it tells about Euclid's works and lemma.

2. Born within the range of 325 B.C in Greece and died around 265
B.C. Euclid most likely came from affluent family because it was
widely known that he actually enrolled and finished from the school
of Plato in the old Greece. Following his education, he got teaching
appointment in Alexandria, Egypt. He was in there when he wrote his
popular book “The elements”. To buttress this fact Euclid of
Alexandria is usually mistaken or confused as Euclid of Megara,
another Socratic philosopher. His book “The elements” is what really
distinguished Euclid from the rest. Euclidean Geometry is the content
of the book and is very useful in the elementary and middle class.

3. GEOMETRY AROUND US
 Our daily life is filled with geometry—the pure
mathematics of points, lines, curves and surfaces. We
can observe various shapes and angles in the objects
that surround us. Observe, for example, this table and
its rectangular surface; the boomerang and its angular
shape; the bangle and its circular shape.
Euclid, an ancient Greek mathematician, observed the
various types of objects around him and tried to define
the most basic components of those objects. He
proposed twenty-three definitions based on his studies
of space and the objects visible in daily life. Let us go
through this lesson to learn each of Euclid’s definitions.
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Euclid`s Geometry

4. PREFACE
Euclidean geometry is a mathematical system attributed to the
Alexandrian Greek mathematician Euclid , which he described in his
textbook on geometry : the Elements Euclid's method consists in
assuming a small set of intuitively appealing axioms , and deducing
many other propositions (theorems ) from these. Although many of
Euclid's results had been stated by earlier mathematicians, Euclid
was the first to show how these propositions could fit into a
comprehensive deductive and logical system . The Elements begins
with plane geometry, still taught in secondary school as the first
axiomatic system and the first examples of formal proof . It goes on
to the solid geometry of three dimensions . Much of the Elements
states results of what are now called algebra and number theory
,explained in geometrical language .This entire project aims
at the explanation of the very complicated Euclid geometry , so that the
students are able to see and understand it in a better way.
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Euclid`s Geometry

5. INTRODUCTION TO
EUCLID`S GEOMETRY
Euclid`s Geometry
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Mathematics presentation

6. CONTENTS
 Definitions of Euclid
 Introduction to axioms
 Axioms-
 Axiom i and ii
 Axiom iii
 Axiom iv and v
 Axiom vi and vii
 Introduction to
Postulates
 Postulates-
 Postulate 1
 Postulate 2
 Postulate 3
 Postulate 4
 Postulate 5
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Euclid`s Geometry

7. DEFINITIONS OF EUCLID
Euclid gave the definitions of a few very basic attributes of objects that are normally around us. These
definitions are listed below.
 1. A point is that which has no part.
 2. A line is a breadth-less length.
 3. The extremities of a line are called points.
 4. A straight line is one that 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 one that lies evenly with the straight lines on itself.
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Euclid`s Geometry

8.  8. A plane angle is the inclination to each other of two lines in a
plane, which meet each other and do not lie in a straight line.
 9. When the lines containing the angle are straight, the angle is
called rectilinear.
 10. When a straight line set up on another straight line makes the
adjacent angles equal to each other, each of the equal angles is
right and the straight line standing on the other is called a
perpendicular to that on which it stands.
 11. An obtuse angle is an angle greater than the right angle.
 12. An acute angle is an angle less than the right angle.
 13. A boundary points out the limit or extent of something.
 14. A figure is that which is contained by any boundary or
boundaries.
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Euclid`s Geometry

9. INTRODUCTION TO AXIOMS
 Euclid’s Axioms
 Euclid assumed certain properties to be universal truths that
did not need to be proved. He classified these properties as
axioms and postulates. The properties that were not specific
to geometry were referred to as common notions or axioms.
 He compiled all the known mathematical works of his time into
the Elements. Each book of the Elements contains a series of
propositions or theorems, varying in number from about ten to
hundred. These propositions or theorems are preceded by
definitions. In Book I, twenty-three definitions are followed by
five postulates. Five common notions or axioms are listed
after the postulates.
 In this lesson, we will study some of Euclid’s axioms.
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Euclid`s Geometry

10. AXIOMS:
 Things that are equal to the same thing are also
equal to one another (Transitive property of
equality).
 If equals are added to equals, then the wholes are
equal.
 If equals are subtracted from equals, then the
remainders are equal.
 Things that coincide with one another are equal to
one another (Reflexive Property).
 The whole is greater than the part.
Euclid`s Geometry
10

11. AXIOMS I AND II
 Let us start with the first axiom which states that things that
are equal to the same thing are also equal to one another.
 Let us suppose the area of a rectangle is equal to the area
of a triangle and the area of that triangle is equal to the area
of a square. Then, according to the first axiom, the area of
the rectangle is equal to the area of the square. Similarly, if
a = b and b = c, then we can say that a = c.
 Now, the second axiom states that if equals are added to
equals, then the wholes are equal.
 Let us take a line segment AD in which AB = CD.
 Let us add BC to both sides of the above relation (‘equals
Euclid`s Geometry
are added’). Then, according to the second axiom, we can
say that AB + BC = CD + BC, i.e., AC = BD. 11

12. AXIOM III
 The third axiom states that if equals are subtracted from
equals, then the remainders are equal.
 Let us consider the following rectangles ABCD and PQRS.
 Suppose the areas of the rectangles are equal. Now, let us
remove a triangle XYZ (as shown in the figure) from each
rectangle. Then, according to the third axiom, we can say that
the area of the remaining portion of rectangle ABCD is equal
to the area of the remaining portion of rectangle PQRS.
Euclid`s Geometry
12

13. AXIOM IV AND V
 The fourth axiom states that things that coincide with one another are
equal to one another.
 This axiom is sometimes used in geometrical proofs.
 Let us consider a point Q lying between points P and R of a line
segment PR, as is shown in the figure.
 We can see that (PQ + QR) coincides with the line segment PR. So,
as per the fourth axiom, we can say that PQ + QR = PR.
 Now, the fifth axiom states that the whole is greater than the part.
 Let us again consider the line segment PR shown above. We can
see that PQ is a part of PR. So, as per the fifth axiom, we can say
that PR (i.e., the whole) is greater than PQ (i.e., the part).
Mathematically, we write it as PR > PQ.
Euclid`s Geometry
13

14. AXIOM VI AND VII
 The sixth and seventh axioms are interrelated. The former states that things that are
double of the same things are equal to one another, while the latter states that things that
are halves of the same things are equal to one another.
 Let us consider two identical circles with radii r1 and r2. Also, suppose their diameters
are d1 and d2 respectively.
 As the circles are identical, their radii are equal.
 ∴ r1 = r2
 Now, as per the sixth axiom, we can say that 2r1 = 2r2
 ∴ d1 = d2
 Hence, we can say that if two circles have equal radii, then their diameters are also
equal.
 Now, instead of taking the radii as equal, let us say that the diameters of the two circles
are equal. Then, as per the seventh axiom, we can say that the radii of the two circles are
also equal.
Euclid`s Geometry
14

15. INTRODUCTION TO POSTULATES
 Certain things are considered universal truths that
need not be proved. Consider, for example, the
following: the sun rises from the east; Sunday
comes after Saturday; March has 31 days. These
things are universally true; hence, they do not need
to be proven.
Similarly, certain geometrical properties are
regarded as universal truths. Euclid identified and
presented such properties in the Elements. The
properties specific to geometry were classified by
him as postulates. In Book I, twenty-three
definitions are followed by five postulates. Let us
learn these postulates in this lesson.
Euclid`s Geometry
15

16. POSTULATES
 "To draw a straight line from any point to any
point."
 "To produce [extend] a finite straight line
continuously in a straight line."
 "To describe a circle with any centre and distance
[radius]."
 "That all right angles are equal to one another."
 The parallel postulates : "That, if a straight line
falling on two straight lines make the interior
angles on the same side less than two right
angles, the two straight lines, if produced
indefinitely, meet on that side on which are the
angles less than the two right angles."
Euclid`s Geometry
16

17. 1ST POSTULATE
 Postulate 1: A straight line may be drawn from any point to any other point.
Proof: Finally proved only yesterday, we must refer to the third and second postulate in order to fully prove this one. In order
to prevent accusations of lack of rigor, I will use the still incomplete third postulate only in those cases where it may be
applied.
Take any two collinear points A and B, where collinear means it is possible to draw a straight line between them. It is possible
therefore to draw a straight line between them.
Now any points on the line AB must also be collinear, for otherwise a straight line could not have been drawn. Hence, it is
also possible to draw a line from A to any point upon the line.
Now, let us rotate the line, such that the collinear point A is the centre of the circle so produced.
Now, it is possible to draw a straight line from A to any point in the circle. This is because the radius of the circle is a straight
line, and upon rotation, it covers all the points in the circle, implying that a straight line can be drawn from all the points
covered by the radius to the center of the circle, which is A.
Therefore all points in the circle are collinear to A i.e. they produce a straight line to A.
It is easy to show that all points in the plane are collinear: merely extend the radius infinitely, so the resultant circle
encompasses the entire region.
Repeating the above for any point in the circle, we see that it is possible to draw a straight line from that point to any other
point in its circle, and so on.
From the information above, we can deduce that all points are collinear to each other, or
It is possible to draw a straight line from any one point to any other point.
Hence proved.
Euclid`s Geometry
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18. 2ND POSTULATE
 Postulate 2: A finite straight line may be extended indefinitely.
Proof: There are an infinite number of points in a region.
This implies that there are an infinite number of collinear
points, as any operation with infinity that does not involve
another infinity results in infinity. By collinear, I mean points
between which a straight line may be drawn. ( I clarify this in
order to prevent accusations of using a circular argument with
the first postulate)
This implies that a line may be extended infinitely.
Euclid`s Geometry
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19. 3RD POSTULATE
 Postulate 3: A circle may be drawn with any center and any radius.
Note: By the term "collinear", I mean that it is possible to draw straight line from it
another specific point.
Proof: This is a little trickier to prove, so I divided the problem down into two parts.
I will first prove that a circle may have any radius.
Taking point A as centre, we may look at the radius as a line. By Postulate 2, we
know that line may be extended indefinitely.
Therefore, the radius may be extended indefinitely.
This implies that a circle may have any radius.
The second part is to prove that a circle may have any center.
Taking any collinear point, we see that it is possible to draw a straight line between
this and any other straight line.
By rotating the line by 360 degrees, we obtain a circle.
This implies that any collinear point may be the center of a circle, as the straight
line that can be drawn may be considered a radius, and rotating the radius
produces a circle.
Euclid`s Geometry
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20. 4TH POSTULATE
 Postulate 4: All right angles are equal to each other.
Proof: Let us assume that this is not true, and all right angles are not
equal to each other.
This instantly leads to a contradiction, as it implies that a triangle may
have more than one right angle.
Therefore, by reduction ad absurdum, we see that all right angles must be
equal to each other.
Another proof is by looking at the definition of a right angle. A right angle
is any angle equal to 90 degrees.
We know that 90 = 90 = 90 ...
We see therefore that all right angles are equal to 90 degrees and as 90
degrees is equal to 90 degrees,
this implies that all right angles are equal to each other.
Euclid`s Geometry
Hence proved. 20

21. 5TH POSTULATE
 Postulate 5: If a straight line falling on two straight lines makes the interior angles
on the same side less than two right angles, the two straight lines, if produced
indefinitely, meet on that side on which the angles are less than two right angles.
Proof: This means to say that the two lines meet on the side whose sum is less
than 180 degrees.
From the diagram, we see that if we extend the lines indefinitely, we eventually get
a triangle.
It has been proved that the sum of the angles of a triangle sum to 180 degrees.
This implies that the two angles formed by the third line which goes through the
other two lines) cannot be equal to 180 degrees, as it would then violate the angle
sum property of that triangle.
The triangle is only possible if the two angles are not equal to 180 degrees or
more. This implies that the lines may only meet on those sides where the angles
together sum up to less than 180 degrees.
Therefore, this implies that no line which produces an obtuse or acute angle may be
parallel to l. Therefore, the only other angle possible is a right angle, which taken in
conjunction with the other interior angle forms 180 degrees, implying that any line
which produces 90 degrees with the perpendicular is always parallel to line l.
Euclid`s Geometry
21

23. Euclid’s division algorithm is a technique to compute the Highest Common Factor
(HCF) of two given positive integers. Recall that the HCF of two positive integers a
and b is the largest positive integer d that divides both a and b.
Let us see how the algorithm works, through an example first. Suppose we need
to find the HCF of the integers 455 and 42. We start with the larger integer, that is,
455. Then we use Euclid’s lemma to get
455 = 42 × 10 + 35
Now consider the divisor 42 and the remainder 35, and apply the division lemma
to get
42 = 35 × 1 + 7
Now consider the divisor 35 and the remainder 7, and apply the division lemma
to get
35 = 7 × 5 + 0
Notice that the remainder has become zero, and we cannot proceed any further.
We claim that the HCF of 455 and 42 is the divisor at this stage, i.e., 7. You can easily
verify this by listing all the factors of 455 and 42. Why does this method work? It works
because of the following result.

24. To obtain the HCF of two positive integers, say c and d, with c > d, follow
the steps below:
Step 1 : Apply Euclid’s division lemma, to c and d. So, we find whole numbers, q
and
r such that c = dq + r, 0 ≤ r < d.
Step 2 : If r = 0, d is the HCF of c and d. If r ≠ 0, apply the division lemma to d and
r.
Step 3 : Continue the process till the remainder is zero. The divisor at this stage will
be the required HCF
This algorithm works because HCF (c, d) = HCF (d, r) where the symbol
HCF (c, d) denotes the HCF of c and d, etc.

25. Example 1 : Use Euclid’s algorithm to find the HCF of 4052 and 12576.
Solution :
Step 1 : Since 12576 > 4052, we apply the division lemma to 12576 and 4052, to get
12576 = 4052 × 3 + 420
Step 2 : Since the remainder 420 ≠ 0, we apply the division lemma to 4052 and 420, to
get
4052 = 420 × 9 + 272
Step 3 : We consider the new divisor 420 and the new remainder 272, and apply the
division lemma to get
420 = 272 × 1 + 148
We consider the new divisor 272 and the new remainder 148, and apply the division
lemma to get
272 = 148 × 1 + 124
We consider the new divisor 148 and the new remainder 124, and apply the division
lemma to get
148 = 124 × 1 + 24
We consider the new divisor 124 and the new remainder 24, and apply the division
lemma to get
124 = 24 × 5 + 4
We consider the new divisor 24 and the new remainder 4, and apply the division lemma to get

26. 24 = 4 × 6 + 0
The remainder has now become zero, so our procedure stops. Since the divisor at this
stage is 4, the HCF of 12576 and 4052 is 4.
Notice that 4 = HCF (24, 4) = HCF (124, 24) = HCF (148, 124) =
HCF (272, 148) = HCF (420, 272) = HCF (4052, 420) = HCF (12576, 4052).
Euclid’s division algorithm is not only useful for calculating the HCF of very
large numbers, but also because it is one of the earliest examples of an algorithm that a computer
had been programmed to carry out.

27. 1. Euclid’s division lemma and algorithm are so closely
interlinked that people often
call former as the division algorithm also.
2. Although Euclid’s Division Algorithm is stated for only
positive integers, it can be
extended for all integers except zero, i.e., b ≠ 0.
However, we shall not discuss this
aspect here.