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 - Geometry - 1868 - 340 pages
...remaining terms will be ill 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 E.... | |
| Elias Loomis - Algebra - 1868 - 386 pages
...ma me nb ~- nd 1 or ma: nb :: me: nd. n 309. If any number of quantities are proportional, any one **antecedent is to its consequent as the sum of all...of all the consequents. Let a: b:: c : d :: e: f;** then, since a: b:: c: d, ad — be; A (1.) and, since a: b :: e: /, «/=fe; (2.) also ab ~ ba. (3.)... | |
| Horatio Nelson Robinson - Geometry - 1868 - 276 pages
...proportional, any one of the antecedents will be to its consequent as the sum of all thf tnlfcedents **is to the sum of all the consequents. Let A, B, C, D,** 13, etc., represent the several magm tudes whi ih give the proportions A : B :: C : J) A : B :: E :... | |
| Benjamin Greenleaf - 1869 - 516 pages
...remaining terms will be in proportion. PROPOSITTON XI. — THEOREM. 147. If any number of magnitudes **are proportional, any antecedent is to its consequent...antecedents is to the sum of all the consequents.** feet A:B::C:D::E:F; then will A:B::A + C + E:B + D + F. For, from the given proportion, we have AXD... | |
| Horatio Nelson Robinson - 1869 - 276 pages
...Q. THEOREM VII. X If any number of magnitudes are proportional, 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,** (7, D, E, etc., represent the several magnitudes which give the proportions To which we may annex the... | |
| William Frothingham Bradbury - Geometry - 1872 - 262 pages
...proved. 23. If any number of quantities are proportional, any antecedent is to its consequent as tl;e **sum of all the antecedents is to the sum of all the...ad=bc (B) and also af=be (C) Adding (A), (B), (C) a (b** + d +/) = b (a + c + e) Hence, by (14) a :b = a -\-c-\-e:b -\-d-\-f THEOREM X. 21. If there are two... | |
| William Frothingham Bradbury - Algebra - 1872 - 268 pages
...proportion. Let a : b = c : d ac 1.0. 7 = -, I d T-, a" c» Hence, = ie a" : bn = c" : ef THEOREM XII. 21 3. **If any number of quantities are proportional', any...antecedents is to the sum 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),... | |
| Benjamin Greenleaf - Geometry - 1873 - 202 pages
...same in both, the remaining terms will be in proportion. THEOREM X. 115. If atiy 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 E. By... | |
| Elias Loomis - Algebra - 1873 - 396 pages
...ma _mc nb ~ nd1 or ma : nb : : me : nd. 309. If any number of quantities are proportional, any one **antecedent is to its consequent as the sum of all...antecedents is to the sum of all the consequents. Let** a:b::c:d::e:f; then, since a:b::c:d, ad=bc; (1.) and, since a : b : : e : ft af=be; (2.) also ab =... | |
| Edward Olney - Algebra - 1873 - 354 pages
...: b—dl У 2. COR. — If there be a series of equal ratios in the form of a continued proportion, **the sum of all the antecedents is to the sum of all the consequents,** as any one antecedent is to its consequent. DEM. — If a : b : : e : d : : e :f: : g : h, etc., a... | |
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