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. An Elementary Geometry - Page 30by William Frothingham Bradbury - 1872 - 110 pagesFull view - About this book
| 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... | |
| William Frothingham Bradbury - Algebra - 1877 - 302 pages
...Let a : b = c : d ac l - e - r = -, bd Hence, ^ = ^ a» c» ie a" : b" = c" : d" THEOREM XII. 213. **If any number of quantities are proportional, any...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 Theorem II.... | |
| 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
...That is. a : b = e : d 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...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;... | |
| Edward Olney - Algebra - 1878 - 516 pages
...: (6 + d+/+ ^ + fc+,ete.) : : a : b, or c : d, or e : f, 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 SOLUTION. =- = r or a& = ba, oo ac , , — = -j or ad = be,... | |
| Edward Olney - 1878 - 360 pages
...Ъ— dt 72. Сов. — If there be a series of equal ratios in the form of a continued proportion, **the sum of all the antecedents is to the sum of all the consequents,** as any one antecedent is to its consequent. DEM. — If a :b : : с : d : : e :f: :g :n, etc., a +... | |
| Benjamin Greenleaf - Algebra - 1879 - 322 pages
...ratios, -=- = j, ce and - = -. Therefore, by Art 38, Ax. 7, | = ^, or, a : b : : c : d. THEOREM X. 324i **If any number of quantities are proportional, any...all the consequents. Let a : b : : c : d : : e : f;** then a : b : : a-\-c-\-e: b-\-d-\-f. For, by Theo. I., arf=4c, an daj — be; also, ab = b a. Adding,... | |
| Benjamin Greenleaf - Algebra - 1879 - 376 pages
...Theorem TX. . ce and -d=j. Therefore, by Art. 38, As^ 7, -r = -,, or. a : b : : c : d. THEOREM X. 324 **1 If any number of quantities are proportional, any...antecedents is to the sum of all the consequents. Let** « : 4 : : c : d : : e : f; then g:t: : m -\- c -\- e : l-\-d-\- f. For, by Theo. I., ad = bc, and... | |
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