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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. "
An Elementary Geometry - Page 30
by William Frothingham Bradbury - 1872 - 110 pages
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Essentials of Geometry (plane).

Webster Wells - Geometry - 1898 - 264 pages
...From(l), o_ = c- ( 237) ac and o^-ft^Cj-d. ac PROP. VIII. THEOREM. 240. In a series of equal ratios, the sum, of all the antecedents is to the sum of all the consequents as any antecedent 18 to its consequent. Given a:b = c:d=e:f. (1) To Prove a + c + e:b + d +/= a : b....
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Plane and Solid Geometry: Inductive Method

Arthur A. Dodd, B. Thomas Chace - Geometry - 1898 - 468 pages
...: W~C'*— CD : C' D' . Now substitute these values in your first equations. By proportion, 198, the sum of all the antecedents is to the sum of all the consequents as any antecedent is to its consequent. Can you write an equation so that the sum of the AS in the...
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Essentials of Algebra for Secondary Schools: By Webster Wells

Webster Wells - Algebra - 1899 - 444 pages
...c--d a — о с — a Whence, a + b : a — b = c + d: с — d. 315. In a series of equal ratios, 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. Then by 306, ad — bc, and а/= ЬeAlso, ab = ba. Adding, a (b + d +/) = b (и + с...
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Plane and Solid Geometry

James Howard Gore - Geometry - 1899 - 266 pages
...like powers of the terms are in proportion. PROPOSITION IX. THEOREM. 209. In a series of equal ratios, 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. To prove a + c + e : b + d+f= a : b = c : d = e :/. Let r be the value of the equal...
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Plane and Solid Geometry

James Howard Gore - Geometry - 1899 - 266 pages
...like powers of the terms are in proportion. PROPOSITION IX. THEOREM. 209. In a series of equal ratios, any antecedent is to its consequent as the sum of...antecedents is to the sum of all the consequents. To prove a + c + e:b + d +/= a:b = c:d = Let r be the value of the equal ratios, that is, acje From...
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The Essentials of Geometry

Webster Wells - Geometry - 1899 - 424 pages
...bc — d .: a + b: a — 6 = c + d:c — d. PROP. VIII. THEOREM. 240. In a series of equal ratios, the sum of all the antecedents is to the sum of all the consequents as any antecedent is to its consequent. Given a:b = c:d = e:f. (1) To Prove a + c + e :b + d+f= a:...
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Secondary Algebra

George Egbert Fisher - 1900 - 444 pages
...From a : b = b : c, we have, by Art. 8, &2 = ac ; whence b = -^/(ac). 19. In a series of equal ratios, any antecedent is to its consequent as the sum of...antecedents is to the sum of all the consequents. Let щ : dl = щ : d = n3 : d3 = ••• = v, ^ = V,^V,^ = V}.... Ctl -,2 , 3 Then, n1 = vd, nj...
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Complete Secondary Algebra

George Egbert Fisher - 1901 - 320 pages
...a : b = b : c, .we have, by Art. 8, b2 = ас ; whence b = ^/(ac). 19. In a series of equal ratios, any antecedent is to its consequent as the sum of...antecedents is to the sum of all the consequents. Let ni : dl = n2: <Z2 = n3 d3=—=v, * = v,b = v,b = v,.... di a, d3 Then, n1 = vdu n = vd, n3 = vdt,...
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Elementary Geometry, Plane and Solid: For Use in High Schools and Academies

Thomas Franklin Holgate - Geometry - 1901 - 462 pages
...h.-.R, or - = ^-^, that P:R = hm: kn. ' kn R kn' 240. THEOREM. If any number of ratios are equal, then the sum of all the antecedents is to the sum of all the consequents as any one antecedent is to its consequent. Let a^ : 6! = 02 : 62 = a3 : 63 = •••. It is required...
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College Algebra

James Harrington Boyd - Algebra - 1901 - 812 pages
...?^*. [{491] ac By dividing (1) by (2), 2-| = e- 493. THEOREM IX. — In a series of equal ratios the sum of all the antecedents is to the sum of all the consequents as any one antecedent is to its consequent. Let the 'ratios be (1) j- = J- = = = r. (2) a = Ar,...
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