| Giovanni Ferraro - Mathematics - 2007 - 392 pages
...number of magnitudes are proportional, then one of the antecedents is to one of the consequents as **the sum of the antecedents is to the sum of the consequents** (see Euclid [E]). In modern symbols, if sn = X/«=i ai, then ai : o2 = (sn - an) . (sn - ai)Hence,... | |
| William Betz, Harrison Emmett Webb, Percey Franklyn Smith - Geometry, Modern - 1912 - 356 pages
...d = a : c. 3. d : b = c : a. 7. c : a = d : b. 4. d : c = b : a. 8. c : d = a : b. 361. Theorem IV. **In a series of equal ratios the sum of the antecedents...consequents as any antecedent is to its consequent.** aceo Proof. If T — i — "7.— 7 ? let # represent the value of each bdfh of the ratios. Then a... | |
| Edinburgh Mathematical Society - Electronic journals - 1888 - 154 pages
...the terms D, E, F are proportional. Since A : B = B : C, by composition A + B:B = B + C:G; therefore **the sum of the antecedents is to the sum of the consequents** in the same ratio, that is, A + 2B + C:B + C = B + C:C. Now D = A + 2B + C, E = B + C, and F = C ;... | |
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