Flatland: Chapter 16 Summary & Analysis

Unable to clearly see his unexpected visitor, A Square feels the Stranger and believes him to be a perfect Circle. The Stranger announces that he has come from Space, a land of three dimensions. He speaks of height, breadth, and length, but A Square does not understand these words. The Stranger tries to prove the third dimension by stating that he has seen the entirety of A Square’s family and household from above. However, A Square is not convinced, and argues that any Circle would have to the power to obtain that kind of personal information.

The Stranger then attempts to convince A Square with a different argument concerning A Square’s Wife. The Stranger argues that, although, theoretically, the Wife is a one-dimensional line, in actuality, she is really a very thin parallelogram with an additional dimension, that is, breadth. In the same line of reasoning, then, the Wife must also have a “height.” But A Square confuses “height” with “brightness” and does not grasp the Stranger’s meaning.

Since dimension has a direction and is something that can be measured, A Square asks the Stranger to measure his “height.” The Stranger decides to use plain words and a visual example to convince his pupil. He argues that Flatland is a plane and that he is not a figure, but a solid that is made up of an infinite number of circles that vary in size from a point to a circle of 13 inches in diameter. He proclaims that he is called a sphere.

The Sphere makes an analogy between the way A Square appears as a line to the Monarch of Lineland and the way he himself looks like a circle in Flatland, since each lower realm (that is, Lineland and Flatland) is but a slice representing the whole (that is, the Sphere’s Land of Three Dimensions). When A Square still expresses doubt, the Sphere physically demonstrates the third dimension by rising in Space and showing that his sections become smaller. A diagram is included.

Although A Square indeed sees that the Sphere decreases in size as he “rose,” he still does not understand the nature of the third dimension, and instead begins believing the Stranger to be a mystical sorcerer. After a moment of silence, the Sphere decides to use analogy as the last resort before convincing A Square through action.

The Sphere begins by asking A Square what a point moving northward creates with its path of motion. A Square answers “a straight line.” He then continues and asks what a straight line moving parallel to itself and leaving a wake of lines creates. A Square answers “a square.” Then, the Sphere requests that A Square consider what a square moving upward parallel to itself. A Square, frustrated, becomes impatient, because he does not understand the word “upward.”

The Sphere claims that he can describe the word “upward” with Flatland language, and proceeds to present another analogy. He begins with a single point, which produces a line with two endpoints. A line creates a square, which has four terminal points. Thus, he demonstrates the geometrical series of 1, 2, and 4, and he asks A Square for the next number, which is 8. The Sphere says that the object with 8 points is called a cube.

A Square asks if this resulting creature has sides. The Sphere answers that what Flatlanders call “sides” are actually called “solids” in his own world. He states that a point has 0 sides, a line 2, and a square 4. This is an arithmetical progression, so A Square answers that the next number is 6. The Sphere states that A Square is correct, and says that a cube is bounded by six sides. However, A Square is enraged by this, and hurls himself at the Sphere.