now 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 adm