For, because AB is to BE as CD is to DF; [Hypothesis. therefore, by division, AE is to EB as CF is to FD; [V. 17. and, by inversion, EB is to AE as FD is to CF. [V. B. Therefore, by composition, AB is to AE as CD is to CF. [V. 18. Wherefore, if four magnitudes &c. Q.E.D. PROPOSITION 20. THEOREM. If there be three magnitudes, and other three, which have the same ratio, taken two and two, then, if the first be greater than the third, the fourth shall be greater than the sixth; and if equal, equal; and if less, less. Let A, B, C be three magnitudes, and D, E, F other three, which have the same ratio taken two and two; that is, let A be to B as D is to E, and let B be to C as E is to F: if A be greater than C, D shall be greater than F; and if equal, equal; and if less, less. First, let A be greater than C: D shall be greater than F. For, because A is greater than C, and B is any other magnitude, therefore A has to B a greater ratio than C has to B. [V. 8. But A is to B as D is to E; (Hypothesis. therefore D has to E a greater ratio than C has to B. [V. 13. And because B is to C as E is to F, [Hyp. À B therefore, by inversion, C is to B as F is D E to E. [V. B. And it was shewn that D has to E a greater ratio than C has to B; therefore D has to E a greater ratio than F has to E; [V. 13, Cor. therefore D is greater than F. [V. 10. Secondly, let A be equal to C: D shall be equal to F. For, because A is equal to C, and B is any other magnitude, therefore A is to B as C is to B. [V. 7. But A is to B as D is to E, [Hypothesis. and C is to B as Fis to E [Hyp. V. B. therefore D is to E as F is to E; [V. 11. and therefore D is equal to F. [V.9. Lastly, let A be less than C : D shall be less than F. For C is greater than A; and, as was shewn in the first case, C is to Bas Fis to E; and, in the same manner, B is to A as E is to D; therefore, by the first case, F is greater than D; that is, D is less than F. Wherefore, if there be three &c. Q.E.D. PROPOSITION 21. THEOREM. If there be three magnitudes, and other three, which have the same ratio, taken two and two, but in a cross order, then if the first be greater than the third, the fourth shull be greater than the sixth; and if equal, equal; and if less, less. Let A, B, C be three magnitudes, and D, E, F other three, which have the same ratio, taken two and two, but in a cross order; that is, let A be to B as E is to F, and let B be to C as D is to E: if A be greater than C, D shall be greater than F; and if equal, equal ; and if less, less. First, let A be greater than C: D shall be greater than F. For, because A is greater than C, and B is any other magnitude, therefore A has to B a greater ratio than C has to B. [V. 8. But A is to B as E is to F; [Hypothesis. therefore E has to F a greater ratio than C has to B. [V. 13. A B ở And because B is to C as D is DĘ F to E, [Hypothesis. therefore, by inversion, C is to B as E is to D. [V. B. And it was shewn that E has to Fa greater ratio than C has to B; therefore E has to F a greater ratio than E has to D; [V. 13, Cor. therefore F is less than D; [V. 10. that is, D is greater than F. Secondly, let A be equal to C: D shall be equal to F. For, because A is equal to C, and B is any other magnitude, therefore A is to B as C is to B. [V.7. But A is to B as E is to F; [Hyp. DE and C is to B as E is to D; [Hyp. V. B. therefore E is to Fas E is to D; [V. 11. and therefore D is equal to F. [V.9. Lastly, let A be less than C: D shall be less than F. For C is greater than A; and, as was shewn in the first case, C is to B as E is to D; and, in the same manner, B is to A as F is to E; therefore, by the first case, Fis greater than D; that is, D is less than F. Wherefore, if there be three &c. Q.E.D. PROPOSITION 22. THEOREM. If there be any number of magnitudes, and as many others, which have the same ratio, taken two and two in order, the first shall have to the last of the first magnitudes the same ratio which the first of the others has to the last. [This proposition is usually cited by the words ex æquali.] First, let there be three magnitudes A, B, C, and other three D, E, F, which have the same ratio, taken two and two in order; that is, let A be to B as D is to E, and let B be to C as E is to F: A shall be to Cas D is to F. Take of A and D any equimultiples whatever G and H; and of B and E any equimultiples whatever K and L; and of C and F any equimul- Á B C D E Ė tiples whatever M and N. Then, because A is to Bas D Ģ Ķ M Ļ Ņ is to E; [Hypothesis. and that G and H are equimultiples of A and D, and K and L equimultiples of B and E; [Construction. therefore G is to K as H is to [V. 4. For the same reason, K is to M as L is to N. And because there are three magnitudes G, K, M, and other three H, L, N, which have the same ratio taken two and two, therefore if G be greater than M, H is greater than N; and if equal, equal; and if less, less. [V. 20. But G and H are any equimultiples whatever of A and D, and M and N are any equimultiples whatever of C and F. Therefore A is to C as D is to F. [V. Definition 5. Next, let there be four magnitudes, A, B, C, D, and other four E, F, G, H, which have the same ratio taken two and two in order; 'A. B. O. D. namely, let A be to B as E is to F, and | E. F. G. H. B to C as F is to G, and C to D as G is to H: A shall be to Das E is to H. For, because A, B, C are three magnitudes, and E, F, G other three, which have the same ratio, taken two and two in order, [Hypothesis. therefore, by the first case, A is to C as E is to G. But C is to D as G is to H; [Hypothesis. therefore also, by the first case, A is to D as E is to H. And so on, whatever be the number of magnitudes. PROPOSITION 23. THEOREM. If there be any number of magnitudes, and as many others, which have the same ratio, taken two and two in a cross order, the first shall have to the last of the first magnitudes the same ratio which the first of the others has to the last. First, let there be three magnitudes, A, B, C, and other three D, E, F, which have the same ratio, taken two and two in a cross order; namely, let A be to B as E is to F, and B to C as D is to E: A shall be to Cas D is to F. Take of A, B, D any equimultiples whatever G, H, K; and of C, E, F any equimultiples whatever , À, N. А в съЕ Е Then because G and H are a equimultiples of A and B G L K M N and that magnitudes have the same ratio which their equimultiples have; [V. 15. therefore A is to B as G is to H. And, for the same reason, E is to Fas M is to N. |