Euclid’s Geometry

Euclid, the greatest mathematician of his time, from Alexandria, transformed geometry into a logical discipline in an era when it lacked clarity and structure. In around 300 BCE, he collected all of the groundbreaking works of geometry into the Elements, a pioneering work that provided a framework for proving geometric facts. These facts were standardised into mathematical definitions, axioms and postulates, which are our topic of discussion today, so let’s begin!

1.0Euclid’s Terms and Definitions

Euclidean geometry consists of elements, elements that form the basis of all geometric reasoning. These are basically the accepted, but not proven, descriptions of certain geometric objects. These descriptions are like the universal language of geometry, which includes: 

1. Point: A point is simply a representation of the position or location of an object in space. It has no length, breadth, or thickness. 
2. Line: A line is a length that doesn’t have breadth, meaning it has only one dimension, that is, length. It extends endlessly in both directions. 
3. Straight Line: A straight line is basically the shortest distance between two points. It spreads evenly with the points on itself only. 
4. Surface: A surface is a two-dimensional figure, meaning it only has length and breadth but no thickness. 
5. Angle: An angle is formed by the intersection of two lines at the point of intersection of these lines. 

2.0Euclid’s Axioms

Euclid’s Axioms are the common notions of mathematics, meaning these are the self-evident truths that are not proven but accepted universally. These axioms are based on logical reasoning and are 7 in number: 

Axiom 1: Things that are equal to the same thing are always also equal to each other. 

Understand it like this: imagine you and your friend scored the same in the exam. Now, imagine your other classmate's scores are equal to your friend's, then your score will also be equal to that classmate's. 

Axiom 2: If equals are added to equals, the wholes are equal. 

Understand it like this: Let Rohan and Priya have 50 Rs each, now if they both get Rs 20 extra, then they both will have a total of Rs 70. Hence, the addition of equal quantities results in equal wholes. 

Axiom 3: If equals are subtracted from equals, the remainders are equal.

To understand this, take the above example. In the above example, if we take Rs 20 from both Priya and Rohan, then they will be left with Rs 30 each. Hence, the subtraction of equal quantities also results in equal wholes. 

Axiom 4: Things which coincide with one another are equal.

Imagine two identical keys that perfectly overlap are the same size; this equal size leads the keys to coincide with one another. 

Axiom 5: The whole is greater than the part.

Understand this axiom with the example of pizza. A full pizza is always greater than a slice in volume and size. This axiom also leads to the result that no part can exceed the value of the whole from which it is taken. 

Axiom 6: Things that are double of the same things are equal.

Take two lines AB and CD. Now, imagine both of these lines are twice the length of the same line PQ. Then AB and CD will also be equal. Mathematically, if: 

AB = 2PQ and CD = 2PQ, 

Then, according to axiom 1: AB = CD

Axiom 7: Things that are half of the same things are equal.

As in the above example, if lines AB and CD are equal to half of line PQ, then in this case, AB and CD will be equal. 

3.0Euclid’s Postulates

Euclid’s Postulates are also assumptions, but are specific to geometric building. These are also accepted without proof and are 5 in number, which include: 

Postulate 1: A straight line may be drawn from any point to any other point.

This postulate simply means that a line can only be drawn between points on paper. 

Postulate 2: A finite straight line can be extended indefinitely in a straight line.

According to postulate two, lines can be extended infinitely unless bounded. 

Postulate 3: A circle can be drawn with any centre and any radius. 

A circle is recognised and defined by its centre and radius; hence, it can be drawn from any centre with a certain radius of any length. 

Postulate 4: All right angles are equal. 

The measure of all the right angles is 90°, regardless of their position, size, etc. Hence, it can be said that all right angles are always equal. 

Postulate 5 (Parallel Postulate): If a line intersects two lines such that the interior angles on one side are less than two right angles, those two lines, if extended, will meet on that side.

Parallel lines never meet or intersect. However, according to this postulate, we assume that any two parallel lines may meet at some point at infinity.

4.0Applications of Euclid’s Geometry

Despite being such an ancient concept, Euclid’s Geometry is a highly informative and useful tool in modern-day geometry and many other fields, including:  

• Architecture & Construction: Angle, triangle, and structural balance designs are based on Euclidean principles.
• Navigation & Surveying: Flat-plane assumptions for maps, compasses, and land measurement are based on Euclidean geometry.
• Art & Design: Symmetry, proportion, and perspective rely on geometric principles.
• Engineering: Civil, mechanical, and electrical designs employ Euclidean tools such as lines, angles, and polygons.
• Education: In education, Euclid Geometry, Class 9, forms the basis for the advanced study of geometric concepts in higher classes. 

5.0Difference Between Euclidean and Non-Euclidean Geometry

6.0Solved Examples on Euclid's Geometry

Problem 1: A stick of length 30 cm is broken into three parts: 10 cm, 10 cm, and 10 cm. Then one part is further divided into two equal subparts. Using Euclid’s Axioms, prove:

1. Each subpart is smaller than the original stick.
2. The subparts of the broken stick are equal to each other.

Solution: It is given that the length of the stick is 30cm, which is broken into 3 equal parts of 10cm, and these subparts are further divided into two equal parts. Hence, the length of these parts will be 5 cm (half of 10).

Now, 

1. According to Axiom 5, which says “the whole is greater than the part”, it can be concluded that: 30 cm > 10 cm > 5 cm
2. As per axiom 7, which states, “Things that are half of the same thing are equal”, it can be concluded that: 5cm = 5cm

Problem 2: Two triangles △ABC and △DEF, when placed over one another, all corresponding sides and angles of both triangles coincide. Prove using Euclid’s Axioms that the two triangles are equal. 

Solution: It is given that in two triangles, △ABC and △DEF, it is given that they coincide completely. Hence, 

According to Euclid’s Axiom 4: Things which coincide with one another are equal. 

So if one triangle is superimposed on the other completely, then both triangles will be equal in all aspects — sides and angles. That is: △ABC △DEF

Problem 3: Two lines, M and N, are drawn on a flat plane. A transversal crosses them such that the interior angles on the same side add up to exactly 180°. Will the lines M and N ever meet?

Solution: It is given that the sum of the interior angles on the same side of the transversal, crossing two flat lines M and N, is exactly 180°.

Hence, by the concept of the converse of Postulate 5, the lines M and N are parallel. Which further indicates that these lines will never meet.