The basic principle for euclidian geometry
was that for any given line
there is exactly one line through the given point that is parallel
to the given. Euclid
was, however unable to prove
this, yet felt that it must be true nonetheless. For millenia
, mathematicians attempted to prove this postulate unsuccessfully.
In the early 19th century
some mathematicians began to wonder if perhaps the lack of proof yet apparent truthfullness of Euclid's postulate meant that other truths existed. 2 possibilities
presented them self readily, that either for any given point and given line, there was no line through the point parallel to the given. This relied on the concept that a line through the point begins intersecting
the given line, and the point of intersection begins to move outward on the line in either direction approaching infinity
yet never leaving the line, therefore appearing to be parallel but still at some point intersecting the line. This proved itself to lead to contradictions
in other fundamental
theorems, and so was thrown out.
Lobachevski, as well Gauss and Boylai at around the same time but seperately, devised another possibility that through any given point and given line, there were two
lines parallel through the point to the given line. This was based yet again on a line through the point intersecting the given line. The point of intersection could be moved to infinity in either direction. At some infinite limit in one direction, the line becomes parallel, and in the other direction as well. He saw that as these lines went to opposite limits, they could be seen as two distinct lines. This postulate expanded into a new geometry with different
theorems but none contradictary
to each other.
A basic problem with this new geometry is that it is tough to visualize. This lack of visualizability (heh
)only pertains to the basic axis of x and y at a right angle. The solution to this is hyperbolic
geometry. picture this: draw a five sided figure. now draw more five sided figures such that at each vertex 4 of these meet, and that there is no space
between five sided figures. the figures should get smaller and smaller and smaller from the central one. When drawn correctly, many "curves
" can be seen. These curves are really illusions
from the way the environment for them plays out, and can effectively be seen as lines. 2 curves very often intersect each other at a point and are parallel to another curve. hyperbolic geometry also leads to all manner of infinite patterns from a central figure and repeated smaller and smaller on all sides to infinity.
) was the first to publish his work on the subject in the book
"The Theory of Parallels
" in 1840
. It is an excellent source for the theorems he found, and was translated and republished in 1914
also found similars sets of theorems and mostly coincide with Lobachevski's.