The successive application of these two operations, provided the original figure consisted of points on one straight line or of straight lines through one point, gives a figure projectively indistinguishable from the former figure; and hence, by extension, if any points in one straight line in the original figure lie in one straight line in the derived figure, and reciprocally for straight lines through points, the two operations have given projectively similar figures. This general principle may be regarded as consisting of two parts, according to the order of the operations: if we begin with projection and end with section, we transform a figure of points into another figure of points; by the converse order, we transform a figure of lines into another figure of lines. =115.= Before we can be clear as to the meaning of our principle, we must have some notion as to our definition of points and straight lines. But this definition, in projective Geometry, cannot be given without some discussion of the principle of duality, the mathematical form of the philosophical circle involved in geometrical definitions. Confining ourselves for the moment to two dimensions, the principle asserts, roughly speaking, that any theorem, dealing with lines through a point and points on a line, remains true if these two terms, wherever they occur, are interchanged. Thus: two points lie on one straight line which they completely determine; and two straight lines meet in one point, which they completely determine. The four points of intersection of a transversal with four lines through a point have an anharmonic ratio independent of the particular transversal; and the four lines joining four points on one straight line to a fifth point have an anharmonic ratio independent of that fifth point. So also our general principle of projective transformation has two sides: one in which points move along fixed lines, and one in which lines turn about fixed points. This duality suggests that any definition of points must be effected by means of the straight line, and any definition of the straight line must be effected by means of points. When we take the third dimension into account, it is true, the duality is no longer so simple; we have now to take account also of the plane, but this only introduces a circle of three terms, which is scarcely preferable to a circle of two terms. We now say: Three points, or a line and a point, determine a plane: but conversely, three planes, or a line and plane, determine a point. We may regard the straight line as a relation between two of its points, but we may also regard the point as a relation between two straight lines through it. We may regard the plane as a relation between three points, or between a point and a line, but we may also regard the point as a relation between three planes, or between a line and a plane, which meet in it.