consider, it is certain they are not contained in it; but, it is
evident that I cannot distinguish innumerable parts in any
particular line, surface, or solid, which I either perceive by
sense, or figure to myself in my mind: wherefore I conclude they are
not contained in it. Nothing can be plainer to me than that the
extensions I have in view are no other than my own ideas; and it is no
less plain that I cannot resolve any one of my ideas into an
infinite number of other ideas, that is, that they are not
infinitely divisible. If by finite extension be meant something
distinct from a finite idea, I declare I do not know what that is, and
so cannot affirm or deny anything of it. But if the terms "extension,"
"parts," &c., are taken in any sense conceivable, that is, for
ideas, then to say a finite quantity or extension consists of parts
infinite in number is so manifest a contradiction, that every one at
first sight acknowledges it to be so; and it is impossible it should
ever gain the assent of any reasonable creature who is not brought
to it by gentle and slow degrees, as a converted Gentile to the belief
of transubstantiation. Ancient and rooted prejudices do often pass
into principles; and those propositions which once obtain the force
and credit of a principle, are not only themselves, but likewise
whatever is deducible from them, thought privileged from all
examination. And there is no absurdity so gross, which, by this means,
the mind of man may not be prepared to swallow.

  125. He whose understanding is possessed with the doctrine of
abstract general ideas may be persuaded that (whatever be thought of
the ideas of sense) extension in abstract is infinitely divisible. And
one who thinks the objects of sense exist without the mind will
perhaps in virtue thereof be brought to admit that a line but an
inch long may contain innumerable parts- really existing, though too
small to be discerned. These errors are grafted as well in the minds
of geometricians as of other men, and have a like influence on their
reasonings; and it were no difficult thing to shew how the arguments
from Geometry made use of to support the infinite divisibility of
extension are bottomed on them. At present we shall only observe in
general whence it is the mathematicians are all so fond and
tenacious of that doctrine.

  126. It hath been observed in another place that the theorems and
demonstrations in Geometry are conversant about universal ideas (sect.
15, Introd.); where it is explained in what sense this ought to be
understood, to wit, the particular lines and figures included in the
diagram are supposed to stand for innumerable others of different
sizes; or, in other words, the geometer considers them abstracting
from their magnitude- which does not imply that he forms an abstract
idea, but only that he cares not what the particular magnitude is,
whether great or small, but looks on that as a thing different to
the demonstration. Hence it follows that a line in the scheme but an
inch long must be spoken of as though it contained ten thousand parts,
since it is regarded not in itself, but as it is universal; and it
is universal only in its signification, whereby it represents
innumerable lines greater than itself, in which may be distinguished
ten thousand parts or more, though there may not be above an inch in
it. After this manner, the properties of the lines signified are (by a
very usual figure) transferred to the sign, and thence, through
mistake, though to appertain to it considered in its own nature.

  127. Because there is no number of parts so great but it is possible
there may be a line containing more, the inch-line is said to
contain parts more than any assignable number; which is true, not of
the inch taken absolutely, but only for the things signified by it.
But men, not retaining that distinction in their thoughts, slide
into a belief that the small particular line described on paper
contains in itself parts innumerable. There is no such thing as the
ten-thousandth part of an inch; but there is of a mile or diameter
of the earth, which may be signified by that inch. When therefore I
delineate a triangle on paper, and take one side not above an inch,
for example, in length to be the radius, this I consider as divided
into 10,000 or 100,000 parts or more; for, though the ten-thousandth
part of that line considered in itself is nothing at all, and
consequently may be neglected without an error or inconveniency, yet
these described lines, being only marks standing for greater
quantities, whereof it may be the ten-thousandth part is very
considerable, it follows that, to prevent notable errors in
practice, the radius must be taken of 10,000 parts or more.

  128. From what has been said the reason is plain why, to the end any
theorem become universal in its use, it is necessary we speak of the
lines described on paper as though they contained parts which really
they do not. In doing of which, if we examine the matter thoroughly,
we shall perhaps discover that we cannot conceive an inch itself as
consisting of, or being divisible into, a thousand parts, but only
some other line which is far greater than an inch, and represented
by it; and that when we say a line is infinitely divisible, we must
mean a line which is infinitely great. What we have here observed
seems to be the chief cause why, to suppose the infinite
divisibility of finite extension has been thought necessary in
geometry.

  129. The several absurdities and contradictions which flowed from
this false principle might, one would think, have been esteemed so
many demonstrations against it. But, by I know not what logic, it is
held that proofs a posteriori are not to be admitted against
propositions relating to infinity, as though it were not impossible
even for an infinite mind to reconcile contradictions; or as if
anything absurd and repugnant could have a necessary connexion with
truth or flow from it. But, whoever considers the weakness of this
pretence will think it was contrived on purpose to humour the laziness
of the mind which had rather acquiesce in an indolent scepticism
than be at the pains to go through with a severe examination of