NCERT Class 9 Mathematics Textbook for Blind Students made Screen Readable by Professor T K Bansal.

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
(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 We carefully study these definitions, we 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 We define ‘a part’ to be that which occupies ‘area’, again ‘an area’ needs to be defined. So, to define one thing, we need to define many other things, and We 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 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 :

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 lines. 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.

How many lines passing through P will also pass through Q (see Fig. 5.4)? Only one, that is, the line PQ.

Fig. 5.4

 

How many lines passing through Q will 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.

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).

Fig. 5.5

 

Postulate 3

A circle can be drawn with any centre and any radius.

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 Line PQ. Therefore, the lines AB and CD will eventually intersect on the left side of Line PQ.

Fig. 5.6

 

A brief look at the five postulates brings to our notice that Postulate 5 is far more complex than the other four postulates. 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.

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/validity is checked afterwards. If it is true, then it is accepted as a ‘Postulate’ (as a noun).

A system of axioms is called consistent (see Appendix 1), 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, we 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:

Example 1

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.

Fig. 5.7

 

Solution :

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.

Example 2

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

Solution :

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

Fig. 5.8 (i)

 

Here, we need to do some construction. Using Euclid’s Postulate 3, We 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.

Fig. 5.8 (ii)

 

The two circles meet at some point, say C. Now, draw the line segments AC and BC to form triangle ABC [see Fig. 5.8 (iii)].

Fig. 5.8 (iii)

 

So, we 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, We 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.

Now we prove a theorem, which is frequently used in different results:

Theorem 5.1

Two distinct lines cannot have more than one point in common.

To Prove :

Fig. 5.8 (ii)

 

Here we are given two lines l and m. We need to prove that they have only one point in common.

 

Proof:
For the time being, let us suppose that the two lines intersect in two distinct points, say P and Q. So, we 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.

EXERCISE 5.1

Q1. Which of the following statements are true and which are false? Give reasons for your answers.
(i) Only one line can pass through a single point.
(ii) There are an infinite number of lines which pass through two distinct points.
(iii) A terminated line can be produced indefinitely on both the sides.
(iv) If two circles are equal, then their radii are equal.
(v) In Fig. 5.9, if AB = PQ and PQ = XY, then AB = XY.

Fig. 5.9

 

A1.
(i) False. This can be seen visually by the students.
(ii) False. This contradicts Axiom 5.1.
(iii) True. (Postulate 2)
(iv) True. If We superimpose the region bounded by one circle on that by the other, then they coincide with each other. So, their centres and boundaries coincide. Therefore, their radii will coincide.
(v) True. The first axiom of Euclid.

Q2. Give a definition for each of the following terms. Are there other terms that need to be defined first? What are they, and how might We define them?
(i) parallel lines
(ii) perpendicular lines
(iii) line segment
(iv) radius of a circle
(v) square

Q3. Consider two ‘ postulates ’ given below:
(i) Given any two distinct points A and B, there exists a third point C which is in between A and B.
(ii) There exist at least three points that are not on the same line.

Do these postulates contain any undefined terms?
Are these postulates consistent?
Do they follow from Euclid’s postulates? Explain.

A3. There are several undefined terms which the student should list. They are consistent, because they deal with two different situations -
(i) says that given two points A and B, there is a point C lying on the line in between them;
(ii) says that given A and B, we can take C not lying on the line through A and B.
These ‘postulates’ do not follow from Euclid’s postulates. However, they follow from Axiom 5.1.

Q4. If a point C lies between two points A and B such that AC = BC, then prove that AC = ½ AB. Explain by drawing the figure.

A4. AC = BC
So, AC + AC = BC + AC (Equals are added to equals)
i.e., 2AC = AB (BC + AC coincides with AB)
Therefore, AC = ½ AB

Q5. In Question 4, point C is called a mid-point of line segment AB. Prove that every line segment has one and only one mid-point.

A5. Make a temporary assumption that different points C and D are two mid-points of AB. Now, we show that points C and D are not two different points.

Q6. In Fig. 5.10, if AC = BD, then prove that AB = CD.

Fig. 5.10

 

A6. AC = BD (Given) (1)
AC = AB + BC (Point B lies between A and C) (2)
BD = BC + CD (Point C lies between B and D) (3)

Substituting (2) and (3) in (1), We get

AB + BC = BC + CD

So, AB = CD (Subtracting equals from equals)

Q7. Why is Axiom 5, in the list of Euclid’s axioms, considered a ‘universal truth’?
(Note that the question is not about the fifth postulate.)

A7. Since this is true for any thing in any part of the world, this is a universal truth.