tion; dividing each side by bb, in order to find the sum and difference sought v±v, we will have from whence we might readily derive the rule for the addition and subtraction of fractions not reduced to the same denominator. 124. It would be without doubt more simple to have recourse to property (4) in order to reduce to the same denominator the fractions but our object is to show, that the principle of equality is sufficient to establish all the doctrine of fractions. 125. We have given the rule for multiplying a fraction by a whole number, which will also answer for the multiplication of a whole number by a fraction. Now, let us suppose that two fractions are to be multiplied by one another. 126. Let the two equalities be a=b. v, a'=b' . v' ; multiplying one by the other, the two products will be equal; thus aa'=bb'. vv', and dividing each side by bb', in order to have the product sought vo', we will obtain Therefore the product of two fractions, is a fraction having for its numerator the product of the numerators, and for its denominator that of the denomina tors. 127. It now remains to show how a whole number is to be divided by a fraction; and also, how one fraction is to be divided by another. Let, in the first case, the two equalities be if we divide one by the other, the two quotients will be equal, that is and multiplying both sides by b, in order to have the expression, we shall find Therefore to divide a whole number by a fraction, we must multiply the whole number by the reciprocal of the fraction, or which is the same, by the fraction inverted. Let, in the second case, the two equalities be a=b. v, a'=b'. v' ; if the first equality be divided by the second, we shall have multiplying each side by b' and dividing by b, for the purpose of obtaining the expression we will Therefore, to divide one fraction by another, we must multiply the fractional dividend by the reciprocal of the fractional divisor, or which is the same, by the fractional divisor inverted. 127. These properties and rules should still take place in case that a and b would represent any polynomials whatever. According to the transformation and 1 = demonstrated (Art. 86), we can change a quantity from a fractional form to that of an integral one, and reci b 1 procally. So that, we have 2=6x=bxa ̈1= ba ̈1, 1 1 In like manner any quantity may be da a2 b2 da transferred from the numerator to the denominator, and reciprocally, by changing the sign of its index : a2b b bc-2 c-2 a-3x-22-1 Thus, = = c2 a-2c2 a-2 a-26-19 = and c-by-n cmyn 128. If the signs of both the numerator and denominator of a fraction be changed, its value will not be Which appears evident from the Division of algebraic quantities having like or unlike signs. Also, if a fraction have the negative sign before it, the value of the fraction will not be altered by making the numerator only negative, or by changing the signs of all its terms. And, in like manner, the value of a fraction, having a negative sign before it, will not be altered by making the denominator only negative: Thus, making a- -b a-b a + = -b d-id-c 129. Note. It may be observed, that if the nume rator be equal to the denominator, the fraction is equal to unity; thus, if ab, then α α b a =1: Also, if a is >b, the fraction is greater than unity; and in each of these two cases it is called an improper fraction: But if a is <b, then the fraction is less than unity, and in this case, it is called a proper frac tion. § II. Method of finding the Greatest Common Divi sor of two or more Quantities. 130. The greatest common divisor of two or more quantities, is the greatest quantity which divides each of them exactly. Thus, the greatest common divisor of the quantities 16a2b2, 12a2 be and 4abc2, is 4ab. 131. If one quantity measure two others, it will also measure their sum or difference. Let c measure a by the units in m, and b by the units in n, then a me, and bnc; therefore, a+b=mc+nc= (mn)e, and a-b-mc-nc (m-n); or a±b= (min); consequently c measures ab (their sum) by the units in m-n, and a-b (their difference) by the units in m-n. 182. Let a and b be any two members or quantities, whereof a is the greater; and let p quotient of a divided by b, and c= remainder; q quotient of b divided by c, and d remainder; r quotient of c divided by d, and the remainder =0; thus, = b) a (p pb c) b (q qc rd Then, since in each case the divisor multiplied by the quotient plus the remainder is equal to the dividend; we have c=rd, hence qc=qrd (Art. 50); d) e (rpb=pqrd+pd=(pqr+p)d (Art.61.); 0 Hence, since p, q, and r, are whole numbers or integral quantities, d is contained in b as many times as there are units in qr+1, and in a as many times as there are units in pqr+p+r; consequently the last divisor d is a common measure of a and b; and this is evidently the case, whatever be the length of the operation, provided that it be carried on till the remainder is nothing. This last divisor d is also the greatest common measure of a and b. For let a be a common measure of a and b ; such that a=mx, and b=nx, then pb=pnx; and c=a-pb = mx-pnx = (m-pn)x, also db-qcnx-(gmx-qpnx) = nx-qmx + pqnx=(n-qm+pqn)x; (because qc=qmx-qpnx) therefore a measures d by the units in n-qm+pqn, and as it also measures a and b, the numbers, ór quantities a, b, and d have a common measure. Now the greatest common measure of d is itself; consequently d is the greatest common measure of a and b. 133. To find the greatest common measure of three numbers, or quantities, a, b, c; let d be the greatest common measure of a and b, and x the greatest common measure of d and c; then x is the greatest common measure of a, b, and c. For, as a, b, and d have a common measure; if d and c |