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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. "
An Elementary Treatise on Algebra: Theoretical and Practical ... - Σελίδα 336
των James Ryan - 1826 - 383 σελίδες
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Elements of Geometry: Containing the First Six Books of Euclid, with a ...

John Playfair - 1842 - 317 σελίδες
...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 : :...

Elements of geometry: consisting of the first four,and the sixth, books of ...

Euclides - 1842
...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 in Paragraphs: The Elements of Euclid: Containing the First Six Books ...

Euclid - 1845 - 199 σελίδες
...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...

Euclid's Elements of geometry [book 1-6, 11,12] with explanatory notes ...

Euclides - 1845
...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...

Euclid's Elements: Or, Second Lessons in Geometry,in the Order of Simson's ...

Dennis M'Curdy - 1846 - 138 σελίδες
...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,...

The Elements of Euclid, the parts read in the University of Cambridge [book ...

Euclides - 1846
...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...

The definitions, postulates, axioms, and enunciations of the propositions of ...

Euclides - 1848
...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,...

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University of Cambridge - 1849
...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...

A Collection of Problems and Examples Adapted to the "Elementary Course of ...

Harvey Goodwin - 1851 - 173 σελίδες
...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...

The first six books of the Elements of Euclid, with numerous exercises

Euclides - 1853 - 147 σελίδες
...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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