in Mathematical Reasoning. 327 
must be regulated by definition; and here arises one of the 
great differences which characterise this science, the definitions 
themselves being exceedingly simple, comprising but few ideas, 
whilst in other sciences they are usually much more complicated. 
In Geometry, definition is the beginning of any enquiry; in meta- 
physical science, it is frequently the result of one: thus that a 
triangle is a figure formed by three sides, is a convention on 
which many of Euclid’s propositions rest, and from this, as a point 
of departure, numerous deductions are made: on the other hand, 
our idea, and censequently our definition of beauty, is. only the 
result of considerable thought and enquiry. 
In the language of analysis, it is very rare that any symbol 
possesses more than one meaning; in ordinary language, it is as 
rare to find a word having but one signification: nor is this the only 
difference; when an algebraical symbol has more than one mean- 
ing, they are always well defined and distinct, and should there 
exist several signs for the same operation, the only difference is in 
their external form, not the slightest in their meaning; whilst in 
common language, the meanings of words shade away into each 
other, and it is frequently difficult, even on mature consideration, 
to assign the precise limits of the signification of words which 
are nearly synonymous. 
Now if this be the case when the words themselves are the 
especial objects of our thoughts, how open must all reasoning be 
to inadyertencies when the mind is compelled to occupy itself at 
once on the various meanings of the signs it uses and on the. train 
of consequences which it endeavours to deduce by them. 
The multitude of significations which attach to, many of the 
words that compose our ordinary language, is a disadvantage 
which is completely removed from that of analysis. In our rea- 
soning concerning any objects even of a moderately complicated 
nature, we are obliged to make use. of the words attached to- those’ 
