139. We have already seen that the squares of numbers, negative as well as positive, are always positive, or affected by the sign +; having shewn that −a multiplied by −a gives +aa, the same as the product of +a by +a: wherefore, in the preceding chapter, we supposed that all the numbers, of which it was required to extract the square roots, were positive.

140. When it is required, therefore, to extract the root of a negative number, a great difficulty arises; since there is no assignable number, the square of which would be a negative quantity. Suppose, for example, that we wished to extract the root of −4; we here require such a number as, when multiplied by itself, would produce −4: now, this number is neither +2 nor −2, because the square both of +2 and of −2, is +4, and not −4.

141. We must therefore conclude, that the square root of a negative number cannot be either a positive number or a negative number, since the squares of negative numbers also take the sign plus: consequently, the root in question must belong to an entirely distinct species of numbers; since it cannot be ranked either among positive or among negative numbers.

142. Now, we before remarked, that positive numbers are all greater than nothing, or 0, and that negative numbers are all less than nothing, or 0; so that whatever exceeds 0 is expressed by positive numbers, and whatever is less than 0 is expressed by negative numbers.