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 disadvantages which arise from the limited nature of our faculties, and the errors which may insensibly creep upon us in the process. We are further exposed to the operation of the unevennesses and irregularities that perpetually occur in external nature, the imperfection of our senses, and of the instruments we construct to assist our observations, and the discrepancy which we frequently detect between the actual nature of the things about us and our impressions respecting them.