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. Plane and Solid Geometry - Page 82by James Howard Gore - 1898 - 210 pagesFull view - About this book
| 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... | |
| Benjamin Greenleaf - 1869 - 516 pages
...will be in proportion. PROPOSITTON XI. — THEOREM. 147. If any number of magnitudes are proportional, any antecedent is to its consequent as the sum of...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... | |
| Benjamin Greenleaf - 1870 - 334 pages
...Ax. 7, ^ = ¿, 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 all the antecedent» is to the sum of all the consequents. Let a : b : : с : d : : e : f; then a : b : : a... | |
| Benjamin Greenleaf - Algebra - 1871 - 412 pages
...r = -. ; " J Л J о а whence, a : b : : c : d. 319i If any number of quantities are proportional, any antecedent is to its consequent as the sum of all the antecedents M to the sum of all the consequents. If a : b : : c : d : : e : f, then a : b : : a-\-c-\-e : b-\-d-\-f.... | |
| William Frothingham Bradbury - Geometry - 1872 - 124 pages
...= -76" dn that is a" : b" = c" : dn THEOREM IX. 23 1 If any number of quantities are proportional, any antecedent is to its consequent as the sum of...sum of all the consequents. Let a : b = c : d = e :f Now ab = ab (A) and by (12) ad=bc (B) and also af=."be (C) Adding (A), (B), (C) a (b -fd +/) = b (a... | |
| William Frothingham Bradbury - Algebra - 1872 - 268 pages
...c» Hence, = ie a" : bn = c" : ef THEOREM XII. 21 3. If any number of quantities are proportional', any antecedent is to its consequent as the sum of...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),... | |
| 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 consequents. Let a : b = c : d=. e :f Now ab = ab (A) and by (12) ad=bc (B) and also af=be (C) Adding (A), (B), (C) a (b + d +/) = b (a +... | |
| Benjamin Greenleaf - Geometry - 1873 - 202 pages
...remaining terms will be in proportion. THEOREM X. 115. If atiy number of magnitudes are proportional, any antecedent is to its consequent as the sum of...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... | |
| William Frothingham Bradbury - Geometry - 1873 - 132 pages
...n 6 n rf" that is a" : 6" = c" : d n THEOREM IX. 23. If any number of quantities are proportional, any antecedent is to its consequent as the sum of...antecedents is to the sum of all the consequents. Let a :b = e : d=.e :f Now ab = ab (A) and by (12) ad —be (B) and also af=be (C) Adding (A), (B), (C) a (b... | |
| 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 =... | |
| |