f elements. If, however, there is an infinite number
of individuals in each of the two sets, the notion of counting is
necessarily ruled out. It may be possible, nevertheless, to set up a
one-to-one correspondence between the elements of two sets even when the
number is infinite. Thus, it is easy to set up such a correspondence
between the points of a line an inch long and the points of a line two
inches long. For let the lines (Fig. 1) be _AB_ and _A'B'_. Join _AA'_ and
_BB'_, and let these joining lines meet in _S_. For every point _C_ on
_AB_ a point _C'_ may be found on _A'B'_ by joining _C_ to _S_ and noting
the point _C'_ where _CS_ meets _A'B'_. Similarly, a point _C_ may be
found on _AB_ for any point _C'_ on _A'B'_. The correspondence is clearly
one-to-one, but it would be absurd to infer from this that there were just
as many points on _AB_ as on _A'B'_. In fact, it would be just as
reasonable to infer that there were twice as many points on _A'B'_ as on
_AB_. For if we bend _A'B'_ into a circle with center at _S_ (Fig. 2), we
see that for every point _C_ on _AB_ there are two points on _A'B'_. Thus
it is seen that the notion of one-to-one correspondence is more extensive
than the notion of counting, and includes the notion of counting only when
applied to finite assemblages.

*5. Correspondence between a part and the whole of an infinite
assemblage.* In the discussion of the last paragraph the remarkable fact
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