Notes on Geometry

In Euclidean geometry, two triangles are congruent if one of them can be moved rigidly onto the other. Definitions such as that of congruence, which tell us when two objects should be regarded as being the same, are basic in geometry and are often used to characterize a particular geometry. Two sets A, B are defined to be equivalent if there is an ‘allowed transformation’ f such that fA = B. For Euclidean geometry the allowed transformations are the rigid motions. In his Erlanger programme of 1872, Felix Klein formulated the principle that a geometry is defined by its allowed transformations. The force of this principle is to make a close connection between geometry and group theory.

We start by studying the linear groups. These are probably already familiar to the reader. They play an important role in the study of geometry.

Projective geometry was invented in the 17th century, the first important contributions to the subject being made by a French architect Gérard Desargues. These studies arose from attempts to understand the geometrical properties of perspective drawing. When one draws a pair of parallel lines, such as the sides of a long, straight road, it is usual to draw them as meeting ‘at infinity’. In the drawing one has ‘points at infinity’, these lie on the ‘vanishing line’. The vanishing line consists of the points where pairs of parallel lines on the (flat) surface of the earth are depicted to meet. The need to introduce and study points at infinity led to projective geometry.

Euclid gave certain axioms that were meant to characterize plane geometry. For twenty centuries much controversy surrounded one of these axioms — the parallel axiom. It was felt that this axiom was not as basic as the others and considerable efforts were put into attempts to deduce it from the other axioms. Before the nineteenth century the most successful attempt seems to have been made by Saccheri in 1773. He produced a thorough investigation of the parallel axiom, in fact he claimed to have deduced the parallel axiom from the others, but his otherwise excellent piece of work had one flaw in it. At the beginning of the nineteenth century Gauss, in work that he did not publicise, showed how a large body of theorems could be deduced from a variant of the parallel axiom, thus discovering hyperbolic geometry. Because Gauss did not publish his work, this discovery is usually associated with the names of Bolyai and Lobachevski who published the same discoveries in 1832 and 1836 respectively. Despite the fact that Euclid’s parallel axiom was shown to be necessary several defects were found, during the nineteenth century, both in the axioms and logic used by Euclid. For a discussion of Euclid’s axioms and their defects, the reader may consult M.J. Greenberg’s book “Euclidean and Non-euclidean geometry”.