a с hence, a:c=b:d, in which a and c, the antecedents of the given proportion, are proportional to b and d, the consequents of the given proportion. PROPOSITION IV.-If four quantities be in proportion, they will be in proportion by INVERSION ; that is, the second will be to the first, as the fourth to the third. Let a: b=c:d; 6 d then by formula (A), i clearing of fractions, bc=ad; hence by Prop. II, b:a=d:c. SCHOLIUM.—The last two propositions are but modifications of Prop. II. Thus we learn that from every equation three different forms of proportion may be derived. Let ad=bc; then, a:b=c:d; or, a:c=b:d; or, b:a=d:c. PROPOSITION V.- Quantities which are proportional to the same quantities are proportional to each other. If =m:n, (1) and =m:n, (8) we are to prove that a:b=c: d. 5 From (1), a: n m or, a:6=cido PROPOSITION VI.--1f four magnitudes be in proportion, they must be in proportion by COMPOSITION or DIVISION; that is, the first is to the sum or difference of the first and second, as the third is to the sum or difference of the third and fourth. If a:b=c:d, we are to prove that a: a+b =c:c+d. 6 d By formula (A), ; ; (1) a a: a hence from (4), a: a+b=cic+d; and from (5), -b = C:Md. SCHOLIUM.-In like manner, it be shown that -5:6=(Vd:. a a: PROPOSITION VII.-If four quantities be in proportion, the sum of the first and second is to their difference, as the sum of the third and fourth is to their difference. If 6 =c:d, we are to prove that a+b:0—6=c+d:c C-d. By Prop. VI, a: a+b=cic+d; also, a:a4=c:c-d; (2) a+b (3) from (1), c+d; a с ab = bc, PROPOSITION VIII.-If there be a proportion, consisting of three or more equal ratios, then either antecedent will be to its consequent, as the sum of all the antecedents is to the sum of all the consequents. Suppose a:b=c:d=e:f=g:h=, etc. Then by comparing the ratio, a : b, first with itself, and afterward with each of the following ratios in succession, we obtain ba, ah bg, etc.; a:b=a+c+e+9+ etc. : b+d+f+h+ etc. PROPOSITION IX.- If four quantities be in proportion, the terms of either couplet may be multiplied or divided by any number, and the results will be proportional. Let Ec:d; b. d then, a: a с And since the value of a fraction is not changed by multiplying or dividing both of its terms by the same number, we have nb d (1) o na a b nd or, -; (2) nc in which n may be either integral or fractional. If n be integral, we have, from (1) and (2), na : nb =cid, (3) a:b = nc : nd; in which the terms of the given couplets are multiplied. But put 1 5 (5) n = m a m m PROPOSITION X.-If four quantities be in proportion, either the antecedents or the consequents may be multiplied or divided by any number, and the results in every case will be proportional. Let a:b=c:d; 6 d then, (1) =;; a in which n may be either integral or fractional. If n be integral, , we have from (2) and (3), a : nb=c: nd; (4) na : b = nc:d; (5) in which the given antecedents and consequents are multiplied. Put 1 т in which the given antecedents and consequents are divided. PROPOSITION XI.-If four quantities which are in proportion, be multiplied or divided, term by term, by four other quantities also in proportion, the products, or quotients, taken in order, will be proportional. If a:b=cid, (1) and 2 : y = m:n, (2) then we are to prove that ax : by = cm : dn, d and y From (1) and (2), we obtain ad = bc; (3) xn = ym; (4) a с : m multiplying (3) by (4), (ax)(dn) = (by)(cm); (5) dividing (%) by (4), ))=)); PROPOSITION XII.-If four quantities be in proportion, like powers or roots of the same quantities will be in proportion. Let a: b=c:d; b d then (1) a Raising (1) to the nth power, also taking the nth root of the same, (8) 1 1 a* :Zn - bb = AC = Hence from (2), a" f" c : đ; 1 and from (3), PROPOSITION XIII.-If three quantities be in continued proportion, the product of the extremes is equal to the square of the mean. Let a:b=b:c; then by Prop. I, = 6. SCHOLIUM.—Taking the square root of the last equation, we have b=Vāc; hence, The mean proportional between two quantities is equal to the square root of their product. PROPOSITION XIV.-If three quantities be in continued proportion, the first is to the third, as the square of the first is to the square of the second ; that is, in the duplicate ratio of the first and second. Let a:b=b:c; then multiplying by a, ab' = a'c; whence, by Prop. II, a:c=a': . 7= ac; |