 | Aaron Schuyler - Navigation - 1873 - 536 pages
...give the continued proportions: AB : AE : : BC : BF :: CD : CG. AB : EB :: BC : FC :: CD : GD. Since the sum of the antecedents is to the sum of the consequents as one antecedent is to its consequent, we have, AD : AE+BF + CG : : AB : AE. NAVIGATION. Now let a right... | |
 | Adrien Marie Legendre - Geometry - 1874 - 500 pages
...±2,0 : D ±^D -t qy T f which was to be proved. PEOPOSITION XI. THEOREM. In any continued proportion, the sum of the antecedents is to the sum of the consequents, as any antecedent to its corresponding consequent. From the definition of a continued proportion (D. 3), Adding and factoring,... | |
 | James Bates Thomson, Elihu Thayer Quimby - Algebra - 1880 - 360 pages
...the sum of the third and fourth is to their difference. COR. 3. — In any number of equal ratios, the sum of the antecedents is to the sum of the consequents as any one antecedent is to its consequent. riH a + e + e + &c. a „ . . , , 5 -2 — - — = r = r = &c.... | |
 | George Albert Wentworth - Algebra - 1881 - 406 pages
...Composition. a + c : c ::b -\- d: d. VII. Division. a — c:c::b — d:d. 350. In a series of equal ratios, the sum of the antecedents is to the sum of the consequents as any antecedent is to its consequent. For if a - c - e - ff ' 5 2~/T r may be put for each of these ratios. Then ..rJr-.r.jf.r, .'. a = br,... | |
 | George Albert Wentworth - 1881 - 266 pages
...&ABC+ACD+ADE+AEF АABС ~ - &A'B'C' + A'C'D' + A' D' E' Л- A' Е' F' Д A' B' C' (in a series of equal ratios the sum of the antecedents is to the sum of the consequents as any antecedent is to its consequent). But АABC -А*-, §342 Л A' B' C' Ä1-Bß (similar A are to each other as the squares on their homologous... | |
 | George Albert Wentworth - Geometry, Modern - 1879 - 262 pages
...Now °L = 1. Subtract 1 from each member of the equation. Then f _ ! = c _ 1; that is, fL^6 = e^Ld, bd or, a — b : b : : c — d : d. QED PROPOSITION...consequents as any antecedent is to its consequent. Let a :6 = c : d — e : f = g : h. We are to prove -a + c+e + g: b + d+f+h: : a: b. Denote each ratio*by... | |
 | Franklin Ibach - Geometry - 1882 - 208 pages
...on \ Then or a ± £- a : b ± £- b :: a : b. QED THEOREM XIII. 168. In any continued proportion, the sum of the antecedents is to the sum of the consequents...to its consequent. Let a : b :: c : d :: e : f :: g : h. To prove that a -\- c -\- e -\- y : b -{- d -\- f -\- h :: a : b. Denote the common ratio by r.... | |
 | Franklin Ibach - Geometry - 1882 - 208 pages
...q / \ q ' or a ± — a : 6 ± — 6 : : a : b. QED THEOREM XIII. 168. In any continued proportion, the sum of the antecedents is to the sum of the consequents...as any antecedent is to its consequent. Let a : b :: с : d :: e : f :: g : h. To prove that a -\- с -\- e -\. g : b -\- d -{. f .\. h :' a : b. Denote... | |
 | George Albert Wentworth - Geometry, Modern - 1882 - 268 pages
...PROPOSITION VIIL 266. In a series of equal ratios, the sum of the antecedents is to the sum of tlie consequents as any antecedent is to its consequent. Let a : b = c : d = e :f — g : h. We are to prove a + с + e + g : b + d + f+ h : : a :b. Denote each ratio by r. Then r. •.'«'.!.... | |
| |
In a series of equal ratios, the sum of the antecedents is to the sum of the consequents as any antecedent is to its consequent. Let a: b = c: d — e :/= g: h. 
