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" In a series of equal ratios, any antecedent is to its consequent, as the sum of all the antecedents is to the sum of all the consequents. Let a: 6 = c: d = e :/. Then, by Art. "
Elements of Plane Geometry: For the Use of Schools - Page 54
by Nicholas Tillinghast - 1844 - 96 pages
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Ray's Algebra Part Second: An Analytical Treatise, Designed for High Schools ...

Joseph Ray - Algebra - 1852 - 410 pages
...ART. 278. PROPOSITION XII. — In any number of proportions having the same ratio, 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 : :m :n, &c. Then a : b : : a-\-c-\-m : b-\-d-\-n. Since a : b : : c : d, we have...
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Elements of Geometry and Trigonometry

Adrien Marie Legendre - Geometry - 1852 - 436 pages
...m, or Mx ( Qn) =Px (Nm) : PROPOSITION X. THEOEEM. If any number of magnitudes are proportionals, any one antecedent will be to its consequent, as the sum of all the antecedents to the sum of the consequents. Let M : N :: P : Q :: B : S, c. Then since, M : N :: P : Q, we have...
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Elements of Geometry, Plane and Spherical Trigonometry and Conic Sections

Horatio Nelson Robinson - Conic sections - 1854 - 350 pages
...proportionals. THEOREM 7. 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=C:D' And ... And ... &c.=&c. Then we are to show that A : B= C+H+ G &c. : D+F+H, &c. If...
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Elements of Geometry and Trigonometry from the Works of A.M. Legendre ...

Charles Davies - Geometry - 1854 - 436 pages
...: : Nm : Qn. BOOK II. 55 PROPOSITION X. THEOREM. If any number of magnitudes are proportionals, any one antecedent will be to its consequent, as the sum of all the antecedents to the sum of tl1e consequents. Let M : N : : P : Q : : R : S, &c. Then since, M : N : : P : Q, we...
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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 University Algebra ...

John Fair Stoddard, William Downs Henkle - Algebra - 1859 - 538 pages
...+ 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 all the 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 antcdent 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
...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 all 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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A Course of Mathematics: Composed for the Use of the Royal Military Academy

Charles Hutton - Mathematics - 1860 - 1022 pages
...THEOREM I.XXIl. If any number of quantities be proportional, then any one of tne antecedent* "/I be to its consequent, as the sum of all the antecedents, is to the aim of ли the consfqnents. Let А:В::тА:тпВ::пЛ:пВ, &С.; then will Л : J : : Л -f...
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