erable diffidence.

They are founded, as has already been said, next to the evidence of our
senses, upon the deductions of mathematical knowledge.

Mathematics are either pure or mixed.

Pure mathematics are concerned only with abstract propositions, and have
nothing to do with the realities of nature. There is no such thing in
actual existence as a mathematical point, line or surface. There is no
such thing as a circle or square. But that is of no consequence. We can
define them in words, and reason about them. We can draw a diagram, and
suppose that line to be straight which is not really straight, and that
figure to be a circle which is not strictly a circle. It is conceived
therefore by the generality of observers, that mathematics is the
science of certainty.

But this is not strictly the case. Mathematics are like those abstract
and imaginary existences about which they are conversant. They may
constitute in themselves, and in the apprehension of an infallible
being, a science of certainty. But they come to us mixed and
incorporated with our imperfections. Our faculties are limited; and we
may be easily deceived, as to what it is that we see with transparent
and unerring clearness, and what it is that comes to us through a
crooked medium, refracting and distorting the rays of primitive truth.
We often seem clear, when in reality the twilight of undistinguishing
night has crept fast and far upon us. In a train of deductions, as
in the steps of an arithmetical process, an error may have insinuated
itself imperceptibly at a very early stage, rendering all the subsequent
steps a wandering farther and farther from the unadulterated truth.
Human mathematics, so to speak, like the length of life, are subject to
the doctrine of chances. Mathematics may be the science of certainty to
celestial natures, but not to man.

But, if in the case of pure mathematics, we are exposed to the chances
of error and delusion, it is much worse with mixed mathematics.
The moment we step out of the high region of abstraction, and apply
ourselves to what we call external nature, we have forfeited that sacred
character and immunity, which we seemed entitled to boast, so long as
we remained inclosed in the sanctuary of unmingled truth. As has already
been said, we know what passes in the theatre of the mind; but we cannot
be said absolutely to know any thing more. In our speculations upon
actual existences we are not only subject to the d