130 Dr. A. C. Brown on Sir Benjamin Brodie's 



x 



For if xy = x implies y = l, this can only be because y = —and 



x . x 



- = 1. In a symbolical system in which this is not the case we 



cannot deduce y=-\ from xy—x. Thus in the logical system of 

 the late Professor Boole, xy=x does not lead to y=\, but to 



The second case is x°=l. As we can have no fractional in- 

 dices, cannot be defined as a limit; its only definition can be 



x n 

 n—n=0: but here x° = x n ~ n = — =1. again the same as the 



third form of the symbol 1. 



The subject of the numerical figures 0, 1, 2, 3, &c, considered 

 as chemical symbols, is scarcely treated so fully as to make the 

 author's meaning quite clear; judging from the context, 1 under- 

 stand 0' to represent no space, 1' one litre of space, 2' two litres 

 of space, and so on ; and as the distributive law has been as- 

 sumed, and as the author expressly states that the chemical and 

 arithmetical symbols and 1 are interchangeable, I understand 

 that x .0' = 0, x. V=x, x .2' = x(V + l') = 2x; that is, # + #01* 

 two litres of x ; and so on. But if this be the case, the only 

 practical use of the chemical symbols 2, 3, &c. is lost. For if 

 we should be forced to admit units of substances occupying 

 twice the volume of a unit of hydrogen (as we may be in the 

 case of nitric oxide), we shall be unable to do so without frac- 

 tional indices ; whereas if x . 2' were interpreted to mean one 

 unit weight of x put into two litres of space (at standard tem- 

 perature and pressure), we might represent nitric oxide as a 

 function of the chemical symbol 2'. This seems to me another 

 disadvantage of the assumption of the distributive law. 



In Section VII. the peculiarities of the author's system of 

 notation make their appearance. We have, first, two assumptions, 

 expressed in the words " The units of chemical substances .... 

 are integral compound weights." The two assumptions con- 

 tained in this expression are, (1) that the unit symbol of a 

 chemical substance never contains as a factor a negative power 

 of a prime factor, and (2) that it never contains a fractional 

 power. Both of these assumptions are common to this system 

 and to that in ordinary use. The essential difference between the 

 two systems lies in a third assumption. The author assumes 

 that " the unit of hydrogen is a simple weight," or is to be ex- 

 pressed by a single prime factor. In the ordinary system the 

 corresponding assumption is that the hitherto undecomposed 

 substances (the so-called elements) are, to use the language of 

 the author, simple with regard to one another, or that their sym- 



