 | American School (Chicago, Ill.) - Engineering - 1903 - 392 pages
...THEOREH IX. 139. If any number of quantities are proportional, any antecedent is to its consei/uenl 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 (A) And by (128), ad = be (B) And also, af =. be (C) Adding (A), (B),... | |
 | Webster Wells - Algebra - 1904 - 642 pages
...c" : d". We may also prove Va : Vb = Ус': Vd. 505. 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 conséquents. Let a:b = c:d = e:f. Then by § 491, ad = be, and «/= be. Also, ab = ba. Adding, a(b... | |
 | Henry Burchard Fine - Algebra - 1904 - 612 pages
...(1) and (2), x = 0, 0, or - 7/3. Theorem. In a series of equal ratios any antecedent is to its 687 consequent as the sum of all the antecedents is to the sum of all the consequents. Thus, if ai : bi = O2 : 62 = О» : b»i then аi : bi = ai + a2 + о.t . Ьi + bt + b¡. For let r... | |
 | Henry Burchard Fine - Algebra - 1904 - 616 pages
...(1) and (2), x = 0, 0, or - 7/3. Theorem. In a series of equal ratios any antecedent is to its 687 consequent as the sum of all the antecedents is to the sum of all the consequents. Thus, if Oi : 61 = O2 : 62 = a3 : 63, then ai:b1 = a1 + a2 + a3:bl + bz + b3. For let r denote the... | |
 | 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... | |
 | Fletcher Durell - Geometry - 1911 - 553 pages
...become by composition? also by division? PROPOSITION IX. THEOREM 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. Given a : b = c : d = e : f=g \ h. To prove a + c + e... | |
 | 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... | |
 | Robert Judson Aley, David Andrew Rothrock - Algebra - 1904 - 344 pages
...given proportion and reducing each member to a fractional form. THEOREM VI. 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. Proof. Let the equal ratios be ^!= (?=.#=<* = BDFH '"" Then... | |
 | Isaac Newton Failor - Geometry - 1906 - 440 pages
...find the ratio of x to y. PROPOSITION IX. THEOREM 336 In a continued proportion, any antecedent is to its consequent as the sum of all the antecedents is to the sum of all the consequents. PROOF ab = ba, Iden. ad = be, § 328 and of = be. . § 328 Adding, ab + ad + af= ba + be + be ; Ax.... | |
 | Isaac Newton Failor - Geometry - 1906 - 431 pages
...find the ratio of x to y. PROPOSITION IX. THEOREM 336 In a continued proportion, any antecedent is to its consequent as the sum of all the antecedents is to the sum of all the consequents. PROOF ab = ba, I den. ad = be, § 328 and af = be. § 328 Adding, ab + ad -f <tf= ba + &c + be ; Ax.... | |
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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. 
