the 6th order, but of the quadratic form, is deduced, which gives the values of m and n, and hence, m+n=x='Vert Vp?+q3+. Tv+q; Cor. I. In the given equation, if 39 is negative, and if s2 is lefs than 9 , this expression of the root involves impoffible quantities; while, at the same time, all the roots of that equation are possible. The reason is, that, in this method of solution, it is necessary to suppose that x the root may be divided into two parts, of which the product is q. But it is easy to shew, that, in this, which is called the irreducible case, it cannot be done. For example, the equation, (Ex. 3. Sea. 3. of this chapter), x=156x +560=o, belongs to the irreducible case, and the three roots are +4, +10, -14, and it is plain that none of thete roots can be divided into two parts mi and n, of which the product can be equal to –9= greatest 156 = 52 ; for the greatest product from the division of the greatest root-14, is -7X-7=49, less than 52. If the cube root of the compound surd can be extracted, the impossible parts balance each other, and the true root is obtained. The geometrical problem of the trisection of an arch is resolved algebraically, by a cubic equation of this form ; and hence the foundation of the rule for resolving an equation belonging to this case, by a table of fines. Cor. 2. Biquadratic equations may be reduced to cubics, and may therefore be resolved by this rule. Some other classes of equations too, may be resolved by particular rules ; but these, and every other order of equations, are commonly resolved by the general rules, which may be equally applied to all. II. II. Solution of Equations, whose Roots are commenfurate. zab} =o. RULE I. All the terms of the equation being brought to one side, find all the divisors of the absolute term, and substitute them successively in the equation for the unknown quantity. That divifor which, substituted in this manner, gives the result =0, shall be a root of the equation. . Ex. 1. x-3ax+2a’x-2a2b --6x2 +3abx The simple literal divisors of —2a2b, are a, b, 2a, 2b, any of which may be inserted for x. Supposing x=ta, the equation becomes a'--3a+2a-2a-6 -ba? +3a+b Ex. 2. *—2x2—33x+90=0. The divisors of go are 1, 2, 3, 5, 6, 9; 10, 15, 18, 30, 45, go. The first of these divisors, which being inserted for x, will make the result = 0, is *6 +3; +5 is another, and it is plain the last the given equation may be divided by X-3 =0, and the result will be a quadratic, which being resolved, will give the other two roots, +5 and 6. The reason of the rule appears from the property of the absolute term formerly defined, viz. that it is the product of all the roots. To avoid the inconvenience of trying many divisors, this method is shortened by the following Substitute in place of the unknown quantity successively three or more terms of the progresion, 1, 0,-1, &c. and find all the divisors of the Jums that result, then take out all the arithmetical progressions that can be found among these divisors whose common aifference is I, and the values of x will less -1 will be among those terms of the progress fions which are tbe divisors of the result arising from the substitution of x=o. When the series increases, the roots will be pofitive; and when it decreases, the roots will be negative. E X A M P L E. Let it be required to find a root of the equas tion x'—2—10x+6=0. The operation is thus : Suppost. Result. | Divisors. Ar. pro. x= EI 411,2,4, 4 O*3-*-10x+6= + 61,2,3,6, 3 1+141,2,7,14,2 In this example there is only one progression, 4, 3, 2, and therefore 3 is a root, and it is -3, since the series decreases. It is evident from the rules for transforming equations, (Chap. 2.), that by inserting for x, til=te) the result is the absolute term of an equation, of which the roots are |