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# axiomatic geometry

Axiomatic geometry can be traced back to the time of Euclid. In his book
*Elements*, written back in the 300’s B.C., Euclid gave five rules, or postulates, describing how
points, lines, line segments, etc behave as they are ordinarily
perceived. Based on these postulates, he set out to prove hundreds of
properties. Today, these properties are under the field of study known as
*plane Euclidean geometry*, more popularly known as high school geometry.
The systematic and axiomatic approach to proving geometric facts is what makes his *Elements* one of the most important contributions to mathematics.

One key feature of Euclid’s axioms is the abundance of what are known today as
the *undefined terms*. Geometric notions such as points, lines, and
circles are mentioned in his axioms but never clearly defined. For example, Euclid
called a “point” as “that which has no part”. But what the meaning of “part” was
never clarified. It is because of this abundance of undefined terms, Euclid’s postulates by
today’s mathematical standards lack rigor. While some undefined terms give no serious
problems, others create holes in proofs which are unacceptable. In the late 19th century,
David Hilbert published his classic *Foundations of Geometry*, putting Euclid’s postulates
on a more solid ground. In the book, he broke down Euclid’s postulates into five
groups of axioms:

1. 2. order axioms

3. 4. 5.

These axioms have been shown to be independent of each other, in the sense that no one axiom can be proved from the rest, and consistent, in the sense that no contradictions can be derived from them. These axioms today serve as the foundation of plane Euclidean geometry.

## Mathematics Subject Classification

51-01*no label found*51-00

*no label found*

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