Mathematical philosophy, a study of fate and freedom; lectures for educated laymen, by Cassius J. Keyser.

388 MATHEMATICAL PHILOSOPHY
conceptual lines before passing to the limit. We are
thus led to believe that any two conceptual lines that
cross one another have a point in common. (In this
connection see the book of Poincaré above referred to.)
Such, however, is not always the case if the lines are
" continua" of only first order, that is, if they have
points that correspond only to rational numbers or, as
we say, to rational coordinates. It is sufficient to consider the crossing of a circle and a straight line through
its center, if either or both of them be supposed to have
no points except such as have rational coordinates. For
FIG. 31.
let x2+y2= i and y =x be respectively such a circle and
line. If we undertake to solve for their common points,
we immediately find that they have none; for we obtain
x=1: /2 and y=I: </2; but these numbers are not
rational and hence do not represent a point common
to the crossing line and circle. It might be supposed that
such a circle could have only a finite number of points
and that their ensemble, if plotted, would not look like a
circle. It is easy, however, to prove that the number
of the points is infinite and that they constitute a dense
set, a set, that is, such that between any two of them
there is another one. Obviously, one of the points is the
point (0, I). Suppose x0O. Then since x and y are to
be rational, y must be of the form, y = -rx, where r is