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
| 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" :... | |
| 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... | |
| 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.... | |
| 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 :... | |
| 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,... | |
| 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... | |
| 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,... | |
| 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... | |
| 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 :... | |
| 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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