An Elementary Course in Synthetic Projective Geometry | Page 7

the position of a point
on a line; two, a point in the plane; three, a point in space; etc. In the
same theory a one-to-one correspondence is set up between loci in the
plane and equations in two variables; between surfaces in space and
equations in three variables; etc. The equation of a line in a plane
involves two constants, either of which may take an infinite number of
values. From this it follows that there is an infinity of lines in the plane
which is of the second order if the infinity of points on a line is
assumed to be of the first. In the same way a circle is determined by
three conditions; a sphere by four; etc. We might then expect to be able
to set up a one-to-one correspondence between circles in a plane and
points, or planes in space, or between spheres and lines in space. Such,

indeed, is the case, and it is often possible to infer theorems concerning
spheres from theorems concerning lines, and vice versa. It is
possibilities such as these that, give to the theory of one-to-one
correspondence its great importance for the mathematician. It must not
be forgotten, however, that we are considering only continuous
correspondences. It is perfectly possible to set, up a one-to-one
correspondence between the points of a line and the points of a plane,
or, indeed, between the points of a line and the points of a space of any
finite number of dimensions, if the correspondence is not restricted to
be continuous.

*22. Elements at infinity.* A final word is necessary in order to explain
a phrase which is in constant use in the study of projective geometry.
We have spoken of the "point at infinity" on a straight line--a fictitious
point only used to bridge over the exceptional case when we are setting
up a one-to-one correspondence between the points of a line and the
lines through a point. We speak of it as "a point" and not as "points,"
because in the geometry studied by Euclid we assume only one line
through a point parallel to a given line. In the same sense we speak of
all the points at infinity in a plane as lying on a line, "the line at
infinity," because the straight line is the simplest locus we can imagine
which has only one point in common with any line in the plane.
Likewise we speak of the "plane at infinity," because that seems the
most convenient way of imagining the points at infinity in space. It
must not be inferred that these conceptions have any essential
connection with physical facts, or that other means of picturing to
ourselves the infinitely distant configurations are not possible. In other
branches of mathematics, notably in the theory of functions of a
complex variable, quite different assumptions are made and quite
different 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