OF MAGNITUDE AND DIRECTION. 28r5 



11. I have now concluded all that I wish to say on the principal subject of this paper, 

 but before bringing it to an end I am desirous of making some observations on the general 

 question of the transition of quantities from the + to the - affection, which will, I believe, illus- 

 trate my general design. 



That design may be stated to have been, to shew that there is in the nature of things 



one law according to which causes, which depend solely on magnitude and direction, vary with 



their obliquity from a given direction to the exact opposite, and according to which the cause 



P varies till it becomes — P. Now in considering the general case of the passage of a quantity 



from the + to the - affection, it is to be observed, that if the quantity be necessarily of one 



dimension, as time for example, then future time being denoted by + t, past time will be denoted 



by — t, and t will vary from + to — by simple diminution and passage through zero ; in this 



case the sign \/- 1 can have no place as a sign of affection; I believe there is no conceivable 



interpretation to be put upon t y/ - 1. And in like manner if distance be measured along a 



fixed axis the variation from + to — is by simple diminution ; but, if space be considered in 



two dimensions so that a line may assume all oblique positions in varying from + to - , then 

 e 



the symbol (-1)" indicates the law according to which the change from + to - takes place. 



Here then are two laws according to which the affection of a quantity may be reversed, and 



there may possibly be others, and probably instances might be found in which such chano-e is 



abrupt not continuous; for instance, Dr. Peacock illustrates the properties of ■%/- ' by saying*, 



that supposing + o to represent property possessed, and - a to represent debt, then \/^\ .n 



may represent property deposited " which admits of similar relations when considered as property 



possessed and property owed by another person;" it must however, I think, be admitted, that 



the use of the symbol v — 1 in this case is conventional in a sense in which it is not when 



applied to lines or forces, and it may be doubted whether the symbol so used can be applied 



to the practical purposes of investigation ; and indeed, if I might hazard an opinion, I should 



hold it probable that the symbol v - 1 can only then be successfully used when it expresses 



a particular stage in the continuous change of affection from + to — . 



12. It is not difficult to devise laws, according to which a quantity may change from + 

 to - , other than those which have been specified. Suppose for example /(0) to represent 

 the sign of affection for a quantity P, and suppose 



P/(0) = P(-ir"l; 



this form satisfies the condition that f(9) = 1 and /(tt) = - 1, and therefore so far agrees with 

 the actual law of lines, forces, &c., as that the affection of P passes from + to - while 6 varies 

 from to TT. But if we examine this case, we find that it is widely diverse from the actual 

 law just mentioned ; for we have 



f(d) = cos (TTsin -J + (- 1)^ sin JTrsin - j . 

 Now if = /(O) = 1 



= ~ /(0)=(-i)*; 



• Algebra, 1st Edition, page 366. 

 o 02 



