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Euclidean geometry is a mathematical system attributed to the Greek mathematician Euclid of Alexandria. Euclid's text Elements was the first systematic discussion of geometry. It has been one of the most influential books in history, as much for its method as for its mathematical content. The method consists of assuming a small set of intuitively appealing axioms, and then proving many other propositions (theorems) from those axioms. Although many of Euclid's results had been stated by earlier Greek mathematicians, Euclid was the first to show how these propositions could fit together into a comprehensive deductive and logical system.

The Elements begin with plane geometry, still often taught in secondary school as the first axiomatic system and the first examples of formal proof. The Elements goes on to the solid geometry of three dimensions, and Euclidean geometry was subsequently extended to any finite number of dimensions. Much of the Elements states results of what is now called number theory, proved using geometrical methods.

For over two thousand years, the adjective "Euclidean" was unnecessary because no other sort of geometry had been conceived. Euclid's axioms seemed so intuitively obvious that any theorem proved from them was deemed true in an absolute sense. Today, however, many other self-consistent geometries are known, the first ones having been discovered in the early 19th century. It also is no longer taken for granted that Euclidean geometry describes physical space. An implication of Einstein's theory of general relativity is that Euclidean geometry is only a good approximation to the properties of physical space if the gravitational field is not too strong.

In his historic work Elements, Euclid assumed the existence of parallel lines with his fifth postulate. The fifth postulate or parallel postulate is equivalent to:

Given a line and a point not on that line, exactly one line can be drawn through that point which does not intersect the original line (see 1).

In the 19th century mathematicians began to seriously question the parallel postulate and found that other forms of geometry are possible. For example elliptical geometry:

Given a line and a point not on that line, all lines drawn through that point will intersect the original line (see 2).

And hyperbolic geometry:

Given a line and a point not on that line, an infinite number of lines can be drawn through the point that do not intersect the original line (see 3).

These other forms of geometry, where the parallel postulate does not hold are called Non-Euclidean geometry.

• ...that there are 6 unsolved mathematics problems whose solutions will earn you one million US dollars each?
• ...that there are different sizes of infinite sets in set theory? More precisely, not all infinite cardinal numbers are equal?
• ...that every natural number can be written as the sum of four squares?
• ...that the largest known prime number is over 12 million digits long?
• ...that the set of rational numbers is equal in size to the subset of integers; that is, they can be put in one-to-one correspondence?
• ...that there are precisely six convex regular polytopes in four dimensions? These are analogs of the five Platonic solids known to the ancient Greeks.
• ...that it is unknown whether π and e are algebraically independent?
• ...that a nonconvex polygon with three convex vertices is called a pseudotriangle?
• ...that it is possible for a three dimensional figure to have a finite volume but infinite surface area? An example of this is Gabriel's Horn.

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