Revisiting Euclidean Geometry

Geometry. When I go back and think about the subject in high school that was the most tedious, especially in math, geometric proofs are one of the first things that come to mind. Even though I have always enjoyed studying math, Euclidean geometry is very different from the rest of the math that is studied in high school, which is more computational in nature. It doesn’t deal with common notions like numbers and algebra, but relies mainly upon an arcane system of postulates, definitions, and theorems. The only way knowledge is derived is via proof, which does not seem mathematical to most 14 or 15 year-olds. Even after coming to the realization that the theorems studied in geometry provide a framework which allows us to study coordinate geometry and trigonometry, I still did not have a high regard for this branch of mathematics.

Why has this forgettable topic suddenly come back? This semester, I am taking Math 163, which is the seminar course that all math majors are required to take. This class is meant for underclassmen who are interested in pursuing mathematics, and its purpose is to introduce us to mathematical rigor and reasoning, as well as to show us how to construct solid mathematical proof.The theme of the class is geometry, and we are currently working with the first book of Euclid’s Elements, which includes all the familiar congruence theorems, characterizes parallel lines, and ends with a proof of the Pythagorean Theorem. A recurring theme that our professor reminds us about constantly is the all-important Fifth Postulate, which deals with the angles formed by two lines and a line that falls across them. The axiom is used mainly in dealing with parallel lines. It is also interesting to view the Pythagorean Theorem in a purely geometric context, rather than the common algebraic equation, a^2+b^2=c^2.

Although I was originally complaining about having to look at Euclid again, the methods of proof learned will come in handy in higher-level mathematics courses, where I am already having to construct proofs for homework on a weekly basis. Maybe it is because I now see a purpose for it, but I am glad I was forced to revisit geometry, and I’m looking forward to moving on to more modern geometry and applying my knowledge acquired from studying Euclid.

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