Very few seemingly incontrovertible mathematical statements throughout history have wreaked quite as much havoc as Euclid's controversial fifth axiom.
The fifth and final of Euclid’s fundamental axioms upon which his entire system of geometry was built, is known as the parallel postulate, for the simple reason that it allowed one to determine incontrovertibly whether or not two lines were parallel.
The axiom is worded like this:
If two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough.
On the surface it seems both mundane and rather innocuous… and a bit too wordy.
In Other Words…
What the parallel is doing is providing a test to prove if two lines are parallel. First, a third line is drawn through them both. Then, the angles created by this third line are measured, and if the inner angles (that is, the angles on either side of this third line, in between the two lines in question) add up to either more or less than 180 degrees, the lines are not parallel.
While it is not explicitly stated, it can be deduced rather easily from this postulate that if the angles do add up to exactly 180 degrees, then the lines are indeed parallel.
The Defense of the Axiom
The question originally asked regarding this postulate is why it was regarded as an axiom at all. While on the surface this notion of parallel lines does appear to be common sense, and a fundamental aspect of geometry, does it really fit in with these other, far simpler notions, such as “any two points can be connected by a straight line?”
Surely it was necessary for what was to follow in Euclid’s Elements, did it really deserve a fundamental place with these other four axioms?
In the end, it doesn’t appear that Euclid had any choice in the matter. While he surely would have included it as a proof in and of itself later, building upon the first four axioms, he simply couldn’t
As many, many mathematicians have discovered since, the parallel postulate does not appear to be fundamentally provable!
What does this mean? Simply that Euclid was right. It is the kind of statement that simply must be taken on faith. It appears to be based on logic, and there are (or were, rather) no arguments against it, and it was certainly a necessity. So he included it here, and used it many, many times throughout the course of his proofs over the next thirteen book.
The Fifth Axiom Through History
While a great many mathematicians have been driven nearly mad in an attempt to find a proof to the fifth axiom, thus proving Euclid wrong (and more a few may have been mad from the very beginning), still more have, mostly within the last couple hundred years, attempted to prove it wrong.
This is the basis for what has become known as non-Euclidean Geometry. Beginning in the nineteenth century, mathematicians began to realize that Euclid’s was not the only valid form of geometry. While it was certainly the best in term of three-dimensional and plane geometry, it could not be extended into further dimensions. In fact, it was shown by Carl Gauss and others that in certain geometries, the idea of parallel lines simply didn’t exist, and that right angles needn’t always have angles adding up to 180 degrees and other seemingly preposterous things.
Does this prove Euclid wrong?
Not necessarily.
Euclid’s geometry still holds fast. All five axioms have withstood the test of time, and it still appears that the fifth axiom (within the realm of standard, flat geometry) is both necessary and utterly un-provable – though certainly this will not stop some from trying.
See Also:
References:
Dunham, William. “Journey Through Genius: The Great Theorems of Mathematics.” John Wiley & Sons, Inc. 1990.
Gardner, Martin. “The Colossal Book of Mathematics.” W.W. Norton & Co. 2001.
