Euclid's First Axiom

A Self-Evident, Yet Important Statement concerning Straight Lines

Mar 19, 2008

Euclid's first axiom, concerning points and straight lines, contains hidden depth that one might miss upon a cursory glance, and which is important to his geometry.

The Greek mathematician Euclid of Alexandria published his magnum opus, Elements, around 300 B.C. It would be the culminating work of years of disciplined mathematical thought by one of history’s greatest geniuses.

Within the pages of these thirteen volumes, Euclid laid down a series of geometric proofs concerning such ideas as the definitions of congruency and similarity in geometrical figures, as well as more “abstract” mathematics, such as a brilliant proof of the infinitude of prime numbers.

With all the achievements accomplished in Euclid’s Elements, one would surely expect that the foundation for all of these proofs would be a complex and thorough series of assumptions – of preconceived notions and definitions upon which all the rest is based.

One would be partly correct.

Euclid’s Axioms

Euclid built his thirteen momentous volumes, in fact, on a series of five central axioms. Only five.

Not only this, but several of these first five axioms were about as simplistic as axioms can come. They are statements which, while obviously truthful, are so self-evident that even a child could easily understand them.

And yet, it is what Euclid does with them that is the chief evidence for his genius.

The First Axiom

Take his very first axiom, for example:

“…any two points can be connected by a straight line.”

That’s it.

Make any two points, anywhere in the universe, no matter how far apart, and it is physically possible (though not always technically feasible) to draw a perfectly straight line connecting the two of them.

Of course this is true, as anyone can plainly tell upon first glance.

So why does Euclid feel it necessary to state this?

The first axiom is actually rather telling about how mathematics – specifically mathematical proofs – was performed during these early days of the science. These were the days, some might say, of pure mathematics.

In his book Journey Through Genius: The Great Theorems of Mathematics, William Dunham writes that these ancient mathematicians were committed to performing all of their constructions and proofs using only two tools – a compass and an unmarked straight-edge.

Dunham says that, “…even the seemingly unsophisticated compass and straightedge can produce, in the hands of ingenious geometers, a rich and varied set of constructions, from the bisection of lines and angles, to the drawing of parallels and perpendiculars, to the creation of regular polygons of great beauty.”

Reasons for the Axiom

This first axiom, then, seems to be of fundamental importance in geometry, for in this Euclid is beginning to define his “tools,” one of which is a straight line. And the fact that any two points can be connected via such a line will have major implications to his coming proofs, from the simple to the complex.

After all, by its very definition, a proof must possess incontrovertible evidence. By stating such a self-evident truth as the fact that any two points can be connected by a line, Euclid is providing a major piece of evidence, so that later, when he begins to construct shapes by plotting their corners and then connecting them with straight lines, he can call back to the first axiom as proof that this is indeed mathematically correct.

Does this seem a bit unnecessary? In light of modern knowledge, of course it does. In the fourth century B.C., however, when all of these things were being written for the very first time, things were surely a little bit different, and one can truly be thankful for Euclid’s thoroughness, as it surely saved many people from a lot of extra trouble in later years.