The Brilliance of Elements: How Euclid Became the Father of Geometry

If you have ever taken a class on geometry, you might have heard of Euclid. His name goes down in history as the “Father of Geometry,” but the reason is not always clear. Euclid’s fame is principally due to his authorship of one of the first textbooks of geometry, called Stoicheia (in Ancient Greek), or Elements. I recommend that anyone who wants to understand the origins of modern mathematics, or simply to think more critically and argue better, should pick it up and read it through to the end. Euclid’s Elements consists of a long series of interconnected proofs and demonstrations of mathematics truths regarding geometry, abstract ratios, and arithmetic. In its conclusion, Euclid shows how it is possible to draw various ordinary solids using only straight lines and squares. This includes spheres, cubes, and more complex shapes, such as a decahedron and dodecahedron (with ten and twelve sides respectively). The book begins with some basic definitions of terms that are necessary for the work that follows.

Elements is divided into twelve books. In Book I, Euclid shows how one can construct lines equal to other lines and circles equal to other circles. His geometry is referred to as the geometry of “the ruler and compass” because all of it can be done simply by drawing straight lines and circles. With these simple tools, and the rules that Euclid uses (called his definitions), one can prove for oneself all of the proofs (or propositions) in the book.

There are several different types of propositions in Euclid’s Elements: constructions, demonstrations, and corollaries. Constructions are propositions where the ability to construct a figure of a given kind is proven, while demonstrations are propositions where given some construction of a certain kind, a new fact about it is shown to be true. Corollaries are demonstrations that follow easily from other demonstrations. Here is an example of both: Proposition one states that when given a straight line of any length, one can construct an equilateral triangle on it with all three sides the same length as the given straight line — this is a construction. Proposition Forty-Seven, alternatively, says that given a right triangle, it can be shown that the squares constructed on the two sides of the triangle equal the square constructed on the hypotenuse., making this a demonstration. This proposition is also more commonly known, in our present day, as the Pythagorean Theorem.

Perhaps Euclid is referred to as the Father of Geometry because all of these propositions were either used later on in his book, as parts of their demonstrations or constructions, or because other authors that came after him used his proofs to prove new facts about mathematics. It is not necessarily the case that Euclid invented all the concepts in his book, but what is conspicuous about the Elements is the way in which it is interconnected, all the propositions building one upon another, and the way in which it collects all the facts discovered before Euclid’s time into a single book. Therefore, just as a father brings all his family members together under one name and tribe, Euclid unites the work of Pythagoras, and other early Greek mathematicians such as Thales and Nicomachus under a single textbook. He structures his textbook such that they build one upon another, the former influencing the latter, which subsequently reveals the logical structure underlying all of these seemingly disparate facts about mathematics/. He thus encourages the mathematicians who follow in his footsteps to use his propositions to create new propositions. In this way, he takes on the role of a progenitor, or father, of the geometry to be discovered after him.

Although Ancient Greeks did not invent mathematics, there was no language to describe mathematics before their time. There were no algebraic equations like in modern mathematics and science, and there was nothing to distinguish mathematical operations from ordinary language. Therefore, a third reason that Euclid might be described as the father of geometry is the way in which he speaks about geometry — using syllogistic logic to make all of his premises clearly known and understood or at least agreed upon before drawing any conclusions. The syllogistic reasoning of Euclid’s Elements is what gives them their powerful effect on the reader. In regards to geometry,  I think it was Plato who said that even if one gains nothing from studying it, neither money nor prestige nor any other material gain, those who study it certainly become sharper. For this reason, I recommend dedicating time to reading Euclid, in order to sharpen your mind and strengthen your critical thinking..It is even said that Abraham Lincoln read and memorized the first six books of the Elements for just this purpose, and in his prose and speeches one can see the effect of Euclid on his mind. In the same way, surely the study of geometry is still capable of sharpening the minds of his students today - consider this your invitation to become one of them.

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