Hello my dear students! Welcome to today's mathematics class. I am so happy to be here with you to explore one of the most fascinating chapters in your geometry curriculum — Chapter 5: Introduction to Euclid's Geometry. Now, before we dive into the details, let me ask you a question. Have you ever looked at the buildings around you, the roads, the fields, or even the pyramids you might have seen in pictures? All of these have shapes, sizes, and structures that follow certain rules. Today, we are going to learn about the foundation of all this geometry — the brilliant system that Euclid developed more than two thousand years ago. So let's begin our journey together.
Let us start with understanding where the word "geometry" comes from. The word 'geometry' comes from the Greek words 'geo', meaning the 'earth', and 'metrein', meaning 'to measure'. So geometry essentially means "measuring the earth". Isn't that interesting? Geometry appears to have originated from the need for measuring land. This branch of mathematics was studied in various forms in every ancient civilisation — be it in Egypt, Babylonia, China, India, Greece, the Incas, and so on. The people of these civilisations faced several practical problems which required the development of geometry in various ways.
Let me give you a wonderful example from history. Whenever the river Nile in Egypt overflowed, it wiped out the boundaries between the adjoining fields of different land owners. After such flooding, these boundaries had to be redrawn. For this purpose, the Egyptians developed a number of geometric techniques and rules for calculating simple areas and also for doing simple constructions. The knowledge of geometry was also used by them for computing volumes of granaries, and for constructing canals and pyramids. They also knew the correct formula to find the volume of a truncated pyramid. Now, do you know what a pyramid is? A pyramid is a solid figure, the base of which is a triangle, or square, or some other polygon, and its side faces are triangles converging to a point at the top. When you cut the top portion of a pyramid parallel to its base, you get what is called a truncated pyramid, and the Egyptians could calculate its volume accurately. Isn't that remarkable?
Now, let us come closer home. In the Indian subcontinent, the excavations at Harappa and Mohenjo-Daro show that the Indus Valley Civilisation, which existed about 3000 BCE, made extensive use of geometry. It was a highly organised society. The cities were highly developed and very well planned. For example, the roads were parallel to each other and there was an underground drainage system. The houses had many rooms of different types. This shows that the town dwellers were skilled in mensuration and practical arithmetic. The bricks used for constructions were kiln fired and the ratio of length : breadth : thickness of the bricks was found to be 4 : 2 : 1. Can you imagine? Even in 3000 BCE, our ancestors were using precise geometric ratios in construction!
In ancient India, the Sulbasutras, which were composed between 800 BCE and 500 BCE, were the manuals of geometrical constructions. The geometry of the Vedic period originated with the construction of altars or vedis and fireplaces for performing Vedic rites. The location of the sacred fires had to be in accordance with the clearly laid down instructions about their shapes and areas, if they were to be effective instruments. Square and circular altars were used for household rituals, while altars whose shapes were combinations of rectangles, triangles and trapeziums were required for public worship. The sriyantra, given in the Atharvaveda, consists of nine interwoven isosceles triangles. These triangles are arranged in such a way that they produce 43 subsidiary triangles. Though accurate geometric methods were used for the constructions of altars, the principles behind them were not discussed. So you see, geometry was being developed and applied everywhere in the world, but this was happening in an unsystematic manner.
What is interesting about these developments of geometry in the ancient world is that they were passed on from one generation to the next, either orally or through palm leaf messages, or by other ways. Also, we find that in some civilisations like Babylonia, geometry remained a very practical oriented discipline, as was the case in India and Rome. The geometry developed by Egyptians mainly consisted of the statements of results. There were no general rules of the procedure. In fact, Babylonians and Egyptians used geometry mostly for practical purposes and did very little to develop it as a systematic science. But in civilisations like Greece, the emphasis was on the reasoning behind why certain constructions work. The Greeks were interested in establishing the truth of the statements they discovered using deductive reasoning.
A Greek mathematician, Thales, who lived from 640 BCE to 546 BCE, is credited with giving the first known proof. This proof was of the statement that a circle is bisected, that is, cut into two equal parts, by its diameter. One of Thales' most famous pupils was Pythagoras, whom you must have heard about. Pythagoras and his group discovered many geometric properties and developed the theory of geometry to a great extent. This process continued till 300 BCE. At that time Euclid, a teacher of mathematics at Alexandria in Egypt, collected all the known work and arranged it in his famous treatise called 'Elements'. He divided the 'Elements' into thirteen chapters, each called a book. These books influenced the whole world's understanding of geometry for generations to come. Euclid lived from 325 BCE to 265 BCE. In this chapter, we shall discuss Euclid's approach to geometry and shall try to link it with the present day geometry. So students, this is the legacy we have inherited — a system of geometry that has stood the test of time for over two thousand years!
Now let us move on to Section 5.2, where we will learn about Euclid's Definitions, Axioms and Postulates. The Greek mathematicians of Euclid's time thought of geometry as an abstract model of the world in which they lived. The notions of point, line, plane, or surface, and so on were derived from what was seen around them. From studies of the space and solids in the space around them, an abstract geometrical notion of a solid object was developed. A solid has shape, size, position, and can be moved from one place to another. Its boundaries are called surfaces. They separate one part of the space from another, and are said to have no thickness. The boundaries of the surfaces are curves or straight lines. These lines end in points.
Consider the three steps from solids to points. We go from solids to surfaces, from surfaces to lines, and from lines to points. In each step we lose one extension, also called a dimension. So, a solid has three dimensions, a surface has two, a line has one, and a point has none. Euclid summarised these statements as definitions. He began his exposition by listing 23 definitions in Book 1 of the 'Elements'. A few of them are given below, and I want you to pay close attention to them.
Definition 1: A point is that which has no part.
Definition 2: A line is breadthless length.
Definition 3: The ends of a line are points.
Definition 4: A straight line is a line which lies evenly with the points on itself.
Definition 5: A surface is that which has length and breadth only.
Definition 6: The edges of a surface are lines.
Definition 7: A plane surface is a surface which lies evenly with the straight lines on itself.
Now, if you carefully study these definitions, you find that some of the terms like part, breadth, length, evenly, and so on need to be further explained clearly. For example, consider his definition of a point. In this definition, 'a part' needs to be defined. Suppose if you define 'a part' to be that which occupies 'area', again 'an area' needs to be defined. So, to define one thing, you need to define many other things, and you may get a long chain of definitions without an end. For such reasons, mathematicians agree to leave some geometric terms undefined. However, we do have an intuitive feeling for the geometric concept of a point than what the 'definition' above gives us. So, we represent a point as a dot, even though a dot has some dimension.
A similar problem arises in Definition 2 above, since it refers to breadth and length, neither of which has been defined. Because of this, a few terms are kept undefined while developing any course of study. So, in geometry, we take a point, a line, and a plane — in Euclid's words, a plane surface — as undefined terms. The only thing is that we can represent them intuitively, or explain them with the help of physical models. So students, remember this important point: a point, a line, and a plane are undefined terms in geometry. We understand them intuitively, but we cannot define them perfectly using other geometric terms.
Now, let us talk about axioms and postulates. Starting with his definitions, Euclid assumed certain properties, which were not to be proved. These assumptions are actually 'obvious universal truths'. He divided them into two types: axioms and postulates. He used the term 'postulate' for the assumptions that were specific to geometry. Common notions, often called axioms, on the other hand, were assumptions used throughout mathematics and not specifically linked to geometry. Some of Euclid's axioms, not in his order, are given below. Let me explain each one to you.
Axiom 1: Things which are equal to the same thing are equal to one another. This is like saying if the length of stick A equals the length of stick B, and the length of stick B equals the length of stick C, then automatically the length of stick A equals the length of stick C. This seems obvious, doesn't it? But it is so fundamental that we accept it without proof.
Axiom 2: If equals are added to equals, the wholes are equal. For example, if you have two equal lengths and you add the same length to both, the results will still be equal.
Axiom 3: If equals are subtracted from equals, the remainders are equal. Similar to the previous one, but with subtraction.
Axiom 4: Things which coincide with one another are equal to one another. This means if two things are exactly the same in size and shape, they are equal. This is the justification for a method called superposition, where we place one figure on top of another to check if they are equal.
Axiom 5: The whole is greater than the part. This is a very intuitive idea. If you have a whole pizza and you take a part of it, the whole pizza is definitely bigger than that part. This axiom gives us the definition of 'greater than'. For example, if a quantity B is a part of another quantity A, then A can be written as the sum of B and some third quantity C. Symbolically, A > B means that there is some C such that A = B + C.
Axiom 6: Things which are double of the same things are equal to one another. If two quantities are each twice the same thing, then they are equal to each other.
Axiom 7: Things which are halves of the same things are equal to one another. Similarly, if two quantities are each half of the same thing, then they are equal to each other.
These 'common notions' refer to magnitudes of some kind. The first common notion could be applied to plane figures. For example, if an area of a triangle equals the area of a rectangle and the area of the rectangle equals that of a square, then the area of the triangle also equals the area of the square. Magnitudes of the same kind can be compared and added, but magnitudes of different kinds cannot be compared. For example, a line cannot be compared to a rectangle, nor can an angle be compared to a pentagon. They are different kinds of magnitudes.
The fourth axiom given above seems to say that if two things are identical, that is, they are the same, then they are equal. In other words, everything equals itself. It is the justification of the principle of superposition. So students, these are the seven axioms that Euclid used as the foundation of his geometric system. They are simple, obvious truths that we accept without needing to prove them.
Now, let us discuss Euclid's five postulates. These are specific to geometry, unlike axioms which apply to all of mathematics.
Postulate 1: A straight line may be drawn from any one point to any other point. Note that this postulate tells us that at least one straight line passes through two distinct points, but it does not say that there cannot be more than one such line. However, in his work, Euclid has frequently assumed, without mentioning, that there is a unique line joining two distinct points. We state this result in the form of an axiom as follows:
Axiom 5.1: Given two distinct points, there is a unique line that passes through them. Let me explain this with an example. Suppose you have two distinct points P and Q. How many lines passing through P also pass through Q? Only one, that is, the line PQ. How many lines passing through Q also pass through P? Only one, that is, the line PQ. Thus, the statement above is self-evident, and so is taken as an axiom. This is a very important result that we will use again and again in geometry.
Postulate 2: A terminated line can be produced indefinitely. Note that what we call a line segment nowadays is what Euclid called a terminated line. So, according to the present day terms, the second postulate says that a line segment can be extended on either side to form a line. In simple words, you can always extend a line segment as much as you want in both directions.
Postulate 3: A circle can be drawn with any centre and any radius. This means that given any point as the centre and any distance as the radius, you can draw a circle. This is something we do with a compass, isn't it?
Postulate 4: All right angles are equal to one another. This is a fundamental postulate. Whether you draw a right angle in your notebook or your friend draws one on the blackboard, both represent the same angle measure — 90 degrees. All right angles are always equal, no matter where or how they are drawn.
Postulate 5: If a straight line falling on two straight lines makes the interior angles on the same side of it taken together less than two right angles, then the two straight lines, if produced indefinitely, meet on that side on which the sum of angles is less than two right angles. This is the most complex of all the postulates. Let me explain it with an example. Suppose you have two straight lines AB and CD. Now imagine another line PQ that crosses both these lines. This crossing line PQ creates angles where it meets AB and CD. If the sum of the interior angles on one side of PQ is less than 180 degrees, then if you extend the lines AB and CD indefinitely, they will eventually meet on that side where the sum is less than 180 degrees.
A brief look at the five postulates brings to your notice that Postulate 5 is far more complex than any other postulate. On the other hand, Postulates 1 through 4 are so simple and obvious that these are taken as 'self-evident truths'. However, it is not possible to prove them. So these statements are accepted without any proof. Because of its complexity, the fifth postulate will be given more attention in the next section when we discuss equivalent versions of it.
Now-a-days, 'postulates' and 'axioms' are terms that are used interchangeably and in the same sense. 'Postulate' is actually a verb. When we say "let us postulate", we mean, "let us make some statement based on the observed phenomenon in the Universe". Its truth or validity is checked afterwards. If it is true, then it is accepted as a 'Postulate'.
A system of axioms is called consistent if it is impossible to deduce from these axioms a statement that contradicts any axiom or previously proved statement. So, when any system of axioms is given, it needs to be ensured that the system is consistent.
After Euclid stated his postulates and axioms, he used them to prove other results. Then using these results, he proved some more results by applying deductive reasoning. The statements that were proved are called propositions or theorems. Euclid deduced 465 propositions in a logical chain using his axioms, postulates, definitions and theorems proved earlier in the chain. In the next few chapters on geometry, you will be using these axioms to prove some theorems.
Now, let us see in the following examples how Euclid used his axioms and postulates for proving some of the results. This is very important, so pay close attention.
Example 1: If A, B and C are three points on a line, and B lies between A and C, then prove that AB + BC = AC.
Let me explain this with the help of a figure. We have three points A, B, and C on a straight line, with B lying between A and C. Now, the segment AC is the entire line from A to C. The segment AB is from A to B, and the segment BC is from B to C. When we put AB and BC together, they exactly coincide with AC. In other words, AC is the sum of AB and BC.
Now, how do we prove this using Euclid's axioms? In the figure, AC coincides with AB + BC. Also, Euclid's Axiom 4 says that things which coincide with one another are equal to one another. So, it can be deduced that AB + BC = AC. Note that in this solution, it has been assumed that there is a unique line passing through two points, which we already established as Axiom 5.1. So students, this is a simple but important result. If you have three collinear points with one point between the other two, then the distance from the first to the third equals the sum of the distances from the first to the second and from the second to the third.
Example 2: Prove that an equilateral triangle can be constructed on any given line segment.
In the statement above, a line segment of any length is given, say AB. Here, you need to do some construction. Using Euclid's Postulate 3, you can draw a circle with point A as the centre and AB as the radius. Similarly, draw another circle with point B as the centre and BA as the radius. The two circles meet at a point, say C. Now, draw the line segments AC and BC to form triangle ABC.
So, you have to prove that this triangle is equilateral, that is, AB = AC = BC. Now, AB = AC, since they are the radii of the same circle. That is statement 1. Similarly, AB = BC, because they are also radii of the same circle. That is statement 2. From these two facts, and Euclid's axiom that things which are equal to the same thing are equal to one another, you can conclude that AB = BC = AC. So, triangle ABC is an equilateral triangle.
Note that here Euclid has assumed, without mentioning anywhere, that the two circles drawn with centres A and B will meet each other at a point. This is actually a reasonable assumption, and it works in practice. So students, this is how we can construct an equilateral triangle on any given line segment. This is a very powerful result.
Now, let us prove a theorem, which is frequently used in different results. This is Theorem 5.1.
Theorem 5.1: Two distinct lines cannot have more than one point in common.
Proof: Here we are given two lines l and m. We need to prove that they have only one point in common. For the time being, let us suppose that the two lines intersect in two distinct points, say P and Q. So, you have two lines passing through two distinct points P and Q. But this assumption clashes with the axiom that only one line can pass through two distinct points. So, the assumption that we started with, that two lines can pass through two distinct points, is wrong. From this, what can we conclude? We are forced to conclude that two distinct lines cannot have more than one point in common. This is a very important theorem, students. It means that if two lines intersect, they can only intersect at exactly one point. They cannot intersect at two or more different points. This is fundamental to our understanding of lines in geometry.
Now, let me summarize what we have learned in this chapter.
In this chapter, you have studied the following important points:
First, though Euclid defined a point, a line, and a plane, the definitions are not accepted by mathematicians in their exact form. Therefore, these terms are now taken as undefined terms in geometry. We understand them intuitively, but we cannot define them perfectly using other geometric terms.
Second, axioms or postulates are the assumptions which are obvious universal truths. They are not proved. We accept them as self-evident truths.
Third, theorems are statements which are proved, using definitions, axioms, previously proved statements and deductive reasoning. We build our entire geometric system on these theorems.
Fourth, some of Euclid's axioms were: Things which are equal to the same thing are equal to one another. If equals are added to equals, the wholes are equal. If equals are subtracted from equals, the remainders are equal. Things which coincide with one another are equal to one another. The whole is greater than the part. Things which are double of the same things are equal to one another. Things which are halves of the same things are equal to one another.
Fifth, Euclid's postulates were: Postulate 1 — A straight line may be drawn from any one point to any other point. Postulate 2 — A terminated line can be produced indefinitely. Postulate 3 — A circle can be drawn with any centre and any radius. Postulate 4 — All right angles are equal to one another. And Postulate 5 — If a straight line falling on two straight lines makes the interior angles on the same side of it taken together less than two right angles, then the two straight lines, if produced indefinitely, meet on that side on which the sum of angles is less than two right angles.
We also learned about the important Axiom 5.1: Given two distinct points, there is a unique line that passes through them.
We proved that if three collinear points are such that one point lies between the other two, then the distance from the first to the third equals the sum of the distances from the first to the second and from the second to the third. We proved that an equilateral triangle can be constructed on any given line segment. And we proved that two distinct lines cannot have more than one point in common.
So students, this completes our journey through Euclid's geometry. You have learned about the historical development of geometry, the concept of undefined terms, the difference between axioms and postulates, Euclid's seven axioms, his five postulates, and how to use them to prove important results. This foundation will help you understand the geometry you will learn in the coming chapters. Always remember that geometry is not just about shapes and sizes — it is about logical thinking, reasoning, and proving statements based on fundamental truths. Thank you for your attention, and keep practicing!