Axiom:Euclid's Fifth Postulate

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Euclid's Statement

In the words of Euclid:

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.

(The Elements: Book $\text{I}$: Postulates: Euclid's Fifth Postulate)


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


Parallel Postulate

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

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.[1]


Legendre and W. 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.[2]


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.[3]


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.[4]


Historical Note

During the course of the history of mathematics, many mathematicians have attempted to prove Euclid's fifth postulate from the remaining four, as it seems less intuitive than the rest of Euclid's Postulates.

Euclid himself avoided using the fifth postulate until the $29$th proposition (part $1$): Parallelism implies Equal Alternate Interior Angles in his seminal The Elements.

Eventually, in $1823$ Nikolai Ivanovich Lobachevsky and, independently in $1832$, János Bolyai, realized that self-consistent, non-euclidean geometries could be developed by not accepting this postulate.

It transpired that Gauss himself had already come to a similar conclusion, but had not had the confidence to publish.

In fact, in a letter to Wolfgang Bolyai, after receiving a copy of János' appendix, Gauss wrote:

If I commenced by saying that I am unable to praise this work, you would certainly be surprised for a moment. But I cannot say otherwise. To praise it, would be to praise myself. Indeed the whole contents of the work, the path taken by your son, the results to which he is led, coincide almost entirely with my meditations, which have occupied my mind partly for the last thirty or thirty-five years. So I remained quite stupefied. So far as my own work is concerned, of which up till now I have put little on paper, my intention was not to let it be published during my lifetime. ... It is therefore a pleasant surprise for me that I am spared this trouble, and I am very glad that it is just the son of my old friend, who takes the precedence of me in such a remarkable manner.

Later, in a letter to Bessel in $1829$, Gauss wrote:

It may take very long before I make public my investigations on this issue: in fact, this may not happen in my lifetime for I fear the "clamor of the Boeotians."


Hence it is apparent that the fifth 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.


Also see


References

  1. Johann Friedrich Lorenz: Grundriss der reinen und angewandten Mathematik, 1791.
  2. Gauss, in a letter to W. Bolyai, 1799.
  3. Veronese: Elementi, 1904.
  4. Ingrami: Elementi, 1904.


Sources