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

THE MATHEMATICS OF PSYCHOLOGY 399
magnitudes of the type of the second-order mathematical
continuum. Next let the breadth and thickness diminish
more and more, keeping always a common part without
crossing. It appears that, at the limit, the common part
will be a point at which, however, the line and the curve
do not cross; that is to say, it appears that the line
becomes a tangent to the curve at the point. In some such
way, it came to be believed that a curve, if continuous
at any point, admits a tangent at the point. Nothing
could be more natural and the belief was persistent and
long-lived. Yet we know to-day that in such matters
FIG. 33.
our intuition, precious as it is, cannot be implicitly
trusted for we know today that the belief in question is
false. This fact may be shown by the following classical
example. Consider the locus of the equation.
y =x sin -
x
It is continuous at every point except the point whose
abscissa is x=O. At this point y is not defined, since
division by zero is meaningless. Hence we may define
it as we please. Let us agree that y shall be zero when
x=O. This being done, the curve is now continuous at
the point, x = O, as well as at all other points. Differentiating, we get
dy sin I s
-- cos -
dx xx x