away к +C so G B fhall be equal to E, and HD to F. G+ In like manner we demonstrate, if G B was â Multiple of E, that +H HD is a like Multiple of F. Therefore, if two Magnitudes be Equimultiples of two Magnitudes, and some Magnitudes, Equimultiples of BD EF the same, be taken away; then the Residues are either equal to those Magnitudes, or else Equimultiples of them; which was to be demonstrated. PROPOSITION VII. THEOREM. L ET A, B, be equal Magnitudes, and let C be any other Magnitude. Ilay, A and B have the same Proportion to C; and likewise C has the same Proportion 1o A as to B. For take. D, and E, Equimultiples of A and B ; and let-F be any other Multiple of c. DA Now, becaule D is the same Multiple of A, as E is of B, and A is equal to B, D shall be also equal to E; but F is a Magnitude taken at Pleasure. Therefore if D exceeds F, chen E will exceed F; if D be equal to F, E will ЕВ СЕ be equal to F; and if less, less. But D and E are Equimulciples of A and B; and F is any other Multiple of C. Therefore it will be * as A is Def. s. to C, so is B to C., K I say, * Def. 5. I say, moreover, that C has the same Proportion to A as to B. For the same Construction remaining, we prove, in like manner, that D is equal to E. Therefore, if F exceeds D, it will also exceed E; if it be equal to D, it will be equal to E; and if it be less than D, it will be less than E. But F is a Multiple of C; and D and E, any other Equimultiples of A and B; therefore, as C is to A, fo thall * C be 'to B. Where. fore, equal Magnitudes have the same Proportion to the same Magnitude, and the fame Magnitude io equal ones ; which was to be demonstrated. PROPOSITION VIII. THEOREM. a greater Proportion to some third Magnitude, two Magnitudes, than it has to ibe greater. L ET AB and C be two unequal Magnitudes, whereof A B is the greater ; and let D be any third Magnitude. I say, AB has a greater Proportion to D, than C has to D; and Ď has a greater Proportion to C, than it has to A B. Because A B is greater than C, make B E equal to C, that is, let A B exceed C by AE; then A E, multiplied F fome Number of Times, will be greater than D. Now let A E A be multiplied until it exceeds D, and let that Multiple of A E G+E+ greater than D be F G. Make GH the fame Multiple of K H B C EB, and K of C, as F G is of A E. Also, affume L double to D, M triple, and so on, until such a Multiple of D is had, as is the nearest greater than K; let this be N, and let M be a Multiple of D, the nearest bers than N. - Now, because N is the nearest Multiple of D. greater N M L D than than K; M will not be greater than K; that is, K . 7. PROPOSITION IX. THEOREM. Magnitudes which have the same Proportion to one and the same Mognitude, are equal to one ano- equal to one another. ortion to C. I sav, A is equal to B. K 2 For, • 8 of Ibiso to B. For, if it was not, A and B would not * have the A Ç * have the same Proportion to A as to B ; but it hath: Therefore A is necessarily B equal to B. Therefore, Magnitudes that have the same Proportion to one and the same Magnitude, are equal to one another ; and if a Magnitude, has the same Proportion to other Magnitudes, those Magnitudes are equal to one another; which was to be demonstrated. PROPOSITION X. THEOREM. Magnitude, that which has the greater Pro- LE ET A have a greater Proportion to C, than B has to C. I fay, A is greater than B. For, if it be not greater, it will either be equal or less. But A is not equal to B, because same Proportion to the Magnitude C; not equal to B: Neither is it less than + 8 of ebis. B; for then A would have t a less С Proportion to C, than B would have ; 1 but it hath not a less Proportion. Therefore A is not less than B. But it has B been proved likewise not to be equal to lit: Therefore A shall be greater than P. Again, let C have a greater Proportion to B than to A. 1 say, B is less than A., For, For, if it be not less, it is greater or equal. Now, B is not equal to A, for then C would have * the * 7,0f ibis. Same 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 Propor- + 8 of ibis. tion to B chan 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 jame bears a greater Proportion, is the lefser Magnitude ; which was to be demonstrated. PROPOSITION XI. PROBLEM. are also the same to one another. Eto F. I say, A is to B, as E is to F. E; and L, M, and N, other Equimultiples of B, D, and F. Then, because A is to B, as C is to D, and there are taken G and H, the Equimultiples of A and C, and L and M, any other Equimultiples of B and D; il G exceeds L, *then H will exceed M; and if * Def. 5. cafting G be equal to L, H will be equal to M; and if less, this. less. 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 and N, any other Equimultiples of D and 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 lefs, less. But if H exceeds M, G will also exceed L; if equal, equal; and if less, less. Wherefore, it Géxceeds L, K will also exceed N; and if G be equal to |