 | Joseph Ray - Algebra - 1852 - 422 pages
...bn : : cr : ds. ART. 27§. 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 consequent*Let a :b : :c: d : :m :n, die. Then a:b:\ a-\-c+m : b-\-d-\-n. Since a : b : : c : d, we... | |
 | 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... | |
 | 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,... | |
 | Isaac Todhunter - Algebra - 1866 - 580 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 + af=... | |
 | Benjamin Greenleaf - 1866 - 336 pages
...Therefore, by Art, 38, Ax. 7, .т- = -з, or, a : b : : с : d. THEOREM X. 324. If any number of quantities are proportional, any antecedent is to its consequent as the sum of aU the antecedents is to the sum of att the consequents. « Let a : b : : с : d : : e : f; then a... | |
 | 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— !... | |
 | Benjamin Greenleaf - 1867 - 336 pages
...Therefore, by Art. 38, Ax. 7, j = ^ or, it : l, ::,••. d. THEOBEM X. 324 i If 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 :... | |
 | Benjamin Greenleaf - Geometry - 1868 - 340 pages
...the remaining terms will be in proportion. sequents the same in both, the remaining terms will be ill proportion. PROPOSITION XI. — THEOREM. 147. If any...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 E. By adding... | |
 | William Frothingham Bradbury - Algebra - 1868 - 270 pages
...ac 6=3' -,-r an c" Hence, F = ^ ie a" : J" = c" : <f THEOREM XII. 213. If any number of quantities are proportional, any antecedent is to its consequent...of all the 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... | |
 | William Frothingham Bradbury - Algebra - 1868 - 264 pages
...: b'='c : d ac l=d Hence, ? = 5 ie a" : 5" = c" : cf THEOREM XII. 213. If any number of quantities are proportional, any antecedent is to its consequent...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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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. 
