Axioms of Euclid

Euclid's Postulates

Euclid was one of the greatest mathematicians known as the "father of geometry". Geometry has a depth meaning by considering geo is earth and metry is the measure. He wrote the book is named “Euclid's Elements”. With the help of this book, we can analyze and use the Euclidean geometry which is stated flat surfaces and includes points, lines, and planes.

Before going to axioms, I want to talk about what the point is. A point can be explained as the location of something in the universe, any place in the universe. The point does not have to be measurable, because as we move from nothing to existence, we benefit from the point indicating the location of a place with a holistic explanation.

1. A straight-line segment can be drawn joining any two points.

2. Any straight-line segment can be extended indefinitely in a straight line.

3. Given any straight-line segment, a circle can be drawn having the segment as radius and one endpoint as center.

4. All right angles are congruent.

5. If two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough. This postulate is equivalent to what is known as the parallel postulate.

The 5th axiom is a bit problematic compared to the other 4 axioms. The reason for this is that it is not clear whether these two lines, which are extended forever, will intersect at one point or not. Also, if this problem is removed, since this axiom will become provable, it becomes a theorem and ceases to be postulate.