we may write every proportion as a fractional equation-whereby the many and all the changes that can be rung on proportions, will be reduced to very simple operations on equations. VARIATION. Def. 7. We shall often have occasion to consider quantities, not as proportional simply, but as passing by inappreciable degrees through all magnitudes compatible with certain conditions. Such quantities are denominated VARIABLE-and are represented by the last letters of the alphabet, as x, y, z, while the first letters are used to indicate quantities regarded as constant, or such as are independent of the variables. Thus, that y varies as x, a being their constant ratio, is expressed by y = ax. In this equation it is to be understood that x, in passing from any one given value to another, is regarded as passing through all intermediate values while a remains unchanged; and that consequently y changes, taking new values depending upon the value of x. On this account r is called the independent, and y the dependent variable. Thus, if the rate of interest, r, be constant, and the principal, p, be also constant, we shall have a given sum pr, as the interest for one year on the given principal; then if y be the interest for the time x in years, there will result the variation y = prx. This manner of looking upon quantities, not so much as known and unknown, as constant and variable, is as important as it is simple. y = abx ; i. e., If y vary as z, and z vary as x; then y varies as x. If y and x vary as z, y±x varies as z. (54) (55) It will be observed that the above forms embrace, very briefly and simply, not only essentially the whole doctrine of proportion, but a vastly wider field, by reason of the unlimited number of values of which y and x are susceptible. SECTION THIRD. Analysis of Equations. The following is a principle of the first importance in analytical investigations : PROPOSITION I. Certain equations are so constituted that they necessari- (56) ly resolve into, and are consequently equivalent to, several independent equations. We do not propose, in the present article, to enter into a full developement of the boundless resources which this principle affords, but simply to illustrate it by examples of such particular cases as will be serviceable to us as we proceed. Required two numbers such that if they be diminished severally by a, b, and the remainders squared, the sum of these squares shall be equal to zero. Denoting the numbers by x, y, we have * and it follows from (63) that (x - a)2, (y—b)2, must both be +, whether xa, y≥b; but it is obvious that, since neither of the terms (x − a)2, (y — b)2 can be minus, neither can be greater than 0; for if either term, (x-a)2 for instance, have an additive value, the other (y-b)2 must possess the same value and be subtractive, in order to satisfy the equation, or that their sum may = 0; whence it is necessary that = 0? As a second example, what numbers are those from which if a and b be severally subtracted, the product of the remainders will be Representing the numbers by x, y, there results QUADRATIC EQUATIONS. 51. Every equation of the second degree, or, embracing no other unknown quantities than r2 and x, may, by transposing, uniting terms, and dividing by the coefficient of x2, be readily reduced to the form In order to find x, understanding that a, b, may be either + or we observe that the first member of the equation will become a binomial square (8) by the addition of a2; To solve a QUADRATIC EQUATION: 1o. Reduce it to the form of (57); 20. Add the square of half the coefficient of x to both sides; 3°. Take the square root of the members, prefixing the double sign [+] to the second; 4°. Transpose. It will be observed that (57) resolves into two independent equations (58), x = −a + √(b+ a2), and x = - · a − √ (b + a2), thus illustrating (56). Adding these values of x, we have [− a + √(b+a2)]+[−a−√(b+ a2)] = − 2a, and their product is (?) - [−a+√(b+a2)] [− a − √(b+a2)] = [− a]2 − [√(b+a2)]2 = a2 - (b+a2) = b. Cor. In a quadratic of the form (57) the sum of the values (59) of x, is equal to the coefficient of x1 taken with the contrary sign, and their product to the second member also taken with the contrary sign. Queries. What will (57) and (58) become, when a is changed into a? b into-b? When b is minus, what must be its value compared with a2 in order that the value of x in (58) be impossible? [See (6.)] Will change of value or sign in a ever render x imaginary? Why not? Scholium I. We naturally inquire if the Cubic Equation can be resolved into three independent equations. Every equation of the third degree reduces to the form As we must suppose x to have some value-one at least-let r be that value; then must r satisfy the equation, or (x3 — r3) + a(x2 — r2) + b(x − r) = 0; dividing by x-r, see examples under (16); x2+(a+r)x+r2 + ar+b= 0, (b) (c) whence [(57), (58)] (a) is resolvable into three equations, giving three values for x. Comparing equations (a), (b), (c), and observing that (b) is the same as (a), we learn that, if r be a root of the cubic equation (a), that the equation is divisible by zr, giving a quadratic (c). The student is requested to prove that the equation x2 + px3 +qx2+rx+s= 0, is resolvable into the factors xa, xb, xc, x-d, .4 3 or that (a) (x − b) (x − c) (x − d) = x2 + px3 +9x2+rx+s=0, a, b, c, d, being values of x, or x = a, x = b, x = c, x = d. Scholium II. Many Biquadratic Equations may be reduced as Quadratics; e. g. 1o. When reducible to the form (x2 + Ax + B) (x2 + A'x + B') = 0, where the conditions are (60) x2 + (A+A')x3 + (AA' + B+B')x2 + (AB' + A'B)x + BB' = 0, or s = BB', q= AA' + B+ B', p= A + A', r = AB' + A'B. 2o. When reducible to the form |