| 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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