254 MR. WARREN ON THE SQUARE ROOTS OF NEGATIVE QUANTITIES. 
I saw M. Mourey’s work in December 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 lias 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 curve 
which has certain properties, 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 
respects 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 
i maginary . 
