In this section of Euclid Geometry we will discuss Euclid's axioms,postulate,division lemma etc.
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.
In Euclid Geometry there are given some definitions, which are as follows :
1. A point is that which has no part.
2. A line is breadthless (no width)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.
Starting with Euclid's 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. Some of Euclid’s axioms,are given below :
(1) Things which are equal to the same thing are equal to one another. Example : 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.
(2) If equals are added to equals, the wholes are equal. Example : For example, a line cannot be added to a rectangle,
nor can an angle be compared to a pentagon.
(3) If equals are subtracted from equals, the remainders are equal. Example : For example, a line cannot be subtracted to a rectangle,nor can an angle be compared to a pentagon.
(4) Things which coincide with one another are equal to one another. Example : Segment AB = Segment BA ; ∠A = ∠A .
(5) The whole is greater than the part. 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.
(6) Things which are double of the same things are equal to one another. Example : If 2x = 2y then x = y.
(7) Things which are halves of the same things are equal to one another. Example : If ½ x = ½y then x = y.
Euclid Geometry contains some 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 center 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.