Can anyone prove Euclid's fifth postulate?

Mathematics: About the 'axiom of parallels'
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Written by Hans-Juergen - 5995 x read [Outline] Printer-friendly version - Choose language

\ (\ begingroup \)About the "axiom of parallels"

In the following I would like to say something about the so-called
Parallels axiom report that not everyone in this
Knows detail.

As is well known, it works Euclid (around 300 BC) back,
but it does not appear there under that name.



In his famous geometry work "Elements", the
until the 19th century as a textbook in many schools
was used and long after the Bible on
was most widespread, Euclid begins with a number
of Definitions:

- A point is what has no parts.
- A line is a length without breadth.
- A line is straight when it is against the one in it
Points in one way or another. (Other translation:
a straight line is one that leads to the points on it
lies evenly.)
- What only has length and width is a surface.
. . . .
It also explains what an angle is, especially a right one,
but which will not be discussed here.

What follows are statements that will be made much later1) "Axioms" called:

- Things that are like the same thing are like each other.
- If one adds like like, then the sums are the same.
- If one removes the same from the same, the remains are the same.
- What can be made to coincide with one another is one another
equal.
- The whole is bigger than its part.

1)axiom from gr. axioma = reputation, dignity, undoubted
Theorem. In this meaning since the 17th century. used -
around 2000 years after Euclid. What is sometimes seen as parallels
axiom does not appear in the above statements.
Rather, it is the last of the five that follow Postulates:

1. It should be demanded that each point is after
draw a straight line for each point.
2. Furthermore, that a bounded straight line is continuously in a straight line
Let the line extend.
3. Furthermore, that with each center point and radius
let describe a circle.
4. Furthermore, that all right angles are equal to each other.
5. Finally, when a straight line meets two straight lines and with
forms inner angles to them on the same side as
together are smaller than two right, they should
two straight lines, extended to infinity, finally
meet on the side on which the angles lie,
which together are less than two rights.

According to this, it is not appropriate to speak of parallelsaxiom
to speak; one should rather have parallelspostulate say.

While the first four postulates (Latin for "demands")
are kept short and sweet, the fifth in this one stands out
Relationship clearly from them. That is why the
Attempts made in antiquity to
to replace equivalent statements. Of Poseidonius
(around 135-51 BC) the formulation comes from:

"All points of the plane by a given straight line
have the same distance and on one side of this straight line
lie, form (also) a straight line. "

This sounds more like what we mean by parallelism
understand. "Parallel" is a in the 16th century. from the Greek
Words pará= next to and állos= other educated
Made-up word that means something like "running next to each other".
("Para" has a number of other meanings and
occurs in many technical terms, e.g. parable, paragraph,
Paramagnetism, parody ...)

Another version of the fifth postulate that I don't
knows who it comes from is:

"For a given line g1 is given by a point P
outside of her exactly one Just g2, which in the of g1
and P is the spanned plane and g1 does not cut. "
G2 called parallel train1.

A few more formulations that correspond to the 5th Euclidean
Are equivalent to the postulate (which of course had to be proven):

"For every triangle there is a similar triangle of any size."
( John Wallis, 1616-1703)

"The sum of the angles in the triangle is two rights". (Geronimo
Saccheri
, 1667-1733)

"By a point within an angle less than a
is more stretched, you can always draw a straight line, both of them
Thigh cuts. "(Adrien-Marie Legendre, 1752-1833)

"By three points that are not on a straight line
there is a circle. "(Farkas (Wolfgang) Bólyai, 1775-1856).

The last statement leaves no connection with that
recognize more of the original thoughts of Euclid.

*

The discomfort with the complicated sounding 5th postulate,
that apart from the ancient Greeks also the Arabs felt,
translating Euclid's writings into their language lasted
until modern times. It was asked early on whether
the 5th postulate with the help of the four others to prove
leaves. If so, it would be dispensable and one could
just leave it out.

Attempts to provide evidence of this kind were unsuccessful. Still
in the 19th century the last-named Hungarian tormented himself
Mathematician F. Bólyai around for decades with it and
wrote a letter after this long, frustrating time
to his son János Bólyai, also a mathematician,
in which he urgently advised him not to continue with
to deal with this problem. (That in the letter
included, long Latin quote - with small. Spelling mistake -
"Si paulum ..." means in German: "If she" -
what is meant in this case is poetry - "only
If it falls short of the highest, it sinks into the
Depth down "and comes from Horace (Ars poetica).)

Bólyai junior did not follow his father's advice. He
recognized the unprovability of the 5th postulate from the
four remaining and in 1825 developed a geometry based on
other prerequisites. He named these
"absolute geometry". (Note: "absolut" means in German
"detached". Interesting way is in the clickable
Page of XI. Axiom of Euclid the speech. But what is meant is
the 5th postulate; apparently another count is here
and a different understanding of the term "axiom".)

Created independently from J. Bólyai and from each other C. F. Gauss
(1777-1855) and N. I. Lobachevsky (1792-1856) in the
Years 1816 and 1832 the hyperbolic Geometry,
in which the 5th postulate of Euclid by the following
is replaced:

"If g is a straight line and P is a point not lying on it,
so there are two of P in the plane determined by P, g
outgoing half-line p1, p2, for which the angle ∠ (p1, p2)
is smaller than a stretched one, both g do not intersect,
while each of P is within the angle ∠ (p1, p2)
outgoing half-line intersects the line g. "(Source: Lexicon of
Mathematics, VEB Bibliographisches Institut Leipzig, 1979)

This sounds more complicated than the original one
Euclidean requirement.

It is used to illustrate the hyperbolic geometry
a so-called Pseudosphere (Gr. for "wrong ball").
It is created by rotating a tractrix (Latin for
"Tow curve"), which is interesting in itself, here on the
Maths planets by Artur Koehler (pendragon302) in detail
was treated.

*

Finally, a look at a real, ordinary sphere.
Geometry can also be carried out on its surface.
What about the 5th postulate of Euclid?

To find out, the first thing to do is to determine
which geometric objects of the spherical surface denote
Straight lines correspond to the plane. The
Euclid's definition of the line cited above hardly. I use
therefore a much better known one, which I don't know if it is
also comes from him or someone else:

"A route is the shortest connection between two points.
If these move away from each other to infinity, arises
a straight. This has no beginning and no end. "

On the surface of a Bullet is the shortest connection
between two points on one Great circle, d. H. one
Circle whose center is the center of the sphere. This can
you look at yourself without a bill with a taut
Make the elastic band clear. A whole great circle (without a beginning
and end) corresponds to a straight line in the plane. That I
Always intersect great circles, there is in this sense on the
Spherical surface no "Straight lines" leading to each other
are parallel. Euclid's fifth postulate cannot be fulfilled here;
the geometry on the sphere is therefore, as is the
hyperbolic, called "non-Euclidean".

Hans-Jürgen



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: Mathematics :: Geometry :: Euclid :: Axiom of Parallels :: Axioms :: History :: Students upwards:
About the 'axiom of parallels' [by Hans-Juergen] 
About the axiom of parallels, a historical foray from Euclid via Wallis, Legendre to Bólyai In the following I would like to report something about the so-called axiom of parallels, which not everyone knows in this detail. As is well known, it goes back to Euclid (around 300 BC), but it does not appear there under this name.
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