Euclid's Axioms & Postulates | examples
History
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| Thales (640 BC – 546 BC) |
The word ‘Geometry’ is derived from the Greek words ‘Geo’
means ‘Earth’ and ‘Metrein’ means to ‘ To Measure’. 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 BC), 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 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 (325 BC – 265 BC) |
‘Elements’ into thirteen chapters, each called a book. These books influenced the whole world’s understanding of geometry for generations to come. Euclid listed 23 definition in book 1 of the ‘Elements’.
A Few the Euclid’s Definitions :
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.
Basic Geometrical Concepts:
Axioms: The basic facts which are taken for granted, without
proof, are called axioms.
Examples: 1. Doubles of
equals are equal.
2. The whole is greater than each of its parts
3. A line contains infinitely many points.
2. The whole is greater than each of its parts
3. A line contains infinitely many points.
Postulate: The basic facts related to geometry which are
taken for granted, without proof are called postulate.
Examples: 1. A
terminated line can be produced indefinitely.
2. All
right angles are equal to one another.
Statements: A sentence which can be judged to be true or
false is called a statement.
Examples: 1. The sum of
the angles of a triangle is 1800, is a true statement.
2. The sum of the angles of a quadrilateral is 1800, is a false
statement.
3. x
+ 10 > 12 is a sentence but not a statement.
Theorem: A statement that requires a proof, is called a
theorem. Establishing the truth of a theorem is known as proving the theorem.
Examples: 1. The sum of the angles of a triangle is 1800.
2. The sum of all the angles around a point is 3600.
Corollary: A statement whose truth can easily be deduced
from a theorem, is called its corollary.
Euclid’s Axioms:
(1) Things which are equal to the same thing are
equal to one another.
Example: If a = b and b = c, then a = c
(2) If equals are added to equals, the wholes are
equal.
Examples: If a = b and c = d
On adding
Then, a + c = b + d
(3)
If equals are subtracted from equals, the
remainders are equal.
Examples: If x = y and a = b
On subtracting
Then, x – a = y – b
(4)
Things which coincide with one another are equal
to one another.
Example: X coincide with X, hence X = X
(5)
The whole is greater than the part.
Example: Suppose earth is the part of whole
universe but universe is not a part of earth.
(6)
Things which are double of the same things are
equal to one another.
Example: If 2a = 2b
Then a = b
(7)
Things which are halves of the same things are
equal to one another.
Example: If
½ x = ½ y
Then x = y
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.
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