Elementary Algebra; For the Use of Schools

Book may have numerous typos, missing text, images, or index. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. 1851. Not illustrated. Excerpt: ... difference of two lines is equal to the difference of their squares, We shall have further illustrations of this kind as we proceed. The learner will now, however, see distinctly the nature of algebra. It is a concise language, by aid of which we are enabled with facility to solve questions either in a particular or general manner, and to conduct the investigations necessary to determine the properties and re lations of numbers, or the quantities represented by them. SECTION IX. -- Rules Of Arithmetic. LIII. By solving a question in a general manner, we obtain, as we have seen, a rule, by which we are enabled, not merely to find the answer to the proposed question, but to all others which differ from it only in the particular numbers which are given. We proceed to some additional illustrations of the aid derived from algebra in determining general rules, by investigating a few of the more important practical rules usually found in treatises on arithmetic. RATIO AND PROPORTION. LIV. There are two ways in which magnitudes, or the numbers which represent them, may be compared. 1°. "We may wish to ascertain how much the greater of the two exceeds the less. In this case, we subtract the less from the greater, aud the result is called their ratio by difference. Comparing, for example, the numbers 8 and 6, their ratio by difference is 8 -- 5, or 3. In like manner, the ratio by difference of the quantities a and b will be a -- b or b -- a, according as the one or the other is .the greater. 2°. TVe may wish to ascertain how often one magnitude is contained in the other. In this case, we divide the one by the other, and the result is called their ratio by quotient. Take, for example, the numbers 18 and 6; comparing the 18 18 with the 6, the ratio will be -pr = 3; co...