Introduction
The word ‘geometry’ comes form the Greek words ‘geo’, meaning the ‘earth’, and ‘metrein’, meaning ‘to measure’. Pythagoras and his group discovered many geometric properties and developed the theory of geometry to a great extent. This process continued till 300 BC. At that time Euclid, a teacher of mathematics at Alexandria in Egypt, collected all the known work and arranged it in his famous treatise,
Euclid’s Definitions
He began his exposition by listing 23 definitions in Book 1 of the ‘Elements’ 7 definitions 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.
Euclid’s axioms
(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.
Euclid’s five postulates
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.
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.
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.
Equivalent Versions of Euclid’s Fifth Postulate
‘For every line l and for every point P not lying on l, there exists a unique line m passing through P and parallel to l’.
Summary
In this chapter, you have studied the following points:
1. Though Euclid defined a point, a line, and a plane, the definitions are not accepted by
mathematicians. Therefore, these terms are now taken as undefined.
2. Axioms or postulates are the assumptions which are obvious universal truths. They are not
proved.
3. Theorems are statements which are proved, using definitions, axioms, previously proved
statements and deductive reasining.
4. Some of Euclid’s axioms were :
(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.
5. 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.
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.
6. Two equivalent versions of Euclid’s fifth postulate are:
(i) ‘For every line l and for every point P not lying on l, there exists a unique line m
passing through P and parallel to l’.
(ii) Two distinct intersecting lines cannot be parallel to the same line.
7. All the attempts to prove Euclid’s fifth postulate using the first 4 postulates failed. But they
led to the discovery of several other geometries, called non-Euclidean geometries.
The word ‘geometry’ comes form the Greek words ‘geo’, meaning the ‘earth’, and ‘metrein’, meaning ‘to measure’. Pythagoras and his group discovered many geometric properties and developed the theory of geometry to a great extent. This process continued till 300 BC. At that time Euclid, a teacher of mathematics at Alexandria in Egypt, collected all the known work and arranged it in his famous treatise,
He began his exposition by listing 23 definitions in Book 1 of the ‘Elements’ 7 definitions 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.
Euclid’s axioms
(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.
Euclid’s five postulates
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.
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.
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.
Equivalent Versions of Euclid’s Fifth Postulate
‘For every line l and for every point P not lying on l, there exists a unique line m passing through P and parallel to l’.
Summary
In this chapter, you have studied the following points:
1. Though Euclid defined a point, a line, and a plane, the definitions are not accepted by
mathematicians. Therefore, these terms are now taken as undefined.
2. Axioms or postulates are the assumptions which are obvious universal truths. They are not
proved.
3. Theorems are statements which are proved, using definitions, axioms, previously proved
statements and deductive reasining.
4. Some of Euclid’s axioms were :
(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.
5. 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.
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.
6. Two equivalent versions of Euclid’s fifth postulate are:
(i) ‘For every line l and for every point P not lying on l, there exists a unique line m
passing through P and parallel to l’.
(ii) Two distinct intersecting lines cannot be parallel to the same line.
7. All the attempts to prove Euclid’s fifth postulate using the first 4 postulates failed. But they
led to the discovery of several other geometries, called non-Euclidean geometries.
No comments:
Post a Comment