Euclid’s Axioms

Euclid's axioms are basic assumptions that form the foundations of Euclidean geometry. Euclid's axioms were proposed by the ancient Greek mathematician Euclid of Alexandria in his geometrical work ‘Elements’. These axioms explain how points, lines, and shapes built the entire structure of geometry. In this guide, you will learn about the axioms along with examples to help understand their real-life applications.

Table of Contents

• What is an Axiom?
• What are Euclid's Axioms

What is an Axiom

An axiom is a self-evident or universally recognised truth. They are statements that are assumed to be true without requiring formal proof. Axioms apply to all of mathematics, not just geometry. The axioms deal with general mathematical concepts (equality, addition, subtraction, ...), while the postulates deal specifically with geometry.

What are Euclid's Axioms

Axioms are universal truths that do not require proofs. Euclid stated seven axioms that laid the foundations of geometry that we know now.

Here are the seven axioms stated by Euclid:

• Axiom 1: Things which are equal to the same thing are equal to one another.

If p = q and q = r, then p = r

• Axiom 2: If equals are added to equals to equals, the wholes are equal.

If ∠A = ∠B and ∠C is added to both, then  ∠A + ∠C = ∠B + ∠C 

• Axiom 3: If equals are subtracted from equals, the remainders are equal.

If ∠A = ∠B, ∠C is subtracted from both, then ∠A - ∠C = ∠B - ∠C.

• Axiom 4: Things that coincide with one another are equal to one another.
  
   
  
  The ∆X coincides exactly with the ∆Y. Hence, ∆X = ∆Y.

• Axiom 5: The whole is greater than the part.
  
  
  
  AB > AC.

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

  If radii r1  = r2 , then diameters d1 = d2

• Axiom 7: Things which are halves of the same things are equal to one another.
  If diameters d1 = d2, then radii r1 = r2.