in Mathematical Reasoning. 337 
that more numbers than one fulfil that condition, there is no 
reason why the answer should produce one of these numbers 
rather than another, it must therefore contain them all. 
The reasoning which is carried on in Geometry is of a general 
nature, and applies to a species, although it is impossible that the 
picture presented to the eye can be any thing else than that of an 
individual; hence, it not unfrequently happens that some pecu- 
liarity in the figure which is actually employed, either leads us 
into erroneous conclusions, or when the results are correct, they 
are supposed to be limited by the individual nature of the figure 
we have employed. If a line is made use of to represent number, 
since some other line is the standard unit, it is impossible by 
such means to represent number in the abstract, but if number 
is denoted by a letter, there is nothing in the sign which at all 
indicates the magnitude of that which it represents: it is evident 
therefore that a property which might lead us into error in the 
first case, is removed from our view in the second. It will per- 
haps be objected that the standard unit need not be visible to 
the eye, since the force of the demonstration is in no way affected 
by its magnitude, this observation is perfectly correct, and if 
only one line be considered, and no unity of linear measure be 
stated, that line may represent length in general, and is to all 
purposes an arbitrary sign: but the moment any other line is 
introduced into the diagram, although the unit should not be 
mentioned, the generality of the former sign is diminished, a 
relation is instantly established, and whatever may be the unit 
of length, the ratio of these two lines is fixed and determinate. 
The position of a line is another circumstance in Geometry 
which must always remain particular, and this brings with it that of 
the points formed by its intersection, as well as that of the angles 
formed by it with other lines, and the attention which the mind 
must exert to perceive that no part of the reasoning it is pursuing 
