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
| Benjamin Greenleaf - 1866 - 336 pages
...ce and -j =s -j. Therefore, by Art. 38, Ах. Т, •£ = ¿, or, a : 6 : : с : d. THEOREM X. 324. **If any number of quantities are proportional, any...antecedents is to the sum of all the consequents. Let a : b** : : с : d : : e : f; then a : b : : a -\-c-\- e :b -\-d-\- f. For, by Theo. I., ad = bc, and af= be;... | |
| Joseph Ray - Algebra - 1866 - 252 pages
...— In any continued proportion, tlmt is, any number of proportions having the same ratio, any one **antecedent is to its consequent, as the sum of all...the sum of all the consequents. Let a : b : : c : d** : : m :n, etc. Then will a : b : : a+c+wi : 6+d+n. Since a : b : : c : d, And a : b : : m : n, We have... | |
| Joseph Ray - Algebra - 1866 - 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 sum of all the consequents. Let a :b : : c :d** : :m-.n, etc. Then will a : 6 : : a+c+m : b-\-d-\-n. Since a : b : : c : d, And a :b: :m:n, We have... | |
| 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... | |
| 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
...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 : : a -\-c-\-e : b-\-d-\- f. For, by Theo. I., ad=bc, and af=be; also, ab = b a. Adding,... | |
| William Frothingham Bradbury - Algebra - 1868 - 270 pages
...proportion. Let a : b = c : d ac 6=3' -,-r an c" Hence, F = ^ ie a" : J" = c" : <f THEOREM XII. 213. **If any number of quantities are proportional, any...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 II. a:... | |
| William Frothingham Bradbury - Algebra - 1868 - 264 pages
...will be in proportion. Let a : b'='c : d ac l=d Hence, ? = 5 ie a" : 5" = c" : cf THEOREM XII. 213. **If any number of quantities are proportional, any...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)... | |
| |