CHAP. V. John Kerfey's Method of finding the Roots of fome ADFECTED, CUBICK, BI QUADRATICK,ỨC. EQUATION S. FIRST prepare the given Æquation thus; viz. 1. If the Coefficient of the higheft Power of the unknown Root be greater than 1, divide the whole Æquation by the faid Coefficient: And then, 2. If any of the Terms be Fractions, multiply the unknown Root by fuch a Number as will give an Integer Product, and the Coefficient of its highest Power 1. (By Propofition H. Chap. I.) And the Root of the Equation, (whether it be at firft, or by these Directions thus prepar'd) call a. Then, 3. Reduce all the Terms of this Equation to one fide of it, and the other fide will be o. Then find all the Divifors of the abfolute Number in rhe Equation fo reduc'd; and try whether any of these Divifors connected to the unknown Root a, by or +, will divide the total Sum of the reduc'd Equation, without leaving a Remainder for when fuch Divifion fucceeds, either the known Part of the Binomial or Refidual Divifor, with a contrary Sign, is the defired Value of the Root a, or at leaft the Quotient gives an Equation, whose first Term hath fewer Dimenfions by 1, than the Equation divided; and if this Equation contains three or more Dimenfions, let it be examined by Divifion as before; and fo on. By which Di-, vifions, the Roots of the given Equation may be fometimes made known, or the given Equation fometimes reduc'd to a Quadratick one'; and then the fought Root will be found by the Canons given for folving Adfected Quadraticks, The Divifors of 120, the laft Term, are 1, 2, 4, 8, 3, 6, &c. Then I try whether a 1, a + 1, a −2, a + 2, 43, or or a 4-3, will divide the given Æquation without leaving a Remainder: But finding that neither of them will do, I try next with a 4, which will exactly do; and therefore 4 is one affirmative Value of the Root a; and the Quotient being a2 — 11 a +30= o, the two other Roots will be found to be 5 and 6, by dividing a 6, 11 a +30= 0, by a or by a 50, or by the Canon given for folving the 3d Cafe of Adfected Quadratick Equations. 2 Example II. Let it be required to find the Roots of this Equation 2 + 2 a 72=0. 36 a = 2, * Mr. Waeffenaer's Method. Its laft Term can be exactly divided by 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, and 72; wherefore the propos'd Equation is to be divided by a -- and 1, aand. &c. to find fuch of them as will exactly do it: But fince here are a great mary Divifors, and that (by the Compofition of Equations) there can be, at most, but three fuch Divifors, which will exactly divide the propos'd Equation; you may try with a great many of thofe Divifors before you find any of the three fought: Wherefore, to fave your felf a great deal of this Trouble, transform the given Equation into another, each of whofe Roots fhall be more or less than thofe of the propos'd one by a given Number, 1 is generally the most convenient, fuppofe therefore a x1, then the propos'd Equation will become 3 X 39x2-1-77x III, an Equation whofe Roots are each by 1 more than thofe of the propos'd one, and the Divifors of its laft Term are 1, 3, 37, and 111: But fince ax 1, it is evident that if of x be 1, 3, 37, III, or I, - 3, 37, III, (as it, or they, must be, if it has any Rational one) that of a will 4,38,-112; but (by what was before faid) all. the rational Values of a are inferted among the following ones; viz. 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72, or — 1,2,3,4,&c. Confequently, if a has any Rational Root in the propos'd Equation, it, or they, must be 2, 36, or 2, 4: Wherefore you need now try to divide the propos'd Equation only by a 2= O, α- 36 =0, a + 2 =0, and a + 4: Wherefore I try first to divide it by a 20; but that not exactly fucceeding, I try next to divide it by a 360, which exactly does, the Quotient being a+20, an Equation be 0, 2, 36, 110, or 2, any of the Values - wherein wherein a is+-2, and - ✔-2 and confequently the 3 Roots or Values of a, in the propos'd Equation, are 36, 2, and → √ 2. CHAP. VI. The Solution of EQUATIONS, by Sir If. Newton's Method. THE HERE is an univerfal Method of Extracting Roots, either in Numbers or Symbols, Invented by Sir If.Newton, which you may find in Pages 381, 382, and 383 of Dr. Walli's Algebra, which is to this Effect. First, Find the first or greatest Member of the Root fought (y), and if it be noty, let that Member +p be fuppos'dy; then having fubftituted this Binomial and its refpective Powers for y, and its Powers in the Equation, collect its feveral Terms into one Sum; then find the second Member of the faid Root, which is done (in fome Cafes, but not in all in the beginning of the Operation) by dividing the first Term (or abfolute Number or known Quantity) of the faid Sum, by the Coefficient of p in the fecond Term thereof, and let the Quotient, affected with the contrary Sign to what it has +g, be fuppofed = p; then proceed with this Binomial or Refidual in refpect of p, as you did with the other in respect of y: and fo on. Note, When you have found 3, 4, or more of the first Figures, or Members of the Root, you may find as many, or almost as many more, by dividing the firft Term of the laft Sum, by the Coefficient in the fecond. Exam Example I. If y3 2y5o; What is one of the Values y? 3 Example II. If y3 + a x y + a 2 y — x 3 — 2 a 3 = o ; proximè. y3 + a xy + a3y — x 3 — 2 a 3 = o ( a Quære y The greatest Difficulty in this Example II. is to find the first Member of the Root, which is done thus. Let a ParallelogramABC D be (or fuppos'd to be) defcrib'd; and let it be divided into as many fimilar fmalls as are requifit; then Denominate each of thefe fmalls from the Dimenfions of the two in-. definite Quantities of the Equation, as x and y increafing regularly (as in the annexed Figure) from وا the small, of which y denotes the Root to be extracted, andx the other indefinite Quantity. Then mark each small, which anfwers the Terms of the given Æ quation |