We have already mentioned that in 1899, Hilbert gave a set of axioms which characterized euclidean geometry. For plane geometry, he gave 15 axioms, and took great care to show that they were independent. He gave examples to show that, if an axiom were omitted, then there are geometries other than euclid's which obey the remaining axioms.	hilbert axioms
Any system of axioms for euclidean geometry must include an axiom which guarantees the existence and uniqueness of parallels. In the system given by Hilbert, this is Hilbert's Parallel Axiom There can be drawn through any point A, lying outside of a line, one and only one line that does not intersect the given line. For the purpose of our discussion, we will refer to the rest of Hilbert's axioms as the common axioms. It is possible to show that, if we assume the common axioms, then we can show that there is at least one such line. If the Parallel Axiom is false, then it can be shown that it must fail for every point and line. In other words, our geometry must have the Alternative Parallel Axiom There can be drawn through any point A, lying outside of a line, at least two lines that do not intersect the given line. In fact, we immediately deduce that, in such a case, there is an infinite family of such lines.
Although we have not checked the details, it is the case that hyperbolic geometry satisfies the common axioms. We also know that it satisfies the Alternative Parallel Axiom. These observations prove that the Parallel Axiom is independent of the common axioms. definition A theorem which can be proved from the common axioms is said to be provable in neutral geometry. This terminology is rather unfortunate. It suggests that there is another geometry to be added to our list - neutral geometry. Following Klein, we would expect to be given a set and a group of transformations. In fact, the definition is rather like the definition of a theorem of group theory, it describes a collection of results which will be true for any system which satisfies the axioms. In this case, these are the common axioms. From our earlier discussions, we see that there are essentially only two geometries which satisfy the common axioms, euclidean and hyperbolic. Before embarking on some examples, we will discuss the theory of neutral geometry a little further. As a digression, we will show that we do have a group associated with the common axioms. This could well be skipped at first reading.

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