Download Euclid`s Axioms and his book `The Elements` Euclid is noted as

Euclid’s Axioms and his book ‘The Elements’
Euclid is noted as being the first Mathematician to devise axioms. Before Euclid wrote his famous book
The Elements around 300 BC, many geometric ideas were well understood, but in a disorganized way
that obscured their logical structure. It was often unclear which facts depended on which others for
their proofs, and this vagueness opened the door to circular reasoning and other logical errors. To
systematize the study of geometry, Euclid formulated five axioms, statements so simple he considered
them self-evident, and then attempted to prove all other geometric facts using only these five axioms
and the principles of logical reasoning. Euclid's analysis was so definitive and far-reaching that it laid the
foundation for the study of geometry for the next 2000 years.
Euclid realized that it is impossible to prove anything without starting with a few basic assumptions, or
axioms; the ideal in mathematics is to start with as few and simple axioms as possible, and prove as
many statements from the axioms as possible. Euclid limited himself to the following five axioms:
1) Any two points can be connected by one and only one straight line.
2) Any line segment is contained in a full (infinitely long) line.
3) Given a point and a line segment starting at the point, there is a circle that has the given point as its
center and the given line segment as a radius.
4) All right angles are equal to each other (Euclid defines a right angle to be the angle formed when
two lines intersect each other perpendicularly, that is, forming equal angles on both sides of the
intersection).
5) (known as Euclid's parallel postulate). Given a line and a point P that is not on the line, there is one
and only one line through P that never meets the original line.
Using these five axioms, Euclid was able to prove, for example, that two triangles that have all equal
side-lengths are congruent; that two triangles that have all equal angles are similar; that a tangent line
to a circle is perpendicular to the diameter that it intersects; and that in a right triangle, the sum of the
squares of the lengths of the two legs is equal to the square of the length of the hypotenuse, the
famous Pythagorean theorem.
Reference:
http://www.bookrags.com/research/euclids-axioms-wom/
Things to do:

See if you can visualise each of Euclid’s axioms by sketching each of them.
Of Euclid's five axioms, the fifth is conspicuously more complicated than the others. Through the
centuries many believed, therefore, that the fifth axiom was not fundamental enough to be one of the
basic axioms, and should instead be provable using the other four. Euclid himself avoided any use of the
fifth axiom in the proofs of the first 28 propositions of The Elements. This evasion has led some
historians to the conclusion that even Euclid felt the fifth axiom to be less natural than the others. Many
mathematicians, first the Greeks, then the Arabs in the Middle Ages, then the Europeans in the
Renaissance, tried to show that the fifth axiom was a consequence of the others, but with no success.
The fifth axiom, known as the ‘parallel postulate’ to be proved true would require someone to
accompany the line forever into infinity to ensure it never intersects the first line.
Euclid’s geometry was found to be consistent on an infinite plane surface, however a new geometry was
discovered by Reimann which was non-Euclidean but worked on the surface of a sphere, this was known
as hyperbolic geometry. Reimann’s geometry has found a practical application, the curved space of
Einstein’s theory of general relativity is well described using Reimann’s geometry.
Following this Mathematical truth came to be understood as truth within a certain axiomatic system,
mathematical statements for example could be true within a Euclidean system, or true within a Reimann
system. The only truth test relevant was the coherence test, the consistency of every statement with
every other statement within its own axiomatic system.
Mathematical proofs, no matter how beautiful do not enter the realm of mathematics until the truth of
the claim is justified by the relevant knowledge community.
Things to do:

Find out about Fermat’s Last Theorem. What was it and how was it accepted?