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INTRODUCTION

Euclid’s Definitions, Axioms and Postulates

Euclid’s Definitions, Axioms and Postulates

The word ‘geometry’ comes form the Greek words ‘geo’, meaning the ‘earth’, and ‘metrein’, meaning ‘to measure’. Geometry appears to have originated from the need for measuring land.

For example, whenever the river Nile 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 (see Fig. 5.1).

You know that 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.

In this chapter, we shall discuss Euclid’s approach to geometry and shall try to link it with the present day geometry.

For example, whenever the river Nile 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 (see Fig. 5.1).

You know that 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.

In this chapter, we shall discuss Euclid’s approach to geometry and shall try to link it with the present day geometry.

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.

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 (solids-surfaces-lines-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 summarized 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 :

1. A point is that which has no part.

2. A line is breadthless length.

3. The ends of a line are points.

4. A straight line is a line which lies evenly with the points on itself.

5. A surface is that which has length and breadth only.

6. The edges of a surface are lines.

7. A plane surface is a surface which lies evenly with the straight lines on itself.

If you carefully study these definitions, you find that some of the terms like part, breadth, length, evenly, etc. 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 Euclid Fig. 5.3

some geometric terms undefined. However, we do have a 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’.

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.

For details about axioms and postulates, refer to Appendix 1. Some of Euclid’s axioms, not in his order, are given below :

(1) Things which are equal to the same thing are equal to one another.

(2) If equals are added to equals, the wholes are equal.

(3) If equals are subtracted from equals, the remainders are equal.

(4) Things which coincide with one another are equal to one another.

(5) The whole is greater than the part.

(6) Things which are double of the same things are equal to one another.

(7) Things which are halves of the same things are equal to one another.

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 added to a rectangle, nor can an angle be compared to a pentagon.

The 4th 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. Axiom (5) 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.`

Now let us discuss Euclid’s five postulates. They are :

`bb"Postulate 1 : A straight line may be drawn from any one point to any other point."`

`"Note : "` 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:

`bb"Axiom 5.1 : Given two distinct points, there is a unique line that passes through them."`

How many lines passing through P also pass through Q (see Fig. 5.4)? 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.

`bb"Postulate 2 : A terminated line can be produced indefinitely."`

Note that what we call a line segment now-a-days 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 (see Fig. 5.5).

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

`bb"Postulate 4 : All right angles are equal to one another."`

`"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.

For example, the line PQ in Fig. 5.6 falls on lines AB and CD such that the sum of the interior angles 1 and 2 is less than 180° on the left side of PQ. Therefore, the lines AB and CD will eventually intersect on the left side of PQ.

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 (see Appendix 1). Because of its complexity, the fifth postulate will be given more attention in the next section.

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 (see Appendix 1). Because of its complexity, the fifth postulate will be given more attention in the next section.

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:

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 (solids-surfaces-lines-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 summarized 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 :

1. A point is that which has no part.

2. A line is breadthless length.

3. The ends of a line are points.

4. A straight line is a line which lies evenly with the points on itself.

5. A surface is that which has length and breadth only.

6. The edges of a surface are lines.

7. A plane surface is a surface which lies evenly with the straight lines on itself.

If you carefully study these definitions, you find that some of the terms like part, breadth, length, evenly, etc. 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 Euclid Fig. 5.3

some geometric terms undefined. However, we do have a 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’.

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.

For details about axioms and postulates, refer to Appendix 1. Some of Euclid’s axioms, not in his order, are given below :

(1) Things which are equal to the same thing are equal to one another.

(2) If equals are added to equals, the wholes are equal.

(3) If equals are subtracted from equals, the remainders are equal.

(4) Things which coincide with one another are equal to one another.

(5) The whole is greater than the part.

(6) Things which are double of the same things are equal to one another.

(7) Things which are halves of the same things are equal to one another.

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 added to a rectangle, nor can an angle be compared to a pentagon.

The 4th 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. Axiom (5) 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.`

Now let us discuss Euclid’s five postulates. They are :

`bb"Postulate 1 : A straight line may be drawn from any one point to any other point."`

`"Note : "` 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:

`bb"Axiom 5.1 : Given two distinct points, there is a unique line that passes through them."`

How many lines passing through P also pass through Q (see Fig. 5.4)? 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.

`bb"Postulate 2 : A terminated line can be produced indefinitely."`

Note that what we call a line segment now-a-days 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 (see Fig. 5.5).

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

`bb"Postulate 4 : All right angles are equal to one another."`

`"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.

For example, the line PQ in Fig. 5.6 falls on lines AB and CD such that the sum of the interior angles 1 and 2 is less than 180° on the left side of PQ. Therefore, the lines AB and CD will eventually intersect on the left side of PQ.

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 (see Appendix 1). Because of its complexity, the fifth postulate will be given more attention in the next section.

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 (see Appendix 1). Because of its complexity, the fifth postulate will be given more attention in the next section.

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:

Q 3270367216

If `A, B` and C are three points on a line, and `B` lies between `A` and `C` (see Fig. 5.7), then prove that `AB + BC = AC.`

Class 9 Chapter 5 Example 1

Class 9 Chapter 5 Example 1

In the figure given above, 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.

Q 3280367217

Prove that an equilateral triangle can be constructed on any given line segment.

Class 9 Chapter 5 Example 2

Class 9 Chapter 5 Example 2

In the statement above, a line segment of any length is given, say AB [see Fig. 5.8(i)].

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 [see Fig. 5.8(ii)]. 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 `Delta` ABC [see Fig. 5.8 (iii)].

So, you have to prove that this triangle is equilateral, i.e., `AB = AC = BC.`

Now, AB = AC, since they are the radii of the same circle (1)

Similarly, AB = BC (Radii of the same circle) (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, `Delta 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.

`bb"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.

`"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.