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.