Elementary Algebra

This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1858 edition. Excerpt: ...principle: In every proportion, the sum of the first and second terms is to the second as the sum of the third and fourth terms is to the fourth. By a process altogether similar, it may also be shown that in every proportion the difference between the first and second terms is to the second as the difference between the third and fourth is to the fourth. The proportion a: b: c: d itturns to a; c:: b. d; whence, from the first of the preceding properties, a-j-c: c:: b-f-d: d. Comparing this with the proposed, we derive the following principle: In every proportion, the sum of the antecedents u to the sum of the consequents as any one antecedent is to its consequent. In like manner it may be shown that the difference of the antecedents is to the difference of the consequents as any one antecedent is to its consequent. Let there be next a series of quantities, a, b, c, d, e, f, g k... forming, two and two, a series of equal ratios, so that we have a: b; c: d:: e: f:: g: h... The two first ratios, viz., a: b:: c: d, give, from the principle last demonstrated, a--c: b--d:: c: d; but c: d:: e: f; whence a, -f-c: b--d:: e: f; applying to this last the same principle, we have and thus in order, whatever the number of equal ratioS. Therefore, in a series of equal ratios, the sum of the antecedents is to the sum of the consequents as any one antecedent is to its consequent. Let us take next the two proportions a: b: c: d, ande: f:: g: h; a c, e g i = 5'and7 = l If we now multiply these equations member by member, we obtain ae _ eg. lfJh' whence ae: b f:: c g: dh. Comparing.his with the proposed proportions, the result is the same as if we had multiplied these proportions term by term. This is called multiplying the proportions in order. Hen