74 
SIR B. C. BRODIE ON THE CALCULUS OF CHEMICAL OPERATIONS. 
not inconsistent with the use and interpretation of the chemical symbol (1), but is 
entirely at variance with the numerical interpretation of that symbol, so that the greatest 
possible divergence is apparently exhibited between the use which is made in this Cal- 
culus of that symbol and its arithmetical signification*. But if this same substitution 
be effected in a “ normal ” equation the anomaly disappears. No divergence is found 
between the arithmetical and chemical use of the symbol 1, and the equation is true both 
as an arithmetical and chemical equation ; for from the very principle of its construc- 
tion every “normal” equation must necessarily vanish when the prime factors in that 
equation are severally put equal to 1. So that the effect of making this substitution is 
simply to lead us to the assertion that 0 = 0, which is an arithmetical as well as chemical 
truth ; that is to say, every “ normal ” equation is true, not only as a “ chemical equa- 
tion,” but also as an arithmetical equation for the only arithmetical value which the 
symbols can possibly assume, namely the value 1 ; and wdien this value is assigned to 
every prime factor by which the equation is expressed, the chemical equation is turned 
into an arithmetical equation, which is both interpretable and true. The reason of this 
is that the chemical symbol 1 is absolutely identical, as regards its algebraical properties, 
with the arithmetical symbol 1, although having a totally different interpretation from 
that symbol, a point which has been fully demonstrated and discussed in Part I. 
Sec. III., and also that a “normal” equation is subject to precisely the same rules of 
algebraical treatment as an arithmetical equation, as has been just demonstrated. 
The real and valid character of the preceding reasoning will be evident from the fol- 
lowing example, which illustrates the way in which a chemical function passes into a 
numerical function when chemical symbols assume the numerical value 1, and the infer- 
ences which may be thus drawn. 
(6) In the first part of this memoir the symbols of the units of chemical substances 
are determined by the aid of certain indeterminate numerical equations, at which we 
arrive by the direct application to chemical equations of the fundamental properties of 
chemical symbols so frequently referred to xy=x-\-y (I. Sec. VII.). 
Take, for example, the equation given, Part I. Sec. VII. (8), 
2a m v m ' = 3a -f - a n v n ' ; 
from this equation we are able immediately to infer that 
(a m v m 'f=a\ aV‘, 
whence 
2m=3+w, 2 m x -=n v 
* The apparent paradox involved in this assertion may be removed hy assigning a special symbol, e p , to the 
unit of space, jo being a positive integer. But on investigating the properties of this symbol we should soon 
find that e— e p , whatever be the value of p (the value 0 included), and that, as we might always replace alge- 
braically the symbol e p by the symbol (1), we were really dealing with the symbol 1 under another name. 
Such paradoxes, however, have no significance when the meaning of the expressions employed is properly under- 
stood. Thus De Morgan, in his ‘ Double Algebra ’ (ed. 1849, p. 114), speaking of the term addition as there 
employed by him, says, “ Nor is there , in one sense, the slightest objection to saying that 12 and 12 make 10,” 
an assertion quite as paradoxical (to say the least) as any here made. 
