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 appears to A Square as a kind of revelator to enlighten his pupil. The resemblance of this situation to A Square’s vision of Lineland is clear, and is another analogy that demonstrates the difficulty of spreading new knowledge. A Square is unwilling to accept any of the Stranger’s claims, and stubbornly tries to explain away the third dimension with what he is already familiar with.
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.
The Stranger utilizes analogy to the fullest. He uses A Square’s Wife to argue that although lines are one-dimensional, they still have a second-dimension, which is breadth. Thus, she must also have a third-dimension, if she can indeed possess a second. In a way, dimensionality represents the capacity for knowledge, since it is latent in everyone. What A Square believes is “brightness” represents deeper truth.
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.
What is interesting is that the characters in Flatland are constantly asking for proof. There is a consistent need for logic to be the arbiter in each situation. So Abbott’s message is even more striking, since he is suggesting that religious revelation and knowledge (symbolized by higher dimensions) should depend on rational thinking, not only “leaps of faith.”
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.
Notice that the Sphere is making the same demonstration that A Square made in Lineland. They both physically move in and out of the lower world in order to make their point about higher knowledge. Pay attention to the difference in responses between the Monarch of Lineland and A Square, though.
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.
A Square’s initial reaction to the Sphere’s movement is the same as the Monarch of Lineland, who associates A Square’s motion with magic. However, Abbott’s deliberate analogy is not just to repeat the same scene, but to convey a more refined idea about the reception of knowledge, which will become more obvious later.
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.”
Note the emotional response of A Square, who feels angry at the Sphere’s incomprehensible lessons. Although throughout Flatland A Square favors reason over emotion, and associates the former with men and their supposed superiority, he clearly demonstrates that reason isn’t inherently masculine, and isn’t inherently superior to emotion. In fact, A Square’s response is very natural. Of course one would feel frustrated when he cannot understand something.
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.
The various methods that the Sphere uses to prove the existence of three dimensions truly illustrate that knowledge can be achieved through many pathways. There is no one right way. This also could apply to faith and understanding. While the Anglican church exerted its authority as the sole path to salvation, Abbott seemed to believe that individual spirituality was more important.
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.
“Sides” mean different things in Flatland and Spaceland, which are lines and figures, respectively. Thus, Abbott affirms the importance of language in determining thought. Without the appropriate words and meanings, such as “upwards” or “sides,” thinking about the third dimension is almost impossible. While A Square was more patient than the Monarch of Lineland, in the end he acts the same way—giving in to his frustration and attacking the person trying to enlighten him.