# Introduction to Geometry

The word *geometry* comes from the Greek words *geo* (meaning *earth*) and *metria* (meaning *measurement*). Literally translated, it means "to measure the earth", and the ancient Greeks used it to describe the methods they used to measure distances and to calculate areas and volumes. The origins of geometry go back to the second millennium BCE. Examples of early writings on the subject include the Egyptian *Rhind Papyrus*, and the Babylonian *Plimpton 322* clay tablet, both of which are believed to have come into existence some four thousand years ago. One of the best known classical works describing geometrical principles, *The Elements*, was written by the Greek mathematician *Euclid* during the third century BCE. Most of the principles described therein were already well-known, but Euclid organised them into a logical framework of definitions, axioms, theorems and mathematical proofs.

Some two to three hundred years before Euclid wrote his *Elements*, the philosopher and mathematician *Pythagoras* (who is generally credited with the theorem of the same name) had founded a philosophical school of thought that took a more abstract, numerically-based view of geometry. The Pythagorean movement was short-lived however, and had little lasting influence. An algebraic approach to geometry re-emerged in the Middle East during the ninth century CE. Although the influence of this region on developments in mathematics was in decline by the end of the twelfth century CE, translations of the writings of many Arab mathematicians had already found their way to Europe and would influence the later work of *Descartes* (see below) and others.

One of the most notable contributions of the seventeenth century CE was the work of *René Descartes*, the celebrated French philosopher and mathematician. In his work *La Géométrie*, he proposed the idea of combining algebra and geometry into a new branch of mathematics called *analytical geometry*. He proposed that mathematical questions involving the properties and behaviour of geometric forms could be resolved algebraically using equations and functions. Descartes was responsible for the *Cartesian* system of coordinates (named after him) in which each point on a two-dimensional plane is defined in terms of its horizontal and vertical distance from a common point of origin. The work of German mathematician *Georg Friedrich Bernhard Riemann* in the nineteenth century CE took geometry beyond the realm of Euclid, and his contributions in the field of *differential geometry* have been influential in areas of modern physics such as Einstein's *general theory of relativity*, and more recently *string theory*.

In these pages we will mainly confine ourselves to Euclidean geometry. The two main areas in which we are interested are *plane geometry* and *solid geometry*. *Plane geometry* is the study of the properties of points, lines and shapes (for example circles, triangles and rectangles) in the two-dimensional plane. *Solid geometry* is the study of three-dimensional objects such as cubes, spheres, pyramids and cones. A working knowledge of both plane and solid geometry is important for many fields of study, including engineering, mathematics and physics.