Euclid’s Definitions, Axioms and Postulates

Euclid assumed some properties which were actually ‘obvious universal truth’. He had bifurcated them in two types: Axioms and postulates.

Axioms

Some common notions which are used in mathematics but not directly related to mathematics are called Axioms.

Some of the Axioms are-

1. If the two things are equal to a common thing then these are equal to one another.

If p = q and s = q, then p = s.

2. If equals are added to equals, the wholes are equal.

If p = q and we add s to both p and q then the result will also be equal.

p + s = q + s

3. If equals are subtracted from equals, the remainders are equal.

This is same as above, if p = q and we subtract the same number from both then the result will be the same.

p – s = q – s

4. Things which coincide with one another are equal to one another.

If two figures fit into each other completely then these must be equal to one another.

5. The whole is greater than the part.

This circle is divided into four parts and each part is smaller than the whole circle. This shows that the whole circle will always be greater than any of its parts.

6. Things which are double of the same things are equal to one another.

This shows that this is the double of the two semicircles, so the two semicircles are equal to each other.

7. Things which are halves of the same things are equal to one another. This is the vice versa of the above axiom.

Postulates

The assumptions which are very specific in geometry are called Postulates.

There are five postulates by Euclid-

1. A straight line may be drawn from any one point to any other point.

This shows that a line can be drawn from point A to point B, but it doesn’t mean that there could not be other lines from these points.

2. A terminated line can be produced indefinitely.

This shows that a line segment which has two endpoints can be extended indefinitely to form a line.

3. A circle can be drawn with any centre and any radius.

This shows that we can draw a circle with any line segment by taking one of its points as a centre and the length of the line segment as the radius. As we have AB line segment, in which we took A as the centre and the AB as the radius of the circle to form a circle.

4. All right angles are equal to one another.

As we know that a right angle is equal to 90° and all the right angles are congruent because if any angle is not 90° then it is not a right angle.

As in the above figure \mathrm{\angle DCA =\angle DCB =\angle HE =\angle HGF= 90^o}

5. Parallel Postulate

If there is a line segment which passes through two straight lines while forming two interior angles on the same side whose sum is less than 180°, then these two lines will definitely meet with each other if extended on the side where the sum of two interior angles is less than two right angles.

And if the sum of the two interior angles on the same side is 180° then the two lines will be parallel to each other.

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