UNDERSTANDING. 
words, founded upon and rising above one 
another. And it is merely association again 
which appropriates the word truth, &c. to 
the coincidence of the words or symbols 
which denote the numbers. 
Tliis coincidence of terms is considered 
as a proof that the visible ideas of the num- 
bers under consideration would coincide as 
much as the visible ideas of twice two and 
four, were the former equally distinct with 
the latter ; and indeed the same thing may 
be fully proved, and often is so, by experi- 
ments with counters, lines, &c. And hence 
thinking persons, who make a distinction 
often unthought of, between the coincidence 
of terms and that of ideas, consider the real 
and absolute truth to be as great in com- 
plex numerical propositions as in the sim- 
plest. Now as it is impossible to gain dis- 
tinct visible ideas of ditferent numbers, 
where at least they are considerable, terms 
rlenoting them are a necessary means of dis- 
tinguishing them one from anotlier, so as to 
reason justly respecting them. 
In geometry there is a like coincidence of 
lines, angles, spaces, and solid contents, to 
prove them equal in simple cases. After- 
wards, in complex cases, we substitute the 
terms whereby equal things are denoted 
for each other, and then the coincidence of 
the terms to denote the coincidence of the 
visible ideas, except in the new step ad- 
vanced in the proposition ; and thus we get 
a new equality, denoted by a new coinci- 
dence of terms ; and this in like manner 
w’e employ in order to obtain a new equa- 
lity. This resembles the addition of unity 
to any number in order to make the next, 
as of 1 to 20 in order to make 21. We 
have no distinct visible idea of 20 or of 21 ; 
but we have of the difference between 
them, by fat'cying to ourselves a confused 
heap of things, supposed or called tw'enty 
in number, and then further fancying one 
thing to be added to it. By a like process 
in geometry we arrive at the demonstration 
of the most complex propositions. — The 
properties of numbers are applied to geo- 
metry in many cases, as when we demon- 
strate a line or space to he half or double of 
any other, or in any other ratio to it. — And 
as in arithmetic w'ords stand for indistinct 
ideas, iir order to help us to reason about 
them as accurately as if they were dis- 
tinct ; as also cyphers stand for words, for 
the same purpose ; and letters for cyphers, 
to render the conclusions less particular ; 
SV letters are put for geometrical quantities 
also, and the agreements of the letters for 
those of the quantities. 
Thus we see the foundation upon which 
the whole doctrine of quantity is built ; for 
all quantity is denoted either by numbers 
or by extension, or by letters denoting ei- 
ther one or the other. The coincidence of 
ideas is the foundation of rational assent in 
simple cases; and that of ideas and of 
terms, or of terras alone, in complex cases. 
This is upon the supposition tliat the quan- 
tities are to be proved equal ; but if they 
are to be proved unequal, the want of coin- 
cidence answers the same purpose. If they 
are in any numerical ratio, this is only intro- 
ducing a new coincidence. — Thus it appears 
that the use of words, (either as visible or as 
audible symbols), is necessary for geome- 
trical and algebraic reasonings, as well as 
for arithmetical. Also that association pre- 
vails in every part of the processes hitherto 
described. 
But these are not the only causes of 
giving rational assent to mathematical pro- 
positions. The recollection of having once 
examined and assented to each step of a de- 
monstration ; the authority of an approved 
writer, &c. are often sufficient to gain our 
assent, though we understand no more than 
the import of the proposition ; nay, even 
though we do not proceed so far as this. 
Now this again is a mere transfer of asso- 
ciation ; the recollection, authority, &c. 
being in a great number of cases associated 
with the before-mentioned coincidence of 
ideas and terms. — But here a new circum- 
stance arises ; for memory and authority 
are sometimes found to mislead ; and the 
recollection of such experience puts the 
mind into a state of doubt, so that some- 
times truth, sometimes falsehood, will recur 
and unite itself with the proposition under 
consideration, according as the recollection, 
authority, &c. in all their peculiar circum- 
stances have been associated witli truffi or 
with falsehood. 
Thus the idea belonging to a mathemati- 
cal proposition, with the rational assent or 
dissent arising in the mind, as soon as it is 
presented to it, is nothing more than a group 
of ideas united by association, and forming a 
very complex idea 53.) And this idea is 
not merely the sum of the ideas belonging to 
the terms ot the proposition, but also in- 
cludes the notions or feelings, whatever 
they be, which belong to tlie words equa- 
lity, coincidence, and truth, and, in some 
cases those of utility, importance, &c.— For 
