 | James Howard Gore - Geometry - 1898 - 232 pages
...proportion. PROPOSITION IX. THEOREM. 209. In a series of equal ratios, 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. To prove a + c + e:b + d +/= a : b = c : d = e : f. Let r be the value of... | |
 | 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: b.... | |
 | George Egbert Fisher - Algebra - 1900 - 438 pages
...Art. 8, &2 = ac ; whence b = 19. In a series of equal ratios, any antecedent is to its consequent as the sum of all the antecedents is to the sum of all the consequents. Let % : d¡ = пi : d? = пя : d3 — ••• = v, П, Wo îi, or -. = v, ~ = v, -f = v, — . di... | |
 | 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... | |
 | James Harrington Boyd - Algebra - 1901 - 818 pages
...= e=± [1491] ac By dividing (1) by (2), J±| = ^ 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 w to its consequent. Let the ratios be (1) •£• = -£• = £ = ..... = r. A... | |
 | 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, b =... | |
 | George Egbert Fisher - 1901 - 320 pages
...= ас ; whence b = ^/(ac). 19. In a series of equal ratios, any antecedent is to its consequent as the sum of all the 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... | |
 | American School (Chicago, Ill.) - Engineering - 1903 - 426 pages
...integral.) IX. 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. Now, ab = ab (A) And,' ad = be (B) And also, af =. be (C) Adding ( A), (B), (C), a (b + d + /) = b... | |
 | Fletcher Durell - Geometry, Plane - 1904 - 382 pages
...become by compositiont also by divisiont PROPOSITION IX. THEOREM 812. 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. Given a : 6 = c : d = e :f=g : h. To prove a + c+ e + g :... | |
 | Fletcher Durell - Geometry, Solid - 1904 - 232 pages
...is to the second as the difference of the last two is to the last. 312. 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. 314. Like powers, or like roots, of the terms of a proportion... | |
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In a series of equal ratios, any antecedent is to its consequent, as the sum of all the antecedents is to the sum of all the consequents. Let a: 6 = c: d = e :/. Then, by Art. 
