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<h2> PART III. OF KNOWLEDGE AND PROBABILITY. </h2>
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<h2> SECT. I. OF KNOWLEDGE. </h2>
<p>There are seven [Part I. Sect. 5.] different kinds of philosophical
relation, viz. RESEMBLANCE, IDENTITY, RELATIONS OF TIME AND PLACE,
PROPORTION IN QUANTITY OR NUMBER, DEGREES IN ANY QUALITY, CONTRARIETY and
CAUSATION. These relations may be divided into two classes; into such as
depend entirely on the ideas, which we compare together, and such as may
be changed without any change in the ideas. It is from the idea of a
triangle, that we discover the relation of equality, which its three
angles bear to two right ones; and this relation is invariable, as long as
our idea remains the same. On the contrary, the relations of contiguity
and distance betwixt two objects may be changed merely by an alteration of
their place, without any change on the objects themselves or on their
ideas; and the place depends on a hundred different accidents, which
cannot be foreseen by the mind. It is the same case with identity and
causation. Two objects, though perfectly resembling each other, and even
appearing in the same place at different times, may be numerically
different: And as the power, by which one object produces another, is
never discoverable merely from their idea, it is evident cause and effect
are relations, of which we receive information from experience, and not
from any abstract reasoning or reflection. There is no single phaenomenon,
even the most simple, which can be accounted for from the qualities of the
objects, as they appear to us; or which we coued foresee without the help
of our memory and experience.</p>
<p>It appears, therefore, that of these seven philosophical relations, there
remain only four, which depending solely upon ideas, can be the objects of
knowledge and certainty. These four are RESEMBLANCE, CONTRARIETY, DEGREES
IN QUALITY, and PROPORTIONS IN QUANTITY OR NUMBER. Three of these
relations are discoverable at first sight, and fall more properly under
the province of intuition than demonstration. When any objects resemble
each other, the resemblance will at first strike the eye, or rather the
mind; and seldom requires a second examination. The case is the same with
contrariety, and with the degrees of any quality. No one can once doubt
but existence and non-existence destroy each other, and are perfectly
incompatible and contrary. And though it be impossible to judge exactly of
the degrees of any quality, such as colour, taste, heat, cold, when the
difference betwixt them is very small: yet it is easy to decide, that any
of them is superior or inferior to another, when their difference is
considerable. And this decision we always pronounce at first sight,
without any enquiry or reasoning.</p>
<p>We might proceed, after the same manner, in fixing the proportions of
quantity or number, and might at one view observe a superiority or
inferiority betwixt any numbers, or figures; especially where the
difference is very great and remarkable. As to equality or any exact
proportion, we can only guess at it from a single consideration; except in
very short numbers, or very limited portions of extension; which are
comprehended in an instant, and where we perceive an impossibility of
falling into any considerable error. In all other cases we must settle the
proportions with some liberty, or proceed in a more artificial manner.</p>
<p>I have already 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.</p>
<p>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.</p>
<p>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.</p>
<p>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 are alone capable. The same notion runs through most
parts of philosophy, and is principally made use of to explain oar
abstract ideas, and to shew how we can form an idea of a triangle, for
instance, which shall neither be an isoceles nor scalenum, nor be confined
to any particular length and proportion of sides. It is easy to see, why
philosophers are so fond of this notion of some spiritual and refined
perceptions; since by that means they cover many of their absurdities, and
may refuse to submit to the decisions of clear ideas, by appealing to such
as are obscure and uncertain. But to destroy this artifice, we need but
reflect on that principle so oft insisted on, that all our ideas are
copyed from our impressions. For from thence we may immediately conclude,
that since all impressions are clear and precise, the ideas, which are
copyed from them, must be of the same nature, and can never, but from our
fault, contain any thing so dark and intricate. An idea is by its very
nature weaker and fainter than an impression; but being in every other
respect the same, cannot imply any very great mystery. If its weakness
render it obscure, it is our business to remedy that defect, as much as
possible, by keeping the idea steady and precise; and till we have done
so, it is in vain to pretend to reasoning and philosophy.</p>
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