which have the same denominator; adding the two equalities (12) and (13), we shall have a+a=bv+bo'=b(v+v'); and dividing both members by b, in order to have the sum sought v+v', it becomes a+a' b ·=v+v'.... (14). Note. In adding the above equalities, the corresponding members are added; that is, the two members on the left-hand side of the sign, are added together, and likewise those on the right. The same thing is to be understood when two equalities are subtracted, multiplied, &c. 122. If two fractions have a common denominator, their difference is equal to the difference of their numerators divided by the common denominator. For, if we subtract equality (13) from (12), we shall have a—a'=bv—bv'=b(v—v'); dividing each side by b, and we will obtain 123. Let us suppose that the two fractions have different denominators, or that we have the equalities a=b. v, a'=b' . v' ; we will multiply the two members of the first by b', and those of the second by b, an operation which will give ab'=bb'v, a'b=bb'v'; then adding and subtracting, we have ab'±ab bb' (v±v'), the double sign which we read plus or minus, indicating at the same time both addition and subtrac tion; dividing each side by bb, in order to find the sum and difference sought vv, we will have ab' ±a'b bb' ·=v±v'. . . . . . (16); 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 frac tion. 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 m=m; a=b. v; 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 ad= ad demon strated (Art. 86), we can change a quantity from a fractional form to that of an integral one, and reciprocally. So that, we have X 1 α a -=bx=bxad bad, and a-2b-2d-2-. =badbad, 1 da = ad 1 In like manner any quantity may be transferred from the numerator to the denominator, and reciprocally, by changing the sign of its index : a2b b bc-2 c-2 Thus, cmyn ca a-2c3 α = a-3x-2-1 and c-mb3y-n 128. If the signs of both the numerator and denominator of a fraction be changed, its value will not be altered. + +b b b C- -d d. C 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, hav ing a negative sign before it, will not be altered by c d d -C d-c 129. Note. It may be observed, that if the numerator be equal to the denominator, the fraction is equal to unity; thus, if a=b, then a α b α 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 Divisor of two or more Quantities. 130. The greatest cómmon 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, 12a 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 amc, and b=nc; therefore, a+b=mc+nc= (m+n)c, and a-b=mc-nc=(m—n); or a±b= (min); consequently c measures a+b (their sum) by the units in m+n, and a-b (their difference) by the units in m―n. 132. 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 remainder; r quotient of b divided by c, and d of e divided by d, and the remainder =0; thus, |