Page 1
60 MATHEMA TICS
CHAPTER 5
INTRODUCTION TO EUCLID’S GEOMETRY
5.1 Introduction
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. This branch of mathematics was studied in various
forms in every ancient civilisation, be it in Egypt, Babylonia, China, India, Greece,
the Incas, etc. The people of these civilisations faced several practical problems
which required the development of geometry in various ways.
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).Y ou 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.
Fig. 5.1 : A T runcated Pyramid
Rationalised 2023-24
Page 2
60 MATHEMA TICS
CHAPTER 5
INTRODUCTION TO EUCLID’S GEOMETRY
5.1 Introduction
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. This branch of mathematics was studied in various
forms in every ancient civilisation, be it in Egypt, Babylonia, China, India, Greece,
the Incas, etc. The people of these civilisations faced several practical problems
which required the development of geometry in various ways.
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).Y ou 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.
Fig. 5.1 : A T runcated Pyramid
Rationalised 2023-24
INTRODUCTION TO EUCLID’S GEOMETRY 61
In the Indian subcontinent, the excavations at Harappa and Mohenjo-Daro, etc.
show that the Indus Valley Civilisation (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 length : breadth : thickness,
of the bricks was found to be 4 : 2 : 1.
In ancient India, the Sulbasutras (800 BCE to 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 V edic rites. The location
of the sacred fires had to be in accordance to 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.
These examples show that 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 (see Appendix 1).
A Greek mathematician, Thales is credited with giving the
first known proof. This proof was of the statement that a circle
is bisected (i.e., cut into two equal parts) by its diameter. One of
Thales’ most famous pupils was Pythagoras (572 BCE), whom
you 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,
Thales
(640 BCE – 546 BCE)
Fig. 5.2
Rationalised 2023-24
Page 3
60 MATHEMA TICS
CHAPTER 5
INTRODUCTION TO EUCLID’S GEOMETRY
5.1 Introduction
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. This branch of mathematics was studied in various
forms in every ancient civilisation, be it in Egypt, Babylonia, China, India, Greece,
the Incas, etc. The people of these civilisations faced several practical problems
which required the development of geometry in various ways.
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).Y ou 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.
Fig. 5.1 : A T runcated Pyramid
Rationalised 2023-24
INTRODUCTION TO EUCLID’S GEOMETRY 61
In the Indian subcontinent, the excavations at Harappa and Mohenjo-Daro, etc.
show that the Indus Valley Civilisation (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 length : breadth : thickness,
of the bricks was found to be 4 : 2 : 1.
In ancient India, the Sulbasutras (800 BCE to 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 V edic rites. The location
of the sacred fires had to be in accordance to 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.
These examples show that 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 (see Appendix 1).
A Greek mathematician, Thales is credited with giving the
first known proof. This proof was of the statement that a circle
is bisected (i.e., cut into two equal parts) by its diameter. One of
Thales’ most famous pupils was Pythagoras (572 BCE), whom
you 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,
Thales
(640 BCE – 546 BCE)
Fig. 5.2
Rationalised 2023-24
62 MATHEMA TICS
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.
In this chapter, we shall discuss Euclid’s approach
to geometry and shall try to link it with the present day
geometry.
5.2 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 (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 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 :
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 (325 BCE – 265 BCE)
Fig. 5.3
Rationalised 2023-24
Page 4
60 MATHEMA TICS
CHAPTER 5
INTRODUCTION TO EUCLID’S GEOMETRY
5.1 Introduction
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. This branch of mathematics was studied in various
forms in every ancient civilisation, be it in Egypt, Babylonia, China, India, Greece,
the Incas, etc. The people of these civilisations faced several practical problems
which required the development of geometry in various ways.
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).Y ou 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.
Fig. 5.1 : A T runcated Pyramid
Rationalised 2023-24
INTRODUCTION TO EUCLID’S GEOMETRY 61
In the Indian subcontinent, the excavations at Harappa and Mohenjo-Daro, etc.
show that the Indus Valley Civilisation (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 length : breadth : thickness,
of the bricks was found to be 4 : 2 : 1.
In ancient India, the Sulbasutras (800 BCE to 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 V edic rites. The location
of the sacred fires had to be in accordance to 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.
These examples show that 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 (see Appendix 1).
A Greek mathematician, Thales is credited with giving the
first known proof. This proof was of the statement that a circle
is bisected (i.e., cut into two equal parts) by its diameter. One of
Thales’ most famous pupils was Pythagoras (572 BCE), whom
you 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,
Thales
(640 BCE – 546 BCE)
Fig. 5.2
Rationalised 2023-24
62 MATHEMA TICS
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.
In this chapter, we shall discuss Euclid’s approach
to geometry and shall try to link it with the present day
geometry.
5.2 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 (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 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 :
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 (325 BCE – 265 BCE)
Fig. 5.3
Rationalised 2023-24
INTRODUCTION TO EUCLID’S GEOMETRY 63
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 compared 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.
Rationalised 2023-24
Page 5
60 MATHEMA TICS
CHAPTER 5
INTRODUCTION TO EUCLID’S GEOMETRY
5.1 Introduction
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. This branch of mathematics was studied in various
forms in every ancient civilisation, be it in Egypt, Babylonia, China, India, Greece,
the Incas, etc. The people of these civilisations faced several practical problems
which required the development of geometry in various ways.
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).Y ou 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.
Fig. 5.1 : A T runcated Pyramid
Rationalised 2023-24
INTRODUCTION TO EUCLID’S GEOMETRY 61
In the Indian subcontinent, the excavations at Harappa and Mohenjo-Daro, etc.
show that the Indus Valley Civilisation (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 length : breadth : thickness,
of the bricks was found to be 4 : 2 : 1.
In ancient India, the Sulbasutras (800 BCE to 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 V edic rites. The location
of the sacred fires had to be in accordance to 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.
These examples show that 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 (see Appendix 1).
A Greek mathematician, Thales is credited with giving the
first known proof. This proof was of the statement that a circle
is bisected (i.e., cut into two equal parts) by its diameter. One of
Thales’ most famous pupils was Pythagoras (572 BCE), whom
you 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,
Thales
(640 BCE – 546 BCE)
Fig. 5.2
Rationalised 2023-24
62 MATHEMA TICS
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.
In this chapter, we shall discuss Euclid’s approach
to geometry and shall try to link it with the present day
geometry.
5.2 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 (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 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 :
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 (325 BCE – 265 BCE)
Fig. 5.3
Rationalised 2023-24
INTRODUCTION TO EUCLID’S GEOMETRY 63
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 compared 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.
Rationalised 2023-24
64 MATHEMA TICS
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 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.
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.
Fig. 5.4
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.
Rationalised 2023-24
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