OF NEGATIVE QUANTITIES. 245 



city which can be acquired by falling from any distance however great ; there- 

 fore the question involves an impossibility : therefore, if we obtain an equation 

 for determining the height from which a body must fall to acquire a velocity 

 equal to the velocity in the circle, the equation must, when n is greater than 3, 

 show, in some way, that the question involves an impossibility. 



Let r be the radius of the circle, and x the height (measuring from the 

 centre) from which the body must fall to acquire the velocity in the circle ; 

 then we obtain the following equation. 



_i ?_ /_i___J_l 



From which we deduce a?"~' . r"~' = 0. 



3 — « 



Now, by the nature of the question, x must be positive ; therefore whenever the 

 above equation has no positive root, the question must involve an impossibility. 



Let n = 5, then j?* + r* = 0, an equation, all whose roots are what are called 

 impossible roots ; therefore, since the equation has no positive root, the question 

 involves an impossibility. 



Next, let « = 6, then x^ + ^ r^ =^ 0, an equation, one of whose roots is ne- 

 gative, and the other four what are called impossible roots ; therefore in this 

 case also, since the equation has no positive root, the question involves an 

 impossibility. 



Therefoi-e a negative root may be a sign of impossibility, as well as what is 

 called an impossible root. 



In like manner other examples might be given, from which it would appear, 

 that, in some cases, fractional roots may also be signs of impossibility. 



Therefore we have no stronger reasons a priori to determine, that, what are 

 called impossible roots, have no real existence, because, in some cases, they are 

 signs of impossibility, than we have to determine that fractional or negative 

 roots have no real existence, because, in some cases, they also are signs of im- 

 possibility. 



To the second objection, viz. that there is no necessary connexion between 

 algebra and geometiy, it may be answered, that there is a connexion between 

 what are called impossible roots and the series for the circumference of the 

 circle, which connexion may be proved on principles purely algebraic, without 

 the intervention of any geometric considerations. 



