in Mathematical Reasoning. 359 
quired were discovered by some artifice without the solution 
of the equation, we could not feel assured that it alone ful- 
filled all the conditions, and we should, by arriving at an equa- 
tion and rejecting its use, have the semblance of generality 
without its reality; nor do I perceive any reasons which should 
induce us to change our course, when we have to consider equa- 
tions of a more comprehensive nature. 
Any enumeration of the ‘causes which contribute to give 
such extensive power to the employment of algebraic signs, 
would be justly considered incomplete, if no notice were be- 
stowed on the symmetry that ought always to prevail, where 
the calculations in which it is employed are in any degree 
complicated. 
In its least restricted signification, symmetry is applied to 
two things, which although sometimes connected, are yet, in 
many instances, totally independent: it either refers to a resem- 
blance between the systems of characters assumed’ to represent 
the data of a question, or it implies a similarity of situation, 
between certain of the letters, which are found in an analytical 
formula. An attention to it in either of these senses, has a direct 
and very beneficial influence in relieving the memory from a 
considerable burthen. In the first case, its precepts would direct 
us to assume similar letters as the representatives of similar 
things: thus, if we have two series, and propose to find another, 
which consists of the product of the corresponding terms of the 
two former; if we assume for the first of two series 
at+b+c+d+.. 
the terms of the second ought to be 
4+B4+C+D+.. 
or which is still more convenient 
ad+b4+ce+d + 
