 | John Playfair - Euclid's Elements - 1842 - 332 pages
...E ; and nB, raF any whatever of B and F ; therefore A : B : : E : F (def. 5. 5.). PROP. XII. THEOR. If any number of magnitudes be proportionals, as one of the antecedents is to its consequent, so are all the antecedents, taken together, to all the consequents. If A : B : : C : D, and C : D : :... | |
 | Euclides - 1842 - 316 pages
...equals, — - — = bd ' o or a + e : b :: c +f : d, which represents A + E : B :: c + F : r>. PROP. IX. IF any number of magnitudes be proportionals, as one of the antecedents is to its consequent, so, are all the antecedents taken together, to all the consequents. Let A : B :: c : D :: E : F, then A:B::A... | |
 | Euclid - Geometry - 1845 - 218 pages
...as A is to B, so is ts Def. s. E to F J. Wherefore, ratios that, &c. QED PROPOSITION XII. THEOR. — If any number of magnitudes be proportionals, as one...antecedents taken together be to all the consequents. as A is to B, so shall A, C, E together be to B, D, F together. Take of A, C, E any equimultiples whatever... | |
 | Euclides - 1845 - 546 pages
...as A is to B, so is E to F. (v. def. 5.) Wherefore, ratios that, &c. QBD PROPOSITION XII. THEOREM. If any number of magnitudes be proportionals, as one of the antecedents is to Us consequent, so shall all the antecedents taken together be to all the consequents. Let any number... | |
 | Dennis M'Curdy - Geometry - 1846 - 168 pages
...less, less. Wherefore, ratios that are the same, &c. Q- ED Kecite (a) dlf. 5, 5 ; (4) ax. 1, 1. 12 Th. If any number of magnitudes be proportionals, as one...of the antecedents is to its consequent, so shall the sum of all the antecedents be to the sum of all the consequents. Given the magnitudes A, B; C,... | |
 | Euclides - 1846 - 292 pages
...any whatever of B, F ; therefore, AistoBasEistoF. Wherefore, Ratios that S$c. QEI>. PROP. XII. THEOR. If any number of magnitudes be proportionals, as one of the antecedents is to its conseifuent, so shall all the antecedents taken together be to all the consequents. Let any number... | |
 | Euclides - 1848 - 52 pages
...ratio, are the same to one another. PROP. XII. THEOREM. If any number of magnitudes be proportipnals, as one of the antecedents is to its consequent, so...antecedents taken together be to all the consequents. PROP. XIII. THEOREM. If the first has to the second the same ratio which the third has to the fourth,... | |
 | University of Cambridge - 1849 - 560 pages
...triangle. How may a circle be described touching one side and the produced parts of the other two? 6. If any number of magnitudes be proportionals, as one...antecedents taken together be to all the consequents. What restriction is here implied as to the species of the magnitudes t 7. The sides about the equal... | |
 | Harvey Goodwin - Mathematics - 1851 - 196 pages
...triangle. How may a circle be described touching one side and the produced parts of the other two ? 6. If any number of magnitudes be proportionals, as one...antecedents taken together be to all the consequents. What restriction is here iniplied as to the species of the magnitudes? 7. The sides about the equal... | |
 | Euclides - Geometry - 1853 - 176 pages
...therefore, as a is to b, so is e to f (v. def. 5). Wherefore, ratios, &c. QED PROPOSITION XII. — THEOREM. If any number of magnitudes be proportionals, as one...antecedents taken together be to all the consequents. LET any number of magnitudes a, b, С, d, e, f be proportionals ; that is, as a is to b, so is С to... | |
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IF any number of magnitudes be proportionals, as one of the antecedents is to its consequent, so shall all the antecedents taken together be to all the consequents. 
