Seven Axioms
Seven Axioms
Euclid seven axioms :
These 'common notions' refer to magnitidues of some kind. The first common notion could be applied to plane figures. For example, if an area of a triangle equals the area of a rectangle and the areas of the rectangle equals that of a square, then the area of the triangle also equal the area of the square.
Magnitudes of the same kind can be compared and added, but magnitudes of different kinds cannot be added to a rectangle, nor can angle be compared to a pentagon.
The 4th axiom given above seems to say that if two things are identical (that is, they are the same), then they are equal. In other words, everything equals itself. It is the justification of the principle of superposition. Axiom (5) gives us the definition of greater than. For example , if a quantity B is a part of another quantity A, then A can be written as the sum of B and so third quantity C. Symbolically, A>B means that there is some C such that A = B + C.
Example : Two district lines cannot have more than one point in common.
Proof: Here we are given two lines L and m. we need to prove that they have only one point in common.
For the time being, let us suppose that the two lines intersect in two distinct points, say P and Q. So, youhave two lines passing through two distinct points P and Q . But this assumption clashes with the axiom that only one line can pass throught two distinct. points. So, the assumption that we started with , that two lines can pass thorugh two distinct points is wrong.
From this, what can we conclude? We are forced to conclude that two distinct lines cannot have more than one point in common.
Students / Parents Reviews [10]
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