13. The sum of four numbers in arithmetical progression is 56, and the sum of their squares is 864; what are the numbers? Ans. 8, 12, 16, and 20. 14. To find four numbers in geometrical progression, whose sum is 15, and the sum of their squares 85. CUBIC AND HIGHER EQUATIONS. A CUBIC EQUATION, or equation of the third degree or power, is one, that contains the third power of the unknown quantity as x —ax+bx=c. A biquadratic, or double quadratic, is an equation, that contains the fourth power of the unknown quantity -ɑx3+bx2-cx—d. as x4 An equation of the fifth power, or degree, is one, that contains the fifth power of the unknown quantity: as x-ax*+ bx3-cx2+dx=e. 5 An equation of the sixth power, or degree, is one, that contains the sixth power of the unknown quantity: as xt—ax +bx*—cx3+dx3-ex=f. And so on, for all other higher powers. Where it is to be noted, however, that all the powers, or terms in the equation, are supposed to be freed from surds, or fractional exponents. There are various particular rules for the resolution oi cubic and higher equations; but they may be all easily resolved by the following rule of Double Position. RULE.* 1. Find by trial two numbers, as near the true root as possible, and substitute them separately in the given equa tion, instead of the unknown quantity; marking the errors, which arise from each of them. 2. Multiply the difference of the two numbers, found by trial, by the least error, and divide the product by the difference of the errors, when they are alike, but by their sum, when they are unlike. Or say, as the difference or sum of the errors is to the difference of the two numbers^ so is the least error to the correction qf its supposed number. 3. Add the quotient last found tp the number belonging to the least error, when that number is too little, but subtract ij, when top great; and the result will give the true root nearly. 4. Take this root and the nearest of the two former, or any other, that may be found nearer; and, by proceeding in like manner as above, a root will be had still nearer than before; and so on, to any degree of exactness required. Each new operation commonly doubles the number of true figures in the root; 1 NOTE 1; It is best to employ always two assumed numbers, that shall differ from each other only by unity in the last figure on die right; because thep the difference, or multiplier, is only 1. EXAMPLES. 1. To find the root of the cubic equation x2+x*+x=IOOj 6r the value of x in it. Here it is soon found, that x lies between 4 and 5. Assume, * This rule may be used for solving the questions of Double Position, as wejl as that given in the Arithmetic, and is preferable for the present purpose. Its truth is easily deduced from the same supposition. For by the supposition, r: s :: x-ɑ xb, therefore, by division, r—-s : * :: bɑ : x; which is the rule. therefore, these two numbers, and the operation will be as Again, suppose 4'2 and 4'3, and repeat the work as fol 6'369. The sum of which is As 6'369 : 1 :: 2' 297 : 0'036 This taken from 4'300 Leaves x nearly = 4'264 Again, suppose 4'264, and 4°265, and work as follows. 2. To find the root of the equation #*—iSx+63x=50, or the value of x in it. Here it soon appears, that x is very little above 1. Suppose, therefore, l'O and 1*1, and work as follows. Again, suppose the two numbers 1'03 and 1'02, and work as follows. NOTE 2. Every equation has as many roots as it contains dimensions, or as there are units in the index of its highest power. That is, a simple equation has only one value or root; but a quadratic equation has two values or roots; a eubic equatiop has three roots; a biquadratic equation has four roots, and so on. And when one of the roots of an equation has been found by approximation, as before, the rest may be found as follows Take for a dividend the given equation, with the known term transposed, its sign being changed, to the unknown side of the equation; and for the divisor take x minus the root just found. Divide the said dividend by the divisor, and the quotient will be the equation depressed a degree lower than the given one. Find a root of this new equation by approximation, as before, and It Will be a second root of the original equation. Then, by means of this root, depress the second equation. one degree lower, and thence find a third root, and so on, till the equation be reduced to a quadratic; then the two roots of this being found, by the method of completing the square, they will make up the remainder of the roots. Thus, in the foregoing equation, having found one root to be 1'02804, connect it by minus with x for a divisor, and take the given equation with the known term transposed for a dividend: thus, x—1°02804)xs-15x*+63x-50(x*-lS 97196x+48 6362r=0. Then the two roots of this quadratic equation, or x313'97196x-48'63627, by completing the square, are 6 57653 and 7 39543, which are also the other two roots of the given cubic equation. So that all the three roots of thaj equation, viz. x2—15x2+63x50, |