Euclid's fourth axiom, like all the others, is clearly a very simple assertion. It is how Euclid ingeniously utilizes these axioms, however, that holds importance.

The fourth axiom presented in Euclid’s magnum opus, Elements (13 books of brilliant geometric deduction), is often differentiated from the first, second, third and fifth axioms based on the fact that it is the only one of the five axioms not based on a construction.

What does this mean?

It means that, while the first and second postulates dealt with the basic principles behind the construction of a simple straight line, the third dealt with the construction of a circle, and the fifth dealt with the construction of two parallel lines, the fourth does not necessarily add any information that allows a new geometrical construction to be assembled.

So what does it do?

It merely gives a hard and fast rule.

The Axiom

All right angles are equal to one another.

A right angle, of course, is any angle equal to exactly 90 degrees (or, in other words a fully open angle of 180 degrees exactly divided into halves).

So any 90 degree angle (or any other two angles that share the feature of having the same degree) lying on a plane, is exactly the same as any other 90 degree angle.

This axiom, like the others, should seem obvious on first glance (and, to be honest, with further glances as well). It is one of the first simple truths that one learns in any geometry course – any two angles of the same degree are considered to be congruent, and thus any constructions based off these angles may be examined in light of this fact.

The Third Axiom as an Example of Euclid’s Purpose

This is exactly Euclid’s point in each of his five axioms. They are not intended, by any means, to break any new ground or to cause his readers to take any sort of notice. If they did happen to cause any sort of controversy, in fact, they would not have been serving their purpose (which, in the case of the fifth axiom, has actually become the case to a limited extent in recent centuries).

The purpose of the axioms, after all, was to provide a listings of some “unquestionables.” Statements that seemed impossible for anyone to disagree with, no matter how much they attempted to find fault with them.

This way, as Euclid delved into greater and greater proofs containing increasing degrees of complexity, each building upon the others which had come before it, he could point to the fact that each and every proof could, in the end, be seen as needing nothing more than a combination of these simple, self evident, axioms.

To use the axioms, Euclid combined them with the other “fundamentals” proposed in Elements before the proofs – that is, a series of 23 definitions of geometry terms such as “point,” “line,” “right angle” and “circle,” a list of five “common notions” which stated such truths as, “Things which are equal to the same thing are also equal to one another,” and “The whole is greater than the part.

These definitions, common notions and axioms could be combined to form proofs, beginning with the most simple (Proposition 1.1 deals with the construction of a right triangle) to the complex (such as the proof of the infinitude of primes).

The fact that it all begins with five simple axioms should be seen as it is: An impressive mathematical feat for a person living in any era.

See Also:

Euclid's First Axiom

Euclid's Second Axiom

Euclid's Third Axiom

Euclid’s Fifth Axiom

References:

Dunham, William. “Journey Through Genius: The Great Theorems of Mathematics.” John Wiley and Sons. 1990.

Euclid. “Elements.”