ranger. But the very fact that a Line is visible implies that it
possesses yet another Dimension.

I. My Lord, I have just acknowledged that a Woman is broad as well as
long. We see her length, we infer her breadth; which, though very
slight, is capable of measurement.

Stranger. You do not understand me. I mean that when you see a Woman,
you ought--besides inferring her breadth--to see her length, and to SEE
what we call her HEIGHT; although the last Dimension is infinitesimal
in your country. If a Line were mere length without "height," it would
cease to occupy Space and would become invisible. Surely you must
recognize this?

I. I must indeed confess that I do not in the least understand your
Lordship. When we in Flatland see a Line, we see length and
BRIGHTNESS. If the brightness disappears, the Line is extinguished,
and, as you say, ceases to occupy Space. But am I to suppose that your
Lordship gives the brightness the title of a Dimension, and that what
we call "bright" you call "high"?

Stranger. No, indeed. By "height" I mean a Dimension like your
length: only, with you, "height" is not so easily perceptible, being
extremely small.

I. My Lord, your assertion is easily put to the test. You say I have
a Third Dimension, which you call "height." Now, Dimension implies
direction and measurement. Do but measure my "height," or merely
indicate to me the direction in which my "height" extends, and I will
become your convert. Otherwise, your Lordship's own understand must
hold me excused.

Stranger. (TO HIMSELF.) I can do neither. How shall I convince him?
Surely a plain statement of facts followed by ocular demonstration
ought to suffice. --Now, Sir; listen to me.

You are living on a Plane. What you style Flatland is the vast level
surface of what I may call a fluid, or in, the top of which you and
your countrymen move about, without rising above or falling below it.

I am not a plane Figure, but a Solid. You call me a Circle; but in
reality I am not a Circle, but an infinite number of Circles, of size
varying from a Point to a Circle of thirteen inches in diameter, one
placed on the top of the other. When I cut through your plane as I am
now doing, I make in your plane a section which you, very rightly, call
a Circle. For even a Sphere--which is my proper name in my own
country--if he manifest himself at all to an inhabitant of
Flatland--must needs manifest himself as a Circle.