252 MR. WARREN ON THE SQUARE ROOTS 



cette meme quantite, ils peuvent designer deux qualites oppos^es ayaiit pour 

 sujet les unites dont cette quantite est composee. 



" Dans r algebre ordinaire, c'est a dire, dans 1' alg^bre consideree comnie 

 arithm^tique universelle, ou Ton fait abstraction de toute espece de quality, 

 les signes -f et — ne peuvent avoir que la premiere de ces significations". . . . 

 toutes les fois qu'on a pour r<jsultat d'une operation une quantite precedee du 

 signe — , 11 faut, pourque ce r^sultat ait un sens, y consid6rer quelque qualite. 

 Alors I'algfebre ne doit plus etre regardee simplement comme une arithmetique 

 universelle, mais comme une langue math^matique." 



He then proceeds to the sign i>J — \ •■ this he considers a sign of perpendi- 

 cularity ; he argues that it is a mean proportional between + 1 and — 1 , and 

 therefore must be a perpendicular ; he also gives another proof that it is a per- 

 pendicular ; he makes a square to revolve through 90° about one of its angular 

 points, and observes, that if the square is positive in its first situation, it will 

 be negative after having moved through 90° ; therefore if the square in its first 

 situation be represented by + 1, it will in its second situation be represented 

 by — 1, and its side will in the first case be represented by -j- 1 or — 1, and 

 in the second by -|- ^ — 1 or — ^ — 1 ; but the side of the square has 

 moved through 90° ; therefore he concludes that ,^ — 1 is a sign of perpendi- 

 cularity. In the above demonstration M. Bue'e applies his method of reason- 

 ing as well to areas as to lines ; but as in my treatise I have confined myself to 

 the algebraical representation of lines, I will not make any observation respect- 

 ing the force of this proof. M. Bue'e afterwards proceeds to say, that though 

 perpendicularity is properly the only quality indicated by */ — 1, yet ^ — \ 

 may be made to signify any other quality, provided we can reason respecting 

 that quality in the same manner as we reason respecting perpendicularity ; he 

 then gives examples illustrative of his theory : some of these examples I cannot 

 understand ; others are more clear ; but in almost all there is one great defect, 

 viz. he is obliged to introduce some arbitrary limitation into the question, in 

 order to make the answer agree with the root of tlie equation : this arises from 

 the want of a general geometrical definition of proportion or multiplication, 

 which is necessary to render the theory complete : he also endeavoiu'S to prove 

 that (^ — l)" = n »y — I; but this I cannot comprehend. However, notwith- 

 standing these defects or errors, the general principles on which he reasons 



