in Mathematical Reasoning. 335 
required, all questions that are made to depend on them may be 
considered as solved. 
The power which language gives us of generalizing our 
reasonings concerning individuals by the aid of general terms, is 
no where more eminent than in the mathematical sciences, nor 
is it carried to so great an extent in any other part of human 
knowledge. In the transition from Arithmetic to Algebra, when 
letters began to be substituted for numbers, the first step consisted 
rather in the circumstance of the possibility of operating on a 
quantity determined but unknown. Thus if it were proposed to 
discover such a number, that its square added to three should be 
equal to four times the number itself; we commence by supposing 
the number to be represented by x: now it is quite certain, as soon 
as the question is stated, that there can only exist two numbers 
fulfilling the condition; x therefore must in reality mean either 
of these two, and the rest of the process is 
“+3 =42, 
@—42+4=1, 
Ge 2 — 2ty Ii, 
Th uOr Ne 
To point out more clearly the force of this observation, we adopt 
the plan which Vieta introduced into Algebra, that of denoting 
known quantities by letters: instead of the numbers 3 and 4, 
let us use the letters a and b; then the process is as follows: 
“+ a= bez, 
x — br=—a, 
bb Bb 
2 _—_—_— 
ipl greg a, 
b 5° 
e==+ ——— & 
Bes 
here it is true that a and b meant 3 and 4, but as no part of 
the reasoning employed in any manner depended on _ their 
