*6. Infinitely distant point.* We have illustrated above a simple method of setting the points of two lines into one-to-one correspondence. The same illustration will serve also to show how it is possible to set the points on a line into one-to-one correspondence with the lines through a point. Thus, for any point _C_ on the line _AB_ there is a line _SC_ through _S_. We must assume the line _AB_ extended indefinitely in both directions, however, if we are to have a point on it for every line through _S_; and even with 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.