Euclid Axioms
Euclid Axioms
Euclid's Axioms:
Starting with his definitions, Euclid assumed certain properties, which were not to be proved. These assumptions are actually ' obvious universal truths' . He divided them into two types: axioms and postulates. He used the term 'postulate' for the assumptions that were specific to geometry. Common notions (often called axioms), on the other hand, were assumptions used throughtout mathematics and not specifically linked to geometry. Now -a- days 'postulates' and 'axioms' are terms that are used interchangeably and in the same sense. 'Postulates' is actually verb. When we say " let us postulate", we mean, "let us make some statement based on the observed phenomenon in the Universe". Its truth/ validity is checked afterwards. If it is true, then it is accepted as a 'postulate'.
A system of axioms is called consistent, if it is impossible to deduce from these axioms a statement that contradicts any axioms or previously proved statement. So, when any system of axioms is given, it needs to be ensured that the system is consistent.
After Euclid stated his postulates and axioms, he used them to prove other results. Then using these results, he proved some more results by applying deductive reasoning. The statements that were proved are called propositions or theorems. Euclid deduced 465 propositions in a logical chain using his axioms, postulates, definitions and theorems proved earlier in the chain. In the next few chapters on geometry, you will be using these axioms to prove some theorems.
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