and consider what has been premised, we may perhaps entertain a low
opinion of those high flights and abstractions, and look on all
inquiries, about numbers only as so many difficiles nugae, so far as
they are not subservient to practice, and promote the benefit of life.

  120. Unity in abstract we have before considered in sect. 13, from
which and what has been said in the Introduction, it plainly follows
there is not any such idea. But, number being defined a "collection of
units," we may conclude that, if there be no such thing as unity or
unit in abstract, there are no ideas of number in abstract denoted
by the numeral names and figures. The theories therefore in
Arithmetic. if they are abstracted from the names and figures, as
likewise from all use and practice, as well as from the particular
things numbered, can be supposed to have nothing at all for their
object; hence we may see how entirely the science of numbers is
subordinate to practice, and how jejune and trifling it becomes when
considered as a matter of mere speculation.

  121. However, since there may be some who, deluded by the specious
show of discovering abstracted verities, waste their time in
arithmetical theorems and problems which have not any use, it will not
be amiss if we more fully consider and expose the vanity of that
pretence; and this will plainly appear by taking a view of
Arithmetic in its infancy, and observing what it was that originally
put men on the study of that science, and to what scope they
directed it. It is natural to think that at first, men, for ease of
memory and help of computation, made use of counters, or in writing of
single strokes, points, or the like, each whereof was made to
signify an unit, i.e., some one thing of whatever kind they had
occasion to reckon. Afterwards they found out the more compendious
ways of making one character stand in place of several strokes or
points. And, lastly, the notation of the Arabians or Indians came into
use, wherein, by the repetition of a few characters or figures, and
varying the signification of each figure according to the place it
obtains, all numbers may be most aptly expressed; which seems to
have been done in imitation of language, so that an exact analogy is
observed betwixt the notation by figures and names, the nine simple
figures answering the nine first numeral names and places in the
former, corresponding to denominations in the latter. And agreeably to
those conditions of the simple and local value of figures, were
contrived methods of finding, from the given figures or marks of the
parts, what figures and how placed are proper to denote the whole,
or vice versa. And having found the sought figures, the same rule or
analogy being observed throughout, it is easy to read them into words;
and so the number becomes perfectly known. For then the number of
any particular things is said to be known, when we know the name of
figures (with their due arrangement) that according to the standing
analogy belong to them. For, these signs being known, we can by the
operations of arithmetic know the signs of any part of the
particular sums signified by them; and, thus computing in signs
(because of the connexion established betwixt them and the distinct
multitudes of things whereof one is taken for an unit), we may be able
rightly to sum up, divide, and proportion the things themselves that
we intend to number.

  122. In Arithmetic, therefore, we regard not the things, but the
signs, which nevertheless are not regarded for their own sake, but
because they direct us how to act with relation to things, and dispose
rightly of them. Now, agreeably to what we have before observed of
words in general (sect. 19, Introd.) it happens here likewise that
abstract ideas are thought to be signified by numeral names or
characters, while they do not suggest ideas of particular things to
our minds. I shall not at present enter into a more particular
dissertation on this subject, but only observe that it is evident from
what has been said, those things which pass for abstract truths and
theorems concerning numbers, are in reality conversant about no object
distinct from particular numeral things, except only names and
characters, which originally came to be considered on no other account
but their being signs, or capable to represent aptly whatever
particular things men had need to compute. Whence it follows that to
study them for their own sake would be just as wise, and to as good
purpose as if a man, neglecting the true use or original intention and
subserviency of language, should spend his time in impertinent
criticisms upon words, or reasonings and controversies purely verbal.

  123. From numbers we proceed to speak of Extension, which,
considered as relative, is the object of Geometry. The infinite
divisibility of finite extension, though it is not expressly laid down
either as an axiom or theorem in the elements of that science, yet
is throughout the same everywhere supposed and thought to have so
inseparable and essential a connexion with the principles and
demonstrations in Geometry, that mathematicians never admit it into
doubt, or make the least question of it. And, as this notion is the
source from whence do spring all those amusing geometrical paradoxes
which have such a direct repugnancy to the plain common sense of
mankind, and are admitted with so much reluctance into a mind not
yet debauched by learning; so it is the principal occasion of all that
nice and extreme subtilty which renders the study of Mathematics so
difficult and tedious. Hence, if we can make it appear that no
finite extension contains innumerable parts, or is infinitely
divisible, it follows that we shall at once clear the science of
Geometry from a great number of difficulties and contradictions
which have ever been esteemed a reproach to human reason, and withal
make the attainment thereof a business of much less time and pains
than it hitherto has been.

  124. Every particular finite extension which may possibly be the
object of our thought is an idea existing only in the mind, and
consequently each part thereof must be perceived. If, therefore, I
cannot perceive innumerable parts in any finite extension that I