 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 - 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 - 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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