SIE B. C. BRODIE ON THE CALCULUS OF CHEMICAL 0PEEAT10N8. 803 



function, yet conformably to the principles of algebraic notation may be invariably sup- 

 pressed, and every numerical symbol which appears in this calculus may be interpreted 

 witli its usual aritlimetical signification, regard being liad to those special properties 

 which are derived from the subject on which it operates. If, for a moment, we discri- 

 minate between the chemical and arithmetical symbols and 1, marking the former as 

 0' and 1', and the latter as (0) and (1), it is at once evident that (1)=(1)1' = 1', and that 

 (0)=(0)x=0'. Hence we may in every case replace the chemical symbols 0' and 1' 

 by the arithmetical symbols (0) and (1), which are, so far as the purposes of this calculus 

 are concerned, identical with them both in interpretation and in properties. These 

 symbols and 1 may be termed the zero-symbols of the chemical system, being marks 

 by which we denote the absence of ponderable matter. Tliat such symbols may serve 

 most important ends is evident from the use which has been made in arithmetic of 

 the zero-symbol 0, which is the very key-stone of the arithmetical system ; and yet it 

 is not too much to assert that the system of chemical symbols without the zero-symbol 1 

 is as incomplete and as little adapted to the purpose which it is destined to fidfil as the 

 arithmetical system would be deprived of the symbol 0. 



(4) No other known system of symbols is characterized by the same property as that by 

 which chemical symbols are defined, but the equation x-\-y=xy\^ similar inform to the 

 equation connecting the logarithms of numbers ; and the relation which subsists between 

 the absolute weight (or weight in grammes) of the ponderable matter of which x and y 

 are the symbols is the same as the logarithmic relation. For, writing 'w{x) and vj{ij) as 

 the absolute weights of the ponderable matter symbolized by x and y, 



w(x)+w{y)=w(xy), 

 w{l)+w(y)^wiy), 

 w(l)=0, 



similar in form with the logarithmic equations 



l(x)+l(y)=l{xy), 

 l{l)+liy)=liy), 

 1{1) = 0. 



The property of chemical symbols given in the equation x-\-y=:xy may from these 

 analogies appropriately be termed the " logarithmic " property of these symbols. 



(5) It is sufficiently obvious that we may operate between chemical equations by means 

 of addition and subtraction as with numerical equations. This is a consequence of the 

 axiom that if identical weights be added to or removed from identical groups the resulting 

 groups are identical. So that, if x=y, and x^=yi, x+x^=y+i/i. The operations, 

 however, which correspond to the algebraic operations :>f multiplication and division can 

 only be performed under certain conditions, which will be considered in a subsequent 

 part of this memoir. 



