**Transcribed by Marsigit , Yogyakarta State University, Indonesia**

**Email: marsigitina@yahoo.com**

The historical developments of
non-Euclidean geometry were attempts to deal with the fifth axiom.

Mathematicians first tried to
directly prove that the first 4 axioms could prove the fifth. However,
mathematicians were becoming frustrated and tried some indirect methods.

Girolamo Saccheri (1667-1733)
tried to prove a contradiction by denying the fifth axiom. He started with
quadrilateral ABCD (later called the Saccheri Quadrilateral) with right angles
at A and B and where AD = BC. Since he is not using the fifth axiom, he
concludes there are three possible outcomes. Angles at C and D are right
angles, C and D are both obtuse, or C and D are both acute. Saccheri knew that
the only possible solution was right angles.

Saccheri said this was enough
to claim a contradiction and he stopped. His reasoning to stop was based on
faulty logic. He was going on the presumption that lines and parallel lines
worked like those in flat geometry. So his contradiction was only applicable in
Euclidean geometry, which was not a contradiction to what he was actually
trying to prove. Of course Saccheri did not realize this at the time and he
died thinking he had proved Euclid’s fifth axiom from the first four.

A contemporary of Saccheri,
Johann Lambert (1728-1777), picked up where Saccheri left off and took the
problem just a few steps further. Lambert considered the three possibilities
that Saccheri had concluded as consequences of the first four axioms. Instead
of finding a contradiction, he found two alternatives to Euclidean geometry.
The first option represented Euclidean geometry and while the other two
appeared silly, they could not be proven wrong.

Through time (and quite a lot
of criticism), these two other possibilities were now being considered as
“alternative geometries” to Euclid’s geometry. Eventually these alternate
geometries were scholarly acknowledged as geometries, which could stand alone
to Euclidean geometry.

The two non-Euclidean
geometries were known as hyperbolic and elliptic. Hyperbolic geometry was
explained by taking the acute angles for C and D on the Saccheri Quadrilateral
while elliptic assumed them to be obtuse.

Let’s compare hyperbolic,
elliptic and Euclidean geometries with respect to Playfair’s parallel axiom and
see what role parallel lines have in these geometries:

1.)
Euclidean: Given
a line L and a point P not on L, there is exactly one line passing through P,
parallel to L.

2.)
Hyperbolic: Given
a line L and a point P not on L, there are at least two lines passing through
P, parallel to L.

3.)
Elliptic: Given a
line L and a point P not on L, there are no lines passing through P, parallel
to L.

**Elliptic Geometry**

Elliptic geometry also says
that the shortest distance between two points is an arc on a great circle (the
“greatest” size circle that can be made on a sphere’s surface).

As part of the revised parallel
postulate for elliptic geometries, we learn that there are no parallel lines in
elliptical geometry.

This means that all straight
lines on the sphere’s surface intersect (specifically, they all interesect in
two places).

A famous non-Euclidean
geometer, Bernhard Riemann, who dealt mostly with and is credited with the
development of elliptical geometries, theorized that the space (we are talking
about outer space now) could be boundless without necessarily implying that
space extends forever in all directions.

This theory suggests that if we
were to travel one direction in space for a really long time, we would
eventually come back to where we started! This theory involves the existence of
four-dimensional space similar to how the surface of a sphere (which is three
dimensional) represents an elliptic 2 dimensional geometry.

**Hyperbolic Geometry**

Recalling the corresponding
Playfair’s axiom for hyperbolic geometry, we see that in hyperbolic geometry,
there is more than one parallel line to L, passing through point P, not on L.

Hyperbolic geometries comes with
some more restrictions about parallel lines. In Euclidean geometry, we can show
that parallel lines are always equidistant, but in hyperbolic geometries, of
course, this is not the case.

In hyperbolic geometries, we
merely can assume that parallel lines carry only the restriction that they
don’t interesect. Furthermore, the parallel lines don’t seem straight in the
conventional sense. They can even approach each other in an asymptotically
fashion. The surfaces on which these rules on lines and parallels hold true are
on negatively curved surfaces.

In hyperbolic geometry, the
triangle’s angle sum is less than 180 degrees whereas elliptic geometry has
more than 180 degrees. The larger the sides of the triangle, the greater the
distortion of the angle sums on both elliptic and hyperbolic geometries. Much
like elliptic geometries, the area of a triangle is proportional to its angle
sum and of course this implies that there are no similar triangles as well.

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