16 On the Rules for Algebraic MuUiplicaiion. 



Numbers are abstract representations of (juantity, and have no 

 particular meaning till applied to some object. Thus 10 may 

 mean ten feet, ten apples, ten pulsations, ten distances of the 

 sun from the earth ; but they have one clear and distinct idea 

 attached to them, that the unit, whatever it be, is taken ten 

 times. Evei!^- sum is therefore in fact a multiple of unity ; and 

 in common multiplication, the multiplicand being considered as 

 unity, the ])rimary and fundamental idea is the same as in the 

 simplest figure or figures written down. 



But in pure symbolic algebra the abstract is carried much 

 forthcr; for the letters are not only, as in numbers, general signs 

 of quantities, but do not even represent any definite numbers of 

 unities ; and therefore are rather receipts or directions for the 

 performance of operations, than real operations themselves: just 

 as a man wlio has written a perfect receipt for making a pud- 

 ding cannot be said to have made a pudding, but told how that 

 pudding has been or may be made. 



a-\-l only means, that when instead of those letters some de- 

 finite quantities are used, they are to be added together, not that 

 they are now incorporated ; and to call this by the name of ad- 

 dition, seems an abuse of terms, or rather a confusion of defini- 

 tions. In like manner a — b means that b, being less than o, is 

 to be taken from it when by the substitution of some real quan- 

 tity this can be done. And this is so true, that, if improper or 

 dissimilar quantities are used, the expression remaining the 

 same, absolute nonsense ensues. 



If, for example, a be supposed to represent ten shillings, and 

 h seven miles, it is obvious that nothing but absolute nonsense 

 is talked when we speak oi a — b. 



In what has been called geometrical multiplication, the abs- 

 traction, though very analogous to that of algebraic notation, 

 is not however precisely the same. In this a superficies is 

 considered, having its four angles right angles, and its sides the 

 two given lines which are thus considered as multiplied by, or 

 drawn into, each other ; and the results of this operation are ri- 

 gorously true, and may be as logically reasoned on when the 

 values of them are inexpressible by numbers, or are incommen- 

 surable, as when they are commensurable. The properties, for 

 example, of the area formed by the side of a square and its 

 diagonal may be investigated, though no numbers can express 

 them. In this respect the abstraction is very similar to that of 

 pure algebraic symbols, which may be supposed to represent any 

 qiiantities of the same kind, whether expressible by numbers or 

 not. In another point of view the abstraction of geometrical 

 investigation is very similar to algebraic. The .'igure drawn is not 



the 



