Having its origin from the ancient Greek words “geo” and “metron,” which when put together, mean “measurement of the earth,” geometry is among the most primitive branches of mathematics. It mostly deals with the parameters of shape, distance, size, and other such properties of space, including figures’ relative spatial arrangement. Until the 19th century, the discourse of this field of study was mostly confined to the ideas of the curve, line, point, surface, angle, and distance. These concepts were treated as fundamental with respect to its framework. The multiple novel discoveries in the 19th century primarily paved the path for a dramatic broadening of the horizon of geometry.
Ever since the field has found itself divided into multiple subfields based largely on the underlying methods employed. Differential geometry, computational geometry, algebraic geometry, discrete geometry, and algebraic topology are a few examples of geometry subfields. Even though it was principally developed with the vision of modeling elements in the physical world, geometry wound up finding applications in different channels of sciences, architecture, and art. Activities that deal with graphics at their core rely on this field of study too.
Much of geometry, as we know of it today, was formulated predominantly by the Greek mathematician, Euclid of Alexandria, who is thus rightfully credited as the founder and father of geometry. His tireless efforts that were mostly dedicated to reworking, organizing, compiling, and gathering his predecessors’ works led to the eventual birth of the immensely coherent body of work now broadly referred to as Euclidean geometry. The following segments shed light on the Greek mathematician’s most important works and the role he played in defining this branch of mathematics.
Comprising thirteen comprehensive books in all, Elements refers to the logically coherent mathematical treatise that Euclid derived from the collective works of his predecessors. It consists of definitions, theorems, postulates, and the requisite mathematical proofs of these theorems and constructions. As a deductive treatment of mathematics, Elements is the largest body of work that is still in existence. Its compilation and subsequent distribution in the 18th century saw the emergence of geometry as an indispensable part of the English gentleman’s standard education during the period. This reason, coupled with the instrumental role that Elements played in the progress of modern science and logic, has resulted in the treatise being consistently regarded as among the most influential and successful textbooks to be ever written.
Each of the 13 books in the treatise cover concepts, which, when looked at collectively, deal with elementary number theory topics, solid and plane Euclidean geometry, and the field of incommensurable lines. Book 1 of Elements abounds mainly in five common notions and postulates revolving around significant plane geometry themes. Some of these topics include the famous Pythagorean theorem, parallelism, equality of angles and areas, the sum of angles in a triangle, and the construction of different geometric figures. Book 2 consists of lemmas concerning the various tenets of geometric algebra, primarily the equality of rectangles and squares. It also contains a construction of the all-important golden ratio that is ubiquitous in nature. Circles and their numerous properties, such as finding the center, tangents, and inscribed angles, including Thales’ theorem, find their place in Book 3 of Elements. With Book 4 and 6 delving into the various properties of triangles and polygons and applying the theory of proportions to plane geometry, respectively, the first four books in the series along with Book 6 effectively summarise the fundamental principles of plane geometry.
Further, while Books 11 through 13 deal with various concepts that are central to solid geometry, Books 5 and 10 concern themselves with theories of ratios and magnitudes. Lastly, Books 7 through 9 cover the realm of whole numbers and all the properties that they entail. In its entirety, the comprehensive discourse plays host to a staggering 465 propositions, essentially laying the framework for all kinds of academic research and deliberations in the field of geometry.
In compiling his findings, Euclid rested his approach on ten axioms that were, in essence, a set of statements that could be considered truths under all conditions and at all times. Referring to them as postulates, Euclid separated them into two groups of five postulates each. While the first set of postulates deals with mathematics properties in general, the second set is completely geometry-centric. Both sets of postulates have been listed below for the reference of the reader.
- Things which are equal to the same thing are additionally equal to each other as well.
- If equals are added to equals, the sums or the wholes are equal.
- If equals are deducted from equals, the differences or the remainders are equal.
- Things that coincide with one another are equal to each other.
- The whole is greater than the part.
- It is possible to draw a straight line between any two points on the plane.
- It is possible to extend any finite straight line infinitely in a straight line.
- For any line segment, it is possible to create a circle by making use of the segment as the radius and one of its endpoints as the center.
- All right angles are congruent in nature.
- If two lines are drawn that intersect a third so that the sum of the inner angles on one side is less than two Right Angles, then the two lines inevitably must intersect each other on that side if they are extended far enough.
Method and style of preparation
A large proportion of Euclid’s propositions were constructive in nature, demonstrating the presence of a particular entity by making use of the same steps that he employed in the construction of another object with the help of a straightedge and a compass. Euclid’s constructive approach reflects largely across his geometry’s postulates as well, for the first and the third postulates that state the existence of a line and a circle are constructive in nature. Further, it is also believed that whenever he had to use a figure in one of his proofs, he would construct it in an earlier proposition before putting it into use. The clearest example of this can be observed by studying his method of proving the Pythagoras theorem. In this case, he first constructed a square on a given line of one proposition only to subsequently inscribe it on the sides of a right triangle to achieve his objective.
The immense influence that Euclid established for all the ages draws largely from the fact that his extensive body of work represents much more than a standard explanation of mathematics in general and geometry in particular. The manner in which he implemented logic and necessitated proof for every theorem more or less moulded the overall thought process of western philosophers and gave the field the structure that we are familiar with today.
Philosopher mathematicians of colossal repute, such as Newton and Descartes, showcased their philosophical works by making use of the format and structure that Euclid formulated. Their method of transcending from rudimentary first principles to gradually complicated concepts was completely derived from Euclid’s school of thought. So vast were the effects of his influence that the 16th President of the United States, Abraham Lincoln, employed them while formulating the US Declaration of Independence. Apart from his groundbreaking work of Elements, Euclid was also recognised for his contributions to optics, astronomy, mirror, music theory, and perspective.
As the father of geometry, Euclid was prominent not only for his prolific development of postulates but for his stronghold on the practical insight that he maintained while formulating them, as well. With the right guidance, you can inculcate the same values in your child. As your children take their first steps in the vast field of mathematics, it is vital that you teach them to discover and unravel the relevance of the subject in the everyday world around them. Cuemath’s highly immersive and comprehensive basic and advanced geometry curriculum ensures that a child not only gains complete clarity on the various topics but also develops strong problem-solving skills in the process. Further, our team prepares a personalized learning plan for every student, which focuses on holistically building their grasp of concepts.
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