Foundations of Geometry . The course is particularly useful for prospective secondary school teachers, in that it illustrates how a mathematical structure can be built upon an axiom system, and it puts in perspective the euclidean geometry that is traditionally studied in the schools. Given any point and distance, there is a circle centered at the point with radius equal to the distance.
All right angles are equal to each other.
Given a line and a point not on the line, there exists one and only one line passing through the point and parallel to the given line. It seemed as if the parallel postulate should be a consequence of the other postulates, that is, it should be a theorem rather than a postulate. For more than 2000 years mathematicians tried unsuccessfully to deduce the parallel postulate from the other four. In the nineteenth century mathematicians began to consider the possibility that there might be geometries for which the parallel postulate fails, and they began to study the properties of such geometries. At long last some models of geometries for which the parallel postulate fails were discovered.