Axiom:Euclid's Fifth Postulate

Euclid's Statement
If a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.

There are many equivalent ways to state this postulate. See below for a selection of them.

Parallel Postulate
This is the name by which the axiom is usually known. As can be seen, its wording in its modern format gives it an intent very similar to Euclid's.


 * "If two straight lines are drawn which intersect a third in such a way that the sum of the measures of the two interior angles on one side is less than the sum of the measures of two right angles, then the two lines must intersect each other on that side if extended far enough."

Playfair's Axiom
This is the other frequently seen presentation of this axiom. It can easily seen to be equivalent to that given by Euclid, but it can be argued that it is easier to understand:


 * "Exactly one straight line can be drawn through any point not on a given line parallel to the given straight line in a plane."

Or:


 * "Given any straight line and a point not on it, there exists one and only one line which passes through this point and does not intersect the first line no matter how far they are extended."

This unique line is defined as being parallel to the original line in question.

Or:


 * "Two straight lines which intersect one another cannot both be parallel to one and the same straight line."

Other equivalent statements
Many further attempts have been made to formulate equivalent definitions of this axiom, often with a view to finding a proof which relies on the other four axioms. (Such attempts have been universally doomed to failure.) Here are a few examples, in approximate chronological order:

Proclus
If a straight line intersects one of two parallels, it will intersect the other also.

Straight lines parallel to the same straight line are parallel to one another.

Posidonius and Geminus
There exist straight lines everywhere equidistant from one another.

(This can be compared with Proclus' tacit assumption that "Parallels remain, throughout their length, at a finite distance from one another.")

Legendre
There exists a triangle in which the sum of the three angles is equal to two right angles.

Wallis, Carnot, Laplace
Given any figure, there exists a figure similar to it of any size we please.

(Saccheri points out that it is necessary only to postulate that "there exist two unequal triangles with equal angles.")

Legendre (again)
Through any point within an angle less than two-thirds of a right angle a straight line can always be drawn which meets both sides of the angle.

Lorenz
Every straight line through a point within an angle must meet one of the sides of the angle.

Legendre and Bolyai
Given any three points not in a straight line, there exists a circle passing through them.

Gauss
"If I could prove that a rectilineal triangle is possible the content of which is greater than any given area, I am in a position to prove perfectly rigorously the whole of geometry."

Worpitzky
There exists no triangle in which every angle is as small as we please.

Clairault
If in a quadrilateral three angles are right angles, the fourth angle is a right angle also. (1741)

Veronese
If two straight lines are parallel, they are figures opposite to (or the reflex of) one another with respect to the middle points of all their transversal segments.

Ingrami
Two parallel straight lines intercept, on every transversal which passes through the middle point of a segment included within them, another segment the middle point of which is the middle point of the first.

Comment
As can be inferred from all the above, many mathematicians have attempted to prove this postulate, as it seems less intuitive than the rest of Euclid's Postulates. Euclid himself avoided using this postulate until the 29th Proposition in his seminal. Eventually, in 1823 Lobachevsky and Bolyai independently realized that self-consistent, non-euclidean geometries could be developed by not accepting this postulate. Therefore, the postulate is in fact axiomatic for Euclidean geometry.

The literature on this subject is voluminous, but since the birth of the concept of non-Euclidean geometry and (to a certain extent) since the study of analytic geometry developed, most of this literature has little mathematical value now beyond historical interest.