66 
SIR B. C. BRODIE ON THE CALCULUS OF CHEMICAL OPERATIONS. 
It is impossible for a false or absurd result to be the consequence of the application 
to a true statement of a correct principle of reasoning ; and we are thus led to the sin- 
gular conclusion that, although up to a certain point the analogy is perfect between 
the properties of chemical and numerical symbols, here this analogy terminates and 
ceases to be a correct guide ; and we are apparently forced to admit that in the treat- 
ment of chemical equations no processes exist equivalent to the algebraical processes of 
multiplication and division. Now, first, we may observe that such an imperfect calculus 
might really exist, logically correct, but with limited methods ; for there is nothing 
either in the nature of symbolical algebra or in the distributive and commutative laws, 
x{y+z)=xy+xz, 
xy=yx, 
through which the symbols of chemical operations are related to the symbols of numbers 
(Part I. Section II. (4) and (5)), to necessitate the application of these processes to equa- 
tions. Indeed in the actual applications of algebra perfectly similar restrictions are 
found to exist. The symbols of certain operations do not satisfy the commutative law, 
and we cannot infer from the equation 
vxy=vx-\-vy 
that (whatever be the interpretation of the symbols) 
xy=x+.y. 
Thus, for example, in the algebra of logic, while the operation of algebraical multiplica- 
tion, as applied to equations, is permissible, the performance of the operation of 
division upon logical equations is absolutely prohibited. Thus we cannot infer from 
the logical equation 
zy=zx 
that y=x, which would be tantamount to asserting that because those members of a 
class y which possess a certain property z are identical with those members of a class x 
which possess the same property, therefore the class y is identical with the class x 
[Boole, ‘ Laws of Thought,’ p. 36], an inference obviously erroneous. But even in the 
algebra of quantity we cannot always perform the operation of division upon equations 
where such division is formally possible. From the equation x(x—a )= 0 we cannot 
infer that #=0. The operation of division can only be thus performed by the exclu- 
sion of certain symbols upon the assumption that 0 and ^ are not values of the symbol 
by which we divide. Hence the circumstance that such processes are inadmissible 
does not detract in any degree from the truth, reality, or logical perfection of an 
algebraical system. Nevertheless it may be admitted with perfect consistency that a 
calculus from which such important processes were entirely excluded, and in which the 
processes of addition and subtraction were the only processes practically available, 
would be of very limited utility, and could only be regarded as a rudimentary and 
