252 
MR. WARREN ON THE SQUARE ROOTS 
cette ineme quantity ils peuvent designer deux qualites opposees ayant pour 
sujet les unites dont cette quantite est composee. 
“ Dans 1’ algebrc ordinaire, c’est a dire, dans 1’ algebre consideree comme 
arithmetique universelle, ou l’on fait abstraction de toute esp&ce de quality, 
les signes + et — lie peuvent avoir que la premiere de ces significations”. . . . 
toutes les fois qu’on a pour r^sultat d’une operation une quantite preced£e du 
signe — , il faut, pourque ce resultat ait un sens, y considerer quelque qualite. 
Alors l’alg^bre ne doit plus etre regardee simplement comme une arithmetique 
universelle, mais comme une langue matliematique.” 
He tlien proceeds to the sign — 1 : 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 -f 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 — 1 
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 the 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 endeavours to prove 
that (V — l) 7l = w N / — T; but this I cannot comprehend. However, notwith- 
standing these defects or errors, the general principles on which he reasons 
