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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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A treatise on the elements of algebra, by G. Ainsworth and J. Yeats

G. Ainsworth - 1854 - 216 pages
...a+a, + a"+ .... + o<"> :6 + 6, + 6"+ +bw=a:b. That is, if any quantities be in continued proportion, the sum of all the antecedents is to the sum of all the consequents as one of the antecedents is to its consequent. By the last proposition, a+o, : 6 + 6,=a, : b,=a" :...
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The science of arithmetic, by J. Cornwell and J.G. Fitch

James Cornwell - 1855 - 380 pages
...original ratio. Hence they are equal to one another. 329. III. — If there be any number of equal ratios, the sum of all the antecedents is to the sum of all the consequents, as either of the antecedents is to its consequent* 3 : 5 : : 9 : 16 : : is : 30 : : 330 : 550. . 3...
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The Elements of Algebra

Elias Loomis - Algebra - 1856 - 280 pages
...same ratio, the first will have ti> the second the same ratio that the sum of all the antecedents has to the sum of all the consequents. Let a, b, c, d, e, f be any number of proportional quantities, such that a: b: :c:d: : e:f, then will a:b: :a+c+e:b+d+f....
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Elements of Geometry and Conic Sections

Elias Loomis - Conic sections - 1858 - 256 pages
...quantities are proportional, any one ante cedent is to its consequent, as the sum of all the antecedents, it to the sum of all the consequents. Let A : B : : C : D : : E : F, &c. ; then will A : B : : A+C+E : B+D+F For, since A : B : : C : D, we have AxD=BxC. And, since A :...
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The University Algebra ...

John Fair Stoddard, William Downs Henkle - Algebra - 1859 - 538 pages
...Prop. 6, (387) " " a + b :ab : : c+d : c—d Q. K D. PROPOSITION (394.) 13. In a continued proportion, any antecedent is to its consequent as the sum of...antecedents is to the sum of all the consequents. DEMONSTRATION. Let a : b : : с : d : : e :f::y: h : : &c. We are to prove that a : 6 ;:a + c+e+g,...
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A Treatise on Elementary and Higher Algebra

Theodore Strong - Algebra - 1859 - 570 pages
...+ H + etc. BDP Hence, when (numbers or) quantities of the same kind are proportionals, we say that the sum, of all the antecedents is to the sum of all the consequents, as any antÚcÚdent is to it╗ consequent. (as.) If we have ^ = =: , and т> = т=ч> t^611 by adding...
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The Mathematical Monthly, Volume 3

Mathematics - 1860 - 294 pages
...cy-j-az ax-\-by-\-cz a — bb — cc — aa -f- 5 -I- e t ions = — . I Since these ratios are equal, any antecedent is to its consequent as the sum of...the antecedents is to the sum of all the consequents ; therefore either fraction equals the sum of all the numerators divided by the mm of all the denominators,...
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Elements of Geometry, and Plane and Spherical Trigonometry: With Numerous ...

Horatio Nelson Robinson - Geometry - 1860 - 470 pages
...magnitudes are proportional, any one of the antecedents will be to its consequent as the sum of all tht antecedents is to the sum of all the consequents. Let A, B, C, D, JB, etc., represent the several inagm tudes whi )h give the proportions To which we may annex the identical...
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Elements of Geometry and Conic Sections

Elias Loomis - Conic sections - 1861 - 244 pages
...are proportional, any one ante e&dent is to its consequent's the sum of all the antecedents* i& t& the sum of all the consequents. Let A : B : : C : D : : E : F, &c. ; then will A : B : : A+C+E : B+D+F For, since A : B : : C : D, we have AxD=BxC. And, since A :...
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Elements of Geometry and Trigonometry: With Practical Applications

Benjamin Greenleaf - Geometry - 1862 - 518 pages
...remaining terms will be in proportion. PROPOSITION XI. — - THEOREM. 147. If any number of magnitudes are proportional, any antecedent is to its consequent...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, we have AXD = BXC, and AXF = BX...
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