rsuaded 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 show 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 has 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