ifferent conceptions of the elements at
infinity are used. As we can know nothing experimentally about such
things, we are at liberty to make any assumptions we please, so long as
they are consistent and serve some useful purpose.

PROBLEMS

1. Since there is a threefold infinity of points in space, there must be a
sixfold infinity of pairs of points in space. Each pair of points
determines a line. Why, then, is there not a sixfold infinity of lines in
space?

2. If there is a fourfold infinity of lines in space, why is it that there
is not a fourfold infinity of planes through a point, seeing that each
line in space determines a plane through that point?

3. Show that there is a fourfold infinity of circles in space that pass
through a fixed point. (Set up a one-to-one correspondence between the
axes of the circles and lines in space.)

4. Find the order of infinity of all the lines of space that cut across a
given line; across two given lines; across three given lines; across four
given lines.

5. Find the order of infinity of all the spheres in space that pass
through a given point; through two given points; through three given
points; through four given points.

6. Find the order of infinity of all the circles on a sphere; of all the
circles on a sphere that pass through a fixed point; through two fixed
points; through three fixed points; of all the circles in space; of all
the circles that cut across a given line.

7. Find the order of infinity of all lines tangent to a sphere; of all
planes tangent to a sphere; of lines and planes tangent to a sphere and
passing through a fixed point.

8. Set up a one-to-one correspondence between the series of numbers _1_,
_2_, _3_, _4_, ... and the series of even numbers _2_, _4_, _6_, _8_ ....
Are we justified in saying that there are just as many even numbers as
there are numbers altogether?

9. Is the axiom "The whole is greater than one of its parts" applicable to
infinite assemblages?

10. Make out a classified list of all the infinitudes of the first,
second, third, and fourth orders mentioned in this chapter.

CHAPTER II - RELATIONS BETWEEN FUNDAMENTAL FORMS IN ONE-TO-ONE
CORRESPONDENCE WITH EACH OTHER

*23. Seven fundamental forms.* In the preceding chapter we have called
attention to seven fundamental forms: the point-row, the pencil of rays,
the axial pencil, the plane system, the point system, the space system,
and the system of