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.
Euclidean geometry examples and notes
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