PRO P. VII. THEOREM. If four magnitudes be proportional, they will be proportional also when taken inversely. If A has to B the same ratio that c has to D; then, inversely, B will have to a the same ratio that D has to c. For, of B and D take any equimultiples whatever E and F; and of A and c any equimultiples whatever G and h: Then, fince A is to B as c is to d (by Hyp.), and G, H are equimultiples of A, C, and E, F of B, D (by Conft.), if G be greater than E, h will be greater than F; and if equal, equal; and if less, less (V. Def. 5.) And, because G has with E the same agreement of equality, excess, or defect, that h has with F, E will have with the same agreement of equality, excess, or defect, that F has with H. If, therefore, e be greater than G, F will also be greater than h; and if equal, equal; and if less, less. But E and F are any equimultiples whatever of B and D (by Conft.); and G and h are any equimultiples whatever of A and c; therefore, B is to A as D is to c (V. Def. 5.) Q. E. D. PRO P. VIII. VIII. THEOREM. If the first of four magnitudes be the same multiple or part of the second as the third is of the fourth; the first will have the same ratio to the second as the third has to the fourth. Let A the first, be the same multiple of B the second, that c the third is of d the fourth; then will a have to B the same ratio that c has to D. For of A and c take any equimultiples whatever E and G; and of B and D any equimultiples whatever F and h: Then, because A is the same multiple of B that c is of D (by Hyp.), and E is the same multiple of A that G is of c (by Conft.), E will also be the fame multiple of B that G is of D (V. 3.) And, since e is the same multiple of B that g is of D, and F is the same multiple of B that h is of D (by Conft.), if e be greater than F, G will be greater than h; and if equal, equal; and if lefs, less. But E and G are any equimultiples whatever of A and c; and F and h are any equimultiples whatever of B and D; therefore, A will have to B the same ratio that c has to D (V. Def. 5.) Again, let the first B, be the fame part of the second Ag as the third D, is of the fourth c; then will B have to A, the fame ratio that D has to c. For A is the same multiple of B that c is of D (by Hyp.); therefore A will have to B the same ratio that chas to D (V.8.) And, fince A is to B as c is to D, therefore, allo, inversely, B is to A as D is to c (V.7.) Q. E. D. PRO P. IX. THEOREM. Equal magnitudes have the same ratio to the same magnitude, and the fame has the same ratio to equal magnitudes. Let A and B be equal magnitudes, and c any other magnitude whatever ; then A will have to c the fame ratio that B has to c. For of A and B take any equimultiples whatever D and E; and of c any multiple whatever F: Then, because D is the same multiple of a that e is of B, and a is equal to B, D will also be equal to E. And, since D and e are equal to each other, if o be greater than F, E will also be greater than F; and if equal, equal ; and if less, less. But D and E are any equimultiples whatever of A and B, and F is any multiple whatever of c; therefore A is to c as B is to c (V. Def. 5.) Again, let A and B be equal magnitudes, and c any, other magnitude whatever; then c has to a the same ratio that it has to B. For, having made the same construction as before, D may, in like manner, be shewn to be equal to E: And, since D is equal to E, if F be greater than D, it will also be greater than E; or if equal, equal; or if less, less. But F is any multiple whatever of c, and D and E are any equimultiples whatever of A and B; therefore, c is to A as c is to B (V. Def. 5.) Q: E. D. PRO P. X. THEOREM. Magnitudes which have the same ratio to the same magnitude are equal to each other ; and those to which the same magnitude has the same ratio are equal to each other. Let A have to c the same ratio that B has to c; then will a be equal to B. For, if they be not equal, one of them must be greater than the other, L Let Let A be the greater; and of A and B take the equimultiples D and E, and of c the multiple F such, that D may be greater than F, and e not greater than F (V. 4.) Then, since a is to c as B is to c, and D and E are equiinultiples of A and B, and F is a multiple of c, d being greater than F, e will also be greater than F (V. Def. 5.) But, by construction, e is not greater than F; whence it is greater and not greater at the same time, which is absurd. The magnitude A is, therefore, not greater than B; and in the same manner it may be shewn that it is not less ; consequently they are equal to each other. Again, let c have to A the same ratio that it has to B; then will a be equal to B. For, since c is to A as c is to B, therefore, also, inversely, A will be to c as B is to c (V.7.) But magnitudes which have the same ratio to the fame magnitude have been shewn to be equal to each other; therefore A is equal to B. Q. E. D. |