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A Treatise of Human Nature

all other cases we must settle the proportions with some liberty, or proceed in a more
artificial manner.
I have already I observed, that geometry, or the art, by which we fix the proportions of
figures; though it much excels both in universality and exactness, the loose judgments of
the senses and imagination; yet never attains a perfect precision and exactness. It's first
principles are still drawn from the general appearance of the objects; and that appearance
can never afford us any security, when we examine, the prodigious minuteness of which
nature is susceptible. Our ideas seem to give a perfect assurance, that no two right lines
can have a common segment; but if we consider these ideas, we shall find, that they
always suppose a sensible inclination of the two lines, and that where the angle they form
is extremely small, we have no standard of a I @ right line so precise as to assure us of
the truth of this proposition. It is the same case with most of the primary decisions of the
mathematics.
There remain, therefore, algebra and arithmetic as the only sciences, in which we can
carry on a chain of reasoning to any degree of intricacy, and yet preserve a perfect
exactness and certainty. We are possest of a precise standard, by which we can judge of
the equality and proportion of numbers; and according as they correspond or not to that
standard, we determine their relations, without any possibility of error. When two
numbers are so combined, as that the one has always an unite answering to every unite of
the other, we pronounce them equal; and it is for want of such a standard of equality in
extension, that geometry can scarce be esteemed a perfect and infallible science.
But here it may not be amiss to obviate a difficulty, which may arise from my asserting,
that though geometry falls short of that perfect precision and certainty, which are peculiar
to arithmetic and algebra, yet it excels the imperfect judgments of our senses and
imagination. The reason why I impute any defect to geometry, is, because its original and
fundamental principles are derived merely from appearances; and it may perhaps be
imagined, that this defect must always attend it, and keep it from ever reaching a greater
exactness in the comparison of objects or ideas, than what our eye or imagination alone is
able to attain. I own that this defect so far attends it, as to keep it from ever aspiring to a
full certainty: But since these fundamental principles depend on the easiest and least
deceitful appearances, they bestow on their consequences a degree of exactness, of which
these consequences are singly incapable. It is impossible for the eye to determine the
angles of a chiliagon to be equal to 1996 right angles, or make any conjecture, that
approaches this proportion; but when it determines, that right lines cannot concur; that we
cannot draw more than one right line between two given points; it's mistakes can never be
of any consequence. And this is the nature and use of geometry, to run us up to such
appearances, as, by reason of their simplicity, cannot lead us into any considerable error.
I shall here take occasion to propose a second observation concerning our demonstrative
reasonings, which is suggested by the same subject of the mathematics. It is usual with
mathematicians, to pretend, that those ideas, which are their objects, are of so refined and
spiritual a nature, that they fall not under the conception of the fancy, but must be
comprehended by a pure and intellectual view, of which the superior faculties of the soul
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