was brought to light that it is sometimes possible to set the elements of
an assemblage into one-to-one correspondence with a part of those
elements. A moment’s reflection will convince one that this is never
possible when there is a finite number of elements in the
assemblage.—Indeed, we may take this property as our definition of an
infinite assemblage, and say that an infinite assemblage is one that may
be put into one-to-one correspondence with part of itself. This has the
advantage of being a positive definition, as opposed to the usual negative
definition of an infinite assemblage as one that cannot be counted.




*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