more than 16 shillings the man would require. Here we have discovered not only how fractions may be compared so as to ascertain which is the greater, but also how they may be prepared for addition and subtraction. We have only to multiply the terms of each by the denominators of all the others; and they will be all equi-multiples, with equal denominators. multiples, having a common denominator, and their sum is But, in order that we may be able to manage fractions with ease and certainty, we must consider how a fraction which presents itself in a complicated form may be made as simple as possible without altering its value. Now, we cannot have any general means of simplifying the terms of fractions by addition or subtraction; because both the sum and the difference of the terms of two equal fractions are a are equal to each other, and also toor; for, multiplying b d the numerator of each by the denominator of the other, we have (a+c) x (b-d) = ab+bc-ad-cd, and (a-c) xb+d= ab-bc+ad-cd, which are evidently equal to each other, for all the four simple products in both are equal, and two have the sign and the other two the sign in each; and thus, though they express accurately the real ratio of ab or c: d multiplied by the terms of the other equal ratio reversed, they are each equal to nothing. As multiplying the one term and dividing the other by the same quantity have exactly the same effect on the value of a fraction, we can shift a multiplier or divisor from the one to the other at pleasure, provided that it affects the whole of the term in which it at first appears. Thus which is a a 1X6 may, by changing the multiplier b of the denominator to a divisor of a the numerator, be changed to b, or 1 7 8 7 may be made 8 with 1 out altering the value. Generally speaking, this renders the fraction more complicated, but there are cases in which it is of use. The converse is much more useful, for by means of it all division of fractions may be changed into multiplication. Thus a b C d a is evidently an expression for the division of by; and Ђ making the divisor of each term a multiplier of the other, we ad have which is the terms of the dividend or numerator mulbc tiplied by those of the divisor or denominator, inverted; from which we may derive this general rule for dividing one fraction by another: turn the divisor upside down and multiply. As the numerators of fractions are multipliers, and the denominators divisors, it follows that fractions are multiplied by multiplying their numerators for numerator, and their denomi Any quantity may be changed into the form of a fraction by writing 1 for its denominator. Thus 3 may be a 3 and gene rally a may be -; and, from what has been already said, the quantity, whether expressed by a number, or a letter, or combination of letters, is a multiplier, but it may be changed into 3 a divisor by inverting its terms. Thus is 3 as a multiplier, 1 3 a 1 is 3 as a divisor, and generally is a multiplier, and divisor. - a a is any quantity divided by the number a It is evident that 1 1 1, which is just that quantity itself; and that is the number a 1 divided by any quantity a, and as the first of these is the expression for the quantity as a multiplier, and the second the expression for the same quantity as a divisor, they are the opposites of each other; and for this reason 1 divided by any quantity is called the reciprocal of that quantity, and that division by any quantity is the same as multiplication by its reciprocal. We need hardly repeat that the reciprocal of a fraction is that fraction with its terms inverted. Thus, if any fraction, b a is the reciprocal of that fraction. a b is Here it may be proper to inquire into what is the difference between any quantity and its reciprocal, as, for instance, what a b is the difference between and ? or, which is the same a ? According to the prin ciple formerly explained, we must multiply the numerator of each by the denominator of the other for the respective numerators, and the two denominators together for the common The product of any quantity by its reciprocal is always = 1, for it is always a fraction of which the numerator and denominator are equal. The quotient by the reciprocal is the square of the quantity When quantities are stated as the terms of ratios, and not as fractions, the reciprocal is the terms transposed; thus, b: a is the reciprocal of a : b. If two quantities are mutually the reciprocals of each other, any equi-multiple of either of them must also be the reciprocal ma n b of any equi-multiple of the other. Thus, and are reci mb na procals, and a and b stand for any quantities whatever, and m and n for any multipliers. Hence, "If two quantities have the same proportion as other two, the product of the first and fourth, when they are placed in the order of proportion, must be equal to that of the second and third;" that is, If a b c d, or a b::c: d, then, a d= bc; : The principles which have been stated contain the foundation of the management of common fractional quantities, whether expressed by numbers or by letters, and we can better explain the method of treating exponential fractions in another section. It may not be amiss, however, to recapitulate the leading points. 1. Fractions are not, arithmetically, quantities of the same kind, unless they have the same denominators; but they can always be reduced to a common denominator by multiplying both terms of each by all the denominators, except its own, and then the sums or differences of the numerators may be found in the same way as in quantities not fractional.—The denominators are not subjects of addition or of subtraction. As the numerator is always a multiplier, and the denominator a divisor, it follows that, to multiply any number of fractions together, we have only to multiply all the numerators for numerator, and all the denominators for denominator; and if there be any quantities which are not fractions among the factors, they may be put in a fractional form by writing 1 for the denominator of each. To divide fractions, we have only to |