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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. "
An Elementary Geometry - Page 30
by William Frothingham Bradbury - 1872 - 110 pages
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New Elementary Algebra

Benjamin Greenleaf - 1866 - 336 pages
...ce and -j =s -j. Therefore, by Art. 38, Ах. Т, • = , or, a : 6 : : с : d. THEOREM X. 324. If any number of quantities are proportional, any...antecedents is to the sum of all the consequents. Let a : b : : с : d : : e : f; then a : b : : a -\-c-\- e :b -\-d-\- f. For, by Theo. I., ad = bc, and af= be;...
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Ray's Algebra, First Book: Primary Elements of Algebra, for Common ..., Book 1

Joseph Ray - Algebra - 1866 - 252 pages
...— In any continued proportion, tlmt is, any number of proportions having the same ratio, any one antecedent is to its consequent, as the sum of all...the sum of all the consequents. Let a : b : : c : d : : m :n, etc. Then will a : b : : a+c+wi : 6+d+n. Since a : b : : c : d, And a : b : : m : n, We have...
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Primary Elements of Algebra: For Common Schools and Academies

Joseph Ray - Algebra - 1866 - 250 pages
...— In any continued proportion, that is, any number of proportions having the same ratio, any one antecedent is to its consequent, as the sum of all...the sum of all the consequents. Let a :b : : c :d : :m-.n, etc. Then will a : 6 : : a+c+m : b-\-d-\-n. Since a : b : : c : d, And a :b: :m:n, We have...
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New Elementary Algebra: Containing the Rudiments of Science for Schools and ...

Horatio Nelson Robinson - Algebra - 1866 - 328 pages
...PROPOSITION xm. 275, If any number of proportionals have the same ratio, any one of the antecedents will be to its consequent as the sum of all the antecedents is to the sum of all the consequents. t Let a : b = a : b (A) -Also, a : b = с : d (в) a : b =m : n (С) &c. = &c. We are to prove that...
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The University Algebra

John Fair Stoddard, William Downs Henkle - Algebra - 1866 - 546 pages
...c+d : cd QED PROPOSITION (>94.) 13. In a continued proportion, any antecedent it to its sjnscquent as the sum of all the antecedents is to the sum of all the consequents. DEMONSTRATION. Let a : b :: c : d :: e :/:: g : h :: &o. We are to prove that a ib '.\a + c + e+g,...
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Algebra for the Use of Colleges and Schools: With Numerous Examples

Isaac Todhunter - Algebra - 1866 - 576 pages
...quantities are proportionals, as one antecedent is to its consequent, so is the sum of all the antecedents to the sum of all the consequents. Let a : b :: c : d :: e : f; then a : b :: a + c +e : b + d +f. For ad=bc, and af= be, (Art. 386), also ab = ba ; hence ab + ad...
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Elements of Plane and Solid Geometry: And of Plane and Spherical ...

Gerardus Beekman Docharty - Geometry - 1867 - 474 pages
...' THEOREM VII. If any number of quantities be proportional, then any one of the antecedents will be to its consequent as the sum of all the antecedents is to the sum of all the consequents. Let A:B::mA:nB::nA:nB, &c. ; then will A: B:: A : B+mB+B, &c. ^ B+mB+nB (l++n)BB , For -T— !...
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New Elementary Algebra

Benjamin Greenleaf - 1867 - 336 pages
...any number of quantities are proportional, any antecedent is to its consequent as the sum of all tJie antecedents is to the sum of all the consequents. Let a : b : : c : d : : e : f; then a : b : : a -\-c-\-e : b-\-d-\- f. For, by Theo. I., ad=bc, and af=be; also, ab = b a. Adding,...
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Eaton's Elementary Algebra: Designed for the Use of High Schools and Academies

William Frothingham Bradbury - Algebra - 1868 - 270 pages
...proportion. Let a : b = c : d ac 6=3' -,-r an c" Hence, F = ^ ie a" : J" = c" : <f 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 = bc (2) and also af=be (3) Adding (1), (2), (3), Hence, by Theorem II. a:...
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Eaton's Elementary Algebra: Designed for the Use of High Schools and Academies

William Frothingham Bradbury - Algebra - 1868 - 264 pages
...will be in proportion. Let a : b'='c : d ac l=d Hence, ? = 5 ie a" : 5" = c" : cf THEOREM XII. 213. If any number of quantities are proportional, any...sum of all the consequents. Let a : b=; c : d = e if Now ab = ab (1) and by Theorem I. ad = bc (2) and also a/=6 (3) Adding (1), (2), (3), g(b+.d+f)...
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