+ Constr. nott greater than D; which is impossible. Therefore A is equal to B. Wherefore, magnitudes which, &c. Q. E. D. PROP. X. THEOR. See N. That magnitude which has a greater ratio than another has unto the same magnitude, is the greater of the two: and that magnitude to which the same has a greater ratio than it has unto another magnitude, is the lesser of the two. Let A have to C a greater ratio than B has to C: A shall be greater than B. For, because A has a greater ratio to C, than B has * 7 Def. 5. to C, there are* some equimultiples of A and B, and some multiple of C such, that the multiple of A is A cl * 4 Ax. 5. is greater than E; therefore A is* greater B than B. Next, let C have a greater ratio to B than it has to A: B shall be less than A. For* there is some multiple F of C, and some equimultiples E and D of B and A such, that F is greater than E, but not greater than D: therefore E is less than D: and because E and D are equimultiples of B * 4 Ax. 5. and A, and that E is less than D, therefore B is * less than A. Therefore, that magnitude, &c. E * 7 Def. 5. Q. E. D. PROP. XI. THEOR. Ratios that are the same to the same ratio, are the same to one another. Let A be to B as C is to D; and as C to D, so let E be to F: A shall be to B, as E to F. Take of A, C, E, any equimultiples whatever G, H, K; and of B, D, F, any equimultiples whatever L, M, N. Therefore, since A is to B as C to D, and G, are taken equimultiples of A, C, and L, M, of B, D; G -K C if G be greater than L, H is greater than M; and if equal, equal; and if less, less *. Again, because C is * 5 Def. 5. to D, as E is to F, and H, K are taken equimultiples of C, E; and M, N, of D, F; if H be greater than M, K is greater than N; and if equal, equal; and Hif less, less: but if G be A E greater than L, it has D F been shewn that H is L Ngreater than M; and if equal, equal; and if less, less: therefore if G be greater than L, K is greater than N; and if equal, equal; and if less, less: and G, K are any equimultiples whatever of A, E; and L, N any whatever of B, F: therefore* * 5 Def. 5. as A is to B, so is E to F. Wherefore, ratios that, &c. B M R. E. D. PROP. XII. THEOR. H A K- B 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. Let any number of magnitudes A, B, C, D, E, F, be proportionals; that is, as A is to B, so C to D, and E to F: as A is to B, so shall A, C, E together be to B, D, F together. Take of A, C, E any G equimultiples whatever G, H, K; and of B, D D, F any equimultiples L M whatever, L, M, N: then, because A is to B, as C is to D, and as E to F; and that G, H, K are equimultiples of A, C, E, and L, M, N, equimultiples of B, D, F; if G be greater than L, H is greater than M, and K greater than N; and if equal, equal; and if less, less *: wherefore if G be greater * 5 Def. 5. than L, then G, H, K together are greater than L, M, N together; and if equal, equal; and if less, less: but G, and G, H, K together are any equimultiples of A, and A, C, E together; because if there be any number of magnitudes equimultiples of as many, each of each, whatever multiple one of them is of its part, the same multiple is the whole of the whole*: for the same reason * 1.5. L, and L, M, N are any equimultiples of B, and B, D, F: therefore as A is to Bf, so are A, C, E, together † 5 Def. 5. to B, D, F together. Wherefore, if any number, &c. R. E. D. Seo N. C PROP. XIII. THEOR. THEOR.) third has to the fourth, but the third to the fourth a Let A the first have the same ratio to B the second which C the third has to D the fourth, but the third a greater ratio to D the fourth, than E the fifth has to F the sixth : also the first A shall have to the second B a greater ratio than the fifth E has to the sixth F. Because C has a greater ratio to D, than E to F, there are some equimultiples of C and E, and some of D and F such, that the multiple of C is greater than the multiple of D, but the multiple of E is not M G H E B F • r Def.5. ple of F*: let these be N L- same multiple of A; and whatever multiple K is of D, + Hyp. take N the same multiple of B: then, because A is to Bt, as C to D, and of A and C, M and G are equimultiples; and of B and D, N and K are equimulüples; if M be greater than N, G is greater than K; and if equal, * 5 Def. B. equal; and if less, less *. but G is greater than K; + Constr. therefore M is greater than N: but H is not+ greater + Constr. than L: and M, H are equimultiples of A, E; and N, L equimultiples of B, F; therefore A has a greater +7 Def. 5. ratio to B, than E has to F* Wherefore, if the first, &c. Cor. And if the first have a greater ratio to the second, than the third has to the fourth, but the third the same ratio to the fourth, which the fifth has to the sixth; it may be demonstrated, in like manner, that the first has a greater ratio to the second, than the fifth has to the sixth. PROP. XIV. THEOR. If the first have the same ratio to the second which the first be greater than the third, the second shall be greater than the fourthi and if equal, equal ; and if less, less. Q. E. D. 3 * 8. 5. Let the first A have the same ratio to the second B which the third C has to the fourth D: if A be greater than C, B shall be greater than D. Because A is greater than C, and B is any other magnitude, A has to B a greater ratio than C has to B*: but, as A is to Bt, so is C to D; there + Нур. fore also C has to D a ABC D ABCD A BC D greater ratio than C has to B:* but of two magnitudes, that to which the same * 13. 5. has the greater ratio is the lesser* : therefore D is less *10. 5. than B; that is, B is greater than D. Secondly, if A be equal to C, B shall be equal to D. For A is to B, as C, that is, A to D: therefore B is equal to D*. * 9.5. Thirdly, if A be less than C, B shall be less than D. For C is greater than A; and because C is to D, as A is to B, therefore D is greater than B, by the first case; that is, B is less than D. Therefore, if the first, &c. Q. E. D. PROP. XV. THEOR. Magnitudes have the same ratio to one another which their equimultiples have. A Let AB be the same multiple of C, that DE is of F: C shall be to F, as AB to DE. Because AB is the same multiple of C, that DE is of F, there are as many magnitudes in AB equal to C, as there are in DE equal to F: let AB be divided intò magnitudes, G each equal to C, viz. AG, GH, HB; and H BC EF first AG, GH, HB, is equal to the number of the last DK, KL, LE: and because AG, GH, HB are all equal, and that DK, KL, LE, are also equal to one another; therefore * AG is to DK as GH to *7.8. KL, and as HB to LE: but as one of the antecedents to its consequent *, so are all the antecedents together *12. 5. to all the consequents together, wherefore, as AG is to DK, so is AB to DE: but AG is equal to C, and DK to F; therefore, as C is to F, so is AB to DE. Therefore magnitudes, &c. Q. E. D. PROP. XVI. THEOR. A lo lo! 111) EV3 If four magnitudes of the same kind be proportionals, si they shall also be proportionals when taken alternatelyaula Let A, B, C, D be four magnitudes of the same kindy ) which are proportionals, viz. as A to B, so C to Deiis they shall also be proportionals when taken alternately, to that is, A shall be to C, as B to D. Take of A and B any equimultiples whatever E and F: and of C and D take any 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; + Hyp. therefore A is to B, as E is to F: but as A is to B sota is C to D; wherefore as C is to E G cause G, H are equimultiples of A B D -H- less, less; and E, F are any + equimultiples whatever * 5 Def. 5. of A, B; and G, H any whatever of C, D: therefore * A is to C as B to D. 'lf, then, four magnitudes, &c. * 11.5. C F 15.5. SO * 11. 5. * 14. 5. + Constr. Q. E. D. PROP. XVII. THEOR. See N. If magnitudes, taken jointly, be proportionals, they shall also be proportionals when taken separately: that is, if two magnitudes together have to one of them the same ratio which two others have to one of these, the remaining one of the first two shall have to the other the same ratio which the remaining one of the last two has to the other of these. Let AB, BE, CD, DF be the magnitudes taken jointly which are proportionals; that is as AB to BE, so let CD be to DF: they shall also be proportionals taken separately, viz. as AE to EB, so shall CF be to FD |