 | Joseph Ray - Algebra - 1852 - 410 pages
...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 antecedents is to 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... | |
 | Adrien Marie Legendre - Geometry - 1852 - 436 pages
...m, or Mx ( Q±n) =Px (N±m) : PROPOSITION X. THEOEEM. If any number of magnitudes are proportionals, any one antecedent will be to its consequent, as the sum of all the antecedents to the sum of the consequents. Let M : N :: P : Q :: B : S, £c. Then since, M : N :: P : Q, we have... | |
 | Horatio Nelson Robinson - Conic sections - 1854 - 350 pages
...proportionals. THEOREM 7. If any number of quantities be proportional, then any one of the antecedents will be to its consequent as the sum of all the antecedents is to the sum of all the consequents. Let ... A:B=C:D' And ... And ... &c.=&c. Then we are to show that A : B= C+H+ G &c. : D+F+H, &c. If... | |
 | Charles Davies - Geometry - 1854 - 436 pages
...: : N±m : Q±n. BOOK II. 55 PROPOSITION X. THEOREM. If any number of magnitudes are proportionals, any one antecedent will be to its consequent, as the sum of all the antecedents to the sum of tl1e consequents. Let M : N : : P : Q : : R : S, &c. Then since, M : N : : P : Q, we... | |
 | 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 - 382 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
...+ 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 the 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
...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,... | |
 | Charles Hutton - Mathematics - 1860 - 1022 pages
...THEOREM I.XXIl. If any number of quantities be proportional, then any one of tne antecedent* "¡/I be to its consequent, as the sum of all the antecedents, is to the aim of ¡ли the consfqnents. Let А:В::тА:тпВ::пЛ:пВ, &С.; then will Л : JÎ : : Л -f... | |
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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. 
