254 MR. WARREN ON THE SQUARE ROOTS OF NEGATIVE QUANTITIES. 



I saw M. Mourey's work in Decembei- 1828, and found that his method of 

 considering the subject is nearly the same as the method which I have adopted 

 in my treatise : but he has in his work a proof that every equation has as 

 many roots as it has dimensions, which I have not in mine ; this proof with a 

 very slight alteration I communicated to the Philosophical Society at Cam- 

 bridge. My reason for introducing an alteration was this : the author, after 

 having taken (in the figure which he makes use of) as many points as the given 

 equation has dimensions, and proved that round each point there is a cui"ve 

 which has certain propei'ties, and that in each curve there is a line which will 

 satisfy the conditions of the equation, concludes that there are as many lines 

 which will satisfy the conditions of the equation as the equation has dimen 

 sions ; which conclusion does not necessarily follow from the premises ; for one 

 curve may surround two or more of the points in his figure, in which case 

 he ought to have proved, that if any one of the curves surrounds m of the 

 points, there will be m lines in that curve, which satisfy the conditions re- 

 quired, which he has not done, therefore his proof is in that part defective ; 

 consequently an alteration was necessary ; and the alteration was easily made, 

 as it is enough to prove, that an equation of n dimensions has one root, after 

 which it may be depressed to an equation of n — I dimensions. In all other 

 i-espects the proof given by M. Mourey is remarkably clear and satisfactory, 

 and an example of the advantages which mathematicians may derive from a 

 knowledge of the true theory of the quantities improperly called impossible or 

 imaginary. 



