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Euclid or Euclid of Alexandria was a great Greek Mathematician back in 300 BC. He is known as the Father of Geometry. Hes axioms and postulates are often necessary for proving mathematical problems. THE BASIC DIFFERENCE BETWEEN AN AXIOM AND A POSTULATE IS THAT AN AXIOM IS "AN ACCEPTED TRUTH" AND CANNOT BE PROVED WHEREAS A POSTULATE CAN BE BASED ON EXISTING AXIOMS. Here are his axioms -

• Things that are equal to the same thing are equal to one another.
• If equals are added to equals, the wholes are equal.
• If equals are subtracted from equals, the remainders are equal.
• Things that are coincident to each other are equal to one another.
• A whole is greater than the part.
• Things that are doubles of the same thing are equal to one another.
• Things that are halves of the same thing are equal to one another.

Here are his postulates -

• A straight line may be drawn from any point to any other point.
• A terminated line can be produced indefinitely.
• A circle may be drawn with any centre and any radius.
• All right angles are equal to each other.
• If a straight line falling on two other straight lines makes interior angle less than two right angles on one side, then these two straight lines produced indefinitely on the side where the sum of the interior angles is less than two right angles will meet at a point.

The fifth postulate can have alternatives,one such alternative being Playfairs axiom,named after Scottish Mathematician John Playfair in 1729. This is - If there is a line l and a point P not lying on line l, then there is an unique line m passing through p, which is parallel to l.