Therefore F is lefs than D; and fo D fhall be greater than F. After the fame manner we demonftrate, if A be equal to C, D will be alfo equal to F; and if A be lefs than C, D will also be lefs than F. If, therefore, there are three Magnitudes, and others equal to them in Number, which taken two and two, are in the fame Proportion, and the Proportion be perturbate; if the firft Magnitude be greater than the third, then the fourth will be greater than the fixth; but if the first be equal to the third, then is the fourth equal to the fixth; if lefs, lefs; which was to be demonftrated. PROPOSITION XXII. THEOREM. If there be any Number of Magnitudes, and others equal to them in Number, which taken two and two, are in the fame Proportion; then they shall be in the fame Proportion by Equality. ET there be any number of Magnitudes A, B, C, and others D, E, F, equal to them in Number, which taken two and two, are in the fame Proportion, that is, as A is to B, fo is D to E, and as B is to C, fo is E to F. I fay, they are also proportional by Equality, viz. as A is to C, fo is D to F. For let G, H, be Equimultiples of A,D; and K, L, any Equimultiples of B, E; and likewife M, N, any Equi- A B C D E F caufe A is to B, as D is to E, GK MHL N G, K, M, and others H, L, N, equal to them in Num 4 of this, ber, which being taken two and two in each Order, 20 of this. are in the fame Proportion. If G exceeds M, * H will exceed N; if G be equal to M, then H fhall be equal to N; and if G be lefs than M, H fhall be lefs than N. But G, H, are Equimultiples of A, D, and M, N, any other Equimultiples of C and F. Whence as A is to C, fo fhall + D be to F. Therefore, if there be any Number of Magnitudes, and others equal to them in Number, which taken two and two, are in the fame Proportion, then they shall be in the fame Proportion by Equality; which was to be demonstrated. + Def. 5. of this. PROPOSITION XXIII. THE ORE M. If there be three Magnitudes, and others equal to them in Number, which, taken two and two, are in the fame Proportion; and if their Analogy be perturbate, then fhall they be alfo in the Jame Proportion by Equality. LE and o ET there be three Mags For let G, H, L, be Equi- Then becaufe G, H, are and fince Parts have the fame Proportion as their like Multiples when taken corre* 15 of this. fpondently, it fhall be as A is to B, fo is G to H; and by the fame Reafon, as E is to F, fo is M to N. † 11 of this. But A is to B as E is to F. Therefore, † as G is to H, fo is M to N. Again, becaufe B is to C, as D is to E, and H, L are Equimultiples of B and D, as likewife K, M any Equimultiples of C, E; it fhall be as H to K, fo is L to M. But it has been alfo proved, that as G is to H, fo is M to N. Therefore, because three Magnitudes, G, H, K, and others, L, M, N, equal to them, in Number, which taken two and two are in the fame Proportion, and their Analogy is perturbate; then if G exceeds K, alfo L* will exceed * 21 of this. N; and if G be equal to K, then L will be equal to N; and if G be less than K, L will likewife be lefs: than N. But G, L are Equimultiples of A, D; and K, N Equimultiples of C, F. Therefore, as A is to C, fo fhall D be to F. Wherefore, if there be three Magnitudes, and others equal to them in Number, which taken two and two are in the fame Proportion; and if their Analogy be perturbate, then shall they be alfo in the fame Proportion by Equality; which was to be demonftrated. PROPOSITION XXIV. THEORE M. If the first Magnitude has the fame Proportion to G H |