this extension there is one line through S, according to Euclid’s postulate, which does not meet the line AB and which therefore has no point on AB to correspond to it.  In order to smooth out this discrepancy we are accustomed to assume the existence of an infinitely distant point on the line AB and to assign this point as the corresponding point of the exceptional line of S.  With this understanding, then, we may say that we have set the lines through a point and the points on a line into one-to-one correspondence.  This correspondence is of such fundamental importance in the study of projective geometry that a special name is given to it.  Calling the totality of points on a line a point-row, and the totality of lines through a point a pencil of rays, we say that the point-row and the pencil related as above are in perspective position, or that they are perspectively related.

7.  Axial pencil; fundamental forms. A similar correspondence may be set up between the points on a line and the planes through another line which does not meet the first.  Such a system of planes is called an axial pencil, and the three assemblages—­the point-row, the pencil of rays, and the axial pencil—­are called fundamental forms.  The fact that they may all be set into one-to-one correspondence with each other is expressed by saying that they are of the same order.  It is usual also to speak of them as of the first order.  We shall see presently that there are other assemblages which cannot be put into this sort of one-to-one correspondence with the points on a line, and that they will very reasonably be said to be of a higher order.

8.  Perspective position. We have said that a point-row and a pencil of rays are in perspective position if each ray of the pencil goes through the point of the point-row which corresponds to it.  Two pencils of rays are also said to be in perspective position if corresponding rays meet on a straight line which is called the axis of perspectivity.  Also, two point-rows are said to be in perspective position if corresponding points lie on straight lines through a point which is called the center of perspectivity.  A point-row and an axial pencil are in perspective position if each plane of the pencil goes through the point on the point-row which corresponds to it, and an axial pencil and a pencil of rays are in perspective position if each ray lies in the plane which corresponds to it; and, finally, two axial pencils are perspectively related if corresponding planes meet in a plane.