Essay/Term paper: Euclidean geometry

Essay, term paper, research paper:  Science

Euclidean Geometry


Geometry was thoroughly organized in about 300 BC, when the Greek
mathematician Euclid gathered what was known at the time, added original work of
his own, and arranged 465 propositions into 13 books, called 'Elements'. The
books covered not only plane and solid geometry but also much of what is now
known as algebra, trigonometry, and advanced arithmetic.

Through the ages, the propositions have been rearranged, and many of the
proofs are different, but the basic idea presented in the 'Elements' has not
changed. In the work facts are not just cataloged but are developed in a
fashionable way.

Even in 300 BC, geometry was recognized to be not just for
mathematicians. Anyone can benefit from the basic learning of geometry, which
are how to follow lines of reasoning, how to say precisely what is intended, and
especially how to prove basic concepts by following these lines of reasoning.
Taking a course in geometry is beneficial for all students, who will find that
learning to reason and prove convincingly is necessary for every profession. It
is true that not everyone must prove things, but everyone is exposed to proof.
Politicians, advertisers, and many other people try to offer convincing
arguments. Anyone who cannot tell a good proof from a bad one may easily be
persuaded in the wrong direction. Geometry provides a simplified universe, where
points and lines obey believable rules and where conclusions are easily verified.
By first studying how to reason in this simplified universe, people can
eventually, through practice and experience, learn how to reason in a
complicated world.

Geometry in ancient times was recognized as part of everyone's education.
Early Greek philosophers asked that no one come to their schools who had not
learned the 'Elements' of Euclid. There were, and still are, many who resisted
this kind of education. It is said that Ptolemy I asked Euclid for an easier way
to learn the material. Euclid told him there was no "royal road" to geometry
instead he told Ptolemy you will not learn what geometry is all about. What you
will learn is the basic shapes of some of the figures dealt with in geometry and
a few facts about them. It takes a geometry course, with textbook and teacher,
to show the complete and orderly arrangement of the facts and how each is proved.
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