For if it was not, A and B would not * have the * 8 of this, same Proportion to the fame Magnitude C; but they have. Therefore A is equal to B. B. Again, let C have the same Proportion to A as to B. I say, A is equal to B. For if it be not, C will not have the fame Proportion to A as to B ; but it A hath : therefore A is neceffarily equal to B. Therefore, Magnitudes that have the same Proportion to one and the Jame Magnitude, are equal to one another; and if à Magnitude has the same Proportion to other Magnitudes, these Magnitudes are equal to one another; which was to be demonstrated. PROPOSITION X. THEOREM. nitude, that which has the greater Proportion, 7 of For if it be not greater, it will either be equat or less. But A is not equal to B, because then both A and B would have * the same Proportion to the fame Magni A tude C; but they have not. Therefore A is not equal to B: Neither is it less than B; for then A would have t a lefs Proportion to C than B would have; but it hath not a lefs Proportion: There B fore A is not less than B. But it has been proved likewise not to be equal to it: Therefore A shall be greater than B. Again, let C have a greater Proportion to B than to A. I say, B is lefs than A. For if it be not less, it is greater, or equal. Now *7 of tbis. B is not equal to A; for then C would have * the fame Proportion to A as to B; but this it has not. Therefore A is not equal to B ; neither is B greater than A; for if it was, C would have a less Proportion to B than to A ; but it has not: Therefore B is not greater than A. But it has also been proved not to be equal to it. Wherefore B shall be less than A. Therefore of Magnitudes having Proportion to the same Magnitude, that which has the greater Proportion, is the greater Magnitude : And that Magnitude to which the same bears a greater Proportion, is the leffer Magnitude ; which was to be demonstrated. PROPOSITION XI. THEOREM. are also the same to one another. LED ET A be to B as C is to D, and C to D as E to F. I fay, A is to B as E is to F. For take G, H, K, Equimultiples of A, C, E; and . L, M, N, other Equimultiples of B, D, F. Then because A is to B as C is to D, and there are taken G, H, the Equimultiples of A and C, and L, M any * Def, of Equimultiples of B, D; if G exceeds L, * then H will exceed M; and if G be equal to L, H will be equal to M; and if lefs, lesser. `Again, because as C is to D, so is E to F; and H and K are taken Equimultiples of C and E; as likewise M, N, any Equimultiples of D, F; if H exceeds M *, then K will exceed N; and if H be equal to M, K will be equal to N; and if less, leffer. But if H exceeds M, G will also exceed L; if equal, equal; and if less, less. Wherefore if G exceeds L, K will also exceed N; and if G G be equal to L, K will be equal to N; and if less, less. But G, K are Equimultiples of A, E, and L, N, are Equimultiples of B, F. Consequently, as A is to B, fo * is E to F. Therefore, Proportions that are, ş Def. of one and the same to any third, are also the fame to one another; which was to be demonstrated. this PROPOSITION XII. THEOREM. If any Number of Magnitudes be proportional, as one of the Antecedents is to one of the Consequents, fo is all the Antecedents to all the Consequents. L ET there be any Number of proportional Magnitudes, A, B, C, D, E, F, whereof as A is to B, so C is to D, and so E to F. I say, as A is to B, so are all the Antecedents A, C, E, to all the Consequents B, D, F. For let G, H, K, be Equimultiples of A, C, E; and L, M, N, any Equimultiples of B, D, F. Then because as A is to B, so is C to D, and fo E to F; and G, H, K, are Equimultiples of A, C, E, and L, M, N, Equimultiples of B, D, F; if G exceeds L, H * will also exceed M, and K* Def. 5. of will exceed N; if G be equal to L, H will be equal tbis. to M, and K to N; and if less, less. Wherefore also, if G exceeds L, then G, H, K, together, will likewise exceed L, M, N, together; and if G be equal to L, then G, H, K, together, will be equal to L, M, N, together; and if less, less: But G, and G, H, K, are Equimultiples of A; and A, C, E; because, if there are any Number of Magnitudes Equimultiples to a like Number of Magnitudes, each to the other, the fame Multiple that one Magnitude is of one, lo shall + all the Magnitudes be of all. † a gf this, And for the same Reason, L, and L, M, N, are Equi Equimultiples of B, and B, D, F. Therefore, as + 5 Def. of A is to B, sot is A, C, E, to B, D, F. Where fore, if there be any Number of Magnitudes proportional, as one of the Antecedents is to one of the Consequents, so are all the Antecedents to all the Confequents ; which was to be demonstrated. PROPOSITION XIII, THE ORI M. as the third to the fourth, and if the third bas second B, as the third C has to the fourth D; and let the third C have a greater Proportion to the fourth D, than the fifth E to the fixth F. I say, sbis. likewise, that the first A to the second B has a greater Proportion than the fifth E to the fixth F. For because C has a greater Proportion to D, *? Def. of than E has to F; there are * certain Equimultiples of © and E, and others of D and F, such that the Multiple of C may exceed the Multiple of D; but the Multiple of E not that of F. Now let these Equia multiples of C and E, be G and H; and K and L, those of D and F; fo that G exceeds K, and H not L: Make M the same Multiple of A, as G is of C; and N the same of B, as K is of D. Then, because A is to B as C is to D, and M and G are Equimultiples of A, C; and N, K, of B, D: + 5 De If M exceeds N; then + G will exceed K; and if M be equal to N; G will be equal to K, and if less, lels, But G does exceed K. Therefore M will al fo this. so exceed N. But H does not exceed L. And M, H, are Equimultiples of A, E; and N, L, any others of B, F. Therefore A has a * greater Proportion to * 7 Def, of B than E has to F. Wherefore, if the firft has the same Proportion to the second, as the third to the fourth; and if the third has a greater Proportion to the fourth than the fifth to the fixtb; then also shall the first have a greater Proportion to the second, than the fifth has to the fixth; which was to be demonstrated. PROPOSITION XIV. THE OR IM. as the third bas to the fourth; and if the first be second B, as the third C has to the fourth D: And let A be greater than C. I say, B is also greater than D. For because A is greater than C, and B is any other Magnitude: A will have * a greater Proportion to B than * 8 of this. C has to B;' but as A is to B, fo is C to D; therefore, alfo, shall + + 13 of ibis. have a greater Proportion to D than Chath to B. But that Magnitude to which the fame bears a greater Proportion, is the lesser Magnitude : I 10 of tbis. Wherefore D is less than B; and consequently B will be greater than D, In A B C D like Manner we demonstrate, if A bę equal to C, that B will be equal to D; and if A be less than C, that B will be less than D. Therefore, if the first has the same Proportion to the second, as the third has to the fourth, and if the first be greater than the third, then will the second be greater than |