Mr. Woodhouse on the Truth of Conclusions 
My inquiry concerning impossible quantities, has been con- 
fined to their use in representing lines belonging to the circle, 
and to the necessary truth of the conclusions obtained by their 
means ; led by the connection of the subjects, I have made a 
small deviation, to examine the true meaning of certain symbols, 
and the contradictions said to embarrass the doctrines of loga- 
rithms when applied to negative quantities. The use, however, 
of impossible quantities has been extended to all parts of ana- 
lysis. By their aid are determined, the values of formulas that 
occur in the science of the motion of fluids, the numerators of 
partial fractions as ( ~ ^ developement of 
— m 
forms as (r 1 — err' cos. % -j- r' 1 ) , and the integration of 
many differential equations. * If, in these cases, the operation 
• By means of impossible quantities. Cardan’s rule, in the irreducible case, when 
the three roots are real and incommensurable, may be applied. In the equation 
x 3 — px -f r, when is < — the root appears under the form \/ a + b V — i 
+ V a + bV — i ; and, by putting a -f b V — i = r (cos. x* -J- sin. a: 1 . V — i), 
the three roots may easily be shewn to be 2 v/ — COS. — , 2 v/T 
COS. a c ), 
3 3 3 \ 3 
av/ f cos - (“rf 
This method, indeed, only exhibits the linear value of the root ; the algebraic value 
-cannot generally be exhibited. In some particular cases, the algebraic value may be 
obtained, when the series that results from adding the terms of the devdopements of 
V a + b V — i and W-bV— i can be summed; as M. Nicole (Mem. 
de l’Acad. 1738, pages 97, 244,) has shewn, who first proved the expression 
\/ a-\-b\l — 1 +>/ a ^.b^J — 1, when expanded, to be real. 
I am of opinion, that Cardan’s solution, in the irreducible case, cannot be extended 
so as to obtain the general linear value, or in particular cases the algebraic value, 
except by operations with impossible quantities ; and that when, by aid of impossible 
quantities, the general linear value or particular algebraic values are exhibited, such 
