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In its rough outline, Euclidean geometry is the plane and solid geometry commonly taught in secondary schools. Indeed, until the second half of the 19th century, when non-Euclidean geometries attracted the attention of mathematicians, geometry meant Euclidean geometry. It is the most typical expression of general mathematical thinking. Rather than the memorization of simple algorithms to solve equations by rote, it demands true insight into the subject, clever ideas for applying theorems in special situations, an ability to generalize from known facts, and an insistence on the importance of proof.

All right angles are equal. 5. If a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight 60 7 Branches of Geometry 7 lines, if produced indefinitely, will meet on that side on which the angles are less than the two right angles. Hilbert refined axioms (1) and (5) as follows: 1. For any two different points, (a) there exists a line containing these two points, and (b) this line is unique. 5. For any line L and point p not on L, (a) there exists a line through p not meeting L, and (b) this line is unique.

But it was not all. In relativity theory time is considered to be a dimension along with the three dimensions of space. On the closed four-dimensional world thus formed, the history of the universe stands revealed as describable by motion within a vast congeries of geodesics in a nonEuclidean universe. 58 CHAPTER 2 BRAnCHes oF GeoMetRY euclideAn geometry A s discussed in a general way, in chapter 1, several ancient cultures had developed a form of geometry suited to the relationships among lengths, areas, and volumes of physical objects.