 | Horatio Nelson Robinson - Algebra - 1875 - 430 pages
...If there be a proportion, consisting of three or more equal ratios, then either antecedent will be to its consequent, as the sum of all the antecedents is to the sum of all the consequents. Suppose a:b=c:d — e:f = g:h=, etc. Then by comparing the ratio, a : b, first with itself, and afterward... | |
 | 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 = «/•... | |
 | William Guy Peck - Conic sections - 1876 - 412 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... | |
 | 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 Frothingham Bradbury - Algebra - 1877 - 302 pages
...- r = -, bd Hence, ^ = ^ a» c» ie a" : b" = c" : d" THEOREM XII. 213. If any number of quantities are proportional, any antecedent is to its consequent...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), (2), (3), a( Hence, by... | |
 | William Frothingham Bradbury - Geometry - 1877 - 262 pages
...a™ cn Hence v- = -36n <#• that is a" : 6" = cn : dn THEOREM IX. 23 1 If any number of quantities are proportional, any antecedent is to its consequent...the sum of all the consequents. Let a :b = c : d = e :/ Now ab = ab (A) and by (12) ad=be (B) and also af=be (C) Adding (A), (B), (C) q(ft + «l+/)=6(a... | |
 | Elias Loomis - Conic sections - 1877 - 458 pages
...B : : C+D : C— D. PROPOSITION IX. THEOREM. If any number of quantities are proportional, any one antecedent is to its consequent as the sum of all...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, since... | |
 | Edward Olney - Algebra - 1877 - 466 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 - 322 pages
...od Again, 12 : 4 = 6 : 2, and 9:3 = 6:2 .-. 12 : 4 = 9 : 3 THEOREM X. Wlien any number of quantities are proportional, any antecedent is to its consequent,...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, " of = be Also, ai> = 6a Adding... | |
 | 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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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. 
