wou'd be b - z, then b→→ q ·; ; that is, if p bey√, z, then q will be, that is, the fame with the other, only with a contrary Sign to the Imagi nary Term, as is evident from the Compofition of Equations: From which likewise it is manifeft, that the Number of Imaginary Roots in any Equation, (not including Imaginary Terms) is even. 1. If you Multiply each Term in this Original Quadratick Equation, by the Index of a in that Term, and Divide the Sum of the feveral Products by 4, the Quotient will be + b Now if for a in this Quotient, you Substitute any real Quantity greater than the greatest Value of a in the foregoing Equation, that is greater than b -p, as b (for tho' b were Determin'd, b — p wou'd be any Quantity lefs than b, p being Indetermin'd only as beingo) you will have 2b +b−P, that is p + q, which is +b-g manifeftly, fince p and q are eacho. 2. If in the foregoing Original Quadratick Equation, you likewife Subftiture b for a, and from that Sum Subftract the Product of pq Multiplyed by the Excels of b, above the greateft Value of a in the faid Equation, that is by p; the Remainder will be 1b b-q which is manifeftly, fince the Value of p is not Imaginary. 4 iso. II. Again, Since any Original Cubick Equation may also be Defign'd by the following one, viz. In which Equation, the three Roots are bp, b. and P will b-r; and fuppofe p not qorr, and o, then b be the greatest Value of a; which, and each Member thereof, fuppofe to be real Quantities. 1. If you Multiply each Term in this Equation, by the Index of a in that Term, and Divide the Sum of the feveral Products by a, the Quotient will be Now if for a in this Quotient, you Subftitute any Quantity greater than bp as b, you'll have, after due Reduction, pq + prqr, which is evidentlyo, if q and then of confequence be real Quantities; if not, fuppofe q——√√z, and then r will be+-2, and pq + pr+gr will become=px -¿ + p x x + √ - 2+2 =√-3 × 7+ √ — ż =2pxy + y2+, which is likewife evidently. 2. If in the foregoing Original Cubick Equation, you likewise Substitute b for a, and from that Sum Subtract pq + jr + qr xp, you'll have, after due Reduction, -p'q- pr, which is manifeftly, if q and be real Quantities; if not, by Substituting the above Imaginary Values of q and r for them, p2q = p2rs will become p2 x y — √ ~ 2 − p2 x x + √-x= x,、-༢= 2p2y, which is likewile manifeftly. III. Again, Since any Original Biquadratick Equation may be Defign'd by the following one, viz. In which Equation the four Values of a, are bp, b — 9, b—r, and b—s; and fuppofe p not gorrors, then ando; pwill be the greatest Value of a, which, and each Member of bp fuppofe to be real Quantities, then if q and r be real Quantities, s will be fo too, but if the Value of a be Imaginary, that of r ors will be fo too. 1. If prs 1. If you Multiply each Term in this Equation, by the Index of a in that Term, and Divide the Sum of the several Products by a, the Quotient will be Now if for a in this Quotient, you Subftitute any Quantity greater than bp as b, you'll have, after due Reduction, par +pqs o +prs+qrs, which is plainly if q, r and s be -- real Quantities: if not, fuppofe q and r to be the two Imaginary ones, and y√ and√-z refpectively, and then s will be real Quantity; and pgr + pqs + prs + grs, will become=pxy√−2 × 9+√−z + p x x = √ xs+pxy+ √2 × 5 + 9−√ - 2×9+√=zxs= p)2 + pz + 2 pysty's + zs, which is likewise plainly □ 0. R 2. If 2. If in the foregoing Original Biquadratick Equation, you likewife Subftitute b for a, and from that Sum Subftract pqr + pgs + prs+qrs xp, the Remainder will be found, after due Reduction, pqr-p'qs - p2rs, which is manifeftlyo, if q, rands be real Quantities; if not, then q and r being Defign'd as above, pqr pigs pers will become p3 × j2 + z -p's x 2y, which is likewife manifeftly o. 1 Corollary. Now the Sum of what is faid in these three Paragraphs, and what I defign to Deduce from them is, 1. That in any Æquation whatsoever, where the greatest Value of the unknown Roota is not Imaginary, and the firft Term or higheft Power of the faid Root a is Affirmative, and the laft Term an abfolute Number or given Quantity; if you Multiply each Term by the Index of d in that Term, and Divide the Sum of the feveral Products by a, and then inftead of a in the Quotient Substitute any Number or Quantity = b greater by any real Quantity than the greateft Value of a the Sum of this Quantity, which here Note, will beo. 2. And if in the faid Equation, instead of a you Substitute b, and from that Sum Subftract the above Noted Quantity x b the Remainder will be o. See the Use of this Corollary, in Demonftrating the Converging Series. |