 In a series of equal ratios, any antecedent is to its consequent, as the sum of all the antecedents is to the sum of all the consequents. Let a: 6 = c: d = e :/. Then, by Art.  A Treatise on Algebra - Page 220by Elias Loomis - 1868 - 384 pagesFull view - About this book
 | Elias Loomis - Conic sections - 1849 - 252 pages
...AxB = wxAxB, or, AxmB=BXffiA. Therefore, by Prop. II., A : B : : mA : mB. PROPOSITION IX. THEOREM. If any number of quantities are proportional, any...its consequent, as the sum of all the antecedents, it to the sum of all the consequents. Let A : B : : C : D : : E : F, &c.; then will A : B : : A+C+E... | |
 | Stephen Chase - Algebra - 1849 - 348 pages
...by ; al= bL .-. (§ 233) a+e+g-\-k : b+f+h+l—a :b = e:f, &c. Hence, In any number of equal ratios, the sum of all the antecedents is to the sum of all the consequents as any one of the antecedents is to its consequent. Thus, if 1:2 = 3:6 = 4:8 = 5: 10, then 1+3+4+5... | |
 | Joseph Ray - Algebra - 1852 - 408 pages
...:bn: :cr:ds. ART. 278. PROPOSITION XII. — In any number of proportions having the same ratio, any antecedent is to its consequent, as the sum of all...the sum of all the consequents. Let a :b : :c : d : :m :n, &c. Then a : b : : a-\-c-\-m : b-\-d-\-n. Since a : b : : c : d, we have bc=ad (Art. 267).... | |
 | Joseph Ray - Algebra - 1848 - 250 pages
...PROPOSITION XII. — In any continued proportion, that is, any number of proportions having the same ratio, any one antecedent is to its consequent, as the sum of all the antecedents i» to the sum of all the conseqtients. Let a : b : : c : d : : m : n, &c. Then will a:b: : o+e+m :... | |
 | G. Ainsworth - 1854 - 216 pages
...a+a, + a"+ .... + o<"> :6 + 6, + 6"+ +bw=a:b. That is, if any quantities be in continued proportion, the sum of all the antecedents is to the sum of all the consequents as one of the antecedents is to its consequent. By the last proposition, a+o, : 6 + 6,=a, : b,=a" :... | |
 | James Cornwell - 1855 - 380 pages
...original ratio. Hence they are equal to one another. 329. III. — If there be any number of equal ratios, the sum of all the antecedents is to the sum of all the consequents, as either of the antecedents is to its consequent* 3 : 5 : : 9 : 16 : : is : 30 : : 330 : 550. . 3... | |
 | John Fair Stoddard, William Downs Henkle - Algebra - 1859 - 538 pages
...6, (387) " " a + b :ab : : c+d : c—d Q. K D. PROPOSITION (394.) 13. In a continued proportion, any antecedent is to its consequent as the sum of all...antecedents is to the sum of all the consequents. DEMONSTRATION. Let a : b : : с : d : : e :f::y: h : : &c. We are to prove that a : 6 ;:a + c+e+g,... | |
 | Theodore Strong - Algebra - 1859 - 570 pages
...+ H + etc. BDP Hence, when (numbers or) quantities of the same kind are proportionals, we say that the sum, of all the antecedents is to the sum of all the consequents, as any antécédent is to it» consequent. (as.) If we have ^ = =: , and т> = т=ч> t^611 by adding... | |
 | Mathematics - 1860 - 294 pages
...ax-\-by-\-cz a — bb — cc — aa -f- 5 -I- e t ions = — . I Since these ratios are equal, any antecedent is to its consequent as the sum of all...the antecedents is to the sum of all the consequents ; therefore either fraction equals the sum of all the numerators divided by the mm of all the denominators,... | |
 | Horatio Nelson Robinson - Geometry - 1860 - 470 pages
...magnitudes are proportional, any one of the antecedents will be to its consequent as the sum of all tht antecedents is to the sum of all the consequents. Let A, B, C, D, JB, etc., represent the several inagm tudes whi )h give the proportions To which we may annex the identical... | |
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