# Geometry

**Geometry**

Geometry, the study of points, lines, and other figures in **space**, is a very old branch of **mathematics**. Its ideas were undoubtedly used, intuitively if not formally, from earliest times. Walking along a straight line toward a particular destination is the shortest way to get there; lining an arrow up with the target is the way to hit it; sitting in a **circle **around a fire is the most equitable way to share the warmth. Early humans need not have been students of formal geometry to know and to use these ideas.

As early as 2,600 years ago the Greeks had not only discovered a large number of geometric properties, they had begun to see them as abstract ideas to be studied in their own right. By the third century B.C., they had created a formal system of geometry. Their system began with

the simplest ideas and, with these ideas as a foundation, went well beyond much of what is taught in schools today.

**Constructions**

Another lasting influence of Euclid's *Elements *is the emphasis which is placed on **constructions**. Three of the five postulates on which Euclid based his geometry describe simple drawings and the conditions under which they can be made. One such drawing (construction) is the circle. It can be drawn if one knows where its center and one point on it are. Another construction is drawing a line segment between two given points. A third is extending a given line segment. These are the so-called ruler-and-compass constructions upon which Euclidean geometry is based.

Modern courses in geometry are frequently based on other postulates. Some, for example, permit one to use a protractor to draw and measure angles; some allow the use of a scale to measure distances. Even so, the traditional limitations which the Euclidean postulates placed on constructions are often observed. Protractors, scales, and other drawing tools which would be easier and more accurate to use are forbidden. Constructions become puzzles, intriguing but separate from the logical structure of the course, and not overly practical.

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**Riemannian Angles: Concept & Programs**

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