In the Introduction we noted that with every group of transformations there is associated a geometry concerned with those properties of geometric figures which remain unaltered by the transformations of the group in question. Thus classical Euclidean geometry studied in high school is not the only possible geometry; by choosing a group of transformations other than the group of motions (or the group of similarities leading to a geometry very closely related to Euclidean) we are led to a new, “non-Euclidean”, geometry. An instance of such a “non-Euclidean” geometry is projective geometry, concerned with those properties of figures which do not change under projective transformations. Projective geometry is not merely not Euclidean geometry; it is “very much non-Euclidean”. For example, distance between points is not a projective property, for it is possible to carry any segment into any other segment by means of a projective transformation; in other words, in projective geometry all segments are “congruent”. Likewise, all angles are “congruent”, so that the usual notion of angle between lines is without meaning in projective geometry. Also, any two quadrilaterals in the plane are “congruent” in the sense of projective geometry (in this connection see Theorem 1, §2, p. 45) so that it makes no sense to speak of various classes of quadrilaterals (such as parallelograms, trapezoids, and so on).