| Benjamin Greenleaf - Geometry - 1875 - 204 pages
...will be in proportion. THEOREM X. 115. If any number of magnitudes are proportional, awy 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 : : E : F; then will A : B: :A+C+E: B-\-D + F. For, from the given proportion,... | |
| William Frothingham Bradbury - 1875 - 280 pages
...5" = c" : <f THEOREM XII. 213. If any number of quantities are proportional, any antecedent is to us **consequent as the sum of all the antecedents is to the sum of all the consequents.** Let a : 6 = с : d=e : f Now ab = ab (1) and by Theorem I. ad = be (2) and also af=be (3) Adding(l),(2),(3),... | |
| William Guy Peck - Algebra - 1875 - 348 pages
...: b + d+f+h + &c. :: a:b; (11) hence, the following principle : 10°. In any continued proportion, **the sum of all the antecedents is to the sum of all the consequents,** as any antecedent is to the corresponding consequent. ь d " bc = ad. a — c' b a = ê' " be = «/•... | |
| Richard Wormell - 1876 - 268 pages
...B + F. -F; .-. A + E : В + F = E : F = С: D. THEOREM LXX. If there be any number of equal ratios, **the sum of all the antecedents is to the sum of all the consequents** as either antecedent is to its consequent. Let A : В = С : D = E : F. By Theorem LXIX., A + E:B +... | |
| William Guy Peck - Conic sections - 1876 - 394 pages
...be multiplied or divided by the same quantity. PROPOSITION VIII. THEOREM. In a continued proportion, **the sum of all the antecedents is to the sum of all the consequents** as any antecedent is to the corresponding consequent. Assume the continued proportion, z 7 /• * df... | |
| William Frothingham Bradbury - Algebra - 1877 - 302 pages
...a" : b" = c" : d" THEOREM XII. 213. If any number of quantities are proportional, 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 = e :f Now ab = ab (1) and by Theorem I. ad = be (2) and also af=be (3) Adding (1),... | |
| Elias Loomis - Geometry - 1877 - 458 pages
...C— D. PROPOSITION IX. THEOREM. If any number of quantities are proportional, any one 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::E:F,etc.; then will A:B:: A+C+E: B+D+F. For,since A:B::C:D, we have A x D=B x C. And,... | |
| Edward Olney - Algebra - 1877 - 468 pages
...(6 + d +/+ & + & +, etc.) : : a : 6, or c : d, or e : /, etc. That is, in a series of equal ratios, **the sum of all the antecedents is to the sum of all the consequents,** as any antecedent is to its consequent ., aa SOLUTION. =- = r- or 06 = 60, oo ac . — = -T or ad =... | |
| James Bates Thomson - Algebra - 1878 - 324 pages
...6:2 .-. 12 : 4 = 9 : 3 THEOREM X. Wlien any number of quantities are proportional, 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 :: e : f, etc. Then a : b :: a + c + e : b + d+f, etc. For (Th. i), ad = be And,... | |
| Thomas Hunter - Geometry, Plane - 1878 - 142 pages
...as the squares of those sides. 1. Since the polygons are similar, AB: FG:: BC:GK::DC:LK, etc. Now, **as the sum of all the antecedents is to the sum of all the consequents** as any one anteB * a sequent, AB+BC+DC cedent is to any one con+ ED + AE:FG+GK + KL + LH + FH::AB:FG;... | |
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