FOUNDATION OF ALGEBRA. 289 



arithmetical explanation would at first think sufficient. The only con- 

 nexion between the two fundamental operations of + and x is contained 

 in a(b±c) = ab±ac, and though from this it is a demonstrable identity 

 that 



abbreviated. 



a +- a + a + "(1 + 1+1+ •••) x «, 



which establishes a connexion between + and x when one of the factors 

 is derived solely from 1, yet it leaves the general symbol ab, when neither 

 a nor b is so derived, apparently more free of the meaning of a + b than 

 any one would predict it ultimately must be: while a h is still less 

 connected with its predecessor ab. I shall now examine the manner 

 in which this independence of the three operations has acted in the 

 explanations which have appeared. 



Choosing a unit-line of arbitrary length and direction, and signifying 

 by A or {a, a), a line of a units in length inclined to the unit-line at 

 an angle a, it is well known that an explanation can be given, under 

 which the preceding laws of operation become real consequences of real 

 conceptions. And it is worth stopping to note that the art of operation, 

 previously to the explanation of its symbols, is precisely what Dugald 

 Stewart imagined every mathematical science to be, namely, a pure 

 consequence of definitions, which upon other definitions might have 

 been another thing. This opinion was not, and perhaps is not, without 

 its followers : but I think it will hardly, in any mind, stand the test 

 of a comparison of any one mathematical science with the purely technical 

 algebra, which is rigorously founded upon definitions. By itself, this 

 method of operation, this algebra of rules without meaning, is no more 

 of a science than the use of the well-known toy called the Chinese 

 puzzle, in which a prescribed number of forms are given, and a large 

 number of different arrangements, of which the outlines only are drawn, 

 are to be produced. Perhaps a dissected map or picture would be a 

 still better illustration : a person who puts one of these together by 

 the backs of the pieces, and therefore is guided only by their forms, 

 and not by their meanings, may be compared to one who makes the 

 transformations of algebra by the defined laws of operation only : while 

 one who looks at the fronts, and converts his general knowledge of 



