I wrote this for my maths tutor. I’ll also be taking on an extracurricular assignment soon, specifically comparing philosophy and methodology of Aristotle and Euclid. But that will probably not be touched until Sunday. This paper is one of my favorites of the semester so far, not because it’s profound, but because it’s unusual, and because I could never get away with anything like this in high school. The assignment was to write some of your thoughts about Euclid’s parallel straight lines, which we are now studying in depth.
“Essay of Parallels”(Lines are interesting.) People are interesting. Imagine two people, living life in the same way, going in the same direction, making all the same moves, but really they have nothing in common. They don’t know each other. They don’t even know of each other. They will never meet, because they’re both moving forward at the same pace in the same direction, completely the same, completely parallel, and never swaying to right or the left.
(Enter the transversal.) Enter the third party. He is nothing like the other two. His character is truly contrary to the first person he meets, and it is likewise contrary to the second. He manages to meet the two running parallel to each other, and he is exclusively aware that both of them exist while the other two remain equally far removed as they were before. But suddenly, there’s something more.
(The angles are the same.) Their relationships are the same. The third man notices that his relationships to these two people are entirely similar, and entirely opposite. Inevitably, via the comparisons of this third person, the two suddenly recognize a relationship they never knew was there. There’s little to show for it at the time they first discover each other, but they are aware of the other one’s existence. Maybe this is a relationship worth pursuing? Oh, but no, the other person is not moving toward the same point. Their goals aren’t the same. Maybe this relationship is meaningless. But then, what happens to people headed for the same point? They reach it, perhaps even together at the exact same moment, and then they keep going, and that one point they had in common is behind them forever, and they spend eternity growing farther and farther apart; they are as contrary to each other as the supposed third party is to the people on parallel paths.
(The figure forms.) The beautiful romance blossoms. True, they aren’t going to the same one point, but at any time, when they meet someone else in their path, they will meet the fourth, fifth, sixth person together. Looking back, they have spent their whole lives together. They belong with each other. Their meaning, their purpose, is essentially lost without the other. The first, without his partner, is only a plain, boring, straight-laced, automatic person. But with someone else running parallel, the world is suddenly more interesting.AfterwordI feel like a bad geometry student, having told a cute story instead of talking about parallel lines. But the significance I see in parallel lines, honestly, does not relate to geometry or mathematics, but to life. As a concept, they are fascinating.
In literature and in life, too, we find parallels. Euclid’s definition illumines the reality of the parallels around us. He talks about parallel straight lines as those in the same plane, and they will never meet regardless of how far they are produced. These are not simply random lines floating in a three-dimensional universe.
They are limited to the same confines of the plane they’re on; they are, in other words, in the same world and bound by the same rules. If given one straight line there is only a specific kind of line that can be parallel to that line. They are like the same line, the same points, but shifted in space. Though there are manifold straight lines that can satisfy the definition of parallelism, there is only one kind. And as a kind, ironically, they never interact directly with each other. They are always apart and never united.
The fact that they will never meet brings an interesting idea. There is no one point of connection. The points the lines are lying on are ever separate. But can points be parallel to each other? Is parallelism more defined by the even evenness of the respective points under each line than the lines themselves? According to Euclid’s definition, whether two straight lines are parallel seems to hinge not on the lines but on the existence or non-existence of a single point. Do the lines meet at that certain point, or is there no point of meeting? That’s the single question that satisfies the definition for parallelism in a plane. But why must it be about that point (or that not-point)? I like this definition as much: parallel straight lines are two straight lines in the same plane, such that when a line is drawn perpendicular from one, it will be perpendicular to the other, and all such perpendicular lines between them are of equal length. I’m not sure why, but the definition hinging on a transversal makes more sense to me than the definition concerning the indefinite not-point. And this preference relates back to the significance of the abovementioned people and their relationship in the essay. These two lines have the same standing with each other, however far they are produced, and nothing can change that.
Maybe I’m reading too much between the lines.
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