As another inftance hereof, let there be proposed the equation 2x3-3x2 + 16x — 24=0; then expounding x by 2, 1, 0, and 1, fucceffively, and proceeding as in the foregoing examples, we have - Therefore, the quantities to be tried being 4 and 3, I first attempt the divifion by x 4; which does not anfwer: but trying, or (its double) 2x-3, I find it to fucceed, the quotient being + 8, exally. The reason why the divifors, thus found, do not always fucceed, is, because the first progreffion 2, 1, 0, I is not continued far enough, to know whether the correfponding progreffion may not break off, after a certain number of terms; which it never can do when the bufinefs fucceeds. Thus, in the last example, where we had two different progreffions refulting, had the operation, or feries, 2, 1, 0, 1, been continued only two terms farther, you would have found the first of thofe progreffions to fail; whereas, on the contrary, the laft (by which the business fucceeds) will hold, carry on the progreffion, 2, 1, 0, i as far as you will. The grounds of which, as well as of the whole method, upon which the foregoing obfervations are founded, may be explained in the following manner. Let there be affumed any equation, as ax+ + bx3 + cx2 + dx + e = 0, wherein a, b, c, d, and e, reprefent any whole numbers, pofitive or negative, and let px + q denote any binomial divifor by which the faid expreffion ax2 + bx3 +cx2 + dx + e is divifible, and let the quo tient thence arifing be reprefented by rx3+ sx2 + tx+ v, or, which is the fame in effect, let ax + bx + cx2 + dx + e = px + 9 × rx3 + sx2 + tx + v. This being premised, fuppofe x to be now, fucceffively, expounded by the terms of the arithmetical progreffion 2, 1, 0, - I, -2 (as above); and then the correfponding values of our divifor px + 4, will, it is manifeft, be expounded by 2p + q, p + 9, 9, -p + 9, and 2p + q refpectively; which also constitute an arithmetical progreffion, whose common difference is p; which common difference (p) must be fome divifor of the coefficient (a) of the first term, otherwise the divifion could not fucceed, that is, p could not be had in a, without a remainder. Hence it appears that the binomial divifor, by which an expreffion of feveral dimenfions is divifible, muft always vary as x varies, fo as to be, fucceffively, expreffed by the terms of an arithmetical progreffion, whofe common difference is fome divifor of the firft, or highest term of that expreffion. It also appears, that the faid common difference is always the coefficient of the first term of the general divifor; and that the term (q) of the progreffion, which arifes by taking xo, is the fecond term. Therefore, whenever, by proceeding according to the method above prefcribed, a progreffion is found, anfwering to the conditions here specified, the terms of that progreffion are to be confidered only as fo many fucceffive values of some general divifor, as px + q. Whence the reason of the whole process is manifeft. After the fame manner we may proceed to the invention of trinomial divisors, or divifors of two dimenfions: for, let mx2 + px + q, be any quantity of this kind, wherein m, p, and q reprefent whole numbers, pofitive or negative, and let the terms of the progreffion 3, 2, 1, 0, 3, be wrote therein, one I, - 2, by one, inftead of whence it will become 9m + 3P +q, 4m + 2p + q, m + p + q, q, m − p + q, 4m 2p+q, and 9m3p+g, refpectively; where m muft be fome divifor of the coefficient of the firft term of the given expreffion; otherwife, the divifion could not fucceed. Hence it appears, 1o, That the coefficient (m) of the first term of the divifor must always be fome numeral divifor of the coefficient of the firft term of the propofed expreffion. 2o, That the product of that coefficient by the fquare of each of the terms of the affumed progreffion, 3, 2, I, 0,- I, - 2, 3, being fubtracted from the correfponding value of the general divifor, the remainders (3p+ 9, 2p + q, p + q, q, − p + q, — 2p + 9, - 3p +9) will be a series of quantities in arithmetical progreffion, whofe common difference is the coefficient of the fecond term of the divifor. 3°, And that the term (q) of this progreffion, which arifes by taking x = 0, will always be the third, or last term of the faid divifor. From whence we have the following rule. Instead of x in the quantity propofed, fubftitute, fucceffively, four or more adjacent terms of the progreffion 3, 2, 1, 0, 1, -2, -3; and from all the feveral divifors of each of the numbers thus refulting, fubtract the fquares of the correfponding terms of that progreffion multiplied by fome numeral divifor of the highest term of the quantity propofed, and fet down the remainders right against the corresponding terms of the progreffion 3, I, 2, 3; and then feek out a collateral progreffion which runs through these remainders ; which being found, let a trinomial be affumed, whereof the coefficient of the first term is the forefaid numeral divifor; that of the fecond term, the common difference of this collateral progreffion; and whereof the third term is equal to that term of the said progreffion which arifes by taking x = 0; and the expreffion fo affumed will be the divifor to be tried. But it is to be observed that the fecond term must have a negative or pofitive fign, according as the progreffion, found among the divifors, is an increafing or a decreafing one. 2, 1, 0, xa. Thus, let the quantity propofed be — μ3 — 5*2+12x-6; and then, by fubftituting 3, 2, 1, 0, I, 2, fucceffively, instead of x, the numbers refulting will be 39, 6, 1, -6, 21, and 26 refpectively; which, together with all their divifors, both pofitive and negative, A negative, I place right-against the corresponding terms of the progreffion 3, 2, 1, 0, 1, 2, in the follow I21. 7.3 I - I I - 2 3. 226. 13.2.1 -I.-2. 13. - 26 Then, from each of these divisors I fubtract the fquare of the correfponding term of the firft progreffion multiplied by unity (as being the only numeral divifor of the first term), and the work ftands thus : 3130. 4.-6.-8.—10.-12.—22.—48 | +4 2 2.-1.-2.-3.— 5.-6.—7—10 +2 -3 0.-2. O 6. 3. 2. I.— 1.- 2.- 3.- 6 •2 +3° -I 20. 6. 2. 0.42. 4.- 8.-22 -4/+6 -222. 9.—2.—3—5.— 6.—17.—30 -6+9 Here I discover, among the remainders, two collateral progreffions, viz. 4, 2, 0, 2, 4, 6, and -6, -3, 0, +3, +6, + 9; therefore the quantity to be tried is either x2 + 2x both of which the business fucceeds. This invention of trinomial divifors is fometimes of ufe in finding out the roots of an equation when they are irrational, or imaginary. Thus, let the equation given be x44x3 + 5x2 4x + 1 = 0; and let x be fucceffively expounded by the terms of the progreffion 3, 2, 1, 0, and the numbers refulting will be 7, — 1 and 1; which, together with their divifors, being ordered according to the preceding directions, the operation will ftand as follows: Here we have two progreffions, 2, 1, 0, 1 ; and -8, -5, -2, 1; therefore the quantity to be tried is either ax + 1, or x2 - 3x+1; but I take the firft, and having divided 4x3 + 5x2 - 4x + I thereby, find it to fucceed, the quotient coming out x2 - 3x + 1, exactly. Therefore x44x3 + 5x2 --1 2 4x + being univerfally equal to IX x + 3x + 1, let x2 o, and alfo x2-3x+1=0; from the former of which equations we have x; and from the latter x = ± Therefore the four roots of the given equation 2 VI 3 49 x + be taken and whereof the two laft are irrational and the two first imaginary. And in the fame manner, the roots of a literal equation, as 24—4az3 + 5a2z2 — 4a3% + aˆt = 0, where the terms are homogeneous, may be derived for, let the roots be divided by a, that is, let x be put = Z -> a or ax = x; and then, this value being fubftituted for 3 z, the equation will become x4 4x3 + 5x2 - 4x + [ 0; from which x will be found, as above; whence z (ax) is also known. Having treated largely of the manner of managing fuch equations as can be refolved into rational factors, whether binomials, or trinomials, I come now to explain the more general methods, by which the roots of equations, of feveral dimenfions, are determined; and fhall begin with The Refolution of cubic Equations, according to Cardan. If the given equation has all its terms, the fecond term must be taken away, as has been taught at the beginning of this fection; and then the equation will be reduced to this form; viz. x3 + ax = b; where a and b reprefent given quantities. Put x = y + z; and then, this value being fubftituted for x, our equation becomes μ3 + 3y2z + 3yz2 + z3 + a × y + z = b, or y3 + % 3 × 3yz × y + z + a × y + z = b. Assume, now, 3yz =-a; so shall the terms 3yz x y + z and a x y + z deftroy each other, and our equation will be reduced to y3+z3b. From the fquare of which, let four times the cube of the equation yza be fubtracted, and we |