=108.= All geometrical reasoning is, in the last resort, circular: if we start by assuming points, they can only be defined by the lines or planes which relate them; and if we start by assuming lines or planes, they can only be defined by the points through which they pass. This is an inevitable circle, whose ground of necessity will appear as we proceed. It is, therefore, somewhat arbitrary to start either with points or with lines, as the eminently projective principle of duality mathematically illustrates; nevertheless we will elect, with most geometers, to start with points[120]. We suppose, therefore, as our datum, a set of discrete points, for the moment without regard to their interconnections. But since connections are essential to any reasoning about them as a system, we introduce, to begin with, the axiom of the straight line. Any two of our points, we say, lie on a line which those two points completely define. This line, being determined by the two points, may be regarded as a relation of the two points, or an adjective of the system formed by both together. This is the only purely qualitative adjective--as will be proved later--of a system of two points. Now projective Geometry can only take account of qualitative adjectives, and can distinguish between different points only by their relations to other points, since all points, _per se_, are qualitatively similar. Hence it comes that, for projective Geometry, when two points only are given, they are qualitatively indistinguishable from any two other points on the same straight line, since any two such other points have the same qualitative relation. Reciprocally, since one straight line is a figure determined by any two of its points, and all points are qualitatively similar, it follows that all straight lines are qualitatively similar. We may regard a point, therefore, as determined by two straight lines which meet in it, and the point, on this view, becomes the only qualitative relation between the two straight lines. Hence, if the point only be regarded as given, the two straight lines are qualitatively indistinguishable from any other pair through the point. =109.= The extension of these two reciprocal principles is the essence of all projective transformations, and indeed of all projective Geometry. The fundamental operations, by which figures are projectively transformed, are called projection and section. The various forms of projection and section are defined in Cremona's "Projective Geometry," Chapter I., from which I quote the following account. "_To project from a fixed point S_ (the _centre of projection_) a figure (_ABCD_ ... _abcd_ ...) composed of points and straight lines, is to construct the straight lines or _projecting rays SA_, _SB_, _SC_, _SD_, ... and the planes (_projecting planes_) _Sa_, _Sb_, _Sc_, _Sd_, ... We thus obtain a new figure composed of straight lines and planes which all pass through the centre _S_.