Euclid's Second Axiom
A Subtle Definition of the Infinite Nature of Straight Lines
On the second of five axioms upon which was built the foundations of geometry, Euclid further explores the nature of straight lines.
The first of Euclid's Axioms - "A straight line can be drawn between any two points," seems almost laughably self-evident on the surface, yet contains within it a necessary piece of "evidence" that Euclid could call back to in later proofs in order to better define his argument.
While the second serves a similar "functional" purpose in geometry, it also helps to raise some very essential issues at stake when considering the nature of geometry, and of spacial relationships.
The Axiom
Any straight line segment can be extended indefinitely in a straight line.
That's it. Another argument that, on the surface, seems rather self-explanatory.
A person familiar with Euclid's first axiom might note that the second builds directly off the foundation previously built. The line segment that Euclid refers to in this axiom is the same that was constructed using "any two points." Here, he simply takes it a step farther.
In essence, Euclid is saying within the space of his first two axioms (the only two which deal directly with defining the nature of lines, thankfully) that given any two points in space, one can draw a straight line to connect them - creating a line segment. Given this line segment, one can technically extend either end of this line an infinite distance in either direction.
In other words, there is no end to the line.
Second Axiom Applications
Euclid's second axiom would serve many applications in the many proofs which would make up the following thirteen volumes of his masterpiece, Elements.
Geometrical figures, after all, are made up of little more than points in space connected by line segments (in other words - Euclid's first axiom seems to be the building block of geometry itself). Using the second axiom, Euclid could then take any of the lines making up figures and extend it further - and he would thus have a perfectly acceptable logical basis for doing so (for anyone who was to accept his proofs would first have to grant the truth of his axioms).
But what purpose does extending a line from a geometrical figure serve?
The examples of this are numerous - extending a side of a triangle can help to prove that two angles on a line formed by the intersection of a second line always add up to 180%, or by extending two lines, that adjecent angles are always congruent.
The Necessity of the Axiom
Together, Euclid's first and second axioms fully described, in as simple terms as possible, the nature of any straight line. Using only two sentences which anyone with the capacity for logical mathematical reasoning couldn't possibly disagree with, Euclid had provided himself a basis for the geometrical proofs that would comprise the rest of his work - he had developed a foundation upon which he could actually construct the hypothetical figures that he would soon be analyzing.
So, no matter how seemingly innocuous or pointless these most basic of axioms may seem, it is important to understand why they exist, and how an ingeniuos geometer like Euclid, armed with nothing but a compass and a straightedge, could use them to construct the entire system of modern geometry.
See Also:
References:
Dunham, William. “Journey Through Genius: The Great Theorems of Mathematics.” John Wiley and Sons. 1990.
Euclid. “Elements.”
