Tired of the Monarch of Lineland’s narrow-mindedness and ignorance, A Square decides to teach him of the nature of the second dimension and Flatland. He begins by asking the king how he ascertains the shapes and positions of his people. The Monarch replies that it is impossible to do so with the sense of sight. Instead, the length of a person is measured by hearing and comparing the time difference in the arrival of a man’s two voices.
Although A Square is unaware of it, he is trying to reveal the mysteries and knowledge of the second dimension to the Monarch in the same way that a religious leader spreads doctrine to non-believers in the hopes of converting them. The truth of the second dimension is thus an allegory of the mysteries of God.
In defiance, A Square asks then how fraud (that is, disguising one’s voice) is detected and checked for. He asks if feeling is a possible method. The Monarch of Lineland is appalled at A Square’s suggestion, because feeling is an offense punishable by death, to protect the frailty of women. He continues by saying that touching is unnecessary, since hearing is an efficient and unalterable process.
Ignorance is truly self-defeating and limiting. This is shown through the Monarch, who is so content with the way his life is organized (predominately through hearing) that he sees no need to consider any other methods. Hearing is apparently efficient enough.
A Square points out to the Monarch of Lineland that life in Lineland must be very boring since all they can see is a point. A Square explains that as he entered their world, he saw the Monarch moving left and right, having 7 men and 1 woman to his left and 8 men and 2 women on his right. Yet the Monarch does not understand A Square’s use of the words “left” and “right,” and confuses them for “northward” and “southward.”
Abbott introduces an aspect of knowledge that we must keep in mind: there is a limit to what we can understand. God cannot be fully comprehended by human faculties, and we must respect divine mystery. So “left” and “right” represent higher knowledge that is difficult to grasp when one is limited to one’s dimension. However, the potential to understand that knowledge is not non-existent, as exemplified by A Square.
The Monarch of Lineland asks A Square to demonstrate “left” and “right.” But A Square feels limited when he tries to explain them only with words, so he decides to literally move in and out of Lineland. A diagram illustrates his position relative to Lineland. But the Monarch still is not able to grasp left and right motion and instead attributes A Square’s movement to magic. A Square scolds him for his refusal to think beyond his own world. In response, the Monarch and his army threaten to attack A Square, who wakes up from his dream.
New knowledge can sometimes be viewed as threatening. The Monarch dismisses A Square’s demonstration as magic, or something absurd or even dangerous. Thus, Abbott claims that one must be humble and trusting in order to be receptive to new ideas. Eventually, the Monarch responds to A Square’s efforts with violence, since he is not willing to learn and rejects any threat to his worldview.