*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.




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