...then the terms D, E, F are proportional. Since A:B = B:C, by composition A + B:B = B + C:C; therefore the sum of the antecedents is to the sum of the consequents in the same ratio, that is, A + 2B + C:B + C = B + C:C. Now D = A + 2B + C, E = B + C, andF = C; therefore...

...to d. For, if a:b = c:d, HMultiplying by *, ^ = *£, с ос cd o-» с d .'. a: с — b : d. 193. In a series of equal ratios, the sum of the antecedents...consequents as any antecedent is to its consequent. For, if ? = - = - = 3-, bdfh r may be put for each of these ratios. rru a с e (7 1 hen - = r — =...

...<—- = — *— -j a — oc — d or, a-\-b: a — b= c-\-d: c — d. aE. D. PROPOSITION IX. 303. In a series of equal ratios, the sum of the antecedents...consequents as any antecedent is to its consequent. Let a:b = c:d = e :f— g : h. To prove a-\-c + e-\-g :b + d-\-f-\- h = a: b. Denote each ratio by...

...ABE = A BCE _ A CPE i.iata, AA>B>£> AB'C'E' A C'D'E'' &ABE+BCE+CDE _AABE AB* AA'B'E' AB? J3°3 (in o series of equal ratios the sum of the antecedents...consequents as any antecedent is to its consequent). .: S:S' = AB* : A'B''. Q. E, D. 377. COR. 1. The areas of two similar polygons are to each other as...

...ma:mb = nc:nd; (ii.) ma:nb = mc: nd; (iii.) a" : b" = c" : d„, n being any exponent. 196. If we have 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. T ace Let --2-7-...г/ Adding these equations, we obtain a + t...

...Demonstrat1on : d = , and x = [P. I, Cor. 1] aa, L ' Therefore, x = d XVI. In any multiple proportion the sum of the antecedents is to the sum of the consequents as any antecedent is to its consequent. Given a:b::c:d::c:f Given Г ( e :b::с :/:•ff :</ :h (A)) (B)î Prove, 1. aXe: ab bX f: d :cXg:...

...member by member, ab, , , — = -т ; whence а : с : : о : d. û Oí 449. In any multiple proportion, the sum of the antecedents is to the sum of the consequents as any antecedent is to its consequent. Given a :b : : с : d : : e :f : : g :h to prove e+g : b + d+f+h : : a : b Demonstration : Let т=г;-3-...

...с then 7 = -• bd Multiplying by —, — = — * 0 be cd ab or - = _ cd .'. a : с = b : d. 317. In a series of equal ratios, the sum of the antecedents...consequents as any antecedent is to its consequent. •c, т a с à g For, 1f - = - = - = J-, bdfh r may be put for each of these ratios. Then |=r,.|=r,i...

...i_ b ab bc Multiplying by —, 7- = —;' ' с be cd ab or - = -. с d л a : с — b : d. L\. 317. In a series of equal ratios, the sum of the antecedents...sum of the consequents as any antecedent is to its 7 Ju_.'. r may be put for each of these ratios. mi „ Я „ С- -' * _ J <Г i aen - = r, - — r,...

...quotients will form a proportion. SECTION IV.— CONTINUED PROPORTIONS. 299. PROP. XVI. In any proportion, the sum of the antecedents is to the sum of the consequents as any antecedent is to its consequent. PROP. XVII. In any proportion, all the antecedents, or all the consequents, may be multiplied by any...