As, fuppofing the Equation to be x04 -|- c3 xc d2 = 04 y4-4cy36cc yy — 4 c3y + c2 = 004 34-4cy3 + 6 cc yy - 3 c3 y — d2 = 0 PRO P. II. d4 To Multiply or Divide the Roots of an Equas tion by any given Quantity; fuppofe q. Kule. Instead of the unknown Root, fubstitute Part, or q times another unknown Root: Thus, Examples. q 1. To Multiply the Roots of this Equation xB xc2 ccx = g3 by q. 3 Part by q', we have z3+qbzz -q q cc z=q3g, an E quation wherein z is qx in the former. The Reafon is obvious; for if I ≈x, then ≈ q xi 2. To divide by 3 the Roots of this Equation x3- 54 CC2 216 do. Suppose 3x=x, then the former Equation will become 2723-162 ccz-216 d3o: And, by dividing each Part by 27, we have 3-6c cz-8 do an Equa 3 COROLLAR Y. Hence an Equation may be cleared from Fractions without increafing the Coefficient of the highest Term: Thus, NB. See the Use of this Coroll. in the Beginning of Chap. 5. of this Part. Note. The laft Term but one of any Equation wanting the fecond Term (which, if prefent, may be taken away by Prop. 1. Schol. 1.) may be deftroyed, without the Ex traction of Roots, by fuppofing the Root y of that Equation: As for Inftance, 2 If y3× — by — c — 0; suppose 2 =y; then ≈3 -- 3 q 9 2 23 CHA P. II. The Solution of Cubic Equations by Cardan's W Method. HEN any Cubick Equation, having the fecond Term, is propos'd to be refolv'd by Cardan's Method, you must firft deftroy its fecond Term (by the last Chap. Prop. I. Sch. 1.); and then you'll reduce it into one of these four following Forms or Cases, viz. | 1 | a 3 In each of thefe Equations the Quantity fought a is inferted only in two different Terms, in which its Indices are treble to one another: And, in order to folve all Equations thus qualified at once, let us fuppofe the reduc'd Equation to n = be reprefented by x-bc, in which the Values of b and c may be equal to any given Quantities whatsoever, Affirmative or Negative; and that of 2 is indetermin'd, but is in all original Cubick Equations=3, and x" is understood to have the Sign--prefix'd to it. 1, 2323-bz = c Suppofe 4ey = ± 3,4 5 e3-3eey 3 eyyy3-bx: c-ty: Now fuppofe 6—zey—b 5,67 e3-3 e e y -|- 3 ey y ↓ 1 3 3 eeg b ce3 = e6 I 63 27 123 = e 6 — ce 3 27 13 w 2.14 √cc + — b3 : — e3 — { c 27 As the Cube Root of any Quantity k is not only 3/k or 1 x 2 3/k, but alfo : ÷ + √ ~ 3 : × 3k, and: -:×k; fo the Value of z in the Equation ≈3 bz 7 CC 27 : For all three are but the fame Values of differently exprefs'd; the first of them being fimpleft, is therefore the best. That these are the three Values of ≈ in the Equation ≈3 +bc I demonftrate; thus: Suppofe u + √ − 3,0 = −√, and w= + ; then the foregoing Values of 27 b b z will become w spectively; wherefore =0, and 2 re =0, 20 uw-| o: And the Products of these two laft Equations will be found, when reduc'd, to be 1); and these Products multiplied by 2-w +bz-c=0; for3 2763 163 27 63 W3 that is z3 ལ + may be eafily prov'd to W3 bec. 2. E. D. See Part IX. COROLLA R Y. From the third and eighteenth foregoing Steps you may deduce particular Theorems (but not the fame with Cardan's) for folving each of the four preceeding Cafes : I. So; 1. If a3 +paq; then (b in the third Step being p in this Equation, and c in the former q in the latter |