About 300 B.C., Euclid wrote Euclid ’s
Geometry. Euclid stated five postulates on which he
based all his theorems. It was obvious
from the start that Euclid ’s
fifth postulate was different than the other four.

*The Elements,*a book which contained his five postulates on which he based all of his theorems. This became one of the most famous books ever written and this system of mathematics came to be known as
Euclid’s fifth postulate, also
known as the parallel postulate, states:
“That, if a straight line falling on two straight lines make the
interior angles on the same side less than two right angles, if produced
indefinitely, meet on that side on which are the angles less than the two right
angles.” (O’Connor and Robertson 1). From
the beginning, Euclid was not satisfied with the fifth postulate. He tried to avoid using the fifth postulate
and was able to in his first 28 propositions of

*The Elements*. There were several main flaws in Euclid’s development of the parallel postulate: failure to recognize the need for undefined terms (in other words, he defined every term), he relied on diagrams to guide the logic in the construction of his proofs, and he assumed that straight lines were infinite but it is nearly impossible to check for the consistency of the fifth axiom.
In the centuries since Euclid’s fifth postulate,
János was one of the first to doubt the truth that the
parallel postulate was a valid theorem. Bolyai’s
real breakthrough was the belief that a new geometry was possible. While there were numerous mathematicians who
developed the new geometry, János
Boylai’s contributions were significant in changing the view of geometry. Because János
ignored his father’s advice to avoid work on Euclid ’s parallel postulate, he provided a
new view to geometry: a geometry in which the parallel postulate did not hold
true was possible. Boylai’s appendix
setup a series of mathematical proposals whose implications would blossom into
the new field of non-Euclidean geometry.
Boylai’s findings later led to Beltrami’s proof that if Euclidean
geometry was true, then non-Euclidean geometry was also true.

Euclid’s geometry and his five
postulates gave mathematicians a way to study the world around them, but
provided some limitations. János Boylai’s contributions changed the way
mathematicians viewed their world and opened them up to the possibilities of
other geometries besides Euclid ’s. Baylai
provided the means of studying what lies beyond our world.

Citations

Heiede,
Torkil. "The History of Non-Euclidean Geometry." : 201-11. Print.

O'Connor,
J J., and E F. Robertson. "Non-Euclidean Geometry."

*MacTutor History of Mathematics*. N.p., Feb. 1996. Web. 14 Nov. 2013. <http://www-history.mcs.st-and.ac.uk/HistTopics/Non-Euclidean_geometry.html>.
Clear, complete +

ReplyDeletecoherent -seems to start (intro) in a different direction then veer off to Bolyai.

Content -postulate not a theorem. The idea was to have a geometry with a different postulate (hyperbolic) or without (neutral).

Consolidated - what's this matter to you?