26 A Treatise os Algebra.

and this multiply’d by •/ — b, will give \/x z (or x) —*/—cb -f- ab: Which also appears to be true, becausethe Result, this way, comes out exactly the same, as ifthe Operations (for finding *) had been performed alto-gether by real Quantities. But, notwithstanding this,it is not from any Reasoning that I can form, about the

Multiplication of the imaginary Quantitiesy/^_ — and

—by &c. considered independently, that I can provetheir Product ought to be so expressed. For (as has beenobserved before) it would be very absurd, to pretend todemonstrate what the Product of two Expressions mustbe, which are impossible in themselves, and of whoseValues we can form no Idea.

In the foregoing Considerations, the negative Quan-tities — b, — c &c. have been represented (in somerespects) as a Kind of imaginary or impossible Quanti-ties; it may not, therefore, be improper to remark here,that, in many Cafes, such imaginary Quantities, serve asmuch to discover the Impossibility of a Problem, as ima-ginary surd Quantities. For it is plain that in all Ques-tions relating to abstract Numbers, or where the Mag-nitude only of something is proposed to be investigated,in which no Considerations of Position, hor contraryValues can take place, the Solution will be altogetheras impossible, when the Conclusion comes out a nega-tive Quantity, as if it was actually affected with animaginary Surd ; since in one Cafe, it is required that aNumber should be actually less than nothing, and in theother, that the double Rectangle of two Numbers shouldbe greater than the Sum of their Squares; both whichare equally impossible. The only Difference is, that theImpossibility in the former Cafe is much more easily dis-covered, without the Help of an algebraic Process, than inthe latter; whence it is, that Conclusions of that sort arelefc frequent. But as an Instance of the Impossibility ofsome sort of Questions, when the Conclusion comes outnegative, let there be given, in a right-angled Triangle,the Sum of the Hypothenuse and Perpendicular, — a,