 | Robert Potts - 1868 - 434 pagina’s
...any equimultiples whatever G and H. And because E is the same multiple of A, that Fis at B, and that magnitudes have the same ratio to one another which their equimultiples have; (v. 15.) therefore A is to B, as E is to F: but as A is to B so is Cto D; (hyp.) wherefore as C is... | |
 | Henry William Watson - 1871 - 320 pagina’s
...of a triangle is half the area of a parallelogram upon the same base and between the same parallels, and because magnitudes have the same ratio to one another which their equimultiples have, therefore the areas of triangles between the same parallels are to one another as their bases. Corollary... | |
 | Euclides - 1874 - 342 pagina’s
...equimultiples whatever G and 11. Demonstration. Because E is the same multiple of A, that F is of B, and that magnitudes have the same ratio to one another which their equimultiples have (V. 15) ; therefore 1. A is to B, as E is to F; but as A is to B so is C to D (hyp.) ; wherefore 2.... | |
 | Euclides, James Hamblin Smith - 1876 - 376 pagina’s
...to B as A, C, E... together is to B, D, .F... together. V. Def. 5. QBD PROPOSITION XL (Eucl. v. 15.) Magnitudes have the same ratio to one another which their equimultiples have. Let A be the same multiple of C that B is of D. Then must C be to D as A to B. Divide A into magnitudes... | |
 | Robert Potts - 1876 - 446 pagina’s
...equimultiples whatever G and H. And because E is the same multiple of A , that F is of B, and that magnitudes have the same ratio to one another which their equimultiples have; (v. 15.) therefore A is to B, as E is to F: but as A is to B so is Cto D; (hyp.) wherefore as C is... | |
 | Āryabhaṭa - 1878 - 100 pagina’s
...parallelogram CF is to CE. It is plain that ABC is to ACD as CF is to CE. PROP. XVI. COROLLARY. 5. Magnitudes have the same ratio to one another, which their equimultiples have. It is plain that ABC to ACD as parallelogram CF to CE and CF and CE are equimultiples of ABC and ACD.... | |
 | Euclides - 1884 - 434 pagina’s
...be proved that if A = C, B = D ; and if A be less than C, B is less than D. PROPOSITION 15. THEOREM. Magnitudes have the same ratio to one another which their equimultiples have. Let A and B be two magnitudes, and m any number : it is required to prove A : B — mA : mB. Because... | |
 | Joseph Battell - 1903 - 722 pagina’s
...greater than, equal to, or less than the other, its consequent must be equally so. PROPOSITION' XV. ' Magnitudes have the same ratio to one another which their equimultiples have.' " Because the mutual relation of the magnitudes to each other in respect of quantity is the same. "... | |
 | Euclid - 1904 - 488 pagina’s
...if A > B, then C : A< C : B ; v. 5. which is contrary to the hypothesis ; .-. A < B. PROPOSITION 8. Magnitudes have the same ratio to one another which their equimultiples have. Let A, B be two magnitudes ; then shall A : B : : mA : mB. If p, q be any two whole numbers, then in... | |
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PROP. XV. THEOR. Magnitudes have the same ratio to one another which their equimultiples have. Let AB be the same multiple of C, that DE is of F: C shall be to F, as AB to DE. 
