Impossible Quantities, in Algebra, are the even roots of negative quantities; which expressions are imaginary, or impossible, or opposed to real quantities; as $\sqrt{-a^2}$, or $+\sqrt{-a^2}$, &c. For as every even power of any quantity whatever, whether positive or negative, is necessarily positive, or having the sign $+$, because $+ \times + = +$, or $- \times - = +$, hence it follows that every even power, as the square for instance, which is negative, or having the sign $-$, has no possible root; and therefore the even roots of such powers or quantities are said to be impossible or imaginary. The mixed expressions arising from imaginary quantities joined to real ones, are also imaginary; as $a - \sqrt{-a^2}$, or $b + \sqrt{-a^2}$.

Imaginary Roots of an equation, are those roots or values of the unknown quantity, which contain some imaginary quantity. Thus, the roots of the equation $x^2 + a = 0$, are the two imaginary quantities $+\sqrt{-a}$ and $-\sqrt{-a}$, or $+\sqrt{-1}$ and $-\sqrt{-1}$.