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Euclidean GeometryEuclid’s Axioms

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Greek mathematicians realised that to write formal proofs, you need some sort of starting point: simple, intuitive statements, that everyone agrees are true. These are called axioms (or postulates).

A key part of mathematics is combining different axioms to prove more complex results, using the rules of logic.

The Greek mathematician Euclid of Alexandria, who is often called the father of geometry, published the five axioms of geometry:

Euclid of Alexandria

First Axiom
You can join any two points using exactly one straight line segment.

Second Axiom
You can extend any line segment to an
infinite line.

Third Axiom
Given a point P and a distance r, you can draw a circle with centre P and radius r.

Fourth Axiom
Any two right angles are congruent.

Fifth Axiom
Given a line L and a point P not on L, there is exactly one line through P that is parallel to L.

Each of these axioms looks pretty obvious and self-evident, but together they form the foundation of geometry, and can be used to deduce almost everything else. According to none less than Isaac Newton, “it’s the glory of geometry that from so few principles it can accomplish so much”.

Euclid published the five axioms in a book “Elements”. It is the first example in history of a systematic approach to mathematics, and was used as mathematics textbook for thousands of years.

One of the people who studied Euclid’s work was the American President Thomas Jefferson. When writing the Declaration of Independence in 1776, he wanted to follow a similar approach. He begins by stating a few, simple “axioms” and then “proves” more complex results:

“We hold these truths to be self-evident: that all men are created equal, that they are endowed by their Creator with certain unalienable Rights, that among these are Life, Liberty and the pursuit of Happiness.”

We, therefore … declare, that these United Colonies are, and of right ought to be, free and independent states.”

This is just one example where Euclid’s ideas in mathematics have inspired completely different subjects.