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" If any number of magnitudes 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; then will A : B : : A + C + E : B + D + F. "
Elements of Geometry and Trigonometry: With Practical Applications - Page 50
by Benjamin Greenleaf - 1862 - 490 pages
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Elements of Geometry and Conic Sections

Elias Loomis - Conic sections - 1849 - 252 pages
...II., A : B : : mA : mB. PROPOSITION IX. THEOREM. If any number of quantities are proportional, any one antecedent 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...
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Ray's Algebra Part Second: An Analytical Treatise, Designed for High Schools ...

Joseph Ray - Algebra - 1852 - 408 pages
...am :bn: :cr:ds. ART. 278. PROPOSITION XII. — In any number of proportions having the same ratio, any antecedent is to its consequent, as the sum of...the sum of all the consequents. Let a :b : :c : d : :m :n, &c. Then a : b : : a-\-c-\-m : b-\-d-\-n. Since a : b : : c : d, we have bc=ad (Art. 267)....
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Ray's Algebra, Part First: On the Analytic and Inductive Methods of ...

Joseph Ray - Algebra - 1848 - 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 antecedents i» to the sum of all the conseqtients. Let a : b : : c : d : : m : n, &c. Then will a:b: : o+e+m :...
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A Practical Treatise on Algebra: Designed for the Use of Students in High ...

Benjamin Greenleaf - Algebra - 1853 - 370 pages
...proportionals, any antecedent has the same ratio to its consequent that the sum of all the antecedents has to the sum of all the consequents. Let a : b : : c : d : : e : f : : g : h ; then, also, a '. b : : o+c +e+g : b+d+f+k. Since ab=ba, ad=bc, af=be, ah=bg, we have a...
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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 - 382 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 antécédent is to it» consequent. (as.) If we have ^ = =: , and т> = т=ч> t^611 by adding...
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