines. Its bare syntax offers a total definition. Men receive differentiated bodies, professions and prospects; women receive one unchanging geometrical identity. The shared appearance which initially made all figures epistemically comparable now names a sex placed outside the ascending series of male development.

The needle experiment in §423 makes orientation crucial. A line viewed from the side shows length; viewed end-on it approaches a point. The woman can become nearly invisible without leaving the plane or acquiring supernatural power. The nickname of an invisible cap imports the language of marvels into an elementary optical fact. It also presents a woman's ordinary body as a source of alarming strategic advantage.

The consequences are political and domestic. A point can pierce; a line cannot bend around or turn within every enclosure. Law requires audible warnings and visible movement, while apartments are designed to prevent rotation. What begins as a definition becomes a programme for organising another person's behaviour. The body supplies genuine constraints, but the narrator's chosen remedies distribute freedom unequally.

A revealing uncertainty occurs when the Square sees his triangular father side-on and cannot immediately tell whether he is looking at a Woman. The moment does not dissolve bodily difference. It does show that the apparently self-evident sex-classification must be recovered through positioning and inference. The father's recognisable masculinity is not simply present in the visible line.

Section 162345 subsequently corrects the popular nomenclature: Women are scientifically thin Parallelograms, with measurable breadth within the plane. This second dimension must not be confused with the unrecognised height which the revised Preface23 attributes to all Flatlanders. The educated Square knows that a familiar social name is not a complete anatomical description, yet continues to reason through it.

Thus the straight line carries the romance's larger caution about appearances in miniature. One visible form can belong to different underlying figures; one named figure can be burdened with many social meanings. A reader must distinguish geometry, perception and judgement even when the prose confidently moves from one to another.

Places to read

1. §12“Nothing was visible, nor could be visible, to us, except straight Lines”
2. §323“Our Women are Straight Lines”
3. §423“a Woman is a needle”
4. §423“nothing but a highly lustrous point”
5. §6234“a Straight Line, or, in other words, a Woman”
6. §162345“a very thin Parallelogram”

In this edition

Notes 25 and 332 explain the optical and social senses; notes 44–452 discuss the needle and invisibility imagery. Note 159 explains the later Parallelogram correction. These are editorial guides to distinctions in the main text. Capitalisation is useful evidence but not a perfectly consistent logical notation.